Patent Publication Number: US-7715928-B1

Title: Efficient communications and data compression algorithms in sensor networks using discrete wavelet transform

Description:
This application claims the benefit of the U.S. provisional application No. 60/572,136 entitled “FLEXIBLE AND DISTRIBUTED WAVELET COMPRESSION METHODS FOR SENSOR NETWORKS USING LIFTING” and filed on May 17, 2004, which is incorporated herein by reference in its entirety. 

   BACKGROUND 
   This application relates to data communications in sensor networks and digital signal processing using a discrete wavelet transform. 
   A sensor network may include a set of sensors or sensing nodes that are capable of sensing, communicating, and processing. An early example of sensor networks is a network of acoustic sensors deployed at the ocean bottom to detect and keep track of submarines. In other examples, sensors may be used to perform various measurements (e.g., temperature or a presence of a target substance) or capture images for various applications. Disposable sensors with processing capabilities may be deployed in a number of environments to perform tasks such as target tracking (e.g. vehicles, chemical agents, or personnel), traffic control, environment monitoring and surveillance. Such sensors may be, for example, wireless sensors to wirelessly transmit or receive signals. 
   SUMMARY 
   Communication, collection and processing of data from such sensor networks require communication bandwidths to carry the data and consume energy. Hence, it may be desirable to reduce the amount of data to be communicated and transmitted and to reduce the energy consumed. In many applications, sensors may collect data at different locations, such that the information is correlated across locations, e.g., some closely located sensors. As an example, temperatures measured by temperature sensors near one another may be correlated so there is certain redundancy in the individual measurements obtained by these separate sensors. As a result, some unnecessary data is transmitted through the network. 
   The redundancy in the data from sensors may be reduced or removed via signal processing at the sensor level to transform the raw data collected by the sensors. The transformed data with the reduced redundancy may be communicated through the sensor network. This reduction in the amount of data reduces the energy consumed in transmitting the data through the sensor network because the transmitted data is less than the raw data collected by the sensors. However, the processing at the sensor level for reducing the data redundancy consumes energy. It is possible that the total energy consumed in processing the raw data and the transmission of the processed the data with a reduced amount of data bits may not be less than the energy consumed for directly transmitting the raw data. Hence, the data processing mechanism in processing the raw data at the sensor level to reduce data redundancy should be designed according to the specific structure sensor network to reduce the overall consumption of energy. 
   In one implementation, inter-sensor communications are introduced between sensors over short distances to allow for a distributed wavelet transform to decorrelate the data and to reduce the overall energy consumption of the network. A lifting scheme may be used to compute wavelet transforms as a way to decorrelate data. In another implementation, a distributed wavelet algorithm is provided for wireless multihop sensor networks to exploit the data flow direction in the network and to perform partial computations to approximate the wavelet coefficients using the available data that arrives at each sensor. In such multihop sensor networks, an upper bound to the distortion introduced by partial data quantization can be used to design the partial quantizers such that a good trade-off is achieved between additional distortion and increase in cost due to the extra bits to be transmitted. 
   These and other implementations and their variations are described in greater detail in the attached drawings, the detailed description, and the claims. 

   
     BRIEF DESCRIPTION OF DRAWINGS 
       FIG. 1  shows an exemplary circuit architecture for a 3-stage discrete wavelet transform (DWT) decomposition. 
       FIG. 2  illustrates the need of data in processing one block from another block when the input is segmented into consecutive blocks. 
       FIG. 3  shows one embodiment of a size-2 polyphase system that can efficiently perform the filterbank DWT algorithm of each filtering stage illustrated in  FIG. 1 . 
       FIG. 4  shows the operation of the filtering system in  FIG. 3  on a selected block where partially updated data is stored for further processing. 
       FIG. 5  shows one example of implementing the overlap-state technique in sequential DWT processing. 
       FIGS. 6A through 6C  show one example of implementing the overlap-state technique in the parallel DWT processing by using two or more processors to operate simultaneously. 
       FIG. 7  illustrates a split and merge process in the parallel processing based on the overlap-state technique. 
       FIG. 8  shows a sequential system for 2D DWT. 
       FIG. 9A  shows a 2D mesh processor network of multiple processors. 
       FIG. 9B  shows one block partition of data for 2D data. 
       FIG. 10A  shows bus processor network of processors. 
       FIG. 10B  shows one embodiment of the strip partition for the 2D data. 
       FIG. 11  shows one embodiment of a two-channel filterbank where the left side is the decomposition stage and the right side is the reconstruction stage. 
       FIGS. 12A and 12B  show the forward and inverse DWT using lifting factorization. 
       FIG. 13  is a schematic plot of the DWT as a finite state machine (FSM). 
       FIGS. 14A and 14B  illustrate state transitions across a block boundary of blocks  1  and  2 . 
       FIG. 15  illustrates one example of the overlap-state processing in the sequential DWT, where the state in (a) is the input in initial state i, the Block  1  in (b) includes samples up to x i (2m), partially updated samples are shown to be overlapped with next block of input samples in (c), and the completely transformed/updated input from state i to i+1 is shown in (d). 
       FIG. 16  illustrates one example of the overlap-state processing in parallel DWT. (a) Input in initial state i. (b) Input is partitioned over two processors. Block  1  and  2  are transformed separately and each have state information appearing near the block boundary. (c) State information is exchanged between two processors and the partially updated samples are fully updated. (d) Completely transformed (updated) input from state i to i+1. 
       FIG. 17  illustrates the parallel DWT processing based on the overlap-state technique in a three-level wavelet decomposition. Solid lines: completely transformed data; Dashed lines: partially transformed data. Operation 1: each block transforms its own allocated data independently and state information is buffered; Operation 2: state information is communicated to neighboring blocks; Operation 3: complete transform for the boundary data samples. 
       FIGS. 18A and 18B  show an example of the delayed normalization in a three-level octave-band wavelet decomposition. 
       FIGS. 19A and 19B  show a one-level 2D wavelet decomposition with recursive normalization and delayed normalization, respectively. 
       FIG. 20A  shows a system for separate implementation of the transform and quantization operations to reduce computation. 
       FIG. 20B  shows a system with joint transform and quantization. 
       FIG. 21  shows on embodiment of a sequential DWT architecture based on the overlap-state technique. 
       FIG. 22  shows on embodiment of a parallel DWT architecture based on the overlap-state technique. In Split stage, each processor computes its allocated data independently up to the required decomposition level. In Merge stage, a one-way communication is initiated to communicate the state information to neighboring processors. A post processing operation is then started to complete the transform for boundary samples. 
       FIG. 23  shows an example 2D DWT with two level decompositions based on the overlap-state technique. 
       FIG. 24  shows a trip sequential DWT system diagram. The input is segmented into data strips, which are transformed sequentially from top to bottom. 
       FIG. 25  shows a block sequential according to one embodiment. The buffer is divided into two parts, one for holding the state information and one for new input samples in the sliding transform window. 
       FIG. 26  shows merge operations in 2D mesh processor network. (a) transfer of row state information from P i,j  to P i, j+1 ; (b) transfer of column state information from P i,j  to P i+1, j ; (c) transfer newly generated row state information from P i,j  to P i+1, j+1 ; (d) complete transform for boundary samples. Notice the total amount of data in each processor in the final state (d) is different from the original uniform allocation due to the Merge operations. 
       FIG. 27  shows merge operations for strip parallel implementation. (a) transfer of row state information from P 1  to P i+1 ; and (b) complete transforms for boundary samples in each processor. 
       FIG. 28  illustrates state transitions of the Daubechies (9,7) filters using factorization, where the state information includes 4 samples (the overlap buffer size) in shaded boxes. Dashed lines represent operations necessary for the transform of the new input sample pair {x 0 (9),x 0 (10)}. 
       FIG. 29  illustrates state transitions of the (2,10) filters using factorization, where state information includes 4 samples in shaded boxes. Dashed lines represent operations necessary for the transform of the new input sample pair {x 0 (10),x 0 (11)}. 
       FIG. 30  illustrates state transitions of cumulative distribution function CDF (4,2) filters using factorization, where the state information includes 3 samples in shaded boxes. Dashed lines represent operations necessary for the transform of the new input sample pair {x 0 (8),x 0 (9)}. 
       FIG. 31  shows one example of a sensor network. 
       FIG. 32  shows one example of a lifting implementation of a wavelet transform. 
       FIG. 33  shows an example of a two-step implementation of a distributed wavelet transform using lifting. 
       FIG. 34  shows an example of lifting for a 5/3 wavelet. 
       FIG. 35  shows a specific example of a sensor network used for simulations. 
       FIGS. 36A and 36B  show simulation results for the sensor network in  FIG. 35 . 
       FIG. 37  shows one example of a multihop sensor network. 
       FIG. 38  shows an example of a network data flow in a portion of a multihop sensor network and the partial coefficient computations at each sensor. 
       FIG. 39  shows a graphical representation of the quantization effects in the partial coefficient computation at each sensor. 
       FIG. 40  shows MSE ratios with and without the partial coefficient quantization. 
       FIGS. 41 ,  42  and  43  show simulation results of a simple multihop sensor network. 
   

   DETAILED DESCRIPTION 
   This application describes, among others, implementations of data processing and communicating techniques based on distributed wavelet compression algorithms for processing data at the sensor level to decorrelate data of the sensors and to reduce the energy consumption in sensor networks such as sensor networks with wireless sensors. The specific examples of implementations use sensor networks that include multiple, distributed sensors that collect data and send the data to a central node. The sensors may directly transmit data to the central node without transmission through other sensors. The transmitted data from one sensor to the central node may be the data with information solely collected from that sensor alone without information from neighboring or adjacent sensors. However, in sensor networks where some or all sensors are correlated in the information they collect, signals from different but correlated sensors have redundancy in information. As such some sensors may be selected to allow for inter-sensor communications. The original raw data and the newly obtained information via the inter-sensor communications can be processed to produce processed data for each of the selected, correlated sensors with reduced redundancy. The new data for each of the selected, correlated sensors is then directly transmitted to the central node for further processing to retrieve useful information on the area or object under measurement by the sensor network. In this approach, certain sensors in the sensor network may be selected to directly transmit their own data to the central node without inter-sensor communications and the data processing regardless whether such sensors may be correlated with other sensors if such direct transmission without inter-sensor communication and processing is more energy efficient. The allocation of sensors for inter-sensor communication and the data processing and sensors for direct transmission without inter-sensor communication and processing can be designed to reduce the power consumption for the overall sensor network. 
   Alternatively, the sensors may transmit data towards the central node by passing through one or more other sensors in a signal path between a sensor and the central node where data from the sensor hops through other sensors which act as signal relays. In such a sensor network with multiple hops from one sensor to another sensor in transmission of data to the central node, a sensor in a downstream of a signal path usually does not have information from a sensor located upstream in the signal path at a particular time. Hence, the downstream sensor may have to wait for the information from the upstream sensor to arrive. In addition, an upstream sensor may need to receive information from a downstream sensor in order to fully perform the computation for the decorrelation and therefore a communication from the downstream sensor the upstream sensor is needed. To address these and other technical limitations, techniques are provided here to use the natural flows of data in sensor network with multiple hops through sensors to perform partial computation of the data decorrelation with currently available data. Hence, extra inter-sensor communications that are not part of the natural flows of the data in the network are eliminated. 
   In techniques for the above two types and other sensor networks, the lifting factorization in discrete wavelet transform can be used to implement the wavelet compression algorithms. The lifting factorization provides a convenient representation of the transform as it assumes in place computation where each sensor represents a single memory location. Thus, the lifting factorization explicitly breaks down the transform into elementary operations that can be easily evaluated in terms of communication costs. Various lifting factorization schemes in discrete wavelet transform may be used. One example of suitable lifting factorization schemes is U.S. Pat. No. 6,757,343 entitled “DISCRETE WAVELET TRANSFORM SYSTEM ARCHITECTURE DESIGN USING FILTERBANK FACTORIZATION” which is incorporated herein by reference in its entirety as part of the specification of this application. 
   One implementation of the lifting factorization described in the U.S. Pat. No. 6,757,343 is a method for performing DWT computation on an input signal. The input signal is partitioned into consecutive blocks of samples. A multilevel filtering operation is then performed on a first block by using a discrete wavelet transform algorithm, without using information from a second adjacent block. Computations of the multilevel filtering operation that are computable based on information solely from the first block are completed and then partially completed results from some samples in the first block are saved. The multilevel filtering operation cannot be completed on these samples in the first block without specified input from the second adjacent block. After the above operations within the first block, the specified input from the second adjacent block is then used to complete the multilevel filtering operation on the samples with the partially completed results in the first block. 
   Wavelet transforms are developed as an alternative to various Fourier transforms to analyze and extract components of a signal, especially when the spectral composition of the signal changes with time as is the case for certain imaging signals. For example, a time-varying signal can be analyzed by a wavelet transform to decompose the signal into various spectral components and their dependence with time. A wavelet transform uses a wavelet as a window filter with a varying scale and a varying wavelet translation in time to correlate with the time-varying signal. This correlation can obtain the spectral information at multiple different spectral resolutions and the position information of different spectral components in time. 
   The discrete wavelet transform (“DWT”) uses a bank of digital filters with different cutoff frequencies to analyze sequential samples of a time-varying signal at different scales. The filters may include a set of high-pass filters to analyze high-frequency components and a set of low-pass filters to analyze low-frequency components. This allows the multi-resolution analysis. Upsampling and downsampling can be used to change the scale in the discrete wavelet transform. Interpolation filters and decimation filters may be used for these purposes. The discrete wavelet transform is a powerful tool for multiscale time-frequency signal decomposition and analysis. This technique can be used in many applications, such as signal processing, digital communications, numerical analysis, and computer graphics. 
   The system design for implementing the discrete wavelet transform may be limited by a number of practical considerations, such as the available buffer size, processing delay, processing power, chip area, and the impact of the control complexity. In general, two different types of system architectures, namely, sequential and parallel architectures, may be used for the discrete wavelet transform. The sequential architecture may be implemented to compute the DWT by splitting an input signal into sequential blocks in time. A processor such as a microprocessor or other computer processors may be used to operate on one block at a time to process the different blocks at different times. The sequential architecture may be used when there is only a limited amount of memory available for the transform computation of the final product. Alternatively, a parallel architecture, as an example, can split the input into blocks among several processors so that the processors operate on different blocks at the same time. This can speed up the transform computation for applications where a large volume of data has to be processed in a reasonably short time. For instance, seismic data processing or illumination computations in computer graphics are potential applications. Fast DWT computation to meet stringent delay constraints can be critical to the success of any wavelet-based techniques. 
   The discrete wavelet transform may be implemented by a series of filtering stages each having a high-pass filter and a low-pass filter connected in parallel. Hence, an input to a filtering stage is filtered by the low-pass filter to produce one output with the low frequency components of the input. At the same time, this input is also filtered by the high-pass filter to produce another output with high frequency components. This filtering operation repeats at each stage to decompose the input signal. 
     FIG. 1  shows an exemplary circuit architecture for a 3-stage DWT composition. Each stage includes a high-pass filter (g)  110  and a low-pass filter (h)  120 . The output terminal of each filter  110  or  120  of a filtering stage is connected to a downsampling circuit  130  to remove one half of the samples. The four outputs y 0 (n), y 1 (n), y 2 (n), and y 3 (n) represent the DWT output of the input x(n). 
   The above process of filtering and downsampling operations need be performed recursively on the input data at each stage for multilevel decompositions. Since filtering operations in DWT generally can not be implemented as a block transform (with the exception of the trivial Haar transform), this recursive nature of the DWT computation poses special challenges when stringent memory and delay constraints have to be observed for practical DWT system designs. 
   One constraint in performing a DWT is that each block of samples cannot be processed independent of samples from another block such as an adjacent block when finite impulse response (“FIR”) filters are implemented. A FIR filter with a finite length L, when operating on samples in one block near the boundary with another adjacent block, may need the data from the adjacent block to complete the filtering computation on the samples within the block. Hence, two adjacent blocks need to communicate to transfer the needed data. 
     FIG. 2  illustrates such a need of data in processing one block from another block. Three consecutive data blocks  1 ,  2 , and  3  are shown. It is assumed that the FIR filter has a length of L. In performing DWT computation on the samples in the block  1 , the FIR filter first operates on first segment of L samples in the block  1  to produce one DWT output  210  and then shifts to operate on the next segment with L samples, which overlaps the last (L−1) samples in the first segment, to produce another DWT output  220 . When the FIR filter shifts to operate on the samples in the block  1  near the boundary  201  between the blocks  1  and  2 , a portion of the required FIR filter inputs falls within the block  2 . For example, when the FIR filter is at a position  232 , the filter needs data from both blocks  1  and  2  to produce the DWT output  230 . 
   However, the data from the block  2  may not be available to the FIR filter when the filter is at the position  232 . For example, in the sequential architecture, blocks become available sequentially in time. It is possible that the needed data from block  2  has not arrived when the filter is computing the output  230  at the position  232 . A delay occurs since the processor needs to wait until the data from the block  2  becomes available. The DWT computation by block partition may be performed by either splitting the data in blocks that have an overlap or, equivalently, keeping samples from block  1 , that will be used by the DWT computation for the block  2 . In either case, a certain amount of memory is needed to store information from the block  1  that is needed in block  2 . 
   Many sequential architecture designs adopt the standard FFT-based filtering techniques [34], i.e., overlap-add or overlap-save. These include the recursive pyramid algorithm (RPA) by Vishwanath [7], the spatially segmented wavelet transform (SSWT) by Kossentini [28], and the reduced line-based compression system by Chrysafis et al. [12], [35]. Since the SSWT overlaps data only once before the start of the transform, the overlap buffer size increases exponentially with the increase of decomposition levels. An alternative is implemented in [7], [12] where data is overlapped at each level of decomposition and the buffer size is reduced. 
   The need of data from another data block may also rise in the parallel DWT architecture. Assume, for example, that the three-level wavelet decomposition shown in  FIG. 1  is performed using two separate processors  1  and  2 , with each processor being allocated half of the input data. A problem arises when DWT is computed near data boundaries at both processors, e.g., the boundary  201  between blocks  1  and  2  in  FIG. 2 . Because DWT is not a block transform, for correct wavelet coefficients computation near data boundaries, each processor would need to access data allocated to the other processor. In this case, either the two processors exchange data before each level of the decomposition or the two processors are given sufficient overlapped data to carry on the whole computation without communication with each other. 
   The first approach may require frequent data exchanges between processors. This, however, can increase the communication overhead and thus adversely affect the system scalability in a parallel architecture, particularly in the presence of slow communication links, for example, when using a network of workstation (NOWs) or a local area multicomputer (LAM)[25], [1], [26], [27]. The second approach, although avoiding frequent communications between the processors, needs to overlap data at each processor. This overlap, due to the recursive nature of the DWT filtering operations, can be large as the levels of filtering stages increases. Hence, the required memory space can also be large in the multi-level decomposition system and can be very expensive in terms of memory and communication [28]. 
   Some parallel architecture designs are developed to increase the efficiency of the DWT operation by providing communication of the boundary data at each level of decomposition. See, for example, the works by Fridman et al. [11] and by Nielsen et al. [24]. To reduce the overhead caused by frequent inter-processor communication, Yang et al. [25] proposed to use boundary extensions in their DWT system configured from a cluster of SGI workstations. This, however, computes incorrect wavelet coefficients near data boundaries, which causes performance degradation in some applications, for example, low-bit rate image coding [36]. 
   The systems and techniques of the present disclosure include an efficient DWT system design under memory and delay constraints achieved by reducing the memory and interprocessor communication required in the DWT based the segmentation of the input data in either sequential or for parallel architecture designs. Two parameters may be used to measure the performance of a DWT system design: the amount of data to be transmitted between processors (or to be stored in the processor if a sequential computation is used) and the number of times data has to be communicated between processors. 
     FIG. 3  shows one embodiment of a size-2 polyphase system  300  that can efficiently perform the filterbank DWT algorithm of each filtering stage illustrated in  FIG. 1 . The input x[n] is downsampled by the dowmsamplers  310  and  320  to produce a sequence of even samples x 0 [n] and another sequence of odd samples x 1 [n]. Filters h 00 , h 10 , h 01 , and h 11  are then used to filter the signals x 0 [n] and x 1 [n] and two adders are used to produce the DWT outputs y 0 [k] and y 1 [k]. The system  300  can be used as a building block to form a multistage DWT computation system. 
   The operation of the system  300  can be represented by a polyphase matrix form: 
                   [             Y   0     ⁡     (   z   )                   Y   1     ⁡     (   z   )             ]     =       [             H   00     ⁡     (   z   )               H   01     ⁡     (   z   )                   H   10     ⁡     (   z   )               H   11     ⁡     (   z   )             ]     ⁡     [             X   0     ⁡     (   z   )                   X   1     ⁡     (   z   )             ]               (   1   )               
where H i (Z) is the ith polyphase component of the filter H(z)(similarly defined for G(z)). The 2×2 polynomial matrix representing the filtering operation can be factorized into a product of matrices, where each of the factoring matrices is either a prediction, an update or a scaling matrix. These correspond to a filtering implementation where prediction and updating filters alternate, and multiple stages of these prediction/update structures are cascaded. This representation converts the DWT computation by band-pass filtering and downsampling in multiple stages in  FIG. 1  into a computation of updating the polyphase components, one stage at a time, over multiple updating stages. In the polyphase matrix form, this updating process can be expressed as:
 
                   [           X   0     i   +   1                 X   1     i   +   1             ]     =       [         1       0               t   i     ⁡     (   z   )           1         ]     ⁡     [           X   0   i               X   1   i           ]               (   2   )               
where t i (z) represents the updating filter at stage i. Since the input signal x[n] is divided into blocks to be processed either sequentially one at a time, or in parallel by two or more processors concurrently, updating the state values of the data near the boundaries of each block may require data from another adjacent block. Hence, when such data is not available at the time of updating operation, the DWT computations on certain cells are either partially performed or unperformed and hence are not fully updated.
 
   One embodiment of this disclosure implements an overlap-state technique for the DWT computation to carry out all computations on the cells in a block based on the available data within that same block. Then, data from one or more adjacent blocks that is needed, including the partially updated samples, is transferred and the DWT computations in the unfinished cells are completed. The data to be transferred is the information of the cells in another block that overlap with the DWT computation of the current block. 
   One aspect of the overlap-state technique is to keep the partially updated samples for data exchange with an adjacent block to complete the DWT computation.  FIG. 4  shows the operation of the filtering system  300  on a selected block to illustrate the above situation. Assume that a block has nine samples, x 0  through x 8 , and is to be updated at time t 1 . The box cells represent the memory cells in the computer, a storage device, or a circuit for the DWT processing. The numbers in the box cells represent the number of updates completed. Hence, “0” means no update is made to a cell, “1” means a cell is updated once, “2” means a cell is updated twice, and so on. Each update corresponds to having performed the computations corresponding to one of the component matrices of the factorization of the polyphase matrix. Solid lines with arrows represent DWT computations with sufficient data, dashed lines with arrows represent DWT computations that need data input from an adjacent block. Similarly, box cells of solid lines represent cells with sufficient input from cells within the block and box cells with dashed lines represent cells which require input from another block and therefore are never updated. 
   At time t 2  after one computation cycle, the cells with odd-numbered samples are computed and updated. Only x 1 , x 3 ,  5 , and x 7  cells are updated. Other cells remain unchanged. At time t 3 , even-numbered cells x 2 , x 4 , and x 6  are updated, cells x 1 , x 3 , and x 7  remain unchanged. Note that the computation on the cell x 8  requires data from cells x 9  and x 10  from the next block. Lacking of a communication from the next block, this update computation cannot be performed. This process repeats. The states of all cells of the block at time t 6  are shown in  FIG. 4 . Notably, only the cell x 4  completed updated to produce the final DWT coefficient. Other cells still need input from other blocks to complete the DWT computation. The DWT computation based on the factorization technique shown in  FIG. 3  will be explained in more detail in a subsequent section of this disclosure. 
   A number of advantages of this overlap-state technique are readily evident from the above description. The data exchange between different blocks is only carried out once at the end of the computation of each block. This simplifies the computation overhead associated with managing and performing the inter-block communication for data exchange. The memory needed for the DWT computation is minimized since only the data from another block required for the uncompleted DWT computation is stored. In addition, the overhead for performing and managing the data storage is minimized since only one data exchange is carried out. Other advantages and benefits can be appreciated from the more detailed description in the later section of this disclosure. 
   This overlap-state technique is in part based on a lifting framework formulated by Daubechies and Sweldens [33]. The DWT is first modeled as a finite state machine (FSM) using the lifting algorithm. Multilevel partial computations (intermediate states) are performed for samples near block boundaries. One aspect of this disclosure suggests that, to obtain a correct transform near data boundaries, these intermediate states can be preserved in their original storage spaces (an extension of the in-place computation feature of the lifting algorithm) and exchanged only once between neighboring data blocks for any arbitrary J level decompositions. 
     FIG. 5  shows one example of implementing the overlap-state technique in sequential DWT processing. The input signal is partitioned into non-overlapping blocks (Block  1 , Block  2 , etc.) each having 9 sequential data samples. The processor receives and processes the block  1  to complete the DWT computations for the first block wherein the first  5  cells are fully updated and the remaining 4 cells are not fully updated since the computations need data from the first two cells from the next block. Next, the processor is directed to form a new overlap-state block  2 ′ which includes the unfinished cells  6  through  9  from the block  1  and new cells  10  through  15  from the block  2 . The DWT computation is again carried out in this new block  2 ′ where the cells  6  through  11  are completely updated and cells  12  through  15  are either partially updated or unchanged (e.g., cell  15  in the example). The above process repeats to compute the remaining samples in the input signal. 
     FIGS. 6A through 6C  show one example of implementing the overlap-state technique in the parallel DWT processing by using two or more processors to operate simultaneously. In a two-processor parallel system, the processors are controlled to perform DWT computations on two sequential blocks, block  1  and block  2 , at the same time by using only the data within each block, independently from another block. This stage of the DWT computation is referred to as the “split” stage where the independent DWT computations on different blocks are parallel and carried up to the desired decomposition level. After the completion of the split stage, the states of the cells in the blocks  1  and  2  are shown in  FIG. 6B . The DWT computations on cells near the boundary between blocks  1  and  2  are not completed. Next, a communication between the two blocks  1  and  2  is performed and a new block is formed to merge the sequential unfinished cells from both blocks  1  and  2  together. Then the DWT computation is performed over this new block to complete the DWT computation. This second processing stage on the new block is referred to as the “merge” stage.  FIG. 7  further illustrates this split and merge process in the parallel processing. 
   Notably, the present technique allows for partial computations for boundary samples at multiple decomposition levels and preserves these partially computed results (intermediate states) in their original locations in the memory for later processing. In addition, data exchange between different processors in parallel processing or between different sequential blocks in sequential processing is performed only after multilevel decompositions rather than at each decomposition level as in some other DWT systems. 
   The following sections provide more detailed description on exemplary implementations, various features, and associated benefits and advantages of the overlap-state technique. 
   I. Technical Issues in Sequential and Parallel DWT Processing 
   This section describes certain technical issues for DWT processing for one dimensional DWT processing in both sequential and parallel architectures. The issues and solutions can be extended to two-dimensional and other DWT sequential and parallel architecture designs. 
   A. Sequential Architecture Design 
   A sequential system for 2D DWT is shown in  FIG. 8 . 
   The transform working buffer (e.g., on-chip memory or cache memory) is usually small in size compared to the data size. Therefore the original data, stored in a secondary storage space (e.g., hard disk, frame buffer), has to be segmented such that each segment can be loaded to the working buffer and the transform can be computed one segment at a time. Variations of this generic system include: 
   1. The block-based system presented in SSWT by Kossentini [28], which computes the wavelet transform one image block at a time. 
   2. The line-based system presented by Chrysafis et al. [12], [35], which computes the wavelet transform “on the fly” and where the basic input units are image lines. 
     FIG. 8  shows a typical sequential DWT system diagram. Assume blocks of size B (in bytes) are given to the processor, i.e., at most B bytes of input data can be loaded into the working buffer at a time. Define the overlap buffer size B s , in bytes, as the memory space taken by the overlapped data which has to be kept in the working memory so that the correct transform can be computed for the next block of input data. In the case of the line-based systems [12], [35], B s , is the minimum working buffer size needed for the transform. After transform of each block, only (B−B s ) bytes can be freed and wavelet coefficients generated can be transferred to the next processing stage, e.g. quantization. We now define the system efficiency η as 
                 η   =         B   -     B   s       B     =     1   -         B   s     B     .                 (   3   )               
If η=1, this indicates that all of the original data samples can be fully transformed, which corresponds to the case of pure block transforms, such as DCT or the Haar transform. If, using the whole buffer, no complete decomposition can be performed (i.e., data is not enough for J-level of decompositions), then η=0. It is possible that some of the wavelet coefficients in high frequency bands can be generated.
 
   The problem is formulated as: Given a fixed working buffer size B, how to compute the DWT to maximize the system throughput η? Obviously, to increase the system throughput, one has to reduce the overlap buffer size B, as much as possible. 
   B. Parallel Architecture Design 
     FIG. 9A  shows a 2D mesh processor network of multiple processors. In this example, each processor is only connected with its immediately neighboring processors. Communications between processors can be established according to the single port model, in which processors may only send or receive a message over one communication link at a time, or the multi-port model, in which a processor may send or receive multiple messages over multiple links at a time. In this model, the natural partition for 2D data is the block partition strategy shown in  FIG. 9B . The processor P m,n  is allocated with input samples with indices (x,y), mN r ≦x≦(m+1)N r , (m+1)N C . Without loss of generality, assume W=MN r  and H=NN c , where (N r , N c ) are the block row and column length, and (M,N) are the number of processors in row and column direction respectively. 
     FIG. 10A  shows bus processor network of processors. A common bus is used to allow a processor to communicate with every other processor. An example is the LAM (Local Area Multicomputer) system, where locally connected machines are reconfigured into a parallel system. One possible data partitioning approach is the strip partition which is depicted in  FIG. 10B  where processor P, is allocated with input samples of indices (x,y), 0≦x≦W−1, nN c ≦y≦(n+1)N c . 
   The message passing mechanisms in both processor networks are modeled as follows. The communication time T, for a size-m message is
 
 T   c   =t   s   +mt   w   +t   p   (4)
 
where t s  is the time it takes to establish a connection, t p  is the propagation time, and t w , is the time to transmit a size−1 message. If one message unit is an integer, then t h , is the time to transmit one integer. Other cases are defined similarly. Notice that for the bus processor network, t p , is taken as the average propagation time and, for the mesh processor network, t p =lt h  where 1 is the number of links and t h  is the propagation time over one link.
 
   The design problem is formulated as: Given the communication model as defined above, minimize the communication overhead in a parallel DWT system. To this end, clearly we can reduce the overhead by reducing the number of communications and/or reducing the amount of data that has to be exchanged. 
   II. DWT Processing-Lifting Factorization 
   This disclosure is presented based on the tree-structured [29] multilevel octave-band wavelet decomposition system with critical sampling using a two-channel wavelet filterbank. The present techniques can be extended to many other DWT systems, including but not limited to, systems of standard DWTs [41], multichannel wavelet filterbanks, and wavelet packet decompositions. 
   A. The Standard Algorithm 
   Theoretically [42], the wavelet transform is a signal decomposition technique which projects an input signal onto a multiscale space constructed from the dilated and translated versions of a prototype wavelet function, i.e., the mother wavelet. Computationally most wavelet transforms can be implemented as recursive filtering operations using the corresponding wavelet filterbank as shown in  FIG. 1 . This implementation will be referred to as the standard algorithm hereafter. We emphasize that the filtering operations are only performed every other sample for two-channel filterbanks, i.e., a subsample-filtering approach, which is already a fast algorithm [6]. The pseudo code implementation is given in Table I. 
   
     
       
         
             
           
             
               TABLE I 
             
             
                 
             
             
               The Standard Algorithm 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               input: x[n],n ε [0,N] 
             
             
                 
                N: input sequence length 
             
             
                 
                J: decomposition level 
             
             
                 
                L: filter length 
             
             
                 
               output: y[k], k ε [0,N] 
             
             
                 
               begin 
             
             
                 
                for (j = 0;j &lt; J; j++) 
             
             
                 
                for (n = 0;n &lt; 2 J−j ;n++) 
             
             
                 
                 { 
             
             
                 
                  
                 x   j     ⁡     [   n   ]       =       ∑     m   =   0       L   -   1       ⁢           ⁢         x     j   -   1       ⁡     [   m   ]       ⁢     h   ⁡     [       2   ⁢   n     -   m     ]                           y   j     ⁡     [   n   ]       =       ∑     m   =   0       L   -   1       ⁢           ⁢         x     j   -   1       ⁡     [   m   ]       ⁢     g   ⁡     [       2   ⁢   n     -   m     ]                     
 
             
             
                 
                 } 
             
             
                 
               end 
             
             
                 
                 
             
          
         
       
     
   
   For practical applications with memory and delay constraints, the standard algorithm, however, may not be a good choice for three reasons: (i) it requires a buffer of same size as the input to store the intermediate results (the lowest subband) for recursive filtering operations; (ii) it has a large latency since all the outputs of one subband are generated before the output of the next subband; and (iii) the computation cost is high. Define algorithm computation cost as the number of multiplications and additions per output point. Using wavelet filters with L-taps, L multiplications and (L−1) additions are needed for one output point at each level. The cost Cs of the standard algorithm, for a J-level wavelet decomposition, can be computed as [6] 
   
     
       
         
           
             
               
                 
                   C 
                   M 
                   J 
                 
                 = 
                 
                   
                     
                       ( 
                       
                         L 
                         + 
                         L 
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         + 
                         
                           1 
                           2 
                         
                         + 
                         
                           1 
                           4 
                         
                         + 
                         ⋯ 
                         + 
                         
                           1 
                           
                             2 
                             
                               J 
                               - 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           L 
                         
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           2 
                           
                             - 
                             J 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
     
   
   B. The Lifting Algorithm 
   A size-N polyphase transform [42] of a signal x[n] is defined as a mapping that generates N subsequences with each being a shifted and downsampled version of x[n], i.e., x i [n]=x[nN+i]. These subsequences are called the polyphase components of signal x[n]. In the case of N=2, this transform simply divides the input sequence into two polyphase components which consist of samples from the original sequence having odd indices and even indices, respectively. In z-transform domain, the polyphase representation of x[n] is 
                   X   i     =       ∑     i   =   0       N   -   1       ⁢       z     -   i       ⁢       X   i     ⁡     (     z   N     )                   (   6   )               
The DWT computation in the polyphase domain is expressed by Equation (1), where the 2×2 polyphase matrix is P(z).
 
     FIG. 11  shows one embodiment of a two-channel filterbank where the left side is the decomposition stage and the right side is the reconstruction stage. If the determinant of P(z) is a monomial, i.e., det(P(z))=z 1 , then X can be reconstructed, without loss, as X=P −1 (z)Y with a phase shift. In this case, the corresponding filters will be called perfect reconstruction (PR) wavelet filters. 
   One advantage of the polyphase domain transform computation is that the polyphase matrix P(Z) can be further factored and the factorization leads to fast DWT algorithms [30], [31], [32], [33]. Using the Euclidean algorithm, Daubechies and Sweldens [33] have shown that the polyphase matrix P(z) of any PR FIR filterbank can be factored into a product form of elementary matrices as 
                   P   ⁡     (   z   )       =       ∏     i   =   1     m     ⁢         [         1           s   i     ⁡     (   z   )               0       1         ]     ⁡     [         1       0               t   i     ⁡     (   z   )           1         ]       ⁡     [         K       0           0         1   /   K           ]                 (   7   )               
where s i (t), t i (t) are the prediction and updating filters, respectively, at stage i. It has been shown that such a lifting-factorization based DWT algorithm is, asymptotically for long filters, twice as fast as an implementation based on the standard algorithm (Theorem 8 in [33]).
 
     FIGS. 12A and 12B  show the forward and inverse DWT using lifting factorization. 
   The elementary matrices in the lifting factorization are all triangular (upper or lower triangular) with constants in the diagonal entries. Such a choice of elementary matrices enables the implementation of the DWT to be in-place (see next section for details), a key difference with respect to other types of factorizations (e.g., the lattice factorization). While all these factorizations can reduce the DWT computation, the in-place feature can also reduce the transform memory. Consequently, the lifting algorithm is chosen as the baseline DWT algorithm for our proposed architecture designs. 
   C. Practical DWT System Design 
   For practical DWT system design under memory and delay constraints, choosing only a fast algorithm (e.g. the lifting algorithm) may not be sufficient. First, the complexity of the lifting algorithm is still linear with the size N of the input data, i.e., O(N). If a parallel system is used to further speed up the computation, the first problem to solve is that of establishing an efficient sharing of data across processors so that the correct transform can be computed at the boundaries. Second, though the in-place feature of the lifting algorithm eliminates the need for a buffer to store intermediate results, it does not address the problem of extra buffer requirement when the input data has to be transformed on a block-by-block basis. 
   The boundary processing in DWT is illustrated in  FIG. 2 . When the filter (length L) moves to the right boundary of block  1 , it needs input data samples from block  2 . In a sequential architecture, blocks  1  and  2  do not reside in memory at the same time. In a parallel architecture, blocks  1  and  2  are allocated to different processors.  FIG. 2  shows the DWT operation near block boundaries for one level decomposition. Obviously, extra buffer or communication is needed to ensure correct computations near data block boundaries. Such a problem also exists in cases of conventional linear filtering of long data sequences and can be dealt with by using either overlap-add or overlap-save techniques. However, because the DWT consists of recursive filtering operations on multilevel downsampled data, direct applications of these two existing techniques may increase significantly the cost in terms of memory and/or communication. 
   Consider a J-level wavelet decomposition with block size N and filter length L. Both overlap-add and overlap-save approaches require an extra buffer (for boundary filtering operations) of size L−2 for each level of decomposition. If the overlap is done once for all decomposition levels (the SSWT approach by Kossentini [28]), the total overlap buffer size is (2 J −1)(L−2) which increases exponentially with J. This can become significant if deep decomposition and long wavelet filters are used. An alternative is to overlap at each level. In this case, the overlap buffer size is J(L−2) for J-level decompositions. This, however, causes delay in parallel architectures since one processor has to wait the other to send new data after each level of decomposition (an approach described in [11], [24]). A third approach [25], [21] is to use boundary extension (e.g. symmetric extension) to approximate the data in the neighboring blocks. This completely eliminates the overlap buffer and also eliminates the communication for data exchanges between processors. Unfortunately, the DWT coefficients near block boundaries are computed incorrectly. 
   The above analysis thus shows the inefficiencies, in terms of memory and/or communication overhead, of DWT system designs that adopt various overlapping techniques. The overlap-state technique of the present disclosure is designed in part to overcome some of the inefficiencies in these and other techniques. The overlap-state technique performs the DWT computation across block boundaries to reduce the communication overhead in parallel architectures and the overlap buffer size in sequential architectures. 
   III. The Overlap-State Technique 
   Various aspects of the overlap-state technique for DWT computation are described in more detailed in the following. 
   A. The Finite State Machine Model 
   From the lifting point of view [43], [33], the elementary triangular matrices in the factorization in Equation (7) can be further classified as prediction/lifting and updating/dual lifting operations respectively. From a computational point of view, however, there is no big difference among these elementary matrices, each of which essentially updates one polyphase component at a time using linear convolutions. 
   Without loss of generality, a notation e i (z) to represent the elementary matrices. That is, 
               e   i     ⁡     (   z   )       =     [         1       0               t   i     ⁡     (   z   )           1         ]           
or
 
   
     
       
         
           
             
               e 
               i 
             
             ⁡ 
             
               ( 
               z 
               ) 
             
           
           = 
           
             [ 
             
               
                 
                   1 
                 
                 
                   
                     
                       s 
                       i 
                     
                     ⁡ 
                     
                       ( 
                       z 
                       ) 
                     
                   
                 
               
               
                 
                   0 
                 
                 
                   1 
                 
               
             
             ] 
           
         
       
     
   
   Let the input be X(z), with polyphase representations denoted X(z)=[X 0 (z)X 1 (z)] t  in the frequency domain and x(n)=[x 0 (n)x 1 (n)] t , in the time domain. Now define the intermediate states in the process of transformation, {X i (z), i=0, 1, . . . , 2m+1}, as 
                           X   i     ⁡     (   z   )       =       e     i   -   1       ⁢     e     i   -   2       ⁢           ⁢   ⋯   ⁢           ⁢     e   0     ⁢     X   ⁡     (   z   )                     =       ∏     j   =     i   -   1       0     ⁢         e   j     ⁡     (   z   )       ⁢     X   ⁡     (   Z   )                       =         e     i   -   1       ⁡     (   z   )       ⁢       X     i   -   1       ⁡     (   z   )                       (   8   )               
where X i (z) is the resulting signal after the first i elementary matrices have been applied. Consider one lifting stage using a lower triangular elementary matrix e i (z) to update X i (z) into X i+1 (z) as follows:
 
                         [           X   0     i   +   1                 X   1     i   =   1             ]     =       [         1       0               t   i     ⁡     (   z   )           1         ]     ⁡     [             X   0   i     ⁡     (   z   )                   X   1   i     ⁡     (   z   )             ]                   =     [             X   0   i     ⁡     (   z   )                     X   1   i     ⁡     (   z   )       +         t   i     ⁡     (   z   )       ⁢       X   0   i     ⁡     (   z   )                 ]                   (   9   )               
In this transformation step, the polyphase component X 0   i (z) is unchanged while the polyphase component X 1   i (z), is updated by adding a quantity computed from the other polyphase component. In time domain, this means that all even samples are preserved while all odd samples are updated. For an input vector X of size N (assuming N even), the state transition can be written as
 
                   [             x     i   +   1       ⁡     (   0   )                   x     i   +   1       ⁡     (   1   )                   x     i   +   1       ⁡     (   2   )                   x     i   +   1       ⁡     (   3   )               ⋮               x     i   +   1       ⁡     (     2   ⁢   k     )                   x     i   +   1       ⁡     (       2   ⁢   k     +   1     )               ⋮               x     i   +   1       ⁡     (     N   -   2     )                   x     i   +   1       ⁡     (     N   -   1     )             ]     =       [             x   i     ⁡     (   0   )                     x   i     ⁡     (   1   )       +     σ   ⁡     (   1   )                     x   i     ⁡     (   2   )                     x   i     ⁡     (   3   )       +     σ   ⁡     (   3   )                 ⋮               x   i     ⁡     (     2   ⁢   k     )                     x     i   +   1       ⁡     (       2   ⁢   k     +   1     )       +     σ   ⁡     (       2   ⁢   k     +   1     )                 ⋮               x   i     ⁡     (     N   -   2     )                     x   i     ⁡     (     N   -   1     )       +     σ   ⁡     (     N   -   1     )               ]     =     [               x   i     ⁡     (   0   )       +     σ   ⁡     (   0   )                       x   i     ⁡     (   1   )       +     σ   ⁡     (   1   )                       x   i     ⁡     (   2   )       +     σ   ⁡     (   2   )                       x   i     ⁡     (   3   )       +     σ   ⁡     (   3   )                 ⋮                 x   i     ⁡     (     2   ⁢   k     )       +     σ   ⁡     (     2   ⁢   k     )                       x   i     ⁡     (       2   ⁢   k     +   1     )       +     σ   ⁡     (       2   ⁢   k     -   1     )                 ⋮                 x   i     ⁡     (     N   -   2     )       +     σ   ⁡     (       2   ⁢   k     -   2     )                       x   i     ⁡     (     N   -   1     )       +     σ   ⁡     (     N   -   1     )               ]               (   11   )               
Let t i (z)=Σ bi   n=−ai , t i   n z −n  (a i ≧0, b i ≧0), then the updating quantity σ(n) can be computed as
 
                   σ   ⁡     (   n   )       =     {         0         n   =     2   ⁢   k                   ∑   1     ⁢       t   1   i     ⁢       x   0   i     ⁡     (     k   -   1     )                 n   =       2   ⁢   k     +   1             }             (   12   )               
If e(z) is upper triangular, then odd samples are unchanged and even samples are updated. In this case, denote s i (z)=Σ bi   n=−ai , S i   n z −n , then the updating quantity σ(n) for upper triangular matrix e(z) is
 
   
     
       
         
           
             
               
                 
                   σ 
                   ⁡ 
                   
                     ( 
                     n 
                     ) 
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             ∑ 
                             l 
                           
                           ⁢ 
                           
                             
                               s 
                               1 
                               i 
                             
                             ⁢ 
                             
                               
                                 x 
                                 1 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   l 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           n 
                           = 
                           
                             2 
                             ⁢ 
                             k 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         
                           n 
                           = 
                           
                             
                               2 
                               ⁢ 
                               k 
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                   } 
                 
               
             
             
               
                 ( 
                 13 
                 ) 
               
             
           
         
       
     
   
   An important observation is that only one polyphase component is updated at each state transition and the updating quantity σ(n) only depends on samples from the other polyphase component. When updating even samples, only odd samples are needed and vice versa. This leads to the following three conclusions for states updating at each stage: 
   1. Whenever X i  is updated into X i+1 , there is no need to keep the old value of X i  since no other updating will need it any more. In other words, every time X i  is generated, only this set of values need be stored, i.e., we do not need to know any of the previous values X j  (j&lt;i), in order to compute the final wavelet coefficients. 
   2. The updated value of each sample x i+1 (n) can be stored in the same memory space allocated for x i (n) since the old value x i (n) does not contribute to the updating of its neighbors and any later stage updating. For example, x i (1) can be over-written by x i+1 X (1) without affecting the updating of x i (3). This is the so-called in-place property of the lifting algorithm. Then, to transform a block of size of N, only a buffer of size N is needed while the standard algorithm needs a buffer of size 2N, where memory of size N is needed for the original input and the remaining N is needed for the transform outputs. 
   3. The updating of each sample x i (n) can be implemented independently from the updating of other samples. That is, there is no ordering of the updating between samples. For example, one can update x i (3) before or after the updating of x i (1) and obtain the same result. 
   For the polyphase matrix factorization, the necessary and sufficient condition for the above properties is that the elementary matrix e i (z) can only be in the form of lower/upper triangular matrices with constants on the diagonal entries. This key property of the lifting factorization guarantees that the DWT can be computed in-place. That is, each raw input data sample x(n)(initial state) is progressively updated into a wavelet coefficient (final state) using samples in its neighborhood. Thus the wavelet transform based on the polyphase factorization can be modeled as a finite state machine (FSM) in which each elementary matrix e i  updates the FSM state X i  to the next higher level X i+1 . The forward wavelet transform Y(z) can be written as 
                         Y   ⁡     (   z   )       =       P   ⁡     (   z   )       ⁢     X   ⁡     (   z   )                     =       ∏     i   =     2   ⁢   m       0     ⁢           ⁢         e   i     ⁡     (   z   )       ⁢     X   ⁡     (   z   )                       =           e     2   ⁢   m       ⁡     (   z   )       ⁢   ⋯   ⁢           ⁢           e   1     ⁡     (   z   )       ⁢           e   0     ⁡     (   z   )       ⁢       X   0     ⁡     (   z   )           ︸       X   1     ⁡     (   z   )               ︸             X   2     ⁡     (   z   )               ⋮                 ︸       Y   ⁡     (   z   )       =       X       2   ⁢   m     +   1       ⁡     (   z   )                           (   14   )               
and the corresponding inverse transform is
 
                           X   ∼     ⁡     (   z   )       =         P     -   1       ⁡     (   z   )       ⁢     Y   ⁡     (   z   )                     =       ∏     i   =   0       2   ⁢   m       ⁢           ⁢         e     -   i       ⁡     (   z   )       ⁢     Y   ⁡     (   z   )                       =           e     -   0       ⁡     (   z   )       ⁢   ⋯   ⁢           ⁢           e         -   2     ⁢   m     +   1       ⁡     (   z   )       ⁢           e       -   2     ⁢   m       ⁡     (   z   )       ⁢       Y   0     ⁡     (   z   )           ︸       X     2   ⁢   m       ⁡     (   z   )               ︸             X       2   ⁢   m     -   1       ⁡     (   z   )               ⋮                 ︸         X   ⋒     ⁡     (   z   )       =       X   0     ⁡     (   z   )                           (   15   )               
where e −i (z) is the inverse of e i (z). The schematic plot of the DWT as a FSM is depicted in  FIG. 13  [44].
 
   B. Overlap-State 
   Assume there are M elementary matrices {e i , i=0, 1, . . . , M−1} in the factorization of the polyphase matrix P(z), then there are a total of M states in the FSM defined above. The FSM modeling suggests that, to compute the transform, each and every sample x(n) has to complete its state transitions from state 0 up to state M−1 sequentially. This means that one has to compute the updating quantities {σ i (n), i=0, 1, . . . , M−1} as in Equations (13) and (14) at all these stages. Unfortunately this cannot be accomplished for samples near block boundaries when the input has to be transformed on a block-by-block basis, due to buffer size limitation, or when parallel processing is used. 
   Consider one operation across a data boundary between two blocks using an upper triangular elementary matrix. Let the current state be i and the input sequence x i (n) be segmented at point 2k ( FIGS. 14A and 14B ). In this state transition, even-indexed samples are updated using odd-indexed samples. 
   The updating quantity σ(2k) is 
                         δ   ⁡     (     2   ⁢   k     )       =       ∑   l     ⁢       δ   l   i     ⁢       x   1   i     ⁡     (     k   -   1     )                       =       ∑   l     ⁢       δ   l   i     ⁢       x   i     ⁡     (       2   ⁢   k     -     2   ⁢   l     +   1     )                       =           ∑     1   =     -     δ   i           -   1       ⁢       δ   l   i     ⁢       x   i     ⁡     (       2   ⁢   k     -     2   ⁢   l     +   1     )             ︸     C   ⁡     (     2   ⁢   k     )           +         ∑     l   =   0       δ   l       ⁢       δ   l   i     ⁢       x   i     ⁡     (       2   ⁢   k     -     2   ⁢   l     +   1     )             ︸     A   ⁡     (     2   ⁢   k     )                           (   16   )               
where C (2k) and A(2k) are the contributions from the causal and anti-causal parts of filter s i (z), respectively.
 
     FIGS. 14A and 14B  illustrate state transitions across a block boundary of blocks  1  and  2  using e i  with a i =3 and b i =2.  FIG. 14A  show the partial computations near boundaries.  FIG. 14B  shows that boundary samples stay in intermediate states after updating. The “new boundary” separates fully updated output coefficients from partially computed ones. Thus, C (2k) and A(2k) for the samples near the block boundary cannot be computed due to the segmentation. As a result, σ(2k) will not be available. These samples cannot be updated into state i+1.  FIG. 14A  shows, for example, x i (2 k) in block  1  cannot be updated into x i+1 (2k) since {x i+1 (2k+1), x i+1 (2k+3), x i+1 (2k+5)} are in the block  2  and thus are not available at the time block  1  is transformed. 
   Consequently, σ(2 k), the updating factor for sample x i (2k) cannot be computed to obtain the updated x i+1 (2k). Rather than leaving x i (2 k) in state i, it is partially updated as X′ i  (2k)=x i (2k)+C(2k) in the present overlap-state technique since C(2k) can be computed from the causal neighborhood as a function of {x i (2k−1), x i (2k−3), x i (2k−5)}. The significance of this partial updating is that one can free all the samples in the causal past for future processing and save memory. In this case, samples {x i (2k−1), x i (2k−3), x i (2k−5)} do not need to be buffered for the fully updating of x i (2k) since their contribution C(2k) has already been added to the partially updated x′ i (2k) in the form of x i (2k)+C(2k). On the other hand, if x i (2k) is not partially updated, then {x i (2k−1), x i (2k−3), x i (2k−5)} have to be buffered. The same partial updating happens also for samples {x i (2k−2), x i (2k−4)} in the left block and samples {x i (2k+2), x i (2k+4)} in the right block. 
   The complete state transition from i to i+1 requires buffering the following samples in each block: 
   1. Partially updated samples such as {x′ i (2k), x′ i (2k−2), x′ i (2k−4)} in the left block and {x′ i (2k+2), x′ i (2k+4)} in the right block. 
   2. Contributing samples required by partially updated samples (in the other block) to complete the transform, such as {x i (2m−1), x i (2m−3)} in the left block and {x i (2m+1), x i (2m+3), x i (2m+5)} in the right block. 
   Notice that these partially updated samples are not exactly the state information as defined before in the FSM definition. For simplicity, however, these partially updated samples and contributing samples will be all called the state information hereafter. Obviously, as long as the state information is preserved at each stage, the transform can be completed at any later time. That is exactly what a FSM is supposed to do. 
   Such a later processing is possible because partial updating in the right block for updating {x i (2m+2), x i (2m+4)} can be implemented independently from the partial updating of {x i (2m), x i (2m−2), x i (2m−4)} in the left block. The partial updating does not remove any information needed by the other block, since it updates samples that are not inputs at the i-th state transition stage. The end state after application of e i  is shown in  FIG. 14B . Because partially updated samples cannot be used for processing, the size of the segment over which can be computed is reduced, so that the effective boundary is now reached before sample x i+1 (2k-4) in block  1  and sample x j- 41(2k+6) in block  2 . Effectively, the physical boundary splits into two and extends inwards in both blocks. The next state transition via e i+1  will operate only on samples in state i+1. All the samples between the two new boundaries become the state information and the same procedure repeats at each state transition stage. 
   The state information in neighboring blocks has to be exchanged to complete the transform for samples near the block boundary. According to the present overlap-state technique, this can be done by overlapping the states between consecutive blocks. Thus DWT computation can be performed across consecutive data blocks.  FIG. 15  illustrates this overlap-state procedure in the sequential processing.  FIG. 16  illustrates the overlap-state procedure in the parallel processing. Though only one state transition is shown in these two figures, the overlap-state design can be easily generalized to multiple state transitions at multiple decomposition levels because all these state transitions share the same three properties. 
   C. Performance Analysis 
   C.1 Buffer Size Analysis 
   This section evaluates the savings on the overlap buffer size produced by the overlap-state technique in comparison with other DWT processing techniques. Buffer size is an important consideration for memory constrained sequential architecture design. 
   As shown before, at each stage, the partially updated samples and contributing samples need to be stored. Denote the total number of partially updated samples as B 1   i  and the total number of contributing samples as B 2   i . Writing s i (z) and t i (z) as 
   
     
       
         
           
             
               
                 
                   
                     δ 
                     i 
                   
                   ⁡ 
                   
                     ( 
                     z 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         ai 
                       
                       
                         b 
                         i 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           δ 
                           i 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       ⁢ 
                       
                         z 
                         
                           - 
                           n 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           t 
                           i 
                         
                         ⁡ 
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           a 
                           i 
                         
                       
                       
                         b 
                         i 
                       
                     
                     ⁢ 
                     
                       
                         
                           δ 
                           i 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       ⁢ 
                       
                         z 
                         
                           - 
                           n 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 17 
                 ) 
               
             
           
         
       
     
   
   where a i ≧0 and b i ≧0. Then B 1   i =a i  and B 2   i =b i . The number of samples that must be buffered at stage i, Bi, is B i =a i +b i . Assume there are N state transitions in the factorization of P(z), the buffer size B s  for one level decomposition is 
                         B   s     =       ∑     i   =   0       N   -   1       ⁢     B   i                   =       ∑     i   =   0       N   -   1       ⁢           ⁢     (       a   i     =     b   i       )                     (   17   )               
Since the lifting factorization of a given polyphase matrix is not unique, obviously one would choose the factorization which gives the minimum B s  if the amount of memory is limited. An alternative way to find out the buffer size is to graphically plot the state transitions for a given factorization.
 
   C.2 Communication 
   The communication delay is the time used for exchanging data between adjacent processors. In a number of prior parallel algorithms [11], [24], before each level of decomposition, (L−2) boundary samples need to be communicated to the adjacent processors (L is the filter length). The total communication time D old  for a J level wavelet decomposition, can be calculated as
 
 D   old   =J ( t   s +( L− 2) t   w   +t   p )  (18)
 
   In the present parallel algorithm based on the overlap-state technique, the data exchange can be delayed after the independent transform of each block so that only one communication is necessary, the size of the state information at each stage B s  is upper bounded by (L−2). So the communication time is bounded by an upper limit:
 
 D   new   ≦t   s   +J ( L− 2) t   w   +t   p .  (19)
 
Hence, the communication time is reduced.
 
     FIG. 17  shows an example of three level decompositions based on the overlap-state in the parallel processing. Two consecutive blocks  1  and  2  are first processed independently without any exchange between them. The samples represented by solid lines are completely updated. The samples represented by dashed lines are partially updated and need data from the other block to complete the computation. After the computations in each block are completed, the state information of the partially updated samples in the block near the boundary is transferred. Next, the computations on the partially updated samples in the block  1  are completed based on the information from the block  2 . 
   The overlap-state technique can reduce the communication overhead in the proposed parallel algorithm by reducing the number of communications and by reducing the amount of data exchanged between blocks. Essentially, the overlap-state technique uses a single communication setup to exchange all necessary data to complete the DWT computation rather than exchanging a small amount of data in multiple communication setups. It is, however, important to emphasize that how much this communication overhead reduction contributes to the reduction of the total computation time strongly depends on the parallel system communication link design. Clearly, the slower the inter-processor communication, the larger the gain and vice versa. 
   D. Delayed Normalization 
   Although the lifting based DWT algorithm has been shown to be twice as fast as the standard algorithm by Daubechies and Sweldens, this is only true in general asymptotically for long filters [33]. This section describes a simple technique, Delayed Normalization to reduce the computation of multilevel wavelet decompositions. 
   Referring to Equation (7), the last matrix factor in the polyphase factorization is a normalization factor which scales the lowband and highband coefficients, respectively. 
   This normalization factor will appear at each level of decomposition for a multilevel wavelet decomposition. Since the wavelet transform is a linear operation and multiplication is commutative for linear operations, this normalization (multiplication) operation can actually be delayed until the last level decomposition. By doing so, computations can be saved. 
     FIGS. 18A and 18B  show an example of the delayed normalization in a three-level octave-band wavelet decomposition.  FIG. 18A  shows the recursive two-channel lifting and  FIG. 18B  shows the two channel lifting with delayed normalization. Interestingly, normalization operations for all the y 1  coefficients can be all eliminated provided that the same wavelet filterbanks are applied at each stage. If a different wavelet filterbank is used at each level of decomposition, then in general only one normalization (multiplication) operation is necessary for each wavelet transform coefficient. Obviously such a delayed normalization technique can also be used for multidimensional wavelet decompositions and wavelet packet decompositions. 
     FIGS. 19A and 19B  show a one-level 2D wavelet decomposition with recursive normalization and delayed normalization, respectively. 
   Turning to the performance analysis for 1D octave-band wavelet decomposition, let the input data sequence length be N and the decomposition level be J. The computational costs of the standard algorithm, the lifting scheme, and the lifting scheme with delayed normalization are denoted respectively as C M   J , C L   J , and C L′   J . The cost unit is the average number of multiplications and additions per output point. Then 
                     C   M   J     =         C   M   1     ⁡     (     1   +     1   2     +     1   4     +   ⋯   +     1     2     J   -   1           )       =     2   ⁢       C   M   1     ⁡     (     1   -     2     -   J         )             ,           (   20   )               
where C M   J  is the number of multiplications and additions per output point for one level decomposition using the standard algorithm. Accordingly, the lifting cost is
 
                   C   L   J     =     2   ⁢         C   L   1     ⁡     (     1   -     2     -   J         )       .               (   21   )               
For the lifting scheme with delayed normalization, the whole wavelet transform can be decomposed into two parts. One is the normal lifting operation part which lasts for J levels without normalization. For this part the one-level average cost is C L′   1 =C L   1 −1 since one normalization/multiplication is saved for each coefficient. The second part is the final normalization part for all the coefficients. This part incurs cost 1 (one multiplication) per output point. So the total average cost is
 
                   C     L   &#39;     1     =         2   ⁢     (       C   L   1     -   1     )     ⁢     (     1   -     2     -   J         )       +   1     =       C   L   J     +     2     -     (     J   -   1     )         -   1.               (   22   )               
If N is large enough such that J can be large enough, then in the limit C L′   1  is on an average one operation fewer than that of a pure lifting scheme. Table II compares costs for different multilevel wavelet decompositions by using the same filters based on Daubechies and Sweldens [33].
 
   
     
       
         
             
           
             
               TABLE II 
             
           
          
             
                 
             
             
               COSTS COMPARISON FOR MULTILEVEL WAVELET 
             
             
               DECOMPOSITIONS (J &gt; 3) 
             
          
         
         
             
             
             
          
             
                 
                 
               Lifting With 
             
             
                 
                 
               Delayed 
             
             
                 
               Lifting 
               Normalization 
             
          
         
         
             
             
             
             
             
             
          
             
               Wavelet 
               Standard 
               Cost 
               speedup 
               cost 
               speedup 
             
             
                 
             
          
         
         
             
             
             
             
             
             
          
             
               Haar 
               1.5 
               1.5 
                0% 
               1.5 
                0% 
             
             
               D4 
               7 
               4.5 
               56% 
               3.5 
               100% 
             
             
               D6 
               11 
               7 
               57% 
               6 
                83% 
             
             
               (9-7) 
               11.5 
               7 
               64% 
               6 
                92% 
             
             
               (4,2)B-spline 
               8.5 
               5 
               70% 
               4 
               113% 
             
             
                 
             
          
         
       
     
   
   The above performance analysis applies for transforms with different wavelet filters at each stage. It is assumed that J is large enough such that 2 −(J−1)  is negligible. If the same filterbank is used at all decomposition stages, the assumption can be further relaxed. 
   Recall that the normalizations for y 1  coefficients can all be eliminated (see  FIG. 17 ). The savings is 0.25 since one-quarter of the total input data samples do not have to be scaled. Thus average cost of the normalization part should be 0.75 rather than 1 per output point. Taking this into consideration, as long as J is large enough such that 2 −(J−1) ≦0.25, the above cost estimation C L′   1 , is accurate. That is equivalent to having J&gt;3 which is a reasonable assumption for most practical wavelet applications. 
   Further reduction of the normalization operation is possible if the DWT system and the immediate data processing system can be jointly designed for this purpose. For example, in a wavelet data compression system, wavelet coefficients can be quantized immediately after transform.  FIG. 20A  shows one embodiment of such a system. The quantizers Q i  (i=0, 1, 2, 3) are designed for wavelet coefficients in different subbands. 
   The normalization operation can be done jointly with this quantization operation and thus can be completely eliminated from the transform operation.  FIG. 20B  shows a design with a joint transformation and quantization. For other applications, such as noise reduction using thresholding, this computation reduction is also possible. Compared to independent transform and quantization, computation can be saved if designed jointly. 
   IV. Exemplary DWT Architectures 
   This section describes exemplary sequential and parallel architecture designs for 1D DWT using the Overlap-State technique. Variations are then detailed for 2D separable DWT systems. 
   A. 1D Systems 
     FIG. 21  shows a sequential DWT system, with the corresponding pseudo code sequential algorithm given in Table III. The input data sequence is first segmented into non-overlapping blocks of length N and fed into the FSM one block at a time. The state information is saved so that after one block has been computed the next one can use it. After transformation, the wavelet coefficients are concatenated together to give the final result. 
   
     
       
         
             
           
             
               TABLE III 
             
             
                 
             
             
               The Proposed Sequential DWT algorithm 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               begin 
             
          
         
         
             
             
          
             
                 
               initialize state S; 
             
          
         
         
             
             
          
             
                 
               for (k=0;k&lt;N;k++) 
             
          
         
         
             
             
          
             
                 
               { 
             
             
                 
                J-level wavelet transform for block k; 
             
             
                 
                update state S; 
             
             
                 
               } 
             
          
         
         
             
             
          
             
                 
               end 
             
             
                 
                 
             
          
         
       
     
   
   The DWT/FSM acts as a state machine with memory and the state information (partially computed boundary samples from the previous block) at multiple decomposition level are overlapped. This helps to reduce the memory requirement for the transform computation. This overlap leads to output delay in practice, i.e., the n output samples shown in  FIG. 21  are delayed relative to the n input samples. 
   Table IV shows the required overlap buffer sizes (B s ) of different sequential DWT algorithms for comparison. For an N-point input data block, if the lifting algorithm is implemented, the total buffer is N+B s . The system efficiency η is 
           η   =       N     N   +     B   s         .           
Hence, the proposed sequential DWT algorithm, using the overlap-state technique, requires a smaller overlap buffer size B s . This improves the system throughput. However, if N&gt;&gt;O (JL) then the relative improvement becomes small. On the other hand, if N=0 when all completely transformed coefficients are immediately transferred (e.g., the line-based system in [12], [35]), the savings in memory can be significant (details are given in the next section).
 
   
     
       
         
             
           
             
               TABLE IV 
             
           
          
             
                 
             
             
               Overlap Buffer Size B s  in 1D DWT for J-Level Decompositions 
             
             
               Using a L-Tap Wavelet Filterbank 
             
          
         
         
             
             
             
             
          
             
                 
               SSWT[28] 
               RPA[7] 
               Proposed 
             
             
                 
                 
             
          
         
         
             
             
             
             
             
          
             
                 
               L-tap 
               (2 J  − 1)(L-2) 
               J(L-2) 
               ≦J(L-2) 
             
             
                 
               (9,7) 
               7(2 J  − 1) 
               7J 
               4J 
             
             
                 
                (2,10) 
               8(2 J  − 1) 
               8J 
               4J 
             
             
                 
               CDF(4,2) 
               5(2 J  − 1) 
               5J 
               3J 
             
             
                 
                 
             
          
         
       
     
   
     FIG. 22  shows an example of the parallel DWT architecture based on the overlap-state technique. The corresponding pseudo code algorithm is given in Table V. The input data is uniformly segmented non-overlapping blocks and allocated onto p available processors. Each processor computes its own allocated data up to the required wavelet decomposition level. This stage is called Split. The output from this stage includes (i) completely transformed coefficients and (ii) the state information (partially updated boundary samples). In the second stage, Merge, a one-way communication is initiated and the state information is transferred to the neighboring processors. The state information from the neighbor processor is then combined together with its own corresponding state information to complete the whole DWT transform. 
   
     
       
         
             
           
             
               TABLE V 
             
             
                 
             
             
               The Proposed Parallel DWT Algorithm 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               begin {transform in processor p} 
             
          
         
         
             
             
          
             
                 
               for(j=0;j&lt;J;j++) 
             
          
         
         
             
             
          
             
                 
               { 
             
          
         
         
             
             
          
             
                 
               transform at current level j. 
             
             
                 
               store state information. 
             
          
         
         
             
             
          
             
                 
               } 
             
          
         
         
             
             
          
             
                 
               send state information to processor p+1 
             
             
                 
               receive state information from processor p−1; 
             
             
                 
               for (j=0;j&lt;J;j++) 
             
          
         
         
             
             
          
             
                 
               { 
             
          
         
         
             
             
          
             
                 
               transform boundary data samples at current level j. 
             
          
         
         
             
             
          
             
                 
               } 
             
          
         
         
             
             
          
             
                 
               end 
             
             
                 
                 
             
          
         
       
     
   
   The proposed parallel architecture only requires one communication between neighboring processors for J-level decompositions. The amount of data exchanged is also less than that in direct overlapping approaches [11], [24]. Therefore, the communication delay is reduced. 
   B. 2D Systems 
     FIG. 23  shows an example 2D DWT with two level decompositions based on the overlap-state technique. The data is row transformed first and then column transformed. Data samples along block boundaries cannot be fully transformed due to the lack of neighboring samples. These constitute the row and column state information at each level. The shaded areas represent the state information with RO/CO the row/column state information at level 0 and so on. The input block is first row transformed as shown in (a) and column transformed as shown in (b) at 1a eve1 0. The input block is then downsampled (taking the LLO subband) and row transformed at level 1 as shown in (c), and column transformed at level 1 shown in (d). 
   The following notations are used. The width and the height of the data block are represented by N r , N c , respectively. For decomposition level j=0, 1, . . . , J−1, the following are defined: 
   {W ro   j , W ri   j }: the numbers of partially transformed samples near left and right boundaries respectively in a row. {W co   j , W c1   j } are defined similarly for a column. 
   {N r   j , N c   j }: the length of a row and a column respectively before the start of the decomposition at each level. 
   {M r   j , M c   j }: the numbers of completely transformed samples in a row and a column respectively. 
   B a   j : the total number of partially updated samples, i.e., the size of the buffer to hold the state information for further processing. 
   The following relationships can be established between these quantities: 
   
     
       
         
           
             
               
                 
                   
                     M 
                     r 
                     j 
                   
                   = 
                   
                     
                       N 
                       r 
                       j 
                     
                     - 
                     
                       W 
                       
                         r 
                         ⁢ 
                         0 
                       
                       j 
                     
                     - 
                     
                       W 
                       
                         r 
                         ⁢ 
                         1 
                       
                       j 
                     
                   
                 
               
             
             
               
                 
                   
                     M 
                     c 
                     j 
                   
                   = 
                   
                     
                       N 
                       c 
                       j 
                     
                     - 
                     
                       W 
                       
                         c 
                         ⁢ 
                         0 
                       
                       j 
                     
                     - 
                     
                       W 
                       
                         c 
                         ⁢ 
                         1 
                       
                       j 
                     
                   
                 
               
             
           
           ⁢ 
           
             
 
           
         
       
     
     
       
         
           
             
               
                 
                   N 
                   r 
                   j 
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 M 
                                 
                                   j 
                                   - 
                                   1 
                                 
                               
                               / 
                               2 
                             
                             ] 
                           
                         
                       
                       
                         
                           
                             N 
                             r 
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           
                             j 
                             ≥ 
                             1 
                           
                         
                       
                       
                         
                           
                             j 
                             = 
                             0 
                           
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 
                   N 
                   c 
                   j 
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 M 
                                 r 
                                 
                                   j 
                                   - 
                                   1 
                                 
                               
                               / 
                               2 
                             
                             ] 
                           
                         
                       
                       
                         
                           
                             N 
                             c 
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           
                             j 
                             ≥ 
                             1 
                           
                         
                       
                       
                         
                           
                             j 
                             = 
                             0 
                           
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 
                   B 
                   s 
                   j 
                 
                 = 
                 
                   
                     
                       N 
                       r 
                       j 
                     
                     ⁢ 
                     
                       N 
                       c 
                       j 
                     
                   
                   - 
                   
                     
                       M 
                       r 
                       j 
                     
                     ⁢ 
                     
                       M 
                       c 
                       j 
                     
                   
                 
               
             
           
         
       
     
   
   Upon completion of all J-level decompositions, we have 
                   B   s     =       ∑     j   =   0       J   =   1       ⁢           ⁢     B   s   j                     B   e     =     B   -     B   s                   =         N   s     ⁢     N   e       -       ∑     j   =   0       J   -   1       ⁢     B   s   j                     
where B s  is the total buffer size necessary to store the state information at all decomposition levels and B e  is the effective block size, i.e., number of wavelet coefficients that can be transferred to the next stage for processing, thus freeing up memory.
 
   C. Sequential Architectures 
   Sequential architectures can have a strip sequential configuration or a block sequential configuration. 
     FIG. 24  shows an example of the strip sequential configuration. In this case, the buffer is organized to hold one strip of data at a time. Hence, B=WN c  or B=N r H, where W×H is the original data size.  FIG. 24  shows the former case. Because the input data is segmented only in column direction, state information (partially transformed samples) will only appear along the column direction. Certainly, some type of boundary extension technique, such as the symmetric extension, has to be used for the transform near the left and right row boundaries. For the transform of the very first strip, extension is also needed for the upper rows of each column. Each strip takes over the state information left by the previous strip to transform its own data. Upon completion, it also generates the state information for the next strip. Then the strip slides down and the DWT is calculated strip-by-strip with state information overlapped between strips. 
   A blown-up version of the state information is shown in the bottom part of  FIG. 24 . For a J-level decomposition, the state buffer size B s  can be calculated as 
                   B   s     =       ∑     j   =   0       J   =   1       ⁢     B   s   j                   =       ∑     j   =   0       J   -   1       ⁢       WW     e   ⁢   1     j     ⁢     2     -   j                       
Thus, B s  is proportional to the row length W for the case depicted in  FIG. 24 . To reduce the state buffer size, the segmentation should choose the dimension with large data size. That is, if W&gt;H, then segment along row direction and segment along column direction if otherwise.
 
   
     
       
         
             
           
             
               TABLE VI 
             
           
          
             
                 
             
             
               Comparison of Memory Requirements Where W is the Width of the 
             
             
               Image Scaline and α = (2 j  − 1), β = (1 − 2 −j ) 
             
          
         
         
             
             
             
             
          
             
                 
               SSWT[28] 
               RPA[7] 
               Proposed 
             
             
                 
                 
             
          
         
         
             
             
             
             
             
          
             
                 
               L-tap 
               Wα(L-2) 
               2Wβ(L-2) 
               ≦2Wβ(L-2) 
             
             
                 
               (9,7) 
               7Wα 
               14Wβ 
               8Wβ 
             
             
                 
               (2,10) 
               8Wα 
               16Wβ 
               8Wβ 
             
             
                 
               CDF(4,2) 
               5Wα 
               10Wβ 
               6Wβ 
             
             
                 
                 
             
          
         
       
     
   
   Table VI shows comparisons of the present algorithm with other algorithms for the minimum memory requirements. Evidently, the present system can produce significant memory savings. Consider a color image size of 4096×4096 where each color component sample is stored as a 4 bytes floating point number for DWT computation. In this case, one image scanline requires 48 KB. Using the Daubechies (9,7) wavelet filterbank (L=9), for a 3-level decomposition, the total memory would be 588 KB if using the RPA algorithm (the approach given in [12], [35]). Using the overlap-state technique, the buffer size can be reduced to 296 KB. 
     FIG. 25  shows a block sequential according to one embodiment. The buffer is divided into two parts, one for holding the state information and one for new input samples in the sliding transform window. Here, B=B s +N r N c . 
   The data is segmented into blocks of size N r N c , and is transformed one block at a time. Since boundary extensions can be applied for the left and up boundaries of the very first block A, state information {A r , A c } will appear only on the right and down side of the block upon completion of the transform. The {A r , A c } correspond respectively to the partially transformed row and column samples. When the window slides right to the position of block B only the row state information A r  can be overlapped. This shows that A r  can be fully transformed by overlapping while A, has to be buffered for later processing. As for block A, the column state information generated by block B also has to be buffered. This process continues until the completion of transforms of all the blocks in the first block row. By that time, the column state information has accumulated to the size B s  exactly same as that of the sequential strip DWT. 
   The state buffer size B s  does not increase beyond this point. This can be verified by checking the first block C in the second block row. For clarity of illustration, the second row is drawn separately from the first block row in  FIG. 25 . Actually, the two block rows are overlapped with the part of the state information. That is, block C takes over the state information A c  to complete the transform. When the transform stops, state information also appears on the right and down boundaries of block C. However, since A c  has now been fully transformed and hence can be transferred out, C c  can be written into the locations of A c  without increase of the total state buffer size B s . 
   The most general case of sequential block DWT is depicted for block D. The block D overlaps with previously generated state information in both the row and column directions, {C r , B c }. When it finishes its transform, it leaves {D c , D r } for later processing. The transform of block E in the last block row is the same as that of D except that boundary extension can be used in the column direction. 
   The system efficiency can be evaluated by determining the buffer size to completely transform a block of data of size N×N. This is typical in a transformed-based image coding application where images are coded on a block-by-block basis. Assume the buffer is of size N B ×N B . 
   Table VI shows the comparison, where N B  is given for J-level wavelet decompositions using different wavelet filterbanks and overlapping techniques. Take the Daubechies (9,7) filterbank as an example and assume the decomposition level is J=3. If the block size is of 32×32, then N=32. Using SSWT, then N B =32+49=81 which means a buffer size of 81×81 is needed to compute the DWT of a data block 32×32. The efficiency η in this case is approximately 16%. Using RPA, then N B =45 and the efficiency increases to 50%. If we use the overlap-state technique, then N B =39 and the efficiency increases to 64%. 
   
     
       
         
             
           
             
               TABLE VI 
             
           
          
             
                 
             
             
               Comparison of Memory Requirements Where W is the Width of the 
             
             
               Image Scaline and α = (2 j  − 1), β = (1 − 2 −j ) 
             
          
         
         
             
             
             
             
          
             
                 
               SSWT[28] 
               RPA[7] 
               Proposed 
             
             
                 
                 
             
          
         
         
             
             
             
             
             
          
             
                 
               L-tap 
               Wα(L-2) 
               2Wβ(L-2) 
               ≦2Wβ(L-2) 
             
             
                 
               (9,7) 
               7Wα 
               14Wβ 
               8Wβ 
             
             
                 
               (2,10) 
               8Wα 
               16Wβ 
               8Wβ 
             
             
                 
               CDF(4,2) 
               5Wα 
               10Wβ 
               6Wβ 
             
             
                 
                 
             
          
         
       
     
   
   D. Parallel Architectures 
   As shown before, in the first phase Split each processor is allocated with its portion of data and starts the transform all the way to the required decomposition level J. Upon completion, the data configuration at each processor is shown in  FIG. 26(   a ). The center part of each block is completely transformed while the boundaries are left with the partially transformed samples, i.e., the state information. The next stage Merge is to communicate the state information and complete the transform for boundary samples. If the single-port model is used, then three communications are necessary to complete the transform, one for row state, one for column and one for the intersection of row and column state. However, if the multi-port model is used, the row and column state information exchanges can be implemented simultaneously thus reducing one communication. This Merge process is shown in  FIG. 26  from (a) to (d) for the single-port model. If the multi-port model is used, (a) and (b) can be combined to simultaneously transmit/receive the row and column state information to/from neighboring processors. This is in contrast to the observation given in [15] that “The 2D DWT algorithms seem not able to effectively utilize more than a single communication port at a time”, our analysis shows that using multi-port model, the communication overhead can actually be reduced compared to that of a single-port model. 
   In the strip parallel configuration, each processor is allocated with its own strip and transforms up to the required level of decomposition J in the first stage Split. Since no segmentation is done in the row direction, state information obviously will only appear along up and down boundaries in each block. This is shown in  FIG. 27 . Next, in the Merge stage, only one communication is necessary to transfer/receive the column state information from neighboring processors. 
   V. Experimental Results 
   In this section, experimental results are provided to show the computation reduction using the Delayed Normalization technique in sequential lifting algorithms. Results are also given for the parallel DWT system using the Overlap-State technique. The wavelet filterbank used is the one using Daubechies (9,7) filters. The input image is of size 512×512. 
   A. Delayed Normalization 
   In this experiment, three DWT algorithms using the (9,7) filters are implemented. 
   1. The recursive standard algorithm (see Table I). The computation cost is 11.5 mults/adds per output point. 
   2. Lifting DWT algorithm. The computation cost is 7 mults/adds per output point. 
   3. Lifting DWT algorithm, which delays the normalization until the last level of decomposition. 
   The computation cost is approximately 6 mults/adds per output point. 
   
     
       
         
             
           
             
               TABLE VIII 
             
           
          
             
                 
             
             
               DWT CPU Time of Different Sequential Algorithms (In Seconds) 
             
          
         
         
             
             
             
          
             
                 
                 
               Lifting with 
             
             
                 
               Lifting 
               Delayed Normalization 
             
          
         
         
             
             
             
             
             
             
          
             
               Level 
               Standard 
               time 
               speedup 
               time 
               speedup 
             
             
                 
             
             
               1 
               0.2398 
               0.2088 
               15% 
               0.1980 
               21% 
             
             
               2 
               0.2887 
               0.2466 
               17% 
               0.2294 
               26% 
             
             
               3 
               0.2988 
               0.2527 
               18% 
               0.2303 
               30% 
             
             
               4 
               0.3035 
               0.2598 
               17% 
               0.2368 
               28% 
             
             
               5 
               0.3098 
               0.2601 
               19% 
               0.2375 
               30% 
             
             
                 
             
          
         
       
     
   
   In the experiment, 2D separable wavelet transforms are implemented. The algorithms are tested on a ULTRA-1 SUN workstation with clock speed 133 MHz. The algorithm running CPU time is measured using the clock( ) function call in C. The average CPU time over 50 running instances of each algorithm are listed in Table VIII. To compare the performances, the standard algorithm is used as the basis and the relative speedup is calculated as T standard / T neu −1. 
   Two observations can be seen from the experiment results. One is that the lifting scheme coupled with delayed scaling can have about 30% improvement over the standard algorithm for over three-level decompositions while lifting alone only gives about 20% improvement. Second, neither lifting algorithms achieve the performance gain as predicted. The second observation actually suggests that the number of multiplications/additions in an algorithm is not the only factor contributing to the total DWT running time. The algorithm speed may also be affected by how efficiently the pseudo code is written and the memory usage too. Obviously, this is a very important, factor to consider when building a real DWT system beyond that of reduction of numbers of multiplications and additions. 
   B. Strip Parallel 
   In this experiment, three different parallel DWT algorithms are implemented and tested against a sequential DWT algorithm. 
   1. Sequential lifting algorithm. 
   2. Each processor computes the DWT using the standard algorithm. Data exchanges between processors follow the direct overlapping technique, i.e., processors exchange data at each level of decompositions [11], [24]. 
   3. Each processor computes the DWT using the fast lifting algorithm. Data exchanges between processors follow the direct overlapping technique, i.e., processors exchange data at each level of decompositions [11], [24]. 
   4. Each processor computes the DWT using the fast lifting algorithm. Data exchanges between processors follow the proposed overlap-state technique. 
   The first issue in parallel system designs is how to allocate data to different processors. In this experiment, the strip partition strategy [11] is adopted for its simplicity and its appropriateness for the parallel system used in the experiment. The 512×512 image is segmented into two strips with size 256×512, each of which is loaded into one machine for transform. The parallel platform is LAM 6.1 from Ohio Supercomputer Center, which runs over Ethernet connected SUN ULTRA-1 workstations. Two workstations are used to simulate a parallel system with two processors. The algorithm running time is measured using the MPI_Wtime( ) function call from MPI libraries. The C-code algorithm is shown in Table IX. The relative speedup is calculated against the sequential lifting algorithm as T seq /T para −1. The algorithms are tested in 50 running instances and the average DWT running times for different decomposition levels are given in Table X. 
   
     
       
         
             
           
             
               TABLE IX 
             
             
                 
             
             
               The Proposed Parallel DWT Algorithm 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               MPI_Barrier(MPI_COMM_WORLD); 
             
             
                 
               start = MPI_Wtime( ); 
             
             
                 
               begin {transform in processor p} 
             
          
         
         
             
             
          
             
                 
               for (j=0;j&lt;J;j++) 
             
          
         
         
             
             
          
             
                 
               { 
             
          
         
         
             
             
          
             
                 
               transform at current level j. 
             
             
                 
               store state information. 
             
          
         
         
             
             
          
             
                 
               } 
             
          
         
         
             
             
          
             
                 
               send state information to processor p+1; 
             
             
                 
               receive state information from processor p−1; 
             
             
                 
               for (j=0;j&lt;J;j++) 
             
          
         
         
             
             
          
             
                 
               { 
             
          
         
         
             
             
          
             
                 
               transform boundary data samples at current level j. 
             
          
         
         
             
             
          
             
                 
               } 
             
          
         
         
             
             
          
             
                 
               end 
             
             
                 
               MPI_Barrier(MPI_COMM_WORLD); 
             
             
                 
               finish_MPL_Wtime( ); 
             
             
                 
               cputime=finish−start; 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
               TABLE X 
             
           
          
             
                 
             
             
               DWT Running Time of Different Parallel Algorithms (In Seconds) 
             
          
         
         
             
             
             
             
             
          
             
                 
               Sequen- 
               Parallel Standard 
               Parallel Lifting 
               Parallel Proposed 
             
          
         
         
             
             
             
             
             
             
             
             
          
             
               Level 
               tial 
               time 
               speedup 
               time 
               speedup 
               time 
               speedup 
             
             
                 
             
             
               1 
               0.3638 
               0.3115 
               17% 
               0.2745 
               33% 
               0.2045 
               78% 
             
             
               2 
               0.3649 
               0.3275 
               11% 
               0.2899 
               26% 
               0.2338 
               56% 
             
             
               3 
               0.3952 
               0.3490 
               13% 
               0.2938 
               34% 
               0.2359 
               67% 
             
             
               4 
               0.4028 
               0.3513 
               15% 
               0.3041 
               34% 
               0.2383 
               69% 
             
             
               5 
               0.4041 
               0.3675 
                9% 
               0.3165 
               28% 
               0.2417 
               67% 
             
             
                 
             
          
         
       
     
   
   It can be seen from the results that our proposed parallel algorithm can significantly reduce the DWT computation time even compared with the fastest available parallel algorithm, parallel lifting algorithm. Notice that the improvement is not linear with the increase of the decomposition level. The reason is that, though the communication overhead increases with the decomposition level, the total numerical computation also increases. Another interesting observation is that, even at one level decomposition the proposed algorithm still outperforms the parallel lifting algorithm. This is because though the two algorithms both require one data exchange between processors, the amount of data exchanged is different. For the (9,7) filters, the proposed algorithm only needs to exchange approximately half the amount necessary in the parallel lifting algorithm. 
   VI. Some DWT/FSM Examples 
   Daubechies (9,7) Filters 
   This filterbank has been used extensively in the image compression algorithms proposed in the literature. The factorization of the analysis polyphase matrix (adapted from [33]) is 
               P   a     ⁡     (   z   )       =           ⁢             [         Ϛ       0           0         1   /   Ϛ           ]     ⁡     [         1         δ   ⁡     (     1   -     z     -   1         )               0       1         ]       ⁡     [         1       0             γ   ⁡     (     1   +   z     )           1         ]       ⁡     [         1         β   ⁡     (     1   +     z     -   1         )               0       1         ]       [           ⁢         1       0             α   ⁡     (     1   +   z     )           1         ]           
where α=−1.58613434; β=−0.0529801185;
 
γ=0.882911076; δ=0.443506852; and ζ=1.149604398
 
   Based on this factorization, the forward transform is
 
 x   0   0 ( n )= x (2 n )
 
 x   1   0 ( n )= x (2 n+ 1)
 
 x   1   1 ( n )= x   1   0 ( n )+α( x   0   0 ( n )+ x   0   0 ( n+ 1))
 
 x   0   1 ( n )= x   0   0 ( n )+β( x   1   1 ( n )+ x   1   1 ( n− 1))
 
 x   1   2 ( n )= x   1   1 ( n )+γ( x   0   1 ( n )+γ( x   0   1 ( n )+ x   0   1 ( n+ 1))
 
 x   0   2 ( n )= x   0   1 ( n )+δ( x   1   2 ( n )+ x   1   2 ( n− 1))
 
 x   0   3 ( n )=ζ x   0   2 ( n )
 
 x   1   3 ( n )= x   1   2 ( n )/ζ
 
   The (9,7) wavelet filters can be used to transform a raw input sample into a wavelet coefficient by a total of 4 state transitions. This process is shown in  FIG. 28 . Assume there are 9 samples, {x 0 (0), x 0 (1), x 0 (2), . . . , x 0 (8)}, loaded in the memory initially. The first elementary matrix e 0 (z) is lower triangular, so the state transition is to update odd samples with two neighboring even samples. For example, x 0 (1) is updated into x 1 (1)=x 0 (1)+α(x 0 (0)+x 0 (2)). The same updating also occurs for samples {x 0 (3), x 0 (5), x 0 (7)}. Notice that samples {x 0 (0), x 0 (8)} remain un-updated because they are needed by neighboring blocks, e.g., x 0 (8) is needed by x 0 (9) and x 0 (0) is needed by x 0 (−1). Consequently, x 0 (0) and x 0 (8) are preserved as the state information at state 0. βγ 
   The next elementary matrix e 1 (z) is upper triangular so it updates even samples using odd samples. For example, x 1 (2) is updated into x 2 (2)=x 1 (2)+β(x 1 (1)+x 1 (3)) and so are samples {x 1 (4), x 1 (6)}. Again, x 1 (1) and x 1 (7) are preserved as the state information at state 1. The same process continues until x 0 (4) is updated into the final transform coefficient x 4 (4). 
   The state information near the right boundary consists of samples shown in shaded boxes in the figure, i.e., {x 3 (5), x 2 (6), x 1 (7), x 0 (8)}. So the overlap buffer size for one level of wavelet decomposition using the Daubechies (9,7) filters is 4 samples. These partially updated samples constitute the only information one needs to buffer for the transform of the new input data pair {x 0 (9), x 0 (10)}. The operations are shown as dashed lines in the figure. All these operations are based on the state information, which is preserved in the memory buffer. 
   (2,10) Filters 
   This filter has been found to give excellent performance for lossless image compression. The factorization of the analysis polyphase matrix is 
               P   a     ⁡     (   z   )       =         [         1       0                 3   64     ⁢     (       z   2     -     z     -   2         )       +       22   64     ⁢     (     z   -     z     -   1         )             1         ]     ⁡     [         1         1   2             0       1         ]       ⁡     [         1       0             -   1         1         ]             
Based on this factorization, the forward transform is
 
                     x   0   1     ⁡     (   n   )       =       ⁢     x   ⁡     (     2   ⁢   n     )                       x   1   1     ⁡     (   n   )       =       ⁢       x   ⁡     (       2   ⁢   n     +   1     )       -     x   ⁡     (     2   ⁢   n     )                         x   0   2     ⁡     (   n   )       =       ⁢         x   0   1     ⁡     (   n   )       +         x   1   1     ⁡     (   n   )       /   2                       x   1   2     ⁡     (   n   )       =       ⁢       x   1   1     ⁡     (   n   )                       x   0   3     ⁡     (   n   )       =       ⁢       x   0   2     ⁡     (   n   )                       x   1   3     ⁡     (   n   )       =       ⁢         x   1   2     ⁡     (   n   )       +       3   64     ⁢     (         x   0   2     ⁡     (     n   -   2     )       -       x   0   2     ⁡     (     n   +   2     )         )       +       22   64     ⁢     (         x   0   2     ⁡     (     n   -   1     )       -       x   0   2     ⁡     (     n   +   1     )         )                     
The respective inverse transform is:
 
                     x   1   2     ⁡     (   n   )       =       ⁢         x   1   3     ⁡     (   n   )       -       3   64     ⁢     (         x   0   2     ⁡     (     n   -   2     )       -       x   0   2     ⁡     (     n   +   2     )         )       +       22   64     ⁢     (         x   0   2     ⁡     (     n   -   1     )       -       x   0   2     ⁡     (     n   +   1     )         )                         x   0   2     ⁡     (   n   )       =       ⁢       x   0   3     ⁡     (   n   )                       x   1   1     ⁡     (   n   )       =       ⁢       x   1   2     ⁡     (   n   )                       x   0   1     ⁡     (   n   )       =       ⁢         x   0   2     ⁡     (   n   )       -         x   1   1     ⁡     (   n   )       /   2                     x   ⁡     (     2   ⁢   n     )       =       ⁢       x   0   1     ⁡     (   n   )                     x   ⁡     (       2   ⁢   n     +   1     )       =       ⁢         x   1   1     ⁡     (   n   )       +     x   ⁡     (     2   ⁢   n     )                     
The first two state transitions are basically the same as those of the (9,7) filters.
 
     FIG. 29  show the state transitions for 10 initial samples in the memory. The last transition is detailed here. The elementary matrix e 2 (z) is 
   
     
       
         
           
             
               e 
               2 
             
             ⁡ 
             
               ( 
               z 
               ) 
             
           
           = 
           
             [ 
             
               
                 
                   1 
                 
                 
                   0 
                 
               
               
                 
                   
                     
                       
                         3 
                         64 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             z 
                             2 
                           
                           - 
                           
                             z 
                             
                               - 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         22 
                         64 
                       
                       ⁢ 
                       
                         ( 
                         
                           z 
                           - 
                           
                             z 
                             
                               - 
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   1 
                 
               
             
             ] 
           
         
       
     
   
   This is a lower triangular matrix so odd samples get updated. For example, x 2 (5) is updated into 
               x   3     ⁡     (   5   )       =         x   2     ⁡     (   5   )       +       3   64     ⁢     (         x   2     ⁡     (   8   )       -       x   2     ⁡     (   0   )         )       +       22   64     ⁢     (         x   2     ⁡     (   6   )       -       x   2     ⁡     (   2   )         )               
On the other hand, {tilde over (x)} 2 (7) can not be fully updated because x 0 (10) is not available (not in buffer yet). However, it can be partially updated as
 
                 x   ∼     2     ⁡     (   7   )       =         x   2     ⁡     (   7   )       +       3   64     ⁢     (     -       x   2     ⁡     (   2   )         )       +       22   64     ⁢     (         x   2     ⁡     (   8   )       -       x   2     ⁡     (   4   )         )               
This partial updating then frees sample x 2 (2) from the buffer. In other words, to fully update {tilde over (x)} 2 (7), no samples with indices smaller than 7 are needed. Same partially updating is also performed for sample {tilde over (x)} 2 (9) as
 
                 x   ∼     2     ⁡     (   9   )       =         x   2     ⁡     (   9   )       +       3   64     ⁢     (     -       x   2     ⁡     (   6   )         )       +       22   64     ⁢     (     -       x   2     ⁡     (   4   )         )               
The only samples which have to be buffered
 
are {X 2 (6),x 2 (7),x 2 (8),x 2 (9)}. So the overlap buffer size is 4 samples.
 
   When the next new input pair {x 0 (10) x 0 (11)} comes, operations in dashed lines are executed. As a result, samples {x 2 (6),x 2 (7)} are completely transformed thus can be removed from the buffer. However, samples {x 0 (10) x 0 (11)} can only be partially updated and thus have to be buffered. This process continues until all inputs are transformed. 
   CDF (4,2) Filters 
   The scaling function of CDF(4,2) filters is a cubic B-spine which is used frequently in computer graphics for interpolation. The factorization of the analysis polyphase matrix (adapted from [33]) is 
               P   a     ⁡     (   z   )       =           [           1   2         0           0       2         ]     ⁡     [         1           3   16     ⁢     (     1   +     z     -   1         )               0       1         ]       ⁡     [         1       0             -     (     1   +   z     )           1         ]       ⁡     [         1           -     1   4       ⁢     (     1   +     z     -   1         )               0       1         ]             
Based on this factorization, the forward transform is
 
                     x   0   0     ⁡     (   n   )       =       ⁢     x   ⁡     (     2   ⁢   n     )                       x   1   0     ⁡     (   n   )       =       ⁢     x   ⁡     (       2   ⁢   n     +   1     )                       x   0   1     ⁡     (   n   )       =       ⁢         x   0   0     ⁡     (   n   )       -       1   4     ⁢     (         x   1   0     ⁡     (   n   )       +       x   1   0     ⁡     (     n   -   1     )         )                         x   1   1     ⁡     (   n   )       =       ⁢         x   1   0     ⁡     (   n   )       -     (         x   0   1     ⁡     (   n   )       +       x   0   1     ⁡     (     n   +   1     )         )                       x   0   2     ⁡     (   n   )       =       ⁢         x   0   1     ⁡     (   n   )       +       3   16     ⁢     (         x   1   1     ⁡     (   n   )       +       x   1   1     ⁡     (     n   -   1     )         )                     
and the inverse transform is
 
   
     
       
         
           
             
               
                 
                   
                     
                       x 
                       0 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         x 
                         0 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         n 
                         ) 
                       
                     
                     - 
                     
                       
                         3 
                         16 
                       
                       ⁢ 
                       
                         
                           x 
                           1 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         x 
                         1 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           n 
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                 
                 ) 
               
             
           
           
             
               
                 
                   
                     x 
                     1 
                     0 
                   
                   ⁡ 
                   
                     ( 
                     n 
                     ) 
                   
                 
                 = 
                   
                 ⁢ 
                 
                   
                     
                       x 
                       1 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   + 
                   
                     ( 
                     
                       
                         
                           x 
                           0 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       + 
                       
                         
                           x 
                           0 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             n 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                 
                   
                     x 
                     0 
                     0 
                   
                   ⁡ 
                   
                     ( 
                     n 
                     ) 
                   
                 
                 = 
                   
                 ⁢ 
                 
                   
                     
                       x 
                       0 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   + 
                   
                     
                       1 
                       4 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             x 
                             1 
                             0 
                           
                           ⁡ 
                           
                             ( 
                             n 
                             ) 
                           
                         
                         + 
                         
                           
                             x 
                             1 
                             0 
                           
                           ⁡ 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
   
   In this case, the state transition is basically the same as that of the (9,7) filters. The overlap buffer size is 3 samples as shown in  FIG. 30 . 
   The above implementations of the overlap-state technique in parallel or sequential DWT processing can be used to efficiently perform multilevel wavelet decompositions. The DWT processing is modeled as a finite state machine using the factorization of the polyphase matrix. In this model, each raw input data sample (initial state) is progressively updated into a wavelet coefficient (final state) with the help of samples in its neighborhood. The benefit of such a DWT/FSM model is that the transform (or state transition) of each sample can be stopped at any intermediate stage as long as the state information at the break point is preserved. Since the state information rather than the raw data samples needs to be stored or communicated, this technique can reduce the buffer size in a sequential architecture and the communication overhead in a parallel architecture. Detailed analysis on buffer size calculation for a given factorization and communication overhead reduction are also provided. To further reduce the computations, a delayed normalization technique for multilevel wavelet decompositions is also presented. 
   Using the overlap-state technique, several sequential and parallel DWT architectures are designed. A number of system variations for 2D separable DWT are provided and analyzed in detail, which include DWT systems of strip sequential, block sequential, random sequential, block parallel and strip parallel. The performance analyses and the experimental results have shown that the proposed sequential architecture requires less memory and runs faster than existing sequential algorithms. The above described parallel architecture reduces the inter-processor communication overhead by reducing the number of communication times and the amount of data exchanged. As a result, the DWT running time of the parallel architecture is faster than many other parallel algorithms available such as the parallel lifting algorithm. 
   The following sections now describe applications of the above or other lifting factorizations to sensor networks. 
     FIG. 31  shows an exemplary sensor network with multiple individual sensors that individually measure and collect data at different locations of a given area or object. A central node receives and processes data from the sensors in order to extract information at each sensor location and the given area or object. The sensors may be wireless sensors, which have their own individual power supplies and operate on limited power budgets. Assume the individual sensors directly transmit data to the central node without pass data through other sensors. A simple and naive design would be for each sensor in  FIG. 31  to simply transmit a quantized version of its own measurement to the central node. However, this approach would not be exploiting the fact that measurements originated from spatially close sensors are likely to be correlated, and energy would be wasted with the transmission of redundant data to the central node. 
   This application provides a different approach to the data handling in the sensor network in  FIG. 31 . Because data is correlated, the information from different sensors can be decorrelated via a distributed transform to transform at the sensor level the data from different but correlated sensors to reduce the data redundancy. This transformation can be used to represent the measurements made by the sensor network in  FIG. 31  using fewer bits. The distributed transform requires inter-sensor communications to allow sensors to access the data from other sensors necessary to compute the transform coefficients. Hence, the sensors may be designed with the processing capability to perform the sensor-level computation and the communication capability to communicate with other sensors in exchange of sensor data. As such extra power would be consumed in the form of the local processing in individual sensors and additional transmission of information to other sensors. This extra power consumption associated with the distributed transform for reducing the data redundancy represents a trade-off. 
   To illustrate this trade-off, consider the effect of choosing transforms of different sizes. Larger transforms tend to provide better decorrelation, but at the expense of added communication cost between sensors. For example, using a block transform of size N would mean that N sensors would need to exchange information, with an average communication distance greater than for, e.g., an N/2 size transform. 
   An implementation of a distributed wavelet transform based on the lifting scheme is described here for the sensor network in  FIG. 31 . Each sensor represents a single memory location that stores the initial raw sensor data and the processed data from the distributed wavelet transform. Hence, the lifting factorization is implemented as in place computation at the sensor level without auxiliary memory outside each sensor in the sensor network in  FIG. 31 . The lifting factorization explicitly breaks down the transform into elementary operations that can be easily evaluated in terms of communication costs. The overall performance of the system depends mainly on the local processing cost and on the communication cost, which is related to (i) the correlation of the signal, (ii) the distances between sensors and (iii) the distances between sensors and the central node. The present technique takes those costs into account, and has the flexibility to choose the proper level of transformation for each environment. By operating at the best trade-off point between processing and transmission cost, given the network requirements, a sensor network can be designed to reduce the overall energy consumption. 
   In some implementations, sensors in a sensor network may not be treated uniformly in the same way. Different groups of sensors in one sensor network may be assigned to perform different processing tasks. For instance, some sensors (sensor nodes) may be much closer to the central node than others. Those sensor nodes may be good candidates to perform direct transmission without the distributed transform, depending on where the trade-off point is. Some sensor network systems may have a very low energy constraint and thus different levels of transform decomposition could perform better. 
   Hence, in the present distributed wavelet transform, a sensor in a block of sensors is not required to have knowledge about all the measurements of other sensors inside the block and thus the inter-sensor communications are reduced in comparison with techniques that require each sensor to have measurements from all other sensors and thus increase the communication costs. Instead, a sensor communicates with immediate adjacent sensors in the block only. The dependency between inter-sensor transmission costs and their distances is considered. 
   Notably, the management of coefficients close to the block boundaries can be addressed by use of efficient parallelization algorithms for the lifting scheme as described with reference to  FIGS. 1-30 . Consider each sample of data within a data block is captured by one sensor, then the exchange of data in lifting steps can be viewed as an inter-sensor communication between two different sensors, and the properties of the algorithm for operating in blocks can be seen as a mechanism for sensor node clustering or grouping where each block represents a cluster or group of sensor nodes which may be spatially adjacent to one another. 
   In a sensor network, each sensor obtains one sample and different sensors obtain different samples. At each stage of lifting, neighboring sensors can exchange samples and compute DWT coefficients, which would then be stored in the sensor node where the computation took place. Hence, different clusters of sensors can exchange data among themselves in order to compute the DWT corresponding to the data they have collected. When a wavelet is distributed over sensor nodes, then parallelization algorithms for the lifting scheme can be used to perform localized versions of the computation in separate node clusters, with the node clusters playing the role of “blocks” as shown, e.g., in  FIG. 7 . The DWT data, which may be partially computed, from each cluster is sent to neighboring sensor clusters, or to a central node, where the overall DWT computation is finalized. 
     FIG. 32  illustrates a lifting implementation of the wavelet transform with two processing stages in each sensor. The input signal s(n) is first split into two branches and is then processed by the two stages. The block “P” represents the prediction stage which predicts a value based on the original data and the block “U” represents the update stage which restores some properties of the original signal, like the mean value, to have some separation in frequency domain and to reduce aliasing. As illustrated, the input signal s(n) is processed by a low pass filter and a high pass filter to produce a low pass output y(n) and a high pass output d(n) after the two processing stages P and U, respectively. The blocks “↓ 2 ” represent down sampling by a factor of 2. Only one level of decomposition is shown. Two or more levels of decomposition may be performed.  FIG. 1  illustrates a 3-level decomposition. 
   The lifting scheme in  FIG. 32  is an alternative method to compute biorthogonal wavelets. It allows a faster implementation of the wavelet transform, along with a full in-place calculation of the coefficients. 
   As a specific example, the lifting scheme may be used to generate the coefficients for the CDF(2,2) wavelet (also known as the 5/3 wavelet) at each of the sensors. Even-numbered sensors correspond to the even samples and odd-numbered sensors to the odd samples. In-place computation reduces the memory requirements for the sensors. Also, as mentioned before, another attractive property of the lifting scheme in this distributed network scenario is the use of efficient parallelization algorithms. Implementations as described above with reference to  FIGS. 1-30  enable the partial computation of coefficients at the boundaries. In this perspective, the sensors are divided into groups and increasing levels of the wavelet decomposition would be computed, as long as the extra energy spent in the calculations and local transmissions are compensated by the increase in decorrelation. At this point, the partial coefficients would be transmitted to the central node. Since each group is essentially independent of the others, the system could be setup such that each of them could be assigned different tasks (such as different levels decompositions, or even direct transmission), depending on the system configuration, optimizing the overall performance. 
     FIG. 33  shows the two step implementation of the wavelet transform using lifting where each sensor only needs data from its neighbors at a given scale. During the first step, the odd-numbered sensors receive data measurements from their even-numbered neighbors, and compute the correspondent detail coefficient. On the second step, these coefficients are sent to the even-numbered sensors, and to the central node. The even sensors use them (along with their own measurement) to generate the smooth coefficients, which are then transmitted to the central node. 
     FIG. 34  shows a more detailed view of the process in  FIG. 33 , including the predictor and update coefficients for the 5/3 wavelet. Each of the branches of the graph would have an associated transmission cost. The cost for the distributed wavelet transform can be estimated as follows. 
   Both local processing and transmission costs are considered and figured into the cost estimation in order to fairly compare a non-distributed approach (direct transmission) and the distributed wavelet transform algorithm. Referring back to  FIG. 31 , without the distributed wavelet transform, each sensor directly transmits its own quantized measurement to the central node and no energy is spent with local processing or inter-sensor transmissions. In the distributed approach, local processing and additional data transmission over short distances are introduced and configured in a way that the obtained decorrelated data will require significantly less energy to be transmitted to the central node, compensating the extra energy spent with the decorrelation process. 
   Since energy dissipation for both transmission and processing is highly dependent on the processor being used, the StrongARM SA-1100 processor described by Wang and Chandraksan in “Energy-efficient DSPs for Wireless Sensor Networks,” IEEE Signal Processing Magazine, pp. 68-78 (July 2002) is used in the following cost estimate. For this DSP, the energy dissipated with the transmission and reception of a k-bit packet over a distance D is
 
 E   TX   =E   elec   ·k+ε   amp   ·k·D   2 
 
 E   RX   =E   elec   ·k 
 
where E elec =50 nJ/b is the energy dissipated to run the transmit or receive electronics, and ε amp =100 pJ/b/m 2  is the energy dissipated by the transmit power amplifier. The energy dissipation due to computation is a function of the supply voltage E lp =NCV 2   dd , where N is the number of clock cycles per task, C is the average capacitance switched per cycle, and V dd  is the supply voltage. For the StrongARM SA-1100, C is approximately 0.67 nF.
 
   The total energy dissipated at each sensor, therefore, includes four main components:
 
 E=E   lp   +E   lt   +E   lr   +E   rt  
 
where subscripts lp, lt, lr and rt stand for local processing, local (inter-sensor) transmission, local reception and remote (sensor to central node) transmission, respectively. The non-distributed case has only the energy cost E rt  due to the remote transmission from each sensor to the central node, and E lp =E lt =E tr =0. When considering a distributed wavelet transform for a group of sensors in a sensor network, the total energy dissipated at each sensor within the group is estimated and compared to the energy dissipation at each sensor for direct transmission without the distributed wavelet transform in order to determine which mode of the transmission should be used.
 
   The above distributed wavelet transform for the sensor network in  FIG. 31  provides a flexible way of meeting requirements by choosing between different system configurations based on the trade-off point between processing and transmission. To illustrate this point, the simulations have been conducted for a simple 5/3 wavelet. 
     FIG. 35  shows a sensor network for the simulations which includes sensors spatially spaced along a line and a central node. For the sake of simplicity, assume d i ≈D for all sensors in  FIG. 35 . Computation of actual distances in a practical application is straightforward. As an example, d=5 m and D=1000 m. The uniform quantization is used in the estimate without the entropy coding. In the simulations, the input process data was created using a second order autoregressive (AR) model, with poles placed such that a reasonably smooth output would be generated from white noise (poles were at 
               0.99   ⁢     e         +   _     ⁢   j     ⁢               ⁢   π     64           )     .         
The measurements at the sensors corresponded to a sampling of the output of the AR model, and consisted of 100 sensors.
 
   Assuming that a general processor can typically perform 150 instructions per bit communicated energy-wise, and that the computation of a wavelet coefficient using the lifting scheme takes only 2 multiplications and 4 additions, the following can be derived:
 
 E   Tχ   +E   Rχ   &gt;&gt;E   lp ,
 
where, for this particular simulation, E=E lt +E lr +E rt .
 
     FIGS. 36A and 36B  show the computed SNR values that are averaged over a large number of different inputs generated from the same AR model. The distortion curves against the bit rate and the dissipated energy for the cases of a 1 and 2-level wavelet decomposition are shown in  FIGS. 36A and 36B , respectively. The similarity of the graphs comes from the fact that since D&gt;&gt;d, the term E rt  dominates over E lt +E lr . 
     FIGS. 36A and 36B  further show how different trade-off points can be used to choose between 1 and 2-level distributed wavelet transform or the non-distributed case. For high SNRs (e.g., above 65 dB), the non-distributed method might be preferable over the distributed method because the non-distributed method gives better performance with fewer bits. For low SNRs (e.g., below 65 dB), the distributed method might be preferable. SNRs of the order of 60 dB are usually far above typical system requirements in some applications. Hence, the distributed method may be applied to such systems with SNRs below 65 dB. 
   In simulating the 2-level decomposition case, the inter-node communication cost is increased but more sensors have low-energy (detail) data that can be coded using less bits. It can be seen that, in this particular example, for SNRs of above about 35 dB, a 1-level decomposition performs better than a 2-level for the same energy dissipation. On the other hand, for a network with very restrictive energy consumption constraints, a 2-level decomposition would give lower distortion than one with only 1-level. In this example, energy savings can achieve values of up to 40%. 
   In a more general sensor network without the simplified assumptions in the sensor network in  FIG. 35 , many parameters could influence the results. Sensors could be unevenly spread, with random distances between them. Also, data in different regions of the network could be more or less correlated or different areas could have different transmission costs due to environment characteristics. In such cases, the present distributed wavelet transform may be implemented by considering local features, and optimizing each region individually in order to increase the overall network performance. For instance, depending on local transmission costs and correlation, a particular region could achieve better performance using a different number of decomposition levels than other group of sensors in the same sensor network. 
   In the sensor network shown in  FIG. 31 , the data from each sensor, which may be quantized raw data collected by the sensor or transformed data by the distributed wavelet transform based on inter-sensor communications with other sensors, is directly transmitted to the central node without passing through any other sensors. Sensor networks may operate in other modes of transmission of data to the central node. For example, data from one sensor may be relayed via other sensors during the transmission to the central node. Assume that a number of power-constrained sensors are spread over an area, acquiring data, e.g., temperature measurements. An estimate of the temperature in each point of that area is to be made available at a central node (sink) based on transmissions from the sensors. Each sensor is capable of transmitting data only over a small distance. Communication from each sensor to the central node can be achieved via data hoping, where data from each sensor is forwarded using other sensors as relay stations until it reaches the target node. Such a network is referred to as a multihop sensor network. 
     FIG. 37  illustrates one example of a multihop sensor network. Similar to the sensor network in  FIG. 31 , if the parameter under measurement at different sensor locations is correlated, the measurements from different sensors are correlated and can be decorrelated via a distributed wavelet transform to represent the measurements using fewer bits. The distributed wavelet transform can include transmission costs as a function of, e.g., the number of bits and the distances over which bits are transmitted. In various implementations of distributed wavelet algorithms, the amount of necessary data exchanges and the degree of decorrelation achieved can be directly related to the number of levels of decompositions used. As such, a specific number of decompositions may perform better than other number of decompositions depending on the network properties such as data correlation, or distances between sensors. 
   Distributed implementations of the wavelet transform in multihop sensor networks may pose several technical challenges. For example, if the filters contain anticausal terms, sensors would be required to transmit data backwards against the natural flow of data, i.e., away from the central node or sink instead of towards the sink, or sending uncompressed data forward. In another example, any data transmitted back and forth over the network need to be quantized because transmissions at the full precision can substantially increase energy consumption and thus affect the overall performance of the network. 
   Present techniques based on the distributed wavelet transform algorithms for multihop sensor networks are in part designed to address these and other technical issues in decorrelating data as it flows through the network. In the described implementations, unnecessary inter-sensor transmissions are eliminated by calculating partial approximations of the wavelet coefficients based on the available data at each sensor. The partially computed coefficients are further refined at future nodes along the natural flow of the data as the data is forwarded to the sink. The impact of data quantization on the final distortion is also addressed. An upper bound to the resulting extra distortion introduced by quantization in terms of the bits allocated to the partial data is derived and is used as a tool to design the quantizers so the extra distortion is controlled within a given threshold. 
   In the described implementations, a lifting scheme is used here to generate the wavelet coefficients at each of the sensors within a multihop sensor network. Even-numbered sensors correspond to the even samples and odd-numbered sensors to the odd samples. In-place computation reduces the memory requirements for the sensors. Implementations of the lifting described with reference to  FIGS. 1-30  may be used to enable the partial computation of coefficients at the boundaries. In this perspective, the sensors can be divided into groups and increasing levels of the wavelet decomposition would be computed, as long as the extra energy spent in the calculations and transmissions is compensated by the increase in decorrelation. 
   Efficient implementation of a lifting based distributed wavelet transform becomes challenging once inter node communication costs are taken into account. In the non-multihop sensor network in  FIG. 31 , inter-sensor communications are introduced as extra transmissions to give sensors knowledge about their neighbors&#39; data, allowing them to compute the wavelet coefficients. The reduction in the average number of bits per sensor in  FIG. 31  compensates for the cost of the extra transmissions, resulting in a smaller overall energy consumption. However, for a multihop network, where all data already flows in a particular direction, sensors that need to transmit data away from the sink may waste resources. 
   A simple solution could be the introduction of a delay, making the system causal, and having the sensors calculating the transform coefficients only after all the necessary data becomes available at a future node. This approach, however, can be inefficient because raw data has to be transmitted until it reaches the node that will process it, the memory requirements for the sensors can increase substantially and the energy consumption due to the raw data transmission itself can be the potentially large. 
   To avoid the problem associated with the causality, a lifting scheme can be implemented based on the natural data flow in the network to compute a distributed wavelet transform. At each sensor, partial computations are performed with the available data that arrives at the sensor with the network flow. In this design, unnecessary backward (against the data flow) transmissions can be eliminated and there is no need for a delay in each sensor. 
   Let 
               H   ⁡     (   z   )       =       ∑     n   =     -   1       b     ⁢       h   ⁡     (   n   )       ⁢     z     -   n             ,     (       a   &gt;   0     ,     b   &gt;   0       )           
be a general filter defined by the transform to be applied to the multihop sensor network. This filter can also be represented as
 
               H   ⁡     (   z   )       =         A   ⁡     (   z   )       ⁢   z     +   c   +       B   ⁡     (   z   )       ⁢     z     -   1             ,         
where A(z) and B(z) are the anticausal and causal parts of the filter, respectively. In the first step in the partial coefficient approach, the current sensor node receives a quantized version of B(z). Then, the sensor node computes a quantized version of B(z)z −1 +c and sends the computed result to the next sensor node in the network data flow. The subsequent sensor nodes will update the coefficient using their local data until it is fully computed.
 
   The lifting implementation of the wavelet transform facilitates the distributed implementation. The in-place computation greatly reduces the memory requirements for the sensors. Within a lifting implementation it is easy to compute the partial data updates in terms of previous partial coefficients, eliminating the need to transmit extra information. As an example, let D(n) denote the raw (quantized) data and  D (n) the full coefficient for the n-th sensor. The 5/3 wavelet coefficients using lifting at odd and even sensors are given by: 
                     D   _     ⁡     (       2   ⁢   n     +   1     )       =         -     1   2       ⁢     D   ⁡     (     2   ⁢   n     )         +     D   ⁡     (       2   ⁢   n     +   1     )       -       1   2     ⁢     D   ⁡     (       2   ⁢   n     +   2     )                           D   _     ⁡     (     2   ⁢   n     )       =         1   4     ⁢       D   _     ⁡     (       2   ⁢   n     -   1     )         +     D   ⁡     (     2   ⁢   n     )       +       1   4     ⁢       D   _     ⁡     (       2   ⁢   n     +   1     )                       
Because the sensor 2n 1 does not have access to data from an downstream sensor 2n+2, the sensor 2n+1 computes just the partial coefficient
 
                   D   _     p     ⁡     (       2   ⁢   n     +   1     )       ⁢           ⁢   as   ⁢           ⁢         D   _     p     ⁡     (       2   ⁢   n     +   1     )         =       -       D   ⁡     (     2   ⁢   n     )       2       +       D   ⁡     (       2   ⁢   n     +   1     )       .             
When this partial data arrives at the sensor 2n+2, it will be updated to
 
   
     
       
         
           
             
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     FIG. 38  illustrates this partial computation scheme for a 1-level 5/3 wavelet. The computations performed at even and odd sensors for this transform are summarized in the table in  FIG. 38 . Notably, in the specific implementations here, the sensors make their computations only based on previous partial or full coefficients, and not on raw data. This eliminates the need for extra information. This technique can be used in other wavelet lifting factorizations. This representation is straightforward for the 5/3 wavelet lifting factorization. When used with other filters, specific optimization for such filters may be made. 
   In the above partial computation at each sensor, transmissions are made with finite precision in transmitting partially computed coefficients through sensors towards the central node. In standard transform computing, the computations to obtain the transform coefficients are performed at the full precision, and only the final coefficients are quantized. However, in a distributed network scenario, transmissions at the full precision can significantly increase energy costs, making it necessary to also quantize partial coefficients. A poor choice of intermediate coefficient quantization may considerably affect the final distortion. The following sections describe the impact of partial coefficient quantization on the final distortion, and describe a rule to determine how many bits should be used to quantize the partial information so as to achieve a target level of degradation, in the form of added distortion as compared to calculating coefficients without partially quantized data. 
   Assume a uniform quantizer Q L , with a bin size equal to L is used to quantize the result of αX 1 +βX 2 , where α and β are known constants (0≦α, β≦1) and X 1  and X 2  are random variables uniformly distributed on the interval [0, 1]. The resulting quantization Q L (αX 1 +βX 2 ) gives a mean-squared error of ε. It is further assumed that a second quantizer Q 1 , with a different bin size equal to i, is used to quantize X i  and X 2  before the quantizer Q L . The resulting mean-squared error of Q L (αQ t (X 1 )+βQ t (X 2 )) is ε′. The mean squared error (MSE) for both cases as double integrals can be expressed as follows: 
   
     
       
         
           
             
               
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     FIG. 39  shows a graphical interpretation of the distortion calculation for the case where α=β=1. The rectangular grid corresponds to the quantization regions for the pair (X 1 , X 2 ). The diagonal lines define the decision regions for the quantizer Q L . Let 
             l   L     =       1     2   N       .           
It can be shown that the number of extra bits that should be assigned to Q t  when compared to Q L  is given by N. Then, by computing the MSE ratio in terms of N, we can relate the additional extra distortion to the bits used to quantize the partial data. This can be used to define a rule to design the quantizer Q t  for any given value of the MSE ratio
 
   
     
       
         
           
             
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   The above integrals for the case α=β=1 can be computed to derive the MSE ratio: 
               ɛ   ′     ɛ     =     1   +     4     2     2   ⁢   N                 
This result shows that the MSE ratio is independent of the absolute value of the bin sizes of the quantizers and depends only on their ratio. Simulations have shown that the case α=β=1 presents a worst scenario case where the MSE ratio
 
             ɛ   ′     ɛ         
decreases as α and β decrease, and that the results are little affected by the probability distributions of X 1  and X 2 .
 
     FIG. 40  shows the theoretical curve from above MSE ratio plotted together with simulated results for a number of cases where α, β≦1 and X 1 , X 2  are Gaussian r.v.&#39;s. In practical transform computations, typical values for α and β might increase or decrease depending on the specific coefficient, but in general they are always smaller than one, making the curve 
           1   +     4     2     2   ⁢   N               
a still reasonable approximation for an upper bound of the ratio
 
               ɛ   ′     ɛ     .         
By allocating N extra bits when quantizing X 1  and X 2  than when quantizing αX 1 +βX 2 , we can guarantee that the extra distortion introduced by the inter-mediate quantization can be bounded by the values given by the theoretical curve.  FIG. 40  also shows that values of N smaller than 3 introduce a large extra distortion, and the reduction in the extra distortion has a diminishing return as N increases for values above 3.
 
   Simulation results suggest that the partial coefficient quantization has indeed a major impact on the final distortion, and the results obtained above can be used to design the partial quantizers such that a good trade-off point between the allocated bits and the extra distortion introduced is achieved. We also compare the performance of the proposed scheme with the cases of raw data transmission (no encoding) and of two-way transmissions (no partial coefficients) in terms of signal-to-noise ratio against transmission cost. 
     FIG. 41  shows the distortion curves obtained for standard 5/3 wavelet implementations and when partial coefficients are coarsely quantized (N=0) and finely quantized (N=3). The resulting SNR is sensitive to coarsely quantized partial coefficients. In the sensor network scenario, choosing values of N much larger than 3 can lead to a better signal-to-noise ratio, but at the cost of an increase in the overall energy consumption. 
   For the cost comparisons, the simulations are made for a simple 5/3 wavelet. The input process data was created using a second order AR model, with poles placed such that a reasonably smooth output would be generated from white noise (poles were at 
               0.99   ⁢     e         +   _     ⁢   j     ⁢           ⁢     π   64           )     .         
The measurements at the sensors corresponded to a sampling of the output of the AR model. The sensors were assumed to lie at a constant distance d of each other. This restriction, however, is not required in practice. We used uniform quantization and no entropy coding at this point. The cost for the transmission of b bits over the distance d was computed as b·d 2 .
 
     FIG. 42  shows the resulting SNR and cost for the case where 100 sensors are equally spread over a distance of 100 meters. The 2-level wavelet with finely quantized partial coefficients performs better, with raw data transmission being the one spending the most energy. The energy savings are around 30% when compared to raw data transmission for a target SNR. 
     FIG. 43  further shows the resulting costs for the case where only 10 sensors are spread over 100 meters. Methods that require more communication tend to perform worse than the simpler ones. This may be attributed mainly to the fact that in a large network, a large portion of the transmissions (and the energy consumption) are related to simply forwarding already computed coefficients, and not with their computation. Therefore, a method that reduces the average number of bits per sensor also reduces the computational load due to data forwarding, resulting in a smaller overall cost. Smaller networks should benefit from methods that do not introduce large extra local costs, since the energy spent with coefficient computation (including due to finer quantization for partial coefficients) and data forwarding have the same magnitude. A comparison between  FIGS. 42 and 43  suggests that the network configuration can be optimized by breaking the network into regions that would operate using different coding schemes depending on the physical location of sensors, thus reducing the overall energy consumption. 
   In the multihop sensor network, a routing table may be implemented in the system to manage hoping from one sensor node to another for transmitting data from a sensor to the central node. Hence, a network data flow may not be a physical path but a signal path defined by the routing table. The present technique essentially uses the signal paths defined by the routing table without creating special inter-sensor communication paths that are not in the routing table to facilitate the computation of the transform coefficients. 
   In some sensor networks, communication may be severely limited so that each sensor captures multiple measurements over a period of time before it can actually communicate those to the central node or neighboring sensors. If these measurements are sufficiently frequent (e.g., temperature measurements at 1 hour intervals) there is likely to be substantial temporal correlation between successive measurements in one sensor, along with the spatial correlation that exists among all the measurements taken at a given time across sensors. In this case the data captured by a sensor in between transmission opportunities has temporal redundancy, which can be removed in order to achieve a more compact data representation. 
   In one such system each node captures S samples over a certain period of time, then performs a transform (e.g., a wavelet transform) of the S samples and quantizes the resulting coefficients before sending them to a neighboring sensor or the central node. Our proposed method can be used in the manner describe earlier but with those wavelet coefficients (generated by applying a temporal transform) playing the same role the original sensor measurements played in our previous discussion. The result of this approach is a method where both temporal and spatial transforms are applied in a separable manner. The spatial transform is applied to like coefficients generated by different sensors. Thus, for example, the sequence of all the i-th coefficients generated by the temporal wavelet transform in each sensor constitutes a spatial measurement signal that is subject to a spatial wavelet transform, to be implemented in the manner described above. 
   Only a few implementations are described. However, various modifications and enhancements may be made based on what is described in this application.