image compression and denoising via nonseparable wavelet approximation it was shown in acta math sci 22 1 2002 19 that two dimensional nonseparable scaling function interpolation has better convergence results than scalar wavelet systems in some cases based on several experimental results in this paper we present applications of two dimensional nonseparable wavelet approximation the algorithms are developed by using two dimensional nonseparable scaling function interpolation to perform image compression and image denoising comparing with the separable counterparts our results show that there are some improvements and advantages of two dimensional nonseparable scaling function interpolation

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

adaptive entropy coded subband coding of images the authors describe a design approach called 2 d entropy constrained subband coding ecsbc based upon recently developed 2 d entropy constrained vector quantization ecvq schemes the output indexes of the embedded quantizers are further compressed by use of noiseless entropy coding schemes such as huffman or arithmetic codes resulting in variable rate outputs depending upon the specific configurations of the ecvq and the ecpvq over the subbands many different types of sbc schemes can be derived within the generic 2 d ecsbc framework among these the authors concentrate on three representative types of 2 d ecsbc schemes and provide relative performance evaluations they also describe an adaptive buffer instrumented version of 2 d ecsbc called 2 d ecsbc aec for use with fixed rate channels which completely eliminates buffer overflow underflow problems this adaptive scheme achieves performance quite close to the corresponding ideal 2 d ecsbc system

entropy based algorithms for best basis selection adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals it permits efficient compression of a variety of signals such as sound and images the predefined libraries of modulated waveforms include orthogonal wavelet packets and localized trigonometric functions and have reasonably well controlled time frequency localization properties the idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost the method relies heavily on the remarkable orthogonality properties of the new libraries all expansions in a given library conserve energy and are thus comparable several cost functionals are useful one of the most attractive is shannon entropy which has a geometric interpretation in this context

on the minimum description length principle for sources with piecewise constant parameters universal lossless coding in the presence of finitely many abrupt changes in the statistics of the source at unknown points is investigated the minimum description length mdl principle is derived for this setting in particular it is shown that for any uniquely decipherable code for almost every combination of statistical parameter vectors governing each segment and for almost every vector of transition instants the minimum achievable redundancy is composed from 0 5 log n n bits for each unknown segmental parameter and log n n bits for each transition where n is the length of the input string this redundancy is shown to be attainable by a strongly sequential universal encoder i e an encoder that does not utilize the knowledge of a prescribed value of n

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

rate distortion optimized tree structured compression algorithms for piecewise polynomial images this paper presents novel coding algorithms based on tree structured segmentation which achieve the correct asymptotic rate distortion r d behavior for a simple class of signals known as piecewise polynomials by using an r d based prune and join scheme for the one dimensional case our scheme is based on binary tree segmentation of the signal this scheme approximates the signal segments using polynomial models and utilizes an r d optimal bit allocation strategy among the different signal segments the scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying r d behavior d r spl sim c sub 0 2 sup c1r thus improving over classic wavelet schemes we also prove that the computational complexity of the scheme is of o nlogn we then show the extension of this scheme to the two dimensional case using a quadtree this quadtree coding scheme also achieves an exponentially decaying r d behavior for the polygonal image model composed of a white polygon shaped object against a uniform black background with low computational cost of o nlogn again the key is an r d optimized prune and join strategy finally we conclude with numerical results which show that the proposed quadtree coding scheme outperforms jpeg2000 by about 1 db for real images like cameraman at low rates of around 0 15 bpp

prioritized dct for compression and progressive transmission of images an approach is based on the block discrete cosine transform dct the novelty of this approach is that the transform coefficients of all image blocks are coded and transmitted in absolute magnitude order the resulting ordered by magnitude transmission is accomplished without sacrificing coding efficiency by using partition priority coding coding and transmission are adaptive to the characteristics of each individual image and therefore very efficient another advantage of this approach is its high progression effectiveness since the largest transform coefficients that capture the most important characteristics of images are coded and transmitted first this method is well suited for progressive image transmission further compression of the image data is achieved by multiple distribution entropy coding a technique based on arithmetic coding experiments show that the approach compares favorably with previously reported dct and subband image codecs

image compression with variable block size segmentation high quality variable rate image compression is achieved by segmenting an image into regions of different sizes classifying each region into one of several perceptually distinct categories and using a distinct coding procedure for each category segmentation is performed with a quadtree data structure by isolating the perceptually more important areas of the image into small regions and separately identifying larger random texture blocks since the important regions have been isolated the remaining parts of the image can be coded at a lower rate than would be otherwise possible high quality coding results are achieved at rates between 0 35 and 0 7 b p depending on the nature of the original image and satisfactory results have been obtained at 0 25 b p

multiple description wavelet based image coding we consider the problem of coding images for transmission over error prone channels the impairments we target are transient channel shutdowns as would occur in a packet network when a packet is lost or in a wireless system during a deep fade when data is delivered it is assumed to be error free but some of the data may never reach the receiver the proposed algorithms are based on a combination of multiple description scalar quantizers with techniques successfully applied to the construction of some of the most efficient subband coders a given image is encoded into multiple independent packets of roughly equal length when packets are lost the quality of the approximation computed at the receiver depends only on the number of packets received but does not depend on exactly which packets are actually received when compared with previously reported results on the performance of robust image coders based on multiple descriptions on standard test images our coders attain similar psnr values using typically about 50 60 of the bit rate required by these other state of the art coders while at the same time providing significantly more freedom in the mechanism for allocation of redundancy among descriptions

on the number of rectangular tilings adaptive multiscale representations via quadtree splitting and two dimensional 2 d wavelet packets which amount to space and frequency decompositions respectively are powerful concepts that have been widely used in applications these schemes are direct extensions of their one dimensional counterparts in particular by coupling of the two dimensions and restricting to only one possible further partition of each block into four subblocks in this paper we consider more flexible schemes that exploit more variations of multidimensional data structure in the meantime we restrict to tree based decompositions that are amenable to fast algorithms and have low indexing cost examples of these decomposition schemes are anisotropic wavelet packets dyadic rectangular tilings separate dimension decompositions and general rectangular tilings we compute the numbers of possible decompositions for each of these schemes we also give bounds for some of these numbers these results show that the new rectangular tiling schemes lead to much larger sets of 2 d space and frequency decompositions than the commonly used quadtree based schemes therefore bearing the potential to obtain better representation for a given image

on multirate optimality of jpeg2000 code stream arguably the most important and defining feature of the jpeg2000 image compression standard is its r d optimized code stream of multiple progressive layers this code stream is an interleaving of many scalable code streams of different sample blocks in this paper we reexamine the r d optimality of jpeg2000 scalable code streams under an expected multirate distortion measure emrd which is defined to be the average distortion weighted by a probability distribution of operational rates in a given range rather than for one or few fixed rates we prove that the jpeg2000 code stream constructed by embedded block coding of optimal truncation is almost optimal in the emrd sense for uniform rate distribution function even if the individual scalable code streams have nonconvex operational r d curves we also develop algorithms to optimize the jpeg2000 code stream for exponential and laplacian rate distribution functions while maintaining compatibility with the jpeg2000 standard both of our analytical and experimental results lend strong support to jpeg2000 as a near optimal scalable image codec in a fairly general setting

the context tree weighting method basic properties describes a sequential universal data compression procedure for binary tree sources that performs the double mixture using a context tree this method weights in an efficient recursive way the coding distributions corresponding to all bounded memory tree sources and achieves a desirable coding distribution for tree sources with an unknown model and unknown parameters computational and storage complexity of the proposed procedure are both linear in the source sequence length the authors derive a natural upper bound on the cumulative redundancy of the method for individual sequences the three terms in this bound can be identified as coding parameter and model redundancy the bound holds for all source sequence lengths not only for asymptotically large lengths the analysis that leads to this bound is based on standard techniques and turns out to be extremely simple the upper bound on the redundancy shows that the proposed context tree weighting procedure is optimal in the sense that it achieves the rissanen 1984 lower bound

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

multiscale hybrid linear models for lossy image representation in this paper we introduce a simple and efficient representation for natural images we view an image in either the spatial domain or the wavelet domain as a collection of vectors in a high dimensional space we then fit a piece wise linear model i e a union of affine subspaces to the vectors at each downsampling scale we call this a multiscale hybrid linear model for the image the model can be effectively estimated via a new algebraic method known as generalized principal component analysis gpca the hybrid and hierarchical structure of this model allows us to effectively extract and exploit multimodal correlations among the imagery data at different scales it conceptually and computationally remedies limitations of many existing image representation methods that are based on either a fixed linear transformation e g dct wavelets or an adaptive uni modal linear transformation e g pca or a multimodal model that uses only cluster means e g vq we will justify both quantitatively and experimentally why and how such a simple multiscale hybrid model is able to reduce simultaneously the model complexity and computational cost despite a small overhead of the model our careful and extensive experimental results show that this new model gives more compact representations for a wide variety of natural images under a wide range of signal to noise ratios than many existing methods including wavelets we also briefly address how the same hybrid linear modeling paradigm can be extended to be potentially useful for other applications such as image segmentation

sharp constants related to the triangle inequality in lorentz spaces we study the lorentz spaces lp s r in the range 1 p s for which the standard functional f p s 0 t1 pf t s dt t 1 s is only a quasi norm we find the optimal constant in the triangle inequality for this quasi norm which leads us to consider the following decomposition norm f p s inf

comparison of different methods of classification in subband coding of images this paper investigates various classification techniques applied to subband coding of images as a way of exploiting the nonstationary nature of image subbands the advantages of subband classification are characterized in a rate distortion framework in terms of classification gain and overall subband classification gain two algorithms maximum classification gain and equal mean normalized standard deviation classification which allow unequal number of blocks in each class are presented the dependence between the classification maps from different subbands is exploited either directly while encoding the classification maps or indirectly by constraining the classification maps the trade off between the classification gain and the amount of side information is explored coding results for a subband image coder based on classification are presented the simulation results demonstrate the value of classification in subband coding

sobolev embeddings in orlicz and lorentz spaces with measures abstract embedding theorems for orlicz sobolev spaces into orlicz or orlicz lorentz spaces in domains in r n endowed with a frostman measure are offered parallel embeddings for lorentz sobolev spaces into generalized lorentz spaces are also established the relevant embeddings yield an optimal target norm in the relevant classes whenever it does exist in particular various results available in the literature are improved by special choices of the function spaces and of the measure

significance linked connected component analysis for wavelet image coding the success in wavelet image coding is mainly attributed to a recognition of the importance of data organization and representation there have been several very competitive wavelet coders developed namely shapiro s 1993 embedded zerotree wavelets ezw servetto et al s 1995 morphological representation of wavelet data mrwd and said and pearlman s see ieee trans circuits syst video technol vol 6 p 245 50 1996 set partitioning in hierarchical trees spiht we develop a novel wavelet image coder called significance linked connected component analysis slcca of wavelet coefficients that extends mrwd by exploiting both within subband clustering of significant coefficients and cross subband dependency in significant fields extensive computer experiments on both natural and texture images show convincingly that the proposed slcca outperforms ezw mrwd and spiht for example for the barbara image at 0 25 b pixel slcca outperforms ezw mrwd and spiht by 1 41 db 0 32 db and 0 60 db in psnr respectively it is also observed that slcca works extremely well for images with a large portion of texture for eight typical 256 spl times 256 grayscale texture images compressed at 0 40 b pixel slcca outperforms spiht by 0 16 db 0 63 db in psnr this performance is achieved without using any optimal bit allocation procedure thus both the encoding and decoding procedures are fast

multispectral code excited linear prediction coding and its application in magnetic resonance images this paper reports a multispectral code excited linear prediction mcelp method for the compression of multispectral images different linear prediction models and adaptation schemes have been compared the method that uses a forward adaptive autoregressive ar model has been proven to achieve a good compromise between performance complexity and robustness this approach is referred to as the mfcelp method given a set of multispectral images the linear predictive coefficients are updated over nonoverlapping three dimensional 3 d macroblocks each macroblock is further divided into several 3 d micro blocks and the best excitation signal for each microblock is determined through an analysis by synthesis procedure the mfcelp method has been applied to multispectral magnetic resonance mr images to satisfy the high quality requirement for medical images the error between the original image set and the synthesized one is further specified using a vector quantizer this method has been applied to images from 26 clinical mr neuro studies 20 slices study three spectral bands slice 256 spl times 256 pixels band 12 b pixel the mfcelp method provides a significant visual improvement over the discrete cosine transform dct based joint photographers expert group jpeg method the wavelet transform based embedded zero tree wavelet ezw coding method and the vector tree vt coding method as well as the multispectral segmented autoregressive moving average msarma method we developed previously

on the norming constants for normal maxima abstract given n independent standard normal random variables it is well known that their maxima m n can be normalized such that their distribution converges to the gumbel law in a remarkable study hall proved that the kolmogorov distance d n between the normalized m n and its associated limit distribution is less than 3 log n in the present study we propose a different set of norming constants that allow this upper bound to be decreased with d n c m log n for n m 5 furthermore the function c m is computed explicitly which satisfies c m 1 and lim m c m 1 3 as a consequence some new and effective norming constants are provided using the asymptotic expansion of a lambert w type function

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

tiling and adaptive image compression we investigate the task of compressing an image by using different probability models for compressing different regions of the image in this task using a larger number of regions would result in better compression but would also require more bits for describing the regions and the probability models used in the regions we discuss using quadtree methods for performing the compression we introduce a class of probability models for images the k rectangular tilings of an image that is formed by partitioning the image into k rectangular regions and generating the coefficients within each region by using a probability model selected from a finite class of n probability models for an image of size n spl times n we give a sequential probability assignment algorithm that codes the image with a code length which is within o k log nn k of the code length produced by the best probability model in the class the algorithm has a computational complexity of o nn sup 3 an interesting subclass of the class of k rectangular tilings is the class of tilings using rectangles whose widths are powers of two this class is far more flexible than quadtrees and yet has a sequential probability assignment algorithm that produces a code length that is within o k log nn k of the best model in the class with a computational complexity of o nn sup 2 logn similar to the computational complexity of sequential probability assignment using quadtrees we also consider progressive transmission of the coefficients of the image

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

coding for a binary independent piecewise identically distributed source two weighting procedures are presented for compaction of output sequences generated by binary independent sources whose unknown parameter may occasionally change the resulting codes need no knowledge of the sequence length t i e they are strongly sequential and also the number of parameter changes is unrestricted the additional transition redundancy of the first method was shown to achieve the merhav lower bound i e log t bits per transition for the second method we could prove that additional transition redundancy is not more than 3 2 log t bits per transition which is more than the merhav bound however the storage and computational complexity of this method are also more interesting than those of the first method simulations show that the difference in redundancy performance between the two methods is negligible

embeddings for spaces of lorentz sobolev type the purpose of this paper is to characterize all embeddings for versions of besov and triebel lizorkin spaces where the underlying lebesgue space metric is replaced by a lorentz space metric we include two appendices one on the relation between classes of endpoint mikhlin hormander type fourier multipliers and one on the constant in the triangle inequality for the spaces l p r when p 1

asymptotic expansions of the moments of extremes from general error distribution abstract in this paper we derive the asymptotic expansions of the moments of normalized partial maxima for general error distribution a byproduct is to deduce the convergence rates of the moments of normalized maxima to the moments of the corresponding extreme value distribution

wavelet footprints theory algorithms and applications wavelet based algorithms have been successful in different signal processing tasks the wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients discontinuities often carry relevant signal information and therefore they represent a critical part to analyze we study the dependency across scales of the wavelet coefficients generated by discontinuities we start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual theorem 1 we then introduce the notion of footprints which are scale space vectors that model discontinuities in piecewise polynomial signals exactly we show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints this also leads to efficient approximation of piecewise smooth functions finally we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising compression and nonblind deconvolution in the case of compression we also prove that at high rates footprint based algorithms attain optimal performance theorem 3

symmetry based scalable lossless compression of 3d medical image data we propose a novel symmetry based technique for scalable lossless compression of 3d medical image data the proposed method employs the 2d integer wavelet transform to decorrelate the data and an intraband prediction method to reduce the energy of the sub bands by exploiting the anatomical symmetries typically present in structural medical images a modified version of the embedded block coder with optimized truncation ebcot tailored according to the characteristics of the data encodes the residual data generated after prediction to provide resolution and quality scalability performance evaluations on a wide range of real 3d medical images show an average improvement of 15 in lossless compression ratios when compared to other state of the art lossless compression methods that also provide resolution and quality scalability including 3d jpeg2000 jpeg2000 and h 264 avc intra coding

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed