Abstract:
An iterative process to determine the wavelet function and combination of ales of the function which provides data where there is a large separability compared to the separability of the data set prior to processing. Wavelets are selected for inclusion in a library in accordance with predetermined criteria and then applied to a digitized signal by convolution to perform digital filtering. The convolution of each wavelet is performed for the number of times dictated by the coefficients of the wavelet for each of the input signal samples. Separability of the wavelet implemented digital filtration is calculated as a percentage for each wavelet. The separation data is stored in memory until the iterative process is applied to all wavelets. The separability data is then examined to identify the wavelet producing the greatest separation. The data separability is estimated using a likelihood ratio after the probability densities for each of two sets of profile data are estimated. The lower and upper bounds for a Bayes error are determined using resubstitution (R) and leave one out (L) methods, respectively.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates generally to feature extraction from an image for classification purposes. In particular, the present invention relates to feature extraction based on spatial frequency using wavelets. 
     2. Description of the Prior Art 
     The problem addressed by the present invention relates to feature extraction based on spatial frequency content for a statistical pattern recognition system. There is a need to reduce the dimensionality and to remove ambiguities, such as noise, in the data to be classified while maintaining the separability of the data set. Extracting features based on frequency involves a tradeoff between resolution in the time or space domain and resolution of the power spectral density estimate of the original signature. The power spectral density estimate is an indication of the frequency distribution in the signature. Another issue to be considered in the feature extraction process is the amount of required computation time. 
     The prior art technology consists of using orthogonal transforms such as the Fourier Transform and conventional digital filtering techniques (Finite Impulse Response (FIR) or Infinite Impulse Response (IIR)) to extract the frequency information. The discrete wavelet transform offers a tradeoff between spatial and frequency resolution that is desirable in problems such as feature extraction. The Fourier Transform and digital filtering techniques have a &#34;fixed&#34; resolution tradeoff regardless of the frequency content of the original signature. 
     The wavelet transform, however, has short basis functions to detect the high frequency band and long basis functions to detect the low frequency band. This unique characteristic of the wavelet transform allows for noise removal and a reduction in dimensionality of the original signature that is superior to that of classical transform and filtering techniques for many applications. 
     Although there are many applications which would benefit from improvement in spatial frequency feature extraction, the present invention has during testing shown improvements possible in the design of a ship classification system using high resolution radar range profiles. It has been shown that wavelet processing has maintained more separability than the Fourier Transform has for a data set consisting of high resolution radar returns from two separate ships. 
     The first step in the method is to preselect a set of filters having the characteristics desired for the particular application, each filter defined by a particular wavelet function. For the purposes of the present invention it is required that filter action maximize the amount of separability while minimizing processing time. There are many tradeoffs to address and a considerable body of open literature available describing these tradeoffs. For example, and in relation to the present invention, in applications involving ship images obtained from radar profiles the following wavelet was considered for the reasons set forth: 
     Wavelet 1  = Daubechies wavelet with the following four coefficients: 0.48296291, 0.8365163, 0.22414386, -0.1294095. 
     The reasons are set forth as follows: A small number of coefficients are required for computational efficiency, time localization and orthogonal filter. 
     The above is exemplary only and should not be deemed limiting our invention in any way. 
     SUMMARY OF THE INVENTION 
     It is thus an object of the present invention to provide a means for extracting features from an input signal to permit reliable classification of data. 
     It is yet another object of the invention to provide a means for extracting as few features from an input signal as needed to permit reliable and repeatable classification of the target represented by such signal. 
     It is still another object of the present invention to provide a means for extracting features of input signals to permit differentiation between them by determining and maximizing the separability between the filtered input signal data. 
     It is finally another object of the present invention to provide a means for maintaining an improved level of separability between data elements in a data set consisting of data from two separate sources such as high-resolution radar returns from two separate targets consisting of, for example ships or aircraft, simultaneously illuminated by a radar or other illumination source. 
     These and other objects of the present invention are satisfied by an apparatus and method for performing spatial frequency feature extraction using wavelets in order to implement a classification system. 
     The present invention comprises an iterative process to determine the appropriate wavelet function and combination of scales of this function that provide data in which there is a large amount of separability compared to the separability of the data set prior to the wavelet processing. The amount of separability between the input data sets is determined and compared to a pre-established criterion. If the criterion is not met the process continues until the criterion is met. 
     The invention requires the creation of a library of preselected wavelet functions, hereinafter referred to as wavelets. Wavelets are selected for inclusion in the library in accordance with criteria dictated by the particular application. 
     The shapes and numbers of coefficients of the wavelets are selected to produce the filtration of input data to provide computational efficiency and data separability consistent with the feature extraction demands of the application. After input signal digitization wavelets from the library are applied to the digitized signal by convolution digitally to perform digital filtering. The convolution of each wavelet is performed for the number of times dictated by the coefficients of that wavelet for each of the input signal samples, where the number of samples is a function of the scales selected by the analyst. The scales are chosen to obtain the levels of resolution desired and consistent with the quantity of data available to support the choice. Each scale has a resolution that is one-half that of the previous scale. The greater the number of samples, the finer the resolution. The fewer the number of samples, the coarser the resolution. Each wavelet is applied at each scale and coefficient to each set of input data. Separability of the resultant wavelet implemented digital filtration is determined for each of the scales of filtration. Separability is calculated as a percentage for each wavelet. The separation data is stored in memory until all wavelets have been applied in the iterative process of the present invention. 
     When all wavelets have been applied, the separability data is examined to identify the wavelet producing the greatest separation. The data separability is estimated using the likelihood ratio after the probability densities for each of the two sets of profile data are estimated. The lower and upper bounds for the Bayes error are determined using the resubstitution (R) and the leave one out (L) methods, respectively. 
     An appreciation of the objectives of the present invention and a more complete understanding of its structure and method of operation may be had by studying the following description of the preferred embodiment and by referring to the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a functional block diagram of the present invention; 
     FIG. 2 is a functional block diagram of the data separation portion of the present invention; and 
     FIGS. 3A, 3B, 3C and 3D are a series of range profiles including those wavelets processed for multiresolution results. 
     FIG. 4 is a KNN plot used for deriving a Bayes error estimate; and 
     FIG. 5 is a comparative plot of Bayes error versus number of features for two classes of range-only-radar target profiles for wavelet and PSD processing. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The present invention is a method and an apparatus for implementing the method for extracting spatial frequency features from an analog profile or signature information produced by a target, either directly or by reflection. The present invention extracts features in a manner that optimizes the separability of the data processed so that multiple targets simultaneously present in the data are easily distinguished. 
     Referring now to FIG. 1, this figure depicts the apparatus of the present invention as well as the methodology involved. In FIG. 1 the feature extraction apparatus 10 of the present invention is shown to comprise a preassembled wavelet library 14 of a type selected for inclusion on the basis of general inclusion criteria. General inclusion criteria includes consideration of the type of data to be operated upon. Ship, aircraft, ground vehicle, noise content and other factors relating to the application data are considered. The wavelet selection device 18 may be, for example, a keyboard or a computer program subroutine connected to communicate with and between itself and the wavelet library 14 and the digital processor 26. A scale selection device 22 can likewise be one of several means for communicating scale selection for each wavelet selected to the digital processor 26. Thus, a keyboard or digital means such as a computer subroutine may be used to provide the appropriate scale inputs consistent with the wavelet selected. 
     The range profile input device 30 may be a radar receiver or an analog signal storage means. The range profile input 30 is connected to the analog to digital (A/D) converter 34 for digitizing the analog input signal from the range profile input device 30. The digital output of digital processor 26 is connected to an output device 42 which may be a video display, a printer, or a combination of output devices. Digital processor 26 is connected to communicate with a memory 38 which may be internal or external to digital processor 26 or a combination of both internal and external memory as applications environments, and weight and space limitations of a host vehicle dictate. Digital processor 26 also connects to range profile input device 30 via a connecting link 32. A convolver 46 and a separation processor 50 are internal to digital processor 26. Memory 38 is connected to output device 42. 
     Referring to FIGS. 1 and 2, FIG. 2 depicts the data separation processor 50 of FIG. 1 which is depicted as being located within digital processor 26. Data separation processor 50 comprises a probability density estimator 54 which is connected to receive wavelet-filtered input data from convolver 46 in digital processor 26 and to provide its output to a likelihood ratio processor 58. The output of likelihood ratio processor 58 is connected to a resubstitution (R) processor 62 and a leave-one-out (L) processor 66 which are in parallel with each other. The outputs of R processor 62 and L processor 66 are connected to a Bayes error processor 70 which is connected to provide its output to memory 38 (FIG. 1) and output device 42 (FIG. 1). 
     OPERATION OF THE PRESENT INVENTION 
     Referring to FIGS. 1 and 2, feature extraction apparatus 10 requires the preparation of the wavelet library 14. Wavelet functions selected for the particular type of data to be processed and the data separation desired are obtained from sources such as Daubechies, as discussed below. The wavelet shapes, coefficients, and the scales at which they are to be applied to the number of samples of input data are considered. Computational efficiency is a consideration when real versus non-realtime operation is important. In addition, the following considerations and others in the literature guide the user of the present invention in selecting the wavelets for inclusion in library 14. How much and what kind of information can be acceptably filtered out? What information must survive the filtering process and be maintained? 
     By way of example and as used in one preferred embodiment of the present invention, wavelets were selected from those identified by Ingrid Daubechies in her paper titled &#34;Orthonormal Bases of Compactly Supported Wavelets&#34; from Communications on Pure and Applied Mathematics, Vol. XII, 909,996 (1988) John Wiley and Sons, Inc.. An Example of a wavelet included in library 14 of the present invention is: 
     a. a Daubechies wavelet with the following coefficients: 0.48296291, 0.8365163, 0.22414386, --0.1294095. 
     b. The Daubechies wavelet was selected for inclusion for the following reasons (in relation to selection criteria): A small number of coefficients are required for computational efficiency, time localization and orthogonal filter. 
     For an embodiment of the present invention applied to two sets of ship profile data which was input from analog radar receivers and having approximately 1500 data points per set, library 14 was created with Daubechies wavelets having coefficients of 2, 6, 8 and 10. 
     To initiate processing by the present invention range profile data in analog form is obtained by range profile input 30 which may be a radar receiver providing direct input or an analog storage device containing data received from a receiver of some type. The analog input is digitized by analog to digital converter 34 and the digital data is input to digital processor 26. 
     The present invention requires that the user initiate wavelet selection via wavelet selection device 18 calling for a particular wavelet from wavelet library 14. The wavelet selection device 18 may be a keyboard, a preprogrammed digital device or a computer subroutine called directly or indirectly by the user. The user also initiates scale selection device 22 so that the appropriate first scale, and those following in order, serially or in parallel are forwarded to digital processor 26 for use with the first selected wavelet. 
     Digital processor 26 operates on the incoming digitalized profile data in each of the two sets. In the preferred embodiment, the two sets of ship profiles contained approximately 1500 profiles each. The digital processor 26 performs a wavelet transform with the selected wavelet at each of the scales selected on each of the data samples from each profile in each target set. The transform process is convolution. The waveform convolutions with each profile data sample produces in effect a filtered result for each of the scales used for each selected wavelet. Thus, for a coefficient of 2 the number of data points to which the wavelet transform is applied was 128, for a coefficient of 6 the number of data points was 64, for a coefficient of 8 the number of data points was 32, and for a coefficient of 10 the number of data points was 16. The wavelet transform comprising the convolution of wavelet values at the coefficient points with each of the data samples of the input data signal is a filtering operation for which the output is numerous versions of the filtered original signal at different resolutions. Each version of the original signal has a resolution that is one-half of the previous version. Each version is commonly referred to as a scale. The user can choose any number of scales for the wavelet transform if there is enough data to support the choice. The transform for one wavelet at one scale is performed on the profile or signature data in each of the two sets representing individually two separate targets. 
     After completion of the convolution filtering, the filtered data for each scale is operated upon by separation processor 50. The filtered data is input to the probability density estimator 54 for each of the two sets of data being operated upon. The probability density is estimated using a nonparametric K Nearest Neighbor (KNN) estimator. This estimator appears as a computer software subroutine in Appendix A. The KNN density estimate is a non-parametric estimation technique in which the probability density is estimated locally by a small number of neighboring samples in a potentially high-dimensional space. The volume from which the samples are drawn in obtaining this estimate is inversely proportional to the density within the volume. The equation for the KNN density estimate is as follows: ##EQU1## where, X= the location at which the density function is estimated. 
     V(X,k)= the hyperspherical volume of the local region surrounding X, which encompasses all k nearest neighbors. 
     N= the total number of samples drawn. 
     k= the total number of samples that are within the volume V(X,k). 
     After the probability density is estimated for each data set the likelihood ratio classification of the data is performed by the likelihood ratio processor 58. The likelihood ratio is a ratio of one probability density to another and is used to develop an optimal classification given a known probability density function. In the present invention the objective is to quickly find and apply the best of the pre-selected wavelets to permit digital wavelet filtration of two sets of data to achieve maximum separation of that data. Thus, for each wavelet applied to the two sets of data an estimate of the separability is obtained based on the likelihood ratio. 
     The likelihood ratio data is next sent to the resubstitution (R) processor 62 and the leave-one-out (L) processor 66 for controlled input to the Bayes error processor 70. The upper and lower bounds for the Bayes error calculation is thus determined using the resubstitution (R) and leave-one-out (L) methods, respectively. The results of this processing of the two data sets is depicted in FIG.4 where the leave-one-out plot 94 and resubstitution plot 98 are graphically depicted asymptotically approaching the graphic representation (plot) 102 representing the Bayes error for the separability of the two data sets filtered by one wavelet at one scale. 
     Using the present invention method implemented and mechanized by the apparatus of the invention, the results of using the discrete wavelet transform for feature extraction for an actual Range-Only-Radar (ROR) ship classification problem is illustrated in FIG. 5. Approximately 1500 range profiles for each of two different ships were used in testing the performance of the wavelet transform for feature extraction. As described previously, the range profiles were fed into the feature extraction apparatus 10, the wavelet transform applied, a KNN probability density estimation process performed, and a Bayes error estimation obtained in separation processor 50. FIGS. 3b, 3c and 3d illustrate three stages of range profile decomposition for the range profile of FIG. 3a using the present invention. FIG. 4 shows the plot of the output of the L and R processors for different values of K. FIG. 5 shows how the Bayes error is dependent on the number of features used in a classification system using the wavelet technique of the present invention versus a Fourier transform. The separability of two classes is inversely proportional to the Bayes error. It is noted from FIG. 5 that the performance of the wavelet transform 106 using the present invention is far superior to that obtained using the Fourier transform 110 approach. 
     The structures and methods disclosed herein illustrate the principles of the present invention. The invention may be embodied in other specific forms without departing from its spirit or essential characteristics. The described embodiments are to be considered in all respects as exemplary and illustrative rather than restrictive. Therefore, the appended claims rather than the foregoing description define the scope of the invention. All modifications to the embodiments described herein that come within the meaning and range of equivalence of the claims are embraced within the scope of the invention. ##SPC1##