CONSTRUCTING OPTIMAL WAVELET BASIS FOR IMAGE COMPRESSION - Acoustics, Speech, and Signal Processing, 1996. ICASSP-96. Conference Proceedings., 1996 IEEE Inte

We study the problem of choosing an image based o p timal wavelet basis with compact support for image data compression and provide a general algorithm for computing the optimal wavelet basis. We parameterize the mother wavelet and scaling function of wavelet systems through a set of real coefficients of the relevant quadrature mirror filter (QMF) banks. We further introduce decomposition entropy as an information measure to describe the distance between a given digital image and its projection into the subspace spanned by the wavelet basis. The optimal basis for the given image is obtained through minimizing this information measure. The resulting subspace is used for image analysis and synthesis. Experiments show improved compression ratios due to the application of the optimal wavelet basis and demonstrate the potential applications of our methodology in image compression. This method is also useful for constructing efficient wavelet based image coding systems. 1. I N T R O D U C T I O N The last few years have witnessed extensive research interest and activities in wavelet theory and its applications in signal processing, image processing and many other fields [l, 21. The most attractive features of wavelet theory are the multiresolution property and time and frequency localization ability. There are many applications of these prop erties in the fields of signal processing, speech processing and especially in image processing [3, 4, 5, 61. It is well known that a wavelet system is usually determined by one mother wavelet function whose dilations and shifts span the signal space. Unlike sin and cos functions, individual wavelet functions are quite localized in frequency and time and they are not unique. Obviously, different wavelets shall yield different wavelet bases. An appropriate selection of the wavelet for signal representation can result in maximal benefits of this new technique. For example, compact wavelets are suitable for approximating discontinuous functions such as images while smooth wavelets are appropriate for solving integral functions to achieve high numerical accuracy. It is reasonable to think that if a wavelet contains enough information about an image to be represented, the wavelet system is going to be simplified in terms of the levels of required resolution. We are interested in finding an image based wavelet basis and applying the reThe key to choosing an image based optimal wavelet basis lies in the appropriate parameterization and adequate performance measure in image compression processes. A method was proposed for choosing a wavelet for signal r e p resentation based on minimizing an upper bound of the L2 norm of error [7, SI in approximating the signal up to a desired scale. Coifman et al. derived an entropy based algorithm for selecting the best basis from a library of wavelet packets [9]. We also proposed an information measure based approach for constructing an optimal discrete wavelet basis with compact support in our earlier work on adaptive wavelet neural networks [lo] and wavelet basis selection [ll]. We shall illustrate the application of our methodology to image compression. This paper is intended to demonstrate that choosing an image based optimal or suboptimal wavelet basis can improve compression ratios of images rather than to design a complete coding system. In the rest of the paper, we first provide the definition of optimal wavelet basis for a given digital image and parameterize the basis through the corresponding quadrature mirror filter (QMF) banks. We then introduce an algorithm for constructing an optimal wavelet basis. Next, we compare the effects of different mother wavelets on image representation and provide numerical results. Finally, we summarize our conclusions. 2. O P T I M A L WAVELET BASIS We first introduce a distance measure for optimization purpose. Inspired by the work in [9], we define an additive information measure of entropy type and the optimal basis as the following. We use Q(t ) to denote the wavelet basis spanned by dilating and shifting mother wavelet denoted by $(t ) . Deflnition 2..1 A non negative map M from a sequence {f;} to R is called an additive information measure if M(O) = 0 and M ( C , fi) = xi M ( f i ) . Deflnition 2 . 2 Let x E RN be a fixed vector containing digital image data and B denote the collection of all orthonormal bases of dimension N, a basis B E B is said to be optimal if M ( B x ) is minimal for all bases in B with respect to the vector x. The wavelet system is parameterized through using QMF banks. From the multiresolution property of wavelets due 0-7803-3 192-3/96 $5.0001996 IEEE 235 1 t o Mallat [12], the scaling function b(t) and mother wavelet +(t) are expressed as [2] + ( t ) = 4 2 C k d ( 2 t k) (1)