EEE TRANSACTIONS ON IMAGE PROCESSING. VOL I. NO. 2. APRIL 1992 Image Coding using Wavelet Transform Marc Antonini, Michel Barlaud, Member, IEEE, Pierre Mathieu, and Ingrid Daubechies, Member, IEEE together with its implementation as described by Mallat [27], satisfies each of these conditions. wvelet s; this new method involves two steps. First, we use aa wavelet transform and a vector quantization coding der to obtain a set of biorthogonal sub- scheme. The wavelet coefficients are coded considering a asses of images; the original image is decomposed at different noise shaping bit allocation procedure. This technique ex maintains constant the number of pixels required to describe in the image data, enabling bit rate reduction. %N s an Section lI describes the wavelet transforms used in this ory, the wavelet coefficients are vector quantized using a multi- paper. After a quick review of wavelets in general, we resolution codebook. Furthermore, to encode the wavelet coef- explain in more detail the properties and construction of hich assumes that details at high resolution are less visible to re regular biorthogonal wavelet bases. We then extend tI m ize a picture as quickiy as possible at minimum cost we scheme with separable filters. The new coding scher wavelet t ransforesi is tranimission scheme It is shown that the next presented in Section ll. We focus particularly in this transmission ients, on the asymptotic coding gain that can be achieved orthogonal wavelet, multiscale py. using vector quantization in the subimages, and on the algorithm, vector quantization, noise shaping, pro- optimal allocation across the subimages. Experimental re- sults are given in Section IV for images taken within and outside of the training Se ACTIc different fields, digitized images are replacing A. A Short Review of Wavelet Analysis onal analog images as photograph or x-rays Wavelets are functions generated from one single func costly. The information contained in the images must 1/ therefore, be compressed by extracting only the visible elements, which are then encoded, The quantity of data (For this introduction we assume t is a one-dimen- nvolved is thus reduced substantially sional variable). The mother wavelet v has to satisfy A fundamental goal of data compression is to reduce dx v(x)=0, which implies at least the bit rate for transmission or storage while maintaining (Technically speaking, the condition on y should an acceptable fidelity or image quality. Compression can dw |Y(w)l2-< oo, where y is the Fourier trans be achieved by transforming the data, projecting it on a form of y; if v(n) decays faster than t for t-o, then basis of functions, and then encoding this transform. Be- this condition is equivalent to the one above). The defi cause of the nature of the image signal and the mecha- nition of wavelets as dilates of one function means that nisms of human vision, the transform used must accept high frequency wavelets correspond to a I or narrow nonstationarity and be well localized in both the space and width, while low frequency wavelets have a>I or wider frequency domains. To avoid redundancy, which hinders width compression, the transform must be at least biorthogonal The basic idea of the wavelet transform is to represent and lastly, in order to save CPU time, the corresponding any arbitrary function f as a superposition of wavelets algorithm must be fast. The two-dimensional wavelet Any such superposition decomposes f into different scale transform defined by Meyer and Lemarie [31], [24], [25], levels, where each level is then further decomposed with a resolution adapted to the level. One way to achieve such ipt received February 7, 1990: revised March 26 a decomposition writes f as an integral over a and b of chies is with at&T Bell Laboratories Murray Hill. N) 07974. tice, one prefers to write f as a discrete superpose ition(sum IEEE Log Number 9106073 rather than integral). Therefore, one introduces a discre 1057714992s3.00@1992IEEE