Compression Transforms—Hierarchical Including Wavelets

Shannon’s rate distortion theory demonstrates that the optimal distortion rate can be achieved by quantizing blocks of arbitrary large dimensions. In practice, however, most image compression coders rely on the principle of transform coding combined with scalar quantization and lossless coding. Transform coding consists in applying a linear transformation to an N-dimensional block of data. The linear transform should, in principle, decorrelate the N-dimensional input vector. Huang and Schulteiss have studied the performance of transform coding in the following context: A linear transform is combined with a fixed rate scalar quantizer. The rate R is measured by the entropy (in bits) and the distortion D is the mean squared error. The input source is assumed to be a weakly stationary Gaussian random variable. Huang and Schulteiss have shown that among all linear transforms the Karhunen–Loève (KL) transform allows to achieve the smallest distortion D(R) for a given rate R. Unfortunately, this theory does not apply to images that are neither stationary nor Gaussian. Another limitation of the KL transform is its computational complexity. One should therefore explore other transforms combined with other quantization strategies. Most transforms that are used for image compression operate in the frequency domain. For instance, the JPEG standard computes the Discrete Cosine Transform (DCT) of 8 8 blocks taken from the original image. While the DCT provides good energy compaction, the JPEG algorithm is not suited for compression rates higher than 30. One of the main limitation of DCT-based algorithms is their inability to exploit the multiscale structure that most natural images exhibit. As opposed to the Fourier or DCT transform, the wavelet transform performs a multiscale, or multiresolution, analysis of an image. The wavelet transform is a remarkably powerful way to represent images and is not only well suited to image compression but also to a wide variety of image processing problems. The goal of the present article is to provide an introduction to the wavelet transform and its application in image coding. The article is organized as follows. ‘‘The Laplacian Pyramid’’ describes one of the earliest hierarchical transforms: the Laplacian pyramid. In ‘‘Wavelets and Multiresolution Analysis, Biorthogonal Wavelet Bases’’ and ‘‘Multidimensional Wavelet Transforms,’’ we present the concept of multiresolution analysis, and describe the construction of wavelets. Wavelet-based coders are described in ‘‘Wavelet Coders.’’ Some important aspects of the coders, such as the choice of the filters and the quantization techniques, are described in ‘‘Choice of the Filters’’ and ‘‘Distribution of the Coefficients and Optimal Quantization.’’ A short discussion of JPEG-2000 is provided in ‘‘JPEG 2000.’’ The section ‘‘Wavelet Packet-Based Compression’’ describes one extension of wavelets: the wavelet packets. Finally, a list of software packages that provide an implementation of the wavelet transform is presented in ‘‘AVAILABLE SOFTWARE.’’