## Image Compression

Image compression seeks to reduce the number of bits involved in representing an image. Most compression algorithms in practice are digital, beginning with an information source that is discrete in time and amplitude. If an image is initially analog in space and amplitude, one must first render it discrete in both space and amplitude before compression. Discretization in space is generally called sampling — this consists of examining the intensity of the analog image on a regular grid of points called picture elements or pixels. Discretization in amplitude is simply scalar quantization: a mapping from a continuous range of possible values into a finite set of approximating values. The term analog-to-digital (A/D) conversion is often used to mean both sampling and quantization — that is, the conversion of a signal that is analog in both space and amplitude to a signal that is discrete in both space and amplitude. Such a conversion is by itself an example of lossy compression.

A general system for digital image compression is depicted in Fig. 1. It consists of one or more of the following operations, which may be combined with each other or with additional signal processing:

• Signal decomposition: The image is decomposed into several images for separate processing. The most popular signal decompositions for image processing are linear transformations of the Fourier family, especially the discrete cosine transform (DCT), and filtering with a subband or wavelet filter bank. Both methods can be viewed as transforms of the original images into coefficients with respect to some set of basis functions. There are many motivations behind such decompositions. Transforms tend to "mash up" the data so that the effects of quantization error are spread out and ultimately invisible. Good transforms concentrate the data in the lower order transform coefficients so that the higher order coefficients can be coded with few or no bits. Good transforms tend to decorrelate the data with the intention of rendering simple scalar quantization more efficient. The eye and ear are generally considered to operate in the transform domain, so that it is natural to focus on coding in that domain where psychophysical effects such as masking can be easily incorporated into frequency-dependent measures of distortion. Lastly, the transformed data may provide a useful data structure, as do the multiresolution representations of wavelet analysis.

• Quantization: High-rate digital pixel intensities are converted into relatively small numbers of bits. This operation is nonlinear and noninvertible; it is "lossy." The conversion can operate on individual pixels (scalar quantization) or groups of pixels (vector quantization). Quantization can include discarding some of the components of the signal decomposition step. Our emphasis is on quantizer design.

• Lossless compression: Further compression is achieved by an invertible (lossless, entropy) code such as run-length, Huffman, Lempel-Ziv, or arithmetic code.

FIGURE l Image compression system.

### FIGURE l Image compression system.

Many approaches to systems for image compression have been proposed in the literature and incorporated into standards and products, both software and hardware. We note that the methods discussed in this chapter for evaluating the quality and utility of lossy compressed medical images, do not depend on the compression algorithm at all. The reader is referred to the literature on the subject for more information on image compression [23,50].

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