Abstract
The approximation of signals and images with parsimonious representations in a wavelet basis is studied, together with its application to image compression. The decay rate of wavelet coefficients depends on the local regularity of the signal. The wavelet coefficients of a piecewise regular function have a large amplitude in the vicinity of singularities. The rate of decay of the nonlinear approximation error in a wavelet basis is related to the local regularity of the signal, and to the sparsity of its coefficients. Local regularity is expressed by membership of Besov spaces, which are characterized by the lp norms of the wavelet coefficients. For images, the approximation error in a wavelet basis depends on the length of the contours along which the image is discontinuous.
A wavelet compression algorithm begins by quantizing the wavelet coefficients of the signal and then encodes their values in binary form using entropy coding. For parsimonious representations, we demonstrate that the number of bits required is proportional to the number M of non-zero coefficients in the basis. The error introduced by quantization of the coefficients can also be related to M, and therefore to the number of bits used by the code. The JPEG-2000 image compression standard is implemented by such an algorithm in a wavelet basis.
