Abstract
Cosine bases are Fourier transforms that symmetrize edge conditions to define periodic signals that are not discontinuous at the edges. To encode an audio signal, it is sliced into time slices which are represented in an orthogonal cosine basis. The cosine coefficients are compressed by quantization and entropy coding. Perceptual error is minimized by a masking technique. For image compression, the JPEG standard divides images into 8 by 8 windows pixels, which are represented in a separable cosine base. This standard specifies the quantization and entropy coding of the coefficients.
These algorithms usually operate in a high-compression regime, where the high-resolution quantization assumption is no longer valid. This introduces a non-linear approximation phenomenon that governs the efficiency of these compression algorithms. This non-linear approximation term can be calculated as a function of the sparsity of the coefficients in the orthogonal basis. This provides a link between the error produced by quantization and the number of coding bits. Its behavior is totally different from that obtained for a low compression ratio. These results show that compression can be improved by optimizing the basis to increase coefficient sparsity. The JPEG-2000 standard thus improves the performance of the JPEG standard by replacing the cosine basis with an orthogonal wavelet basis.
