The transformation of image data is of critical importance in applications of image compression, feature extraction and noise reduction. It provides an organization of image data by which the data can be better analyzed, prioritized, quantized and/or discarded. In general, the choice of signal expansion provides a fundamental limitation within these applications. For example, the rate-distortion performance of the compression of an image using the JPEG algorithm is governed by the ability of the Discrete Cosine Transform (DCT) to decorrelate the image data. For some images, such as those with textured regions, the correlation between pixels may be low. In these cases, the DCT does not provide the best organization of the image data. The suitability of a particular transformation depends on the signal characteristics. In spite of the deficiencies of non-adaptive transforms, typically, the selection of the signal expansion is made non-adaptively to the image signal.
In this paper we present an image compression system which includes the adaptive selection of the signal expansion. We describe in detail the design and implementation of the system. We also propose a novel approach for the efficient expansion of an image into a joint space and spatial-frequency (s/s-f) library from which the best basis is selected. The expansion is produced by cascading permutations of two fundamental elements: (1) two-channel filter bank, and (2) binary segmentation. Since these operations are commutative, we organize the cascade system into a graph. The best joint space and frequency basis is found from the library by pruning the graph such that the minimum cost embedded graph is found.