| Early image compression standards, such as JPEG, use a cosine basis to obtain a sparse representation of an image signal. In recent years, wavelets have become the basis of choice for compressing natural images (e.g., the new JPEG2000 standard). We consider piecewise continuous cosine approximations of image signals and discuss how to use the properties of these approximations so as to take advantage of the inherent structure in natural images. In particular, we can first carefully form a variable block-size partition of the image domain (in contrast to JPEG's uniform block-size partition) and subsequently construct independent cosine approximations on each block. This approach allows us to minimize the influence of discontinuities (that is, edges) on the total error of the approximation. In this way, an adaptive cosine basis mimics the scaling property of wavelet bases, which implicitly isolate discontinuities in a signal. The adaptive cosine basis yields sparser representations of natural images than the best wavelet bases and significantly outperforms other cosine-based approaches. |
