High-resolution still picture compression

Abstract

We shall consider the problem of storing, transmitting, and manipulating digital electronic images. Because of the file sizes involved, transmitting images will always consume large amounts of bandwidth, and storing images will always require hefty resources. Because of the large number N of pixels in a high-resolution image, manipulation of digital images is infeasible without low-complexity algorithms, i.e., O(N) or 0( N log(N) ) . Our goal is to describe some new methods which are firmly grounded in harmonic analysis and the mathematical theory of function spaces, which promise to combine effective image compression with low-complexity image processing. We shall take a broad perspective, but we shall also compare specific new algorithms to the state of the art. Roughly speaking, most image compression algorithms split into three parts: invertible transformation, lossy quantization or rank reduction, and entropy coding (or redundancy removal). There are a few algorithms which differ fundamentally from this scheme, e.g., the collage coding algorithm [ 41, or pure vector quantization of the pixels. The former uses a deep observation that pictures of natural objects exhibit self-similarity at different scales; we prefer to avoid relying on this phenomenon, since our images may not be “natural.” The latter uses a complex algorithm to build a superefficient empirical vocabulary to describe an ensemble of images; we prefer to avoid training our algorithm with any sample of images, to avoid the problem of producing a sufficiently large and suitable ensemble. There has emerged an international standard for picture compression, promulgated by the Joint Photo-