Framework for image segmentation, compression, and hybrid algorithms via wavelet estimation of local smoothness

The present dissertation provides a common, function analytic framework for image segmentation and image compression. As such, the present work serves the purpose of unification—at the level of mathematical underpinnings—of two active fields within image processing science. While interest attaches to these function analytic connections in se, this segmentation-compression marriage leads also to interesting results at the applied level—including new hybrid segmentation-compression algorithms.; The particular function analytic framework presented herein is one which builds upon, refines, and extends certain function analytic foundations to image compression set in place by DeVore, Jawerth, and Lucier (DJL) in the early and middle 1990's. DJL Theory provides, for wavelet-based image compression algorithms, image approximation error-versus-image size estimates in terms of the Besov space membership of the image. The present dissertation refines the estimates of DJL Theory, providing sharper estimates which depend explicitly upon attributes of the mother wavelet as well as the smoothness space membership of the image.; The “localization” of the DJL Theory is the vehicle for extending that theory toward connections with image segmentation theory. In this localization, a further sharpening of the estimates of DJL Theory is obtained. A surprise result in this area is a new segmentation-compression algorithm whose estimates, for certain classes of images, provide asymptotic performance gains over a standard wavelet compression algorithm.; The central result of this extension of the DJL Compression Theory to cover segmentation as well, is a new generalized Mumford-Shah segmentation. The variance merge criterion of Mumford-Shah is generalized to form a relative smoothness energy criterion, corresponding to the K-functional of the associated Besov space interpolation, having optimal separability in a precise, interpolation spaces sense.; In addition to improvements in the representation-error performance of algorithms, the connections to image segmentation imply improvements in algorithmic complexity and operation count. In particular, a theorem giving the Besov membership estimate in terms of wavelet coefficients is important in that it suggests fast algorithms.