We cover two topics in the broad area of nonlinear multiscale methods. In the first topic, we develop computationally efficient procedures for solving certain restoration problems in 1-D, including the discrete versions of the total variation regularized problem and the constrained total variation minimization problem. They are based on a simple nonlinear diffusion equation and related to the Perona-Malik equation. A probabilistic interpretation for this diffusion equation in 1-D is provided by showing that it produces optimal solutions to a sequence of estimation problems. We extend our methods to 2-D where they no longer have similar optimality properties; however, we experimentally demonstrate their effectiveness for image restoration. In the second topic we introduce a new framework of multitree dictionaries and propose new algorithms for efficiently finding the best representation in a multitree dictionary. We apply our framework to develop novel dynamic programming algorithms for finding the best basis in a dictionary of arbitrary lapped bases in 1-D. We illustrate this using a non-dyadic local cosine dictionary, and show that the resulting representations are more compact and are characterized by lower costs and approximate shift-invariance. We also provide an algorithm which is strictly shift-invariant and several accelerated versions of the basic algorithm which explore various tradeoffs between computational efficiency and adaptability. A novel dictionary which constructs the best local cosine representation in the frequency domain is proposed and shown to be better suited for representing certain types of signals. We apply our framework in 2-D to develop novel tree-pruning algorithms for finding the best basis in an arbitrary multitree dictionary. We illustrate our framework through several examples, including a novel block image coder which significantly outperforms both the standard JPEG and quadtree-based methods, and is comparable to embedded coders such as JPEC2000 and SPIHT
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