Nonlinear image estimation using piecewise and local image models

We introduce a new approach to image estimation based on a flexible constraint framework that encapsulates meaningful structural image assumptions. Piecewise image models (PIMs) and local image models (LIMs) are defined and utilized to estimate noise-corrupted images, PIMs and LIMs are defined by image sets obeying certain piecewise or local image properties, such as piecewise linearity, or local monotonicity. By optimizing local image characteristics imposed by the models, image estimates are produced with respect to the characteristic sets defined by the models. Thus, we propose a new general formulation for nonlinear set-theoretic image estimation. Detailed image estimation algorithms and examples are given using two PIMs: piecewise constant (PICO) and piecewise linear (PILI) models, and two LIMs: locally monotonic (LOMO) and locally convex/concave (LOCO) models. These models define properties that hold over local image neighborhoods, and the corresponding image estimates may be inexpensively computed by iterative optimization algorithms. Forcing the model constraints to hold at every image coordinate of the solution defines a nonlinear regression problem that is generally nonconvex and combinatorial. However, approximate solutions may be computed in reasonable time using the novel generalized deterministic annealing (GDA) optimization technique, which is particularly well suited for locally constrained problems of this type. Results are given for corrupted imagery with signal-to-noise ratio (SNR) as low as 2 dB, demonstrating high quality image estimation as measured by local feature integrity, and improvement in SNR.