Image smoothing with shape invariance and L1 curvature constraints

Michael Blauer; Martin D. Levine

doi:10.1117/12.135153

1 February 1992 Image smoothing with shape invariance and L₁ curvature constraints

Michael Blauer, Martin D. Levine

Proceedings Volume 1610, Curves and Surfaces in Computer Vision and Graphics II; (1992) https://doi.org/10.1117/12.135153
Event: Robotics '91, 1991, Boston, MA, United States

Abstract

We investigate the regularization problem with a robust L1 (absolute value) smoothness metric. This metric provides the best tradeoff between the standard L2 norm, which is nonrobust and strongly smoothing, and the nonconvex metrics associated with line process methods that require very costly stochastic optimization. We show that solution can be obtained from a convex analog neural network by replacing the nondifferentiable L1 function with a smoothed version that corresponds to a Huber estimator. The resulting analog network implements a direct descent process on the original cost function and involves sigmoid nonlinearities that are the derivative of the smoothed L1 norm. However, for digital computation the direct descent approach is very ineffective due to the near nondifferentiability of the objective function and the fact that line search cannot be performed analytically. We address this problem by introducing a shape invariance constraint that may be derived from the original image by assuming that the sign of the response to the regularization kernel remains unchanged. The problem may now be transformed into a sparse quadratic program with linear inequality constraints. In turn, this program may be solved very effectively through its dual by a coordinate relaxation algorithm that involves only nonnegativity constraints. The resulting algorithm is extremely simple and amounts to iterative application of a nonlinear operator consisting of convolution followed by rectification on an auxiliary Lagrangian image. The algorithm admits very simple parallel and asynchronous implementation.

Citation Download Citation

Michael Blauer and Martin D. Levine "Image smoothing with shape invariance and L₁ curvature constraints", Proc. SPIE 1610, Curves and Surfaces in Computer Vision and Graphics II, (1 February 1992); https://doi.org/10.1117/12.135153

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