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Piecewise Linear Approximation of Vector-Valued Images and Curves via Second-Order Variational Model

IEEE TRANSACTIONS ON IMAGE PROCESSING 2017年第9期26卷 4414-4429页

作者： Zanetti, Massimo Bruzzone, Lorenzo Univ Trento Dept Informat Engn & Comp Sci I-38123 Trento Italy

Variational models are known to work well for addressing image restoration/regularization problems. However, most of the methods proposed in the literature are defined for scalar inputs and are used on multiband images (such as RGB or multispectral imagery) by the composition of a simple band-wise processing. This involves suboptimal results and may introduce artifacts. Only in a few cases, variational models are extended to the case of vector-valued inputs. However, the known implementations are restricted to the first-order models, while the secondorder models are never considered. Thus, typical problems of the first-order models, such as the staircasing effect cannot be overtaken. This paper considers a second-order functional model to function approximation with free discontinuities given by Blake-Zisserman (BZ) and proposes an efficient minimization algorithm in the case of vector-valued inputs. In the BZ model, the Hessian of the solution is penalized outside a set of finite length, therefore the solution is forced to be piecewise linear. Moreover, the model allows the formation of free discontinuities and free gradient discontinuities. The proposed algorithm is applied to difficult color image restoration/regularization problems and to piecewise linear approximation of curves in space.

关键词： Multiband image variational methods Blake-Zisserman Mumford-Shah piecewise linear approximation block-coordinate descent method

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Numerical minimization of a second-order functional for image segmentation

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2016年 36卷 528-548页

作者： Zanetti, Massimo Ruggiero, Valeria Miranda, Michele, Jr. Univ Trento Dept Informat Engn & Comp Sci I-38123 Trento Italy Univ Ferrara Dipartimento Matemat & Informat I-44122 Ferrara Italy

In this paper we address the numerical minimization of a variational approximation of the Blake-Zisserman functional given by Ambrosio, Faina and March. Our approach exploits a compact matricial formulation of the objective functional and its decomposition into quadratic sparse convex sub-problems. This structure is well suited for using a block coordinate descent method that cyclically determines a descent direction with respect to a block of variables by few iterations of a preconditioned conjugate gradient algorithm. We prove that the computed search directions are gradient related and, with convenient step sizes, we obtain that any limit point of the generated sequence is a stationary point of the objective functional. An extensive experimentation on different datasets including real and synthetic images and digital surface models, enables us to conclude that: (1) the numerical method has satisfying performance in terms of accuracy and computational time;(2) a minimizer of the proposed discrete functional preserves the expected good geometrical properties of the Blake-Zisserman functional, i.e., it is able to detect first and second order edge-boundaries in images and (3) the method allows the segmentation of large images. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Blake-Zisserman functional Variational segmentation Edge-crease detection block-coordinate descent method

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