A Novel Convex Relaxation for Non-binary Discrete Tomography

Kuske, Jan; Swoboda, Paul; Petra, Stefania

doi:10.1007/978-3-319-58771-4_19

Cited by 7 publications

(13 citation statements)

References 17 publications

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“…Cell tracking Small and large cell tracking problems from the study [24]. Discrete tomography The synthetic discrete tomography dataset introduced in [32] consisting of a few thousand instances with a varying number of projections and object densities.…”

Section: Methodsmentioning

confidence: 99%

Efficient Message Passing for 0-1 ILPs with Binary Decision Diagrams

Lange¹,

Swoboda²

2020

Preprint

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We present an efficient message passing method for 0-1 integer linear programs. Our algorithm is based on a decomposition of the original problem into subproblems that are represented as binary decision diagrams. The resulting Lagrangean dual is solved iteratively by a series of efficient block coordinate ascent steps. Our method combines techniques from dedicated solvers developed for subclasses of combinatorial problems in structured prediction with the generality of 0-1 integer linear programming. In particular, our solver has linear iteration complexity w.r.t. the problem decomposition size and empirically needs a moderate number of iterations to converge. We present experimental results on combinatorial problems from MAP inference for Markov Random Fields, graph matching, discrete tomography and cell tracking for developmental biology and show competitive performance.

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Section: Methodsmentioning

confidence: 99%

Efficient Message Passing for 0-1 ILPs with Binary Decision Diagrams

Lange¹,

Swoboda²

2020

Preprint

Self Cite

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“…Proof of equivalence of (19) and (3). Define an extension (20) does not have a feasible solution, and sof t (x) = +∞.…”

Section: A Proofsmentioning

confidence: 99%

“…In [19] a dual decomposition based solver was proposed for the multi-label discrete tomography problem. The decomposition was optimized with ConicBundle [7].…”

Section: Discrete Tomographymentioning

confidence: 99%

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MAP Inference via Block-Coordinate Frank-Wolfe Algorithm

Swoboda

Kolmogorov

2019

2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)

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We present a new proximal bundle method for Maximum-A-Posteriori (MAP) inference in structured energy minimization problems. The method optimizes a Lagrangean relaxation of the original energy minimization problem using a multi plane block-coordinate Frank-Wolfe method that takes advantage of the specific structure of the Lagrangean decomposition. We show empirically that our method outperforms state-of-the-art Lagrangean decomposition based algorithms on some challenging Markov Random Field, multi-label discrete tomography and graph matching problems.

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“…While these approaches can often be implemented efficiently and apply to large-scale problems, getting them to converge to the correct binary solution can be challenging. Another variant of convex relaxation include the linear-programming based method [23]. This method works well on small-scale images and noise-free data.…”

Section: B Literature Reviewmentioning

confidence: 99%

A Convex Formulation for Binary Tomography

Kadu

Leeuwen

2020

IEEE Trans. Comput. Imaging

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Binary tomography is concerned with the recovery of binary images from a few of their projections (i.e., sums of the pixel values along various directions). To reconstruct an image from noisy projection data, one can pose it as a constrained least-squares problem. As the constraints are non-convex, many approaches for solving it rely on either relaxing the constraints or heuristics. In this paper we propose a novel convex formulation, based on the Lagrange dual of the constrained least-squares problem. The resulting problem is a generalized LASSO problem which can be solved efficiently. It is a relaxation in the sense that it can only be guaranteed to give a feasible solution; not necessarily the optimal one. In exhaustive experiments on small images (2×2, 3 × 3, 4 × 4) we find, however, that if the problem has a unique solution, our dual approach finds it. In case of multiple solutions, our approach finds the commonalities between the solutions. Further experiments on realistic numerical phantoms and an experiment on X-ray dataset show that our method compares favourably to Total Variation and DART.

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A Novel Convex Relaxation for Non-binary Discrete Tomography

Cited by 7 publications

References 17 publications

Efficient Message Passing for 0-1 ILPs with Binary Decision Diagrams

Efficient Message Passing for 0-1 ILPs with Binary Decision Diagrams

MAP Inference via Block-Coordinate Frank-Wolfe Algorithm

A Convex Formulation for Binary Tomography

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