Extending continuous cuts: Anisotropic metrics and expansion moves

Byröd, Martin; Overgaard, Niels Christian; Kahl, Fredrik

doi:10.1109/iccv.2009.5459206

Cited by 15 publications

(6 citation statements)

References 19 publications

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“…To our knowledge, this is the first bound available for the fully spatially convex relaxed problem (13). Related is the work of Olsson et al [22,23], where the authors consider a continuous analogue to the α-expansion method known as continuous binary fusion [27], and claim that a bound similar to (16) holds for the corresponding fixed points when using the separable regularizer…”

Section: Contribution and Main Resultsmentioning

confidence: 79%

Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

Lellmann

Lenzen

Schnörr

2011

Lecture Notes in Computer Science

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We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation methods for finitedimensional problems. While for the latter case several optimality bounds are known, to our knowledge no such bounds exist in the continuous setting. We provide such a bound by analyzing a probabilistic rounding method, showing that it is possible to obtain an integral solution of the original partitioning problem from a solution of the relaxed problem with an a priori upper bound on the objective, ensuring the quality of the result from the viewpoint of optimization. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula.

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Section: Contribution and Main Resultsmentioning

confidence: 79%

Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

Lellmann

Lenzen

Schnörr

2011

Lecture Notes in Computer Science

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“…We observe that the first term of E(u * ) according to (13) Furthermore, the anisotropic coarea formula [38] can be used to express the last total variation term of E(u * ) as an integral over the length of all level lines of u * , measured in the norm induced by both the weighting map g and the anisotropic tensor D, i. e. Clearly, the functional is now merely an integral over the binary characteristic functions of the upper level sets of u * . Now, we can deduce E(u * ) = E(1 Σ ν,u * ).…”

Section: Appendix Amentioning

confidence: 91%

A variational approach to vesicle membrane reconstruction from fluorescence imaging

Kolev

Kirchgeíner

Houben

et al. 2011

Pattern Recognition

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Biological applications like vesicle membrane analysis involve the precise segmentation of 3D structures in noisy volumetric data, obtained by techniques like magnetic resonance imaging (MRI) or laser scanning microscopy (LSM). Dealing with such data is a challenging task and requires robust and accurate segmentation methods. In this article, we propose a novel energy model for 3D segmentation fusing various cues like regional intensity subdivision, edge alignment and orientation information. The uniqueness of the approach consists in the definition of a new anisotropic regularizer, which accounts for the unbalanced slicing of the measured volume data, and the generalization of an efficient numerical scheme for solving the arising minimization problem, based on linearization and fixed-point iteration. We show how the proposed energy model can be optimized globally by making use of recent continuous convex relaxation techniques. The accuracy and robustness of the presented approach are demonstrated by evaluating it on multiple real data sets and comparing it to alternative segmentation methods based on level sets. Although the proposed model is designed with focus on the particular application at hand, it is general enough to be applied to a variety of different segmentation tasks.

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“…The length term is an anisotropic Total Variation term which was also used recently by Olsson et al [20]. D(x) is a symmetric non-singular square matrix which specifies the local metric at a given point x.…”

Section: Length Termmentioning

confidence: 99%

Variational segmentation of elongated volumetric structures

Reinbacher

Pock

Bauer

et al. 2010

2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition

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We present an interactive approach for segmenting thin volumetric structures. The proposed segmentation model is based on an anisotropic weighted Total Variation energy with a global volumetric constraint and is minimized using an efficient numerical approach and a convex relaxation. The algorithm is globally optimal w.r.t. the relaxed problem for any volumetric constraint. The binary solution of the relaxed problem equals the globally optimal solution of the original problem. Implemented on today's user-programmable graphics cards, it allows real-time user interaction. The method is applied to and evaluated on the task of articular cartilage segmentation of human knee joints and segmentation of tubular structures like liver vessels and airway trees.

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Extending continuous cuts: Anisotropic metrics and expansion moves

Abstract: Abstract

Cited by 15 publications

References 19 publications

Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

A variational approach to vesicle membrane reconstruction from fluorescence imaging

Variational segmentation of elongated volumetric structures

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