Level Sets and Distance Functions

Gomes, Jonas; Faugeras, Olivier

doi:10.1007/3-540-45054-8_38

Cited by 31 publications

(16 citation statements)

References 20 publications

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“…In numerical implementations, we found that a very steep slope of the level set functions can even inhibit the flexibility of the boundary to displace. Many people have advocated the use of a redistancing procedure in the evolution of level set functions to constrain the slope of φ to |∇φ| = 1, see also Gomes and Faugeras (2000). In order to reproject the evolving level set function to the space of distance functions, we intermittently iterate several steps of the redistancing equation (Sussman et al, 1994):…”

Section: Redistancingmentioning

confidence: 99%

Motion Competition: A Variational Approach to Piecewise Parametric Motion Segmentation

Cremers

Soatto

2004

Int J Comput Vision

232

189

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Abstract. We present a novel variational approach for segmenting the image plane into a set of regions of parametric motion on the basis of two consecutive frames from an image sequence. Our model is based on a conditional probability for the spatio-temporal image gradient, given a particular velocity model, and on a geometric prior on the estimated motion field favoring motion boundaries of minimal length.Exploiting the Bayesian framework, we derive a cost functional which depends on parametric motion models for each of a set of regions and on the boundary separating these regions. The resulting functional can be interpreted as an extension of the Mumford-Shah functional from intensity segmentation to motion segmentation. In contrast to most alternative approaches, the problems of segmentation and motion estimation are jointly solved by continuous minimization of a single functional. Minimizing this functional with respect to its dynamic variables results in an eigenvalue problem for the motion parameters and in a gradient descent evolution for the motion discontinuity set.We propose two different representations of this motion boundary: an explicit spline-based implementation which can be applied to the motion-based tracking of a single moving object, and an implicit multiphase level set implementation which allows for the segmentation of an arbitrary number of multiply connected moving objects.Numerical results both for simulated ground truth experiments and for real-world sequences demonstrate the capacity of our approach to segment objects based exclusively on their relative motion.

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Section: Redistancingmentioning

confidence: 99%

Motion Competition: A Variational Approach to Piecewise Parametric Motion Segmentation

Cremers

Soatto

2004

Int J Comput Vision

232

189

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“…The two level set functions φ 1 and φ 2 are alternately evolved. At even iterations the segmenting level-set is φ = φ 1 and the prior is given by 2 A significant gain from not enforcing φ to be a distance function is the elimination of the process of re-distancing [10,32]. φ = φ 2 .…”

Section: Mutual Segmentation With Projectivitymentioning

confidence: 99%

Shape-Based Mutual Segmentation

2007

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We present a novel variational approach for simultaneous segmentation of two images of the same object taken from different viewpoints. Due to noise, clutter and occlusions, neither of the images contains sufficient information for correct object-background partitioning. The evolving object contour in each image provides a dynamic prior for the segmentation of the other object view. We call this process mutual segmentation. The foundation of the proposed method is a unified level-set framework for region and edge based segmentation, associated with a shape similarity term. The suggested shape term incorporates the semantic knowledge gained in the segmentation process of the image pair, accounting for excess or deficient parts in the estimated object shape. Transformations, including planar projectivities, between the object views are accommodated by a registration process held concurrently with the segmentation.The proposed segmentation algorithm is demonstrated on a variety of image pairs. The Homography between each of the image pairs is estimated and its accuracy is evaluated.

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“…The goals of this paper are the following: (1) to understand the role of PDE in different image processing applications, particularly in segmentation for still and motion imagery; (2) to understand the relationship between PDE, level sets and regularizers; (3) to understand how one can derive the image segmentation process by fusing the regional-based PDE information with boundary-based PDE models, Catte et al 68 and Sochen et al 69 ), (11) color image segmentation: (see Sapiro 83 ) and (12) 2D and 3D medical image processing: (see the works by Malladi et al, [70][71][72][73][74] Gray-Matter/White-Matter (GM/WM) boundary estimation by Gomes et al, 79 GM/WM boundary estimation with fuzzy models by Suri, 1,84 GM/WM thickness estimation by Zeng et al, 85 leakage prevention in fast level sets using fuzzy models by Suri, 2 also a survey article on brain segmentation by Suri et al 5 and a recent article for cell segmentation by Sarti et al 86 ). For a detailed review of some of the above mentioned applications, the reader must see Sethian.…”

Section: Partial Differential Equations (Pdes)mentioning

confidence: 99%

“…Third, the above equations can be written in terms of differential geometry using divergence as: ∂φ/∂t = ∇ · (∇φ/|∇φ|)|∇φ|, where geometrical properties such as normal curvature (N ) and mean curvature (H) are given as: N = ∇φ/|∇φ| and H = ∇ · (∇φ/|∇φ|). Note the above equation remains the fundamental form but recently, Faugeras and his coworkers from INRIA (see Gomes et al 79 ) modified Eq. (3) into the "preserving distance function" as:…”

Section: Three Analogies Of the Curve Evolution Equationmentioning

confidence: 99%

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Modeling Segmentation via Geometric Deformable Regularizers, Pde and Level Sets in Still and Motion Imagery: A Revisit

Suri

Reden

et al. 2001

Int. J. Image Grap.

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Partial Differential Equations (PDEs) have dominated image processing research recently. The three main reasons for their success are: first, their ability to transform a segmentation modeling problem into a partial differential equation framework and their ability to embed and integrate different regularizers into these models; second, their ability to solve PDEs in the level set framework using finite difference methods; and third, their easy extension to a higher dimensional space. This paper is an attempt to survey and understand the power of PDEs to incorporate into geometric deformable models for segmentation of objects in 2D and 3D in still and motion imagery. The paper first presents PDEs and their solutions applied to image diffusion. The main concentration of this paper is to demonstrate the usage of regularizers in PDEs and level set framework to achieve the image segmentation in still and motion imagery. Lastly, we cover miscellaneous applications such as: mathematical morphology, computation of missing boundaries for shape recovery and low pass filtering, all under the PDE framework. The paper concludes with the merits and the demerits of PDEs and level set-based framework for segmentation modeling. The paper presents a variety of examples covering both synthetic and real world images.

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Level Sets and Distance Functions

Cited by 31 publications

References 20 publications

Motion Competition: A Variational Approach to Piecewise Parametric Motion Segmentation

Motion Competition: A Variational Approach to Piecewise Parametric Motion Segmentation

Shape-Based Mutual Segmentation

Modeling Segmentation via Geometric Deformable Regularizers, Pde and Level Sets in Still and Motion Imagery: A Revisit

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