A Geometric Formulation of Gradient Descent for Variational Problems with Moving Surfaces

Solem, Jan Erik; Overgaard, Niels Christian

doi:10.1007/11408031_36

Cited by 38 publications

(56 citation statements)

References 10 publications

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“…This is often neglected and most authors improperly refer to the gradient of the energy. Thus, the classical gradient flows reported in the literature (mean curvature flow, geodesic active contours [6,14,29], multi-view 3D reconstruction [12,15,13]) are relative to the L 2 inner product.…”

Section: Minimization and Inner Productmentioning

confidence: 99%

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Generalized Gradients: Priors on Minimization Flows

Charpiat

Maurel

Pons

et al. 2006

Int J Comput Vision

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This paper tackles an important aspect of the variational problem underlying active contours: optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literature. Most authors, explicitely or implicitely, assume that the space of admissible deformations is ruled by the canonical L 2 inner product. The classical gradient flows reported in the literature are relative to this particular choice. Here, we investigate the relevance of using (i) other inner products, yielding other gradient descents, and (ii) other minimizing flows not deriving from any inner product. In particular, we show how to induce different degrees of spatial consistency into the minimizing flow, in order to decrease the probability of getting trapped into irrelevant local minima. We report numerical experiments indicating that the sensitivity of the active contours method to initial conditions, which seriously limits its applicability and efficiency, is alleviated by our application-specific spatially coherent minimizing flows. We show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors.

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Section: Minimization and Inner Productmentioning

confidence: 99%

“…This idea, proposed in [29], is applied by the same authors to the alignment of curve in images in [23]: a complicated term in the gradient is safely neglected after checking that the evolution still decreases the energy. The spirit of our work is different.…”

Section: Minimization and Inner Productmentioning

confidence: 99%

Generalized Gradients: Priors on Minimization Flows

Charpiat

Maurel

Pons

et al. 2006

Int J Comput Vision

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show abstract

“…The mathematical framework used in this paper is the one defined by Solem and Overgaard [20] in which shapes are implicitly represented by level set functions [12,13]. For the convenience of the reader, we remind the related notions and notations required for understanding our work.…”

Section: Mathematical Background and Notationmentioning

confidence: 99%

“…Let M denote the manifold of admissible surfaces defined by Solem and Overgaard [20]. Points in this space are surfaces.…”

Section: Functional Derivativesmentioning

confidence: 99%

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Minimizing the Reprojection Error in Surface Reconstruction from Images

Gargallo¹,

Prados²,

Sturm³

2007

2007 IEEE 11th International Conference on Computer Vision

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This paper addresses the problem of image-based surface reconstruction. The main contribution is the computation of the exact derivative of the reprojection error functional. This allows its rigorous minimization via gradient descent surface evolution. The main difficulty has been to correctly take into account the visibility changes that occur when the surface moves. A geometric and analytical study of these changes is presented and used for the computation of derivative.Our analysis shows the strong influence that the movement of the contour generators has on the reprojection error. As a consequence, during the proper minimization of the reprojection error, the contour generators of the surface are automatically moved to their correct location in the images. Therefore, current methods adding additional silhouettes or apparent contour constraints to ensure this alignment can now be understood and justified by a single criterion: the reprojection error.

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A Gradient Descent Procedure for Variational Dynamic Surface Problems with Constraints

Solem

Overgaard

2005

Lecture Notes in Computer Science

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Abstract. Many problems in image analysis and computer vision involving boundaries and regions can be cast in a variational formulation. This means that m-surfaces, e.g. curves and surfaces, are determined as minimizers of functionals using e.g. the variational level set method. In this paper we consider such variational problems with constraints given by functionals. We use the geometric interpretation of gradients for functionals to construct gradient descent evolutions for these constrained problems. The result is a generalization of the standard gradient projection method to an innite-dimensional level set framework. The method is illustrated with examples and the results are valid for surfaces of any dimension.

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A Geometric Formulation of Gradient Descent for Variational Problems with Moving Surfaces

Cited by 38 publications

References 10 publications

Generalized Gradients: Priors on Minimization Flows

Generalized Gradients: Priors on Minimization Flows

Minimizing the Reprojection Error in Surface Reconstruction from Images

A Gradient Descent Procedure for Variational Dynamic Surface Problems with Constraints

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