A direct variational approach to a problem arising in image reconstruction

Ambrosio, Luigi; Masnou, Simon

doi:10.4171/ifb/72

Cited by 75 publications

(90 citation statements)

References 41 publications

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“…The proposed method is a second-order flow which has a fundamental relationship to the second-order penalty function described in Section 4. As noted by Ambrosio and Masnou [36], the relationship between such second-order systems (see also [37]) and the corresponding fourth-order variational problem is not fully understood.…”

Section: Appendix a Mathematical Foundationmentioning

confidence: 99%

Higher-order nonlinear priors for surface reconstruction

Taşdizen

Whitaker

2004

IEEE Trans. Pattern Anal. Machine Intell.

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Abstract-For surface reconstruction problems with noisy and incomplete range data, a Bayesian estimation approach can improve the overall quality of the surfaces. The Bayesian approach to surface estimation relies on a likelihood term, which ties the surface estimate to the input data, and the prior, which ensures surface smoothness or continuity. This paper introduces a new high-order, nonlinear prior for surface reconstruction. The proposed prior can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion. An exact solution would require solving a fourth-order partial differential equation (PDE), which can be difficult with conventional numerical techniques. Our approach is to solve a cascade system of two second-order PDEs, which resembles the original fourth-order system. This strategy is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals. We solve one PDE for processing the normals and one for refitting the surface to the normals. Furthermore, we implement the associated surface deformations using level sets. Hence, the algorithm can accommodate very complex shapes with arbitrary and changing topologies. This paper gives the mathematical formulation and describes the numerical algorithms. We also show results using range and medical data.

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Section: Appendix a Mathematical Foundationmentioning

confidence: 99%

Higher-order nonlinear priors for surface reconstruction

Taşdizen

Whitaker

2004

IEEE Trans. Pattern Anal. Machine Intell.

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“…Furthermore, curvature based inpainting methods have been proposed by Ambrosio and Masnou [2], Morel [24] and Chan et al [8]. They treat the level sets of 2D images as Euler's elastica and minimize their bending energy.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, we have to reformulate this problem solely in terms of quantities such as φ(t), h(t) and its gradients, in particular avoiding the term S(t) 2 and derivatives of the normal. Here, we take advantage of a fairly general gradient flow perspective on geometric evolution problems.…”

Section: Introductionmentioning

confidence: 99%

A level set formulation for Willmore flow

Droske¹,

Rumpf²

2004

Interfaces Free Bound.

101

111

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A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set function in time via the level set equation. This approach in particular allows one to identify the natural dependent quantities of the derived variational formulation. Furthermore, spatial and temporal discretizations are discussed and some numerical simulations are presented.2000 Mathematics Subject Classification: 35K55, 53C44, 65M60, 74S05.

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“…In [12], Ballester et al introduced a variational approach based on the smooth continuation of isophote lines. A variational approach based on level sets and a Perimeter and Willmore energy was presented by Ambrosio and Masnou in [13]. A combination of TV inpainting and wavelet representation was proposed in [14].…”

Section: Introductionmentioning

confidence: 99%

Reconstructing Optical Flow Fields by Motion Inpainting

Berkels

Kondermann

Garbe

et al. 2009

Lecture Notes in Computer Science

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Abstract. An edge-sensitive variational approach for the restoration of optical flow fields is presented. Real world optical flow fields are frequently corrupted by noise, reflection artifacts or missing local information. Still, applications may require dense motion fields. In this paper, we pick up image inpainting methodology to restore motion fields, which have been extracted from image sequences based on a statistical hypothesis test on neighboring flow vectors. A motion field inpainting model is presented, which takes into account additional information from the image sequence to improve the reconstruction result. The underlying functional directly combines motion and image information and allows to control the impact of image edges on the motion field reconstruction. In fact, in case of jumps of the motion field, where the jump set coincides with an edge set of the underlying image intensity, an anisotropic TV-type functional acts as a prior in the inpainting model. We compare the resulting image guided motion inpainting algorithm to diffusion and standard TV inpainting methods.

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A direct variational approach to a problem arising in image reconstruction

Cited by 75 publications

References 41 publications

Higher-order nonlinear priors for surface reconstruction

Higher-order nonlinear priors for surface reconstruction

A level set formulation for Willmore flow

Reconstructing Optical Flow Fields by Motion Inpainting

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