Discrete Geodesic Regression in Shape Space

Berkels, Benjamin; Fletcher, P. Thomas; Heeren, Behrend; Rumpf, Martin; Wirth, Benedikt

doi:10.1007/978-3-642-40395-8_9

Cited by 13 publications

(13 citation statements)

References 25 publications

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“…(4) The same trick does not work with non-local noise: now unknown deformation and non-local noise cannot be separated and the optimal segmentation becomes faulty. (5) Adding the deformation as a Wasserstein mode helps to approximately find the object even in this noisy scenario, also thanks to the robustness of the globally optimal branch and bound scheme. Fig.…”

Section: Numerical Resultsmentioning

confidence: 99%

Globally Optimal Joint Image Segmentation and Shape Matching Based on Wasserstein Modes

Schmitzer

Schnörr

2014

J Math Imaging Vis

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A functional for joint variational object segmentation and shape matching is developed. The formulation is based on optimal transport w.r.t. geometric distance and local feature similarity. Geometric invariance and modelling of object-typical statistical variations is achieved by introducing degrees of freedom that describe transformations and deformations of the shape template. The shape model is mathematically equivalent to contour-based approaches but inference can be performed without conversion between the contour and region representations, allowing combination with other convex segmentation approaches and simplifying optimization. While the overall functional is non-convex, non-convexity is confined to a low-dimensional variable. We propose a locally optimal alternating optimization scheme and a globally optimal branch and bound scheme, based on adaptive convex relaxation. Combining both methods allows to eliminate the delicate initialization problem inherent to many contour based approaches while remaining computationally practical.The properties of the functional, its ability to adapt to a wide range of input data structures and the different optimization schemes are illustrated and compared by numerical experiments.

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Section: Numerical Resultsmentioning

confidence: 99%

Globally Optimal Joint Image Segmentation and Shape Matching Based on Wasserstein Modes

Schmitzer

Schnörr

2014

J Math Imaging Vis

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“…Kurtek et al [KKG*12, KSKL13] introduce Riemannian metrics on spaces of surfaces parametrized over the unit sphere. Berkels et al [BFH*13] introduce an approach for computing geodesic regression curves in shape spaces. Important for the application is the efficient computation of the geodesics between pairs of points in these shape spaces.…”

Section: Related Workmentioning

confidence: 99%

Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

Brandt

Tycowicz

Hildebrandt

2016

Computer Graphics Forum

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We introduce techniques for the processing of motion and animations of non-rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in R n to the processing of motion of non-rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced-order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non-linear "Bézier curves" by executing de Casteljau's algorithm in shape space.

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“…Further, in color image processing they appear in connection with color spaces such as HSI, HCL, as well as in chromaticity based color spaces [27,84,59,60]. Other examples are data with values in the special orthogonal group SO(3) which may express camera positions or orientations of aircrafts [83], Euclidean motion group-valued data [70] representing, e.g., poses as well as shape-space data [64,20]. Another prominent manifold is the space of positive (definite) matrices Pos n of dimension n endowed with the Fisher-Rao metric [69].…”

Section: Introductionmentioning

confidence: 99%

Wavelet Sparse Regularization for Manifold-Valued Data

Storath¹,

Weinmann²

2020

Multiscale Model. Simul.

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In this paper, we consider the sparse regularization of manifold-valued data with respect to an interpolatory wavelet/multiscale transform. We propose and study variational models for this task and provide results on their well-posedness. We present algorithms for a numerical realization of these models in the manifold setup. Further, we provide experimental results to show the potential of the proposed schemes for applications.

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Discrete Geodesic Regression in Shape Space

Cited by 13 publications

References 25 publications

Globally Optimal Joint Image Segmentation and Shape Matching Based on Wasserstein Modes

Globally Optimal Joint Image Segmentation and Shape Matching Based on Wasserstein Modes

Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

Wavelet Sparse Regularization for Manifold-Valued Data

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