A General Framework for Curve and Surface Comparison and Registration with Oriented Varifolds

Kaltenmark, Irène; Charlier, Benjamin; Charon, Nicolas

doi:10.1109/cvpr.2017.487

Cited by 49 publications

(94 citation statements)

References 36 publications

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“…Geometric measure theory provides several embeddings of shape spaces into Banach spaces of distributions [22,9,7,18,13] with corresponding metrics. Varifold embeddings are one instance of this construction and are defined as follows (cf.…”

Section: Varifold Distancesmentioning

confidence: 99%

“…The map f → µ f is reparametrization-invariant and, under suitable assumptions on the kernel of W , injective [13]. Thus, one obtains a well-defined distance on the quotient space B i (M, R d ) by defining for any two immersions f 0 , f 1 ∈ Imm(M, R d ):…”

Section: Varifold Distancesmentioning

confidence: 99%

“…Square root normal fields and varifold distances can be combined in an efficient matching algorithm for unparametrized shapes. This idea has been previously used in combination with large deformation models in [7,13] and with H 2 metrics on the space of curves in [1]. The boundary value problem (2) for geodesics on B i (M, R d ) can be formulated as the program…”

Section: 4mentioning

confidence: 99%

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Inexact Elastic Shape Matching in the Square Root Normal Field Framework

Bauer

Charon²,

Harms

2019

Lecture Notes in Computer Science

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This paper puts forth a new formulation and algorithm for the elastic matching problem on unparametrized curves and surfaces. Our approach combines the frameworks of square root normal fields and varifold fidelity metrics into a novel framework, which has several potential advantages over previous works. First, our variational formulation allows us to minimize over reparametrizations without discretizing the reparametrization group. Second, the objective function and gradient are easy to implement and efficient to evaluate numerically. Third, the initial and target surface may have different samplings and even different topologies. Fourth, texture can be incorporated as additional information in the matching term similarly to the fshape framework. We demonstrate the usefulness of this approach with several numerical examples of curves and surfaces.2010 Mathematics Subject Classification. 68U05, 49Q10, 58D10.

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Section: Varifold Distancesmentioning

confidence: 99%

Section: Varifold Distancesmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

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Inexact Elastic Shape Matching in the Square Root Normal Field Framework

Bauer

Charon²,

Harms

2019

Lecture Notes in Computer Science

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“…It is however highly parallelizable and many recent implementations in CUDA take advantage of GPU architectures. We refer the interested reader to [2,23,40] for more detailed discussions on discrete varifold approximations and computations.…”

Section: 3mentioning

confidence: 99%

Metric registration of curves and surfaces using optimal control

Bauer

Charon

Younes

2019

Handbook of Numerical Analysis

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This chapter presents an overview of recent developments in the analysis of shapes such as curves and surfaces through Riemannian metrics. We show that several constructions of metrics on spaces of submanifolds can be unified through the prism of Riemannian submersions, with shape space metrics being induced from metrics defined on the top spaces. Computing the resulting Riemannian distances involves solving geodesic matching problems with boundary conditions. To deal efficiently with such variational problems, one can rely on an auxiliary family of "chordal" distances to simplify the treatment of boundary conditions, which we use to come up with a relaxed inexact formulation of the matching problem. This also allows to turn shape matching into optimal control problems and give a common framework to address them in practice. We then specify our analysis to the cases of intrinsic shape metrics defined using invariant Sobolev metrics on parametrized immersions, outer shape metrics induced from metrics on diffeomorphism groups of the ambient space and finally a recent hybrid model that combines those two approaches.2000 Mathematics Subject Classification. 68Q25, 68R10, 68U05.

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“…Then, D V ar (c 1 , c 2 ) 2 := µ c1 − µ c2 2 V ar , which we call the varifold fidelity metric, defines a discrepancy term between the two curves c 1 and c 2 modulo reparametrizations. The specific properties of D Var crucially depend on the choice of kernel functions ρ and γ: a more thorough discussion of this topic can be found in [10] and [1]. We also note that for the general class of kernels defined above, the resulting discrepancy term D Var is equivariant to the action of translations and rotations, namely that for any α ∈ [0, 2π) and z ∈ C, we have D Var (e iα c 1 + z, e iα c 2 + z) = D Var (c 1 , c 2 ).…”

Section: Varifold Fidelity Termsmentioning

confidence: 99%

An inexact matching approach for the comparison of plane curves with general elastic metrics

Sukurdeep

Bauer²,

Charon

2019

2019 53rd Asilomar Conference on Signals, Systems, and Computers

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This paper introduces a new mathematical formulation and numerical approach for the computation of distances and geodesics between immersed planar curves. Our approach combines the general simplifying transform for first-order elastic metrics that was recently introduced by Kurtek and Needham, together with a relaxation of the matching constraint using parametrization-invariant fidelity metrics. The main advantages of this formulation are that it leads to a simple optimization problem for discretized curves, and that it provides a flexible approach to deal with noisy, inconsistent or corrupted data. These benefits are illustrated via a few preliminary numerical results.Index Terms-elastic shape analysis, F a,b transform, inexact matching, varifold metrics.

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A General Framework for Curve and Surface Comparison and Registration with Oriented Varifolds

Cited by 49 publications

References 36 publications

Inexact Elastic Shape Matching in the Square Root Normal Field Framework

Inexact Elastic Shape Matching in the Square Root Normal Field Framework

Metric registration of curves and surfaces using optimal control

An inexact matching approach for the comparison of plane curves with general elastic metrics

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