Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles

Charon, Nicolas; Charlier, Benjamin; Glaunès, Joan Alexis; Gori, Pietro; Roussillon, Pierre

doi:10.1016/b978-0-12-814725-2.00021-2

Cited by 23 publications

(29 citation statements)

References 33 publications

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“…Yet, unlike images or landmarks, for objects such as measures and varifolds, deriving adequate fidelity metrics that can be nicely embedded within the type of variational problems considered here is not immediate. In fact, this issue has been the object of several different works in the past such as [9,10,[38][39][40][41][42]. In this paper, we shall rely on fidelity terms obtained from reproducing kernel metrics on the space of varifolds which have proved successful for that purpose in diffeomorphic registration problems.…”

Section: Relaxed Matching Problemmentioning

confidence: 99%

“…All these characteristics make kernel metrics d W * well suited as relaxation terms for our variational problems. We shall not discuss in many more details the different families of kernels k pos and k G that could be selected and the corresponding properties they induce on the distance: these questions have been examined quite thoroughly in previous publications notably [10] and [42]. For some of the upcoming mathematical results and in all numerical applications, we shall specifically restrict k pos to a radial kernel…”

Section: Kernel Metrics On Varifoldsmentioning

confidence: 99%

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Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

Hsieh¹,

Charon²

2021

Preprint

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This paper introduces and studies a metamorphosis framework for geometric measures known as varifolds, which extends the diffeomorphic registration model for objects such as curves, surfaces and measures by complementing diffeomorphic deformations with a transformation process on the varifold weights. We consider two classes of cost functionals to penalize those combined transformations, in particular the LDDMM-Fisher-Rao energy which, as we show, leads to a well-defined Riemannian metric on the space of varifolds with existence of corresponding geodesics. We further introduce relaxed formulations of the respective optimal control problems, study their well-posedness and derive optimality conditions for the solutions. From these, we propose a numerical approach to compute optimal metamorphoses between discrete varifolds and illustrate the interest of this model in the situation of partially missing data.

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Section: Relaxed Matching Problemmentioning

confidence: 99%

Section: Kernel Metrics On Varifoldsmentioning

confidence: 99%

Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

Hsieh¹,

Charon²

2021

Preprint

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“…the 2 metric for meshes with point-to-point correspondence (i.e. the sum of squared differences between point positions), the current metric [14,59] for oriented surface meshes without point-to-point correspondence (details are given in appendix for the reader's convenience).…”

Section: Positioning a Shape With Respect To A Static Atlasmentioning

confidence: 99%

“…The deformation kernel width is set to σ = 10 mm. The current distance is used to compute distances between meshes without point correspondence, with a kernel width of σ E = 5 mm [14,59]. Figure 9 shows the estimated average progression, which consists in an overall atrophy of both the left and right hippocampus with a specific deformation of their shape.…”

Section: Models Of Atrophy Of the Hippocampusmentioning

confidence: 99%

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Learning the spatiotemporal variability in longitudinal shape data sets

2020

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In this paper, we propose a generative statistical model to learn the spatiotemporal variability in longitudinal shape data sets, which contain repeated observations of a set of objects or individuals over time. From all the short-term sequences of individual data, the method estimates a longterm normative scenario of shape changes and a tubular coordinate system around this trajectory. Each individual data sequence is therefore (i) mapped onto a specific portion of the trajectory accounting for differences in pace of progression across individuals, and (ii) shifted in the shape space to account for intrinsic shape differences across individuals that are independent of the progression of the observed process. The parameters of the model are estimated using a stochastic approximation of the expectation-maximization algorithm. The proposed approach is validated on a simulated data set, illustrated on the analysis of facial expression in video sequences, and applied to the modeling of the progressive atrophy of the hippocampus in Alzheimer's disease patients. These experiments show that one can use the method to reconstruct data at the precision of the noise, to highlight significant factors that may modulate the progression, and to simulate entirely synthetic longitudinal data sets reproducing the variability of the observed process.

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Shape Spaces: From Geometry to Biological Plausibility

Charon

Younes

2022

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

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Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles

Cited by 23 publications

References 33 publications

Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

Learning the spatiotemporal variability in longitudinal shape data sets

Shape Spaces: From Geometry to Biological Plausibility

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