Metrics, Quantization and Registration in Varifold Spaces

Hsieh, Hsi-Wei; Charon, Nicolas

doi:10.1007/s10208-020-09484-7

Cited by 7 publications

(16 citation statements)

References 39 publications

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“…This is not traditional measure transport which is mass preserving which is not a desirable feature for biological samples. Rather we have defined a new action on measures that is reminiscent of the action on d -dimensonal varifolds [35, 36]. We call this property “copy and paste”, the notion being that the brain is built on basic structuring elements that are conserved.…”

Section: Discussionmentioning

confidence: 99%

Hierarchical Computational Anatomy: Unifying the Molecular to Tissue Continuum Via Measure Representations of the Brain

Tward

Trouvé

2021

Preprint

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This paper presents a unified representation of the brain based on mathematical functional measures integrating the molecular and cellular scale descriptions with continuum tissue scale descriptions. We present a fine-to-coarse recipe for traversing the brain as a hierarchy of measures projecting functional description into stable empirical probability laws that unifies scale-space aggregation. The representation uses measure norms for mapping the brain across scales from different measurement technologies. Brainspace is constructed as a metric space with metric comparison between brains provided by a hierarchy of Hamiltonian geodesic flows of diffeomorphisms connecting the molecular and continuum tissue scales. The diffeomorphisms act on the brain measures via the 3D varifold action representing "copy and paste" so that basic particle quantities that are conserved biologically are combined with greater multiplicity and not geometrically distorted. Two applications are examined, the first histological and tissue scale data in the human brain for studying Alzheimer's disease, and the second the RNA and cell signatures of dense spatial transcriptomics mapped to the meso-scales of brain atlases. The representation unifies the classical formalism of computational anatomy for representing continuum tissue scale with non-classical generalized functions appropriate for molecular particle scales.

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Section: Discussionmentioning

confidence: 99%

Hierarchical Computational Anatomy: Unifying the Molecular to Tissue Continuum Via Measure Representations of the Brain

Tward

Trouvé

2021

Preprint

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“…The model we shall study in this paper builds on the varifold diffeomorphic registration approach that the authors had introduced in their previous works [11,12]. In the following paragraphs, we give a brief summary of the general framework of those papers and thereby introduce some notations and definitions that will be necessary for the upcoming sections.…”

Section: Weight Metamorphoses On Varifolds 21 Diffeomorphic Varifold ...mentioning

confidence: 99%

“…Indeed, let X be a d-dimensional oriented rectifiable subset of R n . As this will not be of critical importance for the rest of the paper, we refer the reader to [31] or [12] for precise definitions of rectifiable sets; otherwise the reader unfamiliar with those notions can instead restrict X to be an oriented d-dimensional smooth submanifold of R n . Then X can be naturally represented by a varifold µ X ∈ V d defined by:…”

Section: Weight Metamorphoses On Varifolds 21 Diffeomorphic Varifold ...mentioning

confidence: 99%

“…But interestingly, this approach does not need to limit itself to submanifolds as varifolds in fact encompass a much wider category of geometric objects which can be loosely described as spatial distributions of local orientation planes. Thus, in their recent works [11,12], the authors suggested to formulate a more general diffeomorphic registration problem directly on varifolds themselves by considering a proper notion of group action of Diff(R n ) on the varifold space.…”

Section: Introductionmentioning

confidence: 99%

“…The main goal of this paper is precisely to address that issue by defining and studying a new Riemannian metric on varifolds based on a specific model of varifold metamorphosis that generalizes the purely diffeomorphic approach of [12]. Our model essentially augments the diffeomorphic transport of a varifold with a dynamical change of its weight (or mass) at each point.…”

Section: Introductionmentioning

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

Hsieh

Charon²

2022

Calc. Var.

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Metrics, Quantization and Registration in Varifold Spaces

Cited by 7 publications

References 39 publications

Hierarchical Computational Anatomy: Unifying the Molecular to Tissue Continuum Via Measure Representations of the Brain

Hierarchical Computational Anatomy: Unifying the Molecular to Tissue Continuum Via Measure Representations of the Brain

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

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

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