Volume Reconstruction from Slices

Bretin, Élie; Dayrens, François; Masnou, Simon

doi:10.1137/17m1116283

Cited by 15 publications

(10 citation statements)

References 23 publications

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“…The diffuse approximation of the Willmore flow and of more general curvature energies and flows are used for numerical simulations in a huge number of applications. Let us only mention here [13,15,19,26,27,29,30,33,35,39,51,52]. For numerical treatment of Willmore flow in a sharp-interface approach, we refer to [2-4, 14, 25, 32, 48].…”

Section: Standard Diffuse Approximationmentioning

confidence: 99%

A new diffuse-interface approximation of the Willmore flow

Rätz

Röger

2021

ESAIM: COCV

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Standard diffuse approximations of the Willmore flow often lead to intersecting phase boundaries that in many cases do not correspond to the intended sharp interface evolution. Here we introduce a new two-variable diffuse approximation that includes an rather simple but efficient penalization of the deviation from a quasi-one dimensional structure of the phase fields. We justify the approximation property by a Gamma convergence result for the energies and a matched asymptotic expansion for the flow. Ground states of the energy are shown to be one-dimensional, in contrast to the presence of saddle solutions for the usual diffuse approximation. Finally we present numerical simulations that illustrate the approximation property and apply our new approach to problems where the usual approach leads to an undesired behavior.

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Section: Standard Diffuse Approximationmentioning

confidence: 99%

A new diffuse-interface approximation of the Willmore flow

Rätz

Röger

2021

ESAIM: COCV

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“…-Provide regularization prior for tubular segmentation: the generation model could be included as a regularizing loss in variational segmentation methods of vascular structures [33,71,107], to select certain tubular morphologies against others. It could also be combined with 3D reconstruction from 2D slices methods [13,53,64]. -Build a synthetic database of textures: the generation algorithm could provide, at a low cost, a complete panel of synthetic textures on which to test and train vascular shape analysis methods [50,84,89], including topological analysis methods [15,47,79], segmentation algorithms, microvascular blood flow simulations [5,86], but also simulations in porous materials [65,95,109].…”

Section: Importance Of a Unifying Theory And Future Applicationsmentioning

confidence: 99%

“…Mumford [75] considered Euler's elastica functional as a prior curve model in computer vision, and this was subsequently applied to digital inpainting [10,68,99]. Similar ideas were then used for 3D volume reconstruction from 2D slices [13,53,64]. Our work is also related to Poisson reconstruction methods [48,49] that implicitly reconstruct a surface from noisy oriented points.…”

Section: Introductionmentioning

confidence: 99%

Generation of Tubular and Membranous Shape Textures with Curvature Functionals

Song

2021

J Math Imaging Vis

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Tubular and membranous shapes display a wide range of morphologies that are difficult to analyze within a common framework. By generalizing the classical Helfrich energy of biomembranes, we model them as solutions to a curvature optimization problem in which the principal curvatures may play asymmetric roles. We then give a novel phase-field formulation to approximate this geometric problem, and study its Gamma-limsup convergence. This results in an efficient GPU algorithm that we validate on well-known minimizers of the Willmore energy; the software for the implementation of our algorithm is freely available online. Exploring the space of parameters reveals that this comprehensive framework leads to a wide continuum of shape textures. This first step towards a unifying theory will have several implications, in biology for quantifying tubular shapes or designing bio-mimetic scaffolds, but also in computer graphics, materials science, or architecture.

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“…This is indeed clearly the case on Figure 13 which shows that the obtained evolution law corresponds well to the motion by mean curvature. We highlight in this section how the previous schemes derived from our trained networks S N N θ,α , α ∈ {1, 2} are sufficiently stable to be coupled with additional constraints such as volume conservation ( Q u n dx = const) or inclusion constraints (reformulated as an inequality u n ≥ u in (x), see [61]), and sufficiently stable to be extended to the multiphase case. The main idea is not to reproduce the training procedure on new networks according to the intended application but rather to couple the action of our trained networks with an additional constraint, showing the adaptability of our neural network approach.…”

Section: Validationmentioning

confidence: 99%

“…Indeed, we expect that the stationary state of such an evolution is at least a local minimum of the Steiner problem. From a phase field point of view, the strategy is to consider the non-oriented phase field representation q (dis(x, Γ(t)/ε)), to use the non oriented trained network S N N θ,2 to let the interface evolve by mean curvature, and to incorporate the inclusion constraint by using an additional inequality constraint u ≤ u in (x) = N i=1 q (dist(a i , x)/ε), in the spirit of [75,61]. Finally, the scheme reads as follows:…”

Section: Validationmentioning

confidence: 99%

Learning phase field mean curvature flows with neural networks

Bretin,

Denis,

Masnou

et al. 2021

Preprint

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We introduce in this paper new, efficient numerical methods based on neural networks for the approximation of the mean curvature flow of either oriented or non-orientable surfaces. To learn the correct interface evolution law, our neural networks are trained on phase field representations of exact evolving interfaces. The structure of the networks draws inspiration from splitting schemes used for the discretization of the Allen-Cahn equation. But when the latter approximates the mean curvature motion of oriented interfaces only, the approach we propose extends very naturally to the non-orientable case. In addition, although trained on smooth flows only, our networks can handle singularities as well. Furthermore, they can be coupled easily with additional constraints which allows us to show various applications illustrating the flexibility and efficiency of our approach: mean curvature flows with volume constraint, multiphase mean curvature flows, numerical approximation of Steiner trees, numerical approximation of minimal surfaces.

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Volume Reconstruction from Slices

Cited by 15 publications

References 23 publications

A new diffuse-interface approximation of the Willmore flow

A new diffuse-interface approximation of the Willmore flow

Generation of Tubular and Membranous Shape Textures with Curvature Functionals

Learning phase field mean curvature flows with neural networks

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