Mumford–Shah functionals on graphs and their asymptotics

Caroccia, Marco; Chambolle, Antonin; Slepčev, Dejan

doi:10.1088/1361-6544/ab81ee

Cited by 20 publications

(20 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The discreteto-continuum analysis performed in the present paper thus provides a very general framework on the one hand for atomistic systems whose macroscopic behavior can be studied in the context of fracture mechanics and on the other hand for possible discrete approximations of energies used in image reconstruction, such as for instance the approximations studied in [27,28,19,37]. We point out that our analysis is also connected to some recent results in Data Science [39,26,38]. The assumptions on the potentials φ ε i that are needed to restrict the class of possible discreteto-continuum limits to functionals of the form (1.2) are carefully listed in Section 2.…”

Section: Introductionmentioning

confidence: 55%

Discrete-to-continuum limits of multi-body systems with bulk and surface long-range interactions

Bach¹,

Braides²,

Cicalese³

2019

Preprint

View full text Add to dashboard Cite

We study the atomistic-to-continuum limit of a class of energy functionals for crystalline materials via Γ-convergence. We consider energy densities that may depend on interactions between all points of the lattice and we give conditions that ensure compactness and integralrepresentation of the continuum limit on the space of special functions of bounded variation. This abstract result is complemented by a homogenization theorem, where we provide sufficient conditions on the energy densities under which bulk-and surface contributions decouple in the limit. The results are applied to long-range and multi-body interactions in the setting of weak-membrane energies.

show abstract

Section: Introductionmentioning

confidence: 55%

Discrete-to-continuum limits of multi-body systems with bulk and surface long-range interactions

Bach¹,

Braides²,

Cicalese³

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…It remains to consider the case where I 0 = {2}. Embarking from (13) with i 0 = 2, and arguing as we have done for f 1 in Case 1 to show (11), and using that f 2 is actually t-independent, we get in this case that (14) ∂ ∂t ϕ(x 0 , t 0 ) = 0.…”

Section: Proof Of Proposition 24mentioning

confidence: 99%

“…Studying consistency and continuum limits of certain evolution and variational problems on graphs and networks is an active research area; see [27,26,37,33,34,25,24,46,13] for a non-exhaustive list and references therein. In particular, the authors in [8,41] studied continumm limits of Lipschitz learning on graphs.…”

Section: Contributions and Relation To Prior Workmentioning

confidence: 99%

Limits and consistency of non-local and graph approximations to the Eikonal equation

Fadili¹,

Forcadel²,

Nguyen³

et al. 2021

Preprint

View full text Add to dashboard Cite

In this paper, we study a non-local approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the non-local problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and non-local problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the non-local problem and that of the local one, both in continuoustime and Backward Euler time discretization. We then turn to studying continuum limits of non-local problems defined on random weighted graphs with n vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as n grows, then almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices n grows large.

show abstract

“…The level set algorithm that we used in this paper does not depend on the gradient information of the BUS image. Therefore, it is insensitive to noise and has a significant advantage in medical image processing [ 14 , 15 ]. Here we employed the Distance Regularized Level Set Evolution (DRLSE) [ 16 ] model that eliminates the need for reinitialization but employs a distance regularization term and energy functions to propagate the zero-level set function (LSF) towards the desired locations.…”

Section: Lesion Segementation and Feature Calculationmentioning

confidence: 99%

Multi‐Features‐Based Automated Breast Tumor Diagnosis Using Ultrasound Image and Support Vector Machine

Zhuang

Yang

Zhuang

et al. 2021

Computational Intelligence and Neuroscience

View full text Add to dashboard Cite

Breast ultrasound examination is a routine, fast, and safe method for clinical diagnosis of breast tumors. In this paper, a classification method based on multi-features and support vector machines was proposed for breast tumor diagnosis. Multi-features are composed of characteristic features and deep learning features of breast tumor images. Initially, an improved level set algorithm was used to segment the lesion in breast ultrasound images, which provided an accurate calculation of characteristic features, such as orientation, edge indistinctness, characteristics of posterior shadowing region, and shape complexity. Simultaneously, we used transfer learning to construct a pretrained model as a feature extractor to extract the deep learning features of breast ultrasound images. Finally, the multi-features were fused and fed to support vector machine for the further classification of breast ultrasound images. The proposed model, when tested on unknown samples, provided a classification accuracy of 92.5% for cancerous and noncancerous tumors.

show abstract

Mumford–Shah functionals on graphs and their asymptotics

Cited by 20 publications

References 36 publications

Discrete-to-continuum limits of multi-body systems with bulk and surface long-range interactions

Discrete-to-continuum limits of multi-body systems with bulk and surface long-range interactions

Limits and consistency of non-local and graph approximations to the Eikonal equation

Multi‐Features‐Based Automated Breast Tumor Diagnosis Using Ultrasound Image and Support Vector Machine

Contact Info

Product

Resources

About