Optimal approximations by piecewise smooth functions and associated variational problems
1989/07/01 by David Mumford, Jayant Shah · 5,199 citations
Computer Science · Engineering · Mathematics · #Applied mathematics #Citation #Combinatorics #Computer science #Library science #Mathematical Approximation and Integration #Mathematical analysis #Mathematical economics #Mathematics #Optimization and Variational Analysis #Piecewise #Sparse and Compressive Sensing Techniques #Variational inequality
paper · open access · doi:10.1002/cpa.3160420503
published in Communications on Pure and Applied Mathematics 42(5), 577-685 (Wiley)
openalex publication_date 1989/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Cited by
- Approximation of functional depending on jumps by elliptic functional via t‐convergence
- Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations
- A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns
- Topology optimization of particle‐matrix composites for optimal fracture resistance taking into account interfacial damage
- Formulation and implementation of strain rate‐dependent fracture toughness in context of the phase‐field method
- A Smooth Phase-Separation Model for Weak-Boundary Segmentation of Homogeneous Structures
- From level set evolution to threshold optimization: A grayscale level set framework for image segmentation
- A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects
- A Unified Variational Framework for Deep Weakly Supervised Image Segmentation
- Allure of Craquelure: A Variational-Generative Approach to Crack Detection in Paintings
- The latent variable proximal point algorithm for variational problems with inequality constraints
- Edge-guided Non-local Fully Convolutional Network for Salient Object Detection
- Discrete approximation of functionals with jumps and creases
- Deep Variational Instance Segmentation
- Segmentation and Restoration of Images on Surfaces by Parametric Active\n Contours with Topology Changes
- Spectral Graph Cut from a Filtering Point of View
- A Hamilton-Jacobi equation for the continuum limit of non-dominated sorting
- The Variational Approach to Fracture
- Understanding Deep Learning Techniques for Image Segmentation
- A cortical-inspired geometry for contour perception and motion integration
- Robust Variational Model Based Tailored UNet: Leveraging Edge Detector and Mean Curvature for Improved Image Segmentation
- Lesion Net -- Skin Lesion Segmentation Using Coordinate Convolution and Deep Residual Units
- Ricci Curvature Based Volumetric Segmentation of the Auditory Ossicles
- An active curve approach for tomographic reconstruction of binary radially symmetric objects
- Sampling in BV-Type Spaces
- Second-order edge-penalization in the Ambrosio-Tortorelli functional
- Self-Supervision with Superpixels: Training Few-shot Medical Image Segmentation without Annotation
- TA-LSDiff:Topology-Aware Diffusion Guided by a Level Set Energy for Pancreas Segmentation
- Convex variational methods for multiclass data segmentation on graphs
- MH-Net: Model-data-driven hybrid-fusion network for medical image segmentation
- Regularity up to the Crack-Tip for the Mumford-Shah problem
- Image inpainting via fixed-point neighborhood-based reconstruction
- On infinite-dimensional hierarchical probability models in statistical inverse problems
- A Level Set Method for Image Segmentation in the Presence of Intensity Inhomogeneities With Application to MRI
- Nonlinear dynamic fracture problems with polynomial and strain-limiting constitutive relations
- Reclassification formula that provides to surpass K-means method
- A Free Boundary Problem Related to Thermal Insulation
- A simplified threshold dynamics algorithm for isotropic surface energies
- Agent-Based Optimal Control for Image Processing
- Color image segmentation based on a convex K-means approach
- Curvature penalization of strongly anisotropic interfaces models and their phase-field approximation
- Scalable 3D Panoptic Segmentation As Superpoint Graph Clustering
- Evaluating the effects of preprocessing, method selection, and hyperparameter tuning on SAR-based flood mapping and water depth estimation
- A new approach for image segmentation based on diffeomorphic registration and gradient fields
- Automation of Hemocompatibility Analysis Using Image Segmentation and a Random Forest
- Segmentation for radar images based on active contour
- Partial Functional Correspondence
- Stereo integration, mean field theory and psychophysics
- Iterative thresholding meets free discontinuity problems
- Segmentation of Three-dimensional Images with Parametric Active Surfaces and Topology Changes
- A Real-Time Multi-Model Parametric Representation of Point Clouds
- Fast Global Minimization of the Active Contour/Snake Model
- Boundary-Aware Segmentation Network for Mobile and Web Applications
- A Local Active Contour Model for Image Segmentation with Intensity Inhomogeneity
- Global calibrations for the non-homogeneous Mumford-Shah functional
- A Weighted Difference of Anisotropic and Isotropic Total Variation for Relaxed Mumford-Shah Color and Multiphase Image Segmentation
- Dual-Scale Volume Priors with Wasserstein-Based Consistency for Semi-Supervised Medical Image Segmentation
- A Two-stage Classification Method for High-dimensional Data and Point Clouds
- A Structure-Preserving Numerical Method for Harmonic Maps Between High-genus Surfaces
- Low-Rank Regularized Convex-Non-Convex Problems for Image Segmentation or Completion
- Cohesive Dynamics and Brittle Fracture
- Homogenisation of phase-field functionals with linear growth
- Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer
- Reimagining Image Segmentation using Active Contour: From Chan Vese Algorithm into a Proposal Novel Functional Loss Framework
- Fast Image Processing with Fully-Convolutional Networks
- Topology-Preserving 3D Image Segmentation Based On Hyperelastic Regularization
- Two-phase image segmentation by the Allen-Cahn equation and a nonlocal edge detection operator
- A Registration-Based Star-Shape Segmentation Model and Fast Algorithms
- Generalised elastic nets
- An Implemention of Two-Phase Image Segmentation using the Split Bregman Method
- Approximate l0-penalized estimation of piecewise-constant signals on graphs
- Crack initiation in elastic bodies
- A very accurate method to approximate discontinuous functions with a finite number of discontinuities
- Relaxed Total Generalized Variation Regularized Piecewise Smooth Mumford-Shah Model for Triangulated Surface Segmentation
- A Preconditioned Difference of Convex Algorithm for Truncated Quadratic Regularization with Application to Imaging
- Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements
- Effective two-stage image segmentation: a new non-Lipschitz decomposition approach with convergent algorithm
- The differential geometry of perceptual similarity
- An Edge Driven Wavelet Frame Model for Image Restoration
- Dynamic Brittle Fracture as a Small Horizon Limit of Peridynamics
- Convex Histogram-Based Joint Image Segmentation with Regularized Optimal Transport Cost
- A Convex Decomposition Formula for the Mumford-Shah Functional in Dimension One
- Effective Two-Stage Image Segmentation: A New Non-Lipschitz Decomposition Approach with Convergent Algorithm
- Gated CRF Loss for Weakly Supervised Semantic Image Segmentation
- A fast method for simultaneous reconstruction and segmentation in X-ray CT application
- Discretized Gradient Flow for Manifold Learning in the Space of Embeddings
- Multidimensional persistence in biomolecular data
- Re-initialization Free Level Set Evolution via Reaction Diffusion
- Mumford-Shah Mesh Processing using the Ambrosio-Tortorelli Functional
- Stationarity of the crack-front for the Mumford-Shah problem in 3D
- Total Variation and Tight Frame Image Segmentation with Intensity Inhomogeneity
- Image Restoration using Total Variation with Overlapping Group Sparsity
- Endpoint regularity of 2d Mumford-Shah minimizers
- Adaptive finite element convergence analysis of AT1 phase-field model for quasi-static fracture in strain-limiting solids
- Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models
- Phase field modeling of crack propagation under combined shear and tensile loading with hybrid formulation
- An Active Contour Model for Silhouette Vectorization using Bézier Curves
- Offset Curves Loss for Imbalanced Problem in Medical Segmentation
- Segmentation of Time Series: Parameter Dependence of Blake-Zisserman and Mumford-Shah Functionals and the Transition from Discrete to Continuous
- EGNet:Edge Guidance Network for Salient Object Detection
- Exact Relaxation for Classes of Minimization Problems with Binary Constraints
- Active Contour Models Driven by Hyperbolic Mean Curvature Flow for Image Segmentation
- Volume Preserving Image Segmentation with Entropic Regularization Optimal Transport and Its Applications in Deep Learning
- A variational algorithm for the detection of line segments
- MBO Scheme for Local Chan--Vese Segmentation
- From a Modified Ambrosio-Tortorelli to a Randomized Part Hierarchy Tree
- The Γ-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension
- Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory
- A second order minimality condition for the Mumford-Shah functional
- An immersed phase field fracture model for fluid-infiltrating porous media with evolving Beavers-Joseph-Saffman condition
- Multi-Sensor Fusion via Reduction of Dimensionality
- The Stress-Intensity Factor for nonsmooth fractures in antiplane\n elasticity
- The Chan-Vese Algorithm
- Bayesian Field Theory: Nonparametric Approaches to Density Estimation, Regression, Classification, and Inverse Quantum Problems
- A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme
- Convergence Analysis of Adaptive Finite Element Algorithms for a Regularized Variational Model of Quasi-Static Brittle Fracture in "Strain-Limiting" Elastic Solids
- Freedom through Imperfection: Exploiting the flexibility offered by\n redundancy in signal processing
- Image Segmentation Using Weak Shape Priors
- Particle methods enable fast and simple approximation of Sobolev gradients in image segmentation
- Unsupervised Moving Object Detection via Contextual Information Separation
- Hierarchical image simplification and segmentation based on Mumford-Shah-salient level line selection
- Crystal image analysis using 2D synchrosqueezed transforms
- Image Segmentation via Variational Model Based Tailored UNet: A Deep Variational Framework
- Bayesian Wombling
- Fast image segmentation and restoration using parametric curve evolution with junctions and topology changes
- Equilibrium and absolute minimal states of Mumford-Shah functionals and brittle fracture propagation
- Visual Grouping by Neural Oscillators
- A phase-field modeling approach to sea-ice fracturing
- Level Set Based Shape Prior Segmentation
- SegAN: Adversarial Network with Multi-scale L1 Loss for Medical Image Segmentation
- Does median filtering truly preserve edges better than linear filtering?
- Scale-space and edge detection using anisotropic diffusion
- Weberized Mumford-Shah Model with Bose-Einstein Photon Noise
- A Stochastic‐Variational Model for Soft Mumford‐Shah Segmentation
- Euler's Elastica and Curvature-Based Inpainting
- 3D Crystal Image Analysis based on Fast Synchrosqueezed Transforms
- Graph Merriman--Bence--Osher as a SemiDiscrete Implicit Euler Scheme for Graph Allen--Cahn Flow
- Model-Based Learning of Local Image Features for Unsupervised Texture Segmentation
- Image Restoration Using Joint Statistical Modeling in a Space-Transform Domain
- Tomography: mathematical aspects and applications
- Persistence Lenses: Segmentation, Simplification, Vectorization, Scale Space and Fractal Analysis of Images
- Region Tracking in an Image Sequence: Preventing Driver Inattention
- New region force for variational models in image segmentation and high dimensional data clustering
- An overview of the modelling of fracture by gradient damage models
- Cross-Entropy Method in Application to the SIRC Model
- Laws of crack motion and phase-field models of fracture
- Size homotopy groups for computation of natural size distances
- Simultaneous segmentation and reconstruction: a level set method approach for limited view computed tomography. [europepmc]
- A fast and fully automatic method for cerebrovascular segmentation on time-of-flight (TOF) MRA image. [europepmc]
- An image classification approach to analyze the suppression of plant immunity by the human pathogen Salmonella Typhimurium. [europepmc]
- Fick's second law transformed: one path to cloaking in mass diffusion. [europepmc]
- Region-based progressive localization of cell nuclei in microscopic images with data adaptive modeling. [europepmc]
- A conceptual framework of computations in mid-level vision. [europepmc]
- MRI segmentation of the human brain: challenges, methods, and applications. [europepmc]
- Fully automated whole-head segmentation with improved smoothness and continuity, with theory reviewed. [europepmc]
- Robust tracking and quantification of C. elegans body shape and locomotion through coiling, entanglement, and omega bends. [europepmc]
- Unsupervised Retinal Vessel Segmentation Using Combined Filters. [europepmc]
- Phase-field modelling of ductile fracture: a variational gradient-extended plasticity-damage theory and its micromorphic regularization. [europepmc]
- Segmentation of prostate from ultrasound images using level sets on active band and intensity variation across edges. [europepmc]
- An Automatic Segmentation and Classification Framework Based on PCNN Model for Single Tooth in MicroCT Images. [europepmc]
- Life and Understanding: The Origins of "Understanding" in Self-Organizing Nervous Systems. [europepmc]
- Semi-automatic bowel wall thickness measurements on MR enterography in patients with Crohn's disease. [europepmc]
- Segmentation of Left and Right Ventricles in Cardiac MRI Using Active Contours. [europepmc]
- Automated breast tumor detection and segmentation with a novel computational framework of whole ultrasound images. [europepmc]
- Hybrid two-stage active contour method with region and edge information for intensity inhomogeneous image segmentation. [europepmc]
- Towards Reinforced Brain Tumor Segmentation on MRI Images Based on Temperature Changes on Pathologic Area. [europepmc]
- Active contour method for ILM segmentation in ONH volume scans in retinal OCT. [europepmc]
- LinSEM: Linearizing segmentation evaluation metrics for medical images. [europepmc]
- Regional registration of whole slide image stacks containing major histological artifacts. [europepmc]
- A Systematic Approach for MRI Brain Tumor Localization and Segmentation Using Deep Learning and Active Contouring. [europepmc]
- Learning fuzzy clustering for SPECT/CT segmentation via convolutional neural networks. [europepmc]
- Automatic ladybird beetle detection using deep-learning models. [europepmc]