Mathematical problems in image processing - partial differential equations and the calculus of variations

By Gilles Aubert, Pierre Kornprobst. Applied Mathematical Sciences, 2010

1,497 citations recorded by the source corpus.

Subfields: Computer Vision and Pattern Recognition, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/93aeb19a54780c66662000dcfbef238fdcfcc10f

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Citations in the corpus

Topology optimization of functionally graded cellular materials (2013, methodology): “This could be done by exploiting a filtering scheme such as nonlinear diffusion technique [24, 25].”
Variational PDE models in image processing (2002, background): “[13 (1/2), 2002] and also two recent monographs [1], [18].”
Distance Regularized Level Set Evolution and Its Application to Image Segmentation (2010, methodology): “In calculus of variations [28], a standard method to minimize an energy functional is to ﬁnd the steady state solution of the gradient ﬂow equation (8) where is the Gâteaux derivative of the functional .”
A binary level set model and some applications to Mumford-Shah image segmentation (2006, methodology): “This model was adapted to image segmentation in [18], [ 19 ].”
The phase field method for geometric moving interfaces and their numerical approximations (2019, methodology): “Imaging science is a good example where variational methods have been widely used for imaging and data analysis [89, 22].”
A Level Set Method for Image Segmentation in the Presence of Intensity Inhomogeneities With Application to MRI (2011, methodology): “For fixed and , the minimization of with respect to can be achieved by using standard gradient descent method, namely, solving the gradient flow equation (21) where is the Gâteaux derivative [1] of the energy .”
Vector-valued image regularization with PDEs: a common framework for different applications (2005, neutral): “Index Terms —Diffusion PDEs, color image regularization, denoising, inpainting, vector-valued smoothing, anisotropic filtering, flow visualization.”
What Is the Range of Surface Reconstructions from a Gradient Field? (2006, methodology): “Several PDE’s based methods such as anisotropic diffusion [13], shock filters and energy based methods [14] (see [15] for detailed analysis and algorithms) have been proposed that try to restore an image while maintaining edges or sharp features.”
Histology image analysis for carcinoma detection and grading (2012, background): “A review of current denoising methods is given in [4,151].”
Implicit Neural Representations with Periodic Activation Functions (2020, methodology): “2 Representing Shapes with Signed Distance Functions Inspired by recent work on shape representation with differentiable signed distance functions (SDFs) [1, 4, 9], we fit SDFs directly on oriented point clouds using both ReLU-based implicit neural representations and SIRENs.”
Design of graded two-phase microstructures for tailored elasticity gradients (2008, background): “nonlinear diffusion [41, 42] is of considerable significance in suppressing checkerboard patterns and blurred boundaries for the PBC design [14, 15], thereby ensuring proper topo-”
A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model (2002, methodology): “Also, for recent expositions of geometric PDE’s, variational problems and image processing (including snakes, active contours, curve evolution problems), we refer the reader to Sapiro (2001), Aubert and Kornprobst (2001) , and Guichard and Morel (to appear).”
Active contours driven by local image fitting energy (2010, methodology): “010 Using the calculus of variation and the steepest descent method [22], we minimize ELIFðfÞ with respect to f to get the corresponding gradient descent flow (please refer to Appendix A for detailed derivation): @f @t 1⁄4 ðI IÞðm1 m2ÞdeðfÞ ð15Þ”