Critical Thresholds (Eitan Tadmor)

Hierarchical Decompositions (Eitan Tadmor)

HIERARCHICAL DECOMPOSITIONS

A collection of selected references on

Hierarchical decompositions with applications in image processing and PDEs.

E. Tadmor, S. Nezzar and L. Vese A multiscale image representation using hierarchical (BV,L²) decompositions Multiscale Modeling and Simulations 2(4) (2004) 554-579.
D. Paquin, D. Levy, E. Schreibmann and L. Xing, Multiscale Image Registration Mathematical Biosciences and Engineering 3(2) (2006) 389-418.
N. Kruglyak The K-functional and Calderon-Zygmund type decompositions in "Interpolation Theory and Applications", Contemporary Mathematics Series, Proceedings of the conference in honor of Professor M. Cwikel held in Miami, Florida in 2006, Vol. 445 (Laura De Carli and Mario Milman, eds.), Amer. Math. Society (2007) 183-194..
D. Paquin, D. Levy and L. Xing Hybrid multiscale landmark and deformable image registration Mathematical Biosciences and Engineering 4 (2007) 711-737.
Björn Jawerth and Mario Milman Lectures on optimization, image processing, and interpolation theory in "Function Spaces, Inequalities and Interpolation", Lectures on Optimization, Image Processing and Interpolation Theory, Spring School on Analysis held at Paseky nad Jizerou, chalet in the Krkonose Mountains in 2007 (2007).
Dana Paquin, Doron Levy and Lei Xing Multiscale deformable registration of noisy medical images Mathematical Bio-sciences and Engineering 5(1) (2008) 125-144.
E. Tadmor, S. Nezzar and L. Vese Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation Communications in Mathematical Sciences 6(2) (2008) 281-307.
E. Tadmor and P. Athavale Multiscale image representation using integro-differential equations Inverse Problems and Imaging 3(4) (2009) 693-710.
P. Athavale and E. Tadmor Novel integro-differential equations in image processing and its applications in "Computational Imaging VIII", Proceedings of SPIE meeting held in San Jose in 2010, Vol. 7533 (C. A. Bouman, I. Pollak, P. J. Wolfe eds.) (2010) 75330S..
Michael Hintermüller and M Monserrat Rincon-Camacho Expected absolute value estimators for a spatially adapted regularization parameter choice rule in L¹-TV-based image restoration Inverse Problems 26(8) (2010).
Moncef Hidane, Olivier Lézoray, Vinh-Thong Ta and Abderrahim Elmoataz Nonlocal multiscale hierarchical decomposition on graphs in "ECCV2010", Proceeding of 11th European Conference on Computer Vision held in Heraklion, Crete in 2010, Vol. LNCS 6314 (Part IV) (K.Daniilidis, P.Maragos and N.Paragios eds.), Springer-Verlag (2010) 638-650..
Yiqiu Dong, Michael Hintermüller and M. Monserrat Rincon-Camacho Automated regularization parameter selection in multi-scale total variation models for image restoration Journal of Mathematical Imaging and Vision 40(1) (2010) 82-104.
P. Athavale and E. Tadmor Integro-differential equations based on (BV, L¹) image decomposition SIAM journal on Imaging Sciences 4(1) (2011) 300-312.
E. Tadmor and C. Tan Hierarchical construction of bounded solutions of div U=F in critical regularity spaces in "Nonlinear Partial Differential Equations", Proceedings of the 2010 Abel Symposium held in 2010, Vol. 7 - Abel Symposia (H. Holden & K. Karlsen eds.), Springer, Oslo (2012) 255-269..
Jerome Gilles Multiscale texture separation Multiscale Modeling & Simulation 10(4) (2012) 1409-1427.
Moncef Hidane, Olivier Lézoray and Abderrahim Elmoataz Nonlinear multilayered representation of graph-signals Journal of Mathematical Imaging and Vision 45(2) (2013) 114-137.
P. Athavale., R. Xu, P. Radau., A. Nachman and G. Wright Multiscale TV flow with applications to fast denoising and registration in "SPIE Medical Imaging: image processing", Proceedings of SPIE meeting held in Lake Buena Vista, Florida in 2013, Vol. 8669 86692K-1 (S. Ourselin, D. R. Haynor eds.) (2013).
Liming Tang and Chuanjiang He Multiscale Texture Extraction with Hierarchical (BV,G_p,L²) Decomposition Journal of Mathematical Imaging and Vision 45(2) (2013) 148-163.
Xiaohui Wang Multiscale decompositions and optimization ArXiv 1301.5041v2 (2013).
Prashant Athavale, Robert Xu, Perry Radau, Graham Wright and Adrian Nachman Multiscale properties of weighted total variation flow with applications to denoising and registration IEEE Transactions in Medical Imaging 11(4) (2013) 1-12.
Liming Tang and Chuanjiang He Multiscale image representation and texture extraction using hierarchical variational decomposition Journal of Applied Mathematics 107120 (2013).
Juan Carlos Moreno, Surya Prasath, Dmitry Vorotnikov and Hugo Proença Adaptive diffusion constrained Total Variation scheme with application to cartoon+texture+edge image decomposition.
E. Tadmor Hierarchical construction of bounded solutions in critical regularity spaces Communications on Pure and Applied Mathematics 69(6) (2016) 1087-1109.
Prashant Athavale, Robert L. Jerrard, Matteo Novaga and Giandomenico Orlandi Weighted TV minimization and applications to vortex density models Journal of Convex Analysis 24(4) (2017) 1051-1084.
Klas Modin, Adrian Nachman and Luca Rondi A multiscale theory for image registration and nonlinear inverse problems Advances in Mathematics 346 (2019) 1009-1066.
Huan Han, Zhengping Wang and Yimin Zhang MultiScale approach for two-dimensional diffeomorphic image registration Multiscale Modeling & Simulation 19(4) (2021) 1538-1572.
Wen Li, Elena Resmerita and Luminita A. Vese Multiscale hierarchical image decomposition and refinements: qualitative and quantitative results SIAM J. Imaging Sciences 14(2) (2021) 844-877.
Stefan Kindermann, Elena Resmerita and Tobias Wolf Multiscale hierarchical decomposition methods for ill-posed problems Inverse Problems 39(12) (2023) 125013.
Tristan Milne and Adrian Nachman An optimal transport analogue of the Rudin Osher Fatemi model and its corresponding multiscale theory SIAM J. Mathematical Analysis 56(1) (2024) 1114-1148.
A. Cohen, R. DeVore and E. Tadmor Constructions of bounded solutions of div u=f in critical spaces Multiscale, Nonlinear and Adaptive Approximation II (2024) 177-200.
S. Rebegoldi and L. Rondi A counterexample to convergence for multiscale decompositions in "Inverse Problems: Modelling and Simulation", IPMS Conference 2024 (A. Hasanov Hasanoglu, R. Novikov and K. Van Bockstal eds.) Birkhauser, 2025.
S. Rebegoldi and L. Rondi On the optimality of convergence conditions for multiscale decompositions in imaging and inverse problems ArXiv:2503.00827v1 (2025).
Tobias Wolf, Stefan Kindermann, Elena Resmerita and Luminita Vese Applications of multiscale hierarchical decomposition to blind deconvolution ArXiv:2409.08734v3 (2025).