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Algorithms and software for projections onto intersections of convex and non-convex sets with applications to inverse problems [article]

Bas Peters, Felix J. Herrmann
2019 arXiv   pre-print
We propose algorithms and software for computing projections onto the intersection of multiple convex and non-convex constraint sets.  ...  The primary design criterion is working with multiple sets, which allows us to solve inverse problems with multiple pieces of prior knowledge.  ...  ) compared to the straightforward application of operator splitting methods. • a software design specifically for set intersection projection problems.  ... 
arXiv:1902.09699v2 fatcat:yp3t7tfj6jaulcgtvxe5ejqlkm

Generalized Minkowski sets for the regularization of inverse problems [article]

Bas Peters, Felix J. Herrmann
2019 arXiv   pre-print
We derive the projection operation onto the generalized Minkowski sets and construct an algorithm based on the alternating direction method of multipliers.  ...  We generalize the Minkowski set, such that the model parameters are the sum of two components, each of which is constrained to an intersection of sets.  ...  Formulation of inverse problems with generalized Minkowski constraints So far, we proposed a generalization of the Minkowski set (M, equation (5) ), and developed an algorithm to compute projections onto  ... 
arXiv:1903.03942v1 fatcat:4bhwoeehl5cu7h2m4hxcohh4na

Deconvolution using Fourier Transform phase,ℓ1andℓ2balls, and filtered variation

Onur Yorulmaz, A. Enis Cetin
2017 SoftwareX  
ball Projection onto convex sets In this article, we present a deconvolution software based on convex sets constructed from the phase of the Fourier Transform, bounded 2 energy and 1 energy of a given  ...  The iterative deconvolution algorithm is based on the method of projections onto convex sets.  ...  Since all of the sets are closed and convex the iterates converge to the intersection (20) regardless of the initial input image x 1 provided that the set C int is non-empty.  ... 
doi:10.1016/j.softx.2017.11.007 fatcat:vkghbtazhba2jk5fgl6idzme7q

Deconvolution using Fourier transform phase, ℓ 1 and ℓ 2 balls, and filtered variation

Onur Yorulmaz, A. Enis Cetin
2018 Digital signal processing (Print)  
The iterative deconvolution algorithm is based on the method of projections onto convex sets.  ...  ℓ 1 ball Projection onto convex sets a b s t r a c t In this article, we present a deconvolution software based on convex sets constructed from the phase of the Fourier Transform, bounded ℓ 2 energy and  ...  The proposed method and accompanying software aims to deconvolve blurry images using projections onto convex sets (POCS).  ... 
doi:10.1016/j.dsp.2017.11.004 fatcat:p7k2all2jzazrmkuwjetfosysy

Entropy-Functional-Based Online Adaptive Decision Fusion Framework With Application to Wildfire Detection in Video

O. Gunay, B. U. Toreyin, K. Kose, A. E. Cetin
2012 IEEE Transactions on Image Processing  
Decision values are linearly combined with weights which are updated online according to an active fusion method based on performing entropic projections onto convex sets describing sub-algorithms.  ...  Index Terms-Projections onto convex sets, active learning, decision fusion, online learning, entropy maximization, wildfire detection using video.  ...  According to the projection onto convex sets (POCS) theory, when there are a finite number of convex sets, repeated cyclical projections onto these sets converge to a vector in the intersection set [14  ... 
doi:10.1109/tip.2012.2183141 pmid:22249709 fatcat:2apizyvqbjeahkzchxivoxml24

Optimization of intensity modulated beams with volume constraints using two methods: Cost function minimization and projections onto convex sets

Paul S. Cho, Shinhak Lee, Robert J. Marks, Seho Oh, Steve G. Sutlief, Mark H. Phillips
1998 Medical Physics (Lancaster)  
The second technique is based on the theory of projections onto convex sets ͑POCS͒ in which the dose-volume constraint is replaced by a limit on integral dose.  ...  The convex projection method can find solutions in much shorter time with minimal user interaction. © 1998 American Association of Physicists in Medicine. ͓S0094-2405͑98͒01004-9͔  ...  Dose-volume histograms for the rectum obtained with ͑a͒ the cost function minimization method and with ͑b͒ the method of projections onto convex sets.  ... 
doi:10.1118/1.598218 pmid:9571609 fatcat:ozx3bqzor5aqjk52rfv2h7qvs4

Dual Constrained TV-based Regularization on Graphs

Camille Couprie, Leo Grady, Laurent Najman, Jean-Christophe Pesquet, Hugues Talbot
2013 SIAM Journal of Imaging Sciences  
They play a key role in a variety of applications such as image denoising, compressive sensing and inverse problems in general.  ...  The proposed method consequently applies to a variety of other inverse problems including image fusion and mesh filtering.  ...  Dykstra's algorithm [55] to compute the projection onto an intersection of convex sets would not be applicable either, as the matrix DΓD is usually singular.  ... 
doi:10.1137/120895068 fatcat:g5puk3g36jfeplxocdjcgunz7u

Fast and Robust Inversion-Free Shape Manipulation

Tiantian Liu, Ming Gao, Lifeng Zhu, Eftychios Sifakis, Ladislav Kavan
2016 Computer graphics forum (Print)  
Using a schedule of increasing penalty coefficients, we efficiently and robustly converge to an inversion free state by solving a sequence of unconstrained convex minimization problems.  ...  We demonstrate the efficacy of our scheme in a number of 2D and 3D shapes undergoing moderate to drastic deformation.  ...  To address this problem, we start by designing a convex quadratic deformation energy which produces results similar to skinning, but allows us to enhance the algorithm with non-inversion constraints.  ... 
doi:10.1111/cgf.12806 fatcat:hnrlwz7uz5ekzeoltwmkgazeku

Optimized halftoning using dot diffusion and methods for inverse halftoning

M. Mese, P.P. Vaidyanathan
2000 IEEE Transactions on Image Processing  
Furthermore, inverse halftoning of dot diffused images is discussed and two methods are proposed. The first one uses projection onto convex sets (POCS) and the second one uses wavelets.  ...  However, image quality offered by error diffusion is still regarded as superior to most of the other known methods.  ...  Lau for his valuable comments on a conference version of the paper. Finally, they would like to thank Dr. Xiong for the use of his software on inverse halftoning.  ... 
doi:10.1109/83.841944 pmid:18255440 fatcat:ylqmwndodrhdzoprf4xfq2slqe

Optimal Moebius Transformations for Information Visualization and Meshing [article]

Marshall Bern, David Eppstein
2001 arXiv   pre-print
We give linear-time quasiconvex programming algorithms for finding a Moebius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed  ...  We can also use similar methods to maximize the minimum distance among a set of pairs of input points.  ...  The set of (r, R)-good points is then just the projection onto the boundary of H of N R (L) ∩ X. So, the convexity of this set follows immediately from Lemma 2.  ... 
arXiv:cs/0101006v2 fatcat:yvewgrcecvbt3d3mkuolha5sve

Projection Methods in Conic Optimization [chapter]

Didier Henrion, Jérôme Malick
2011 International Series in Operations Research and Management Science  
There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace.  ...  This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization.  ...  Acknowledgements The work of the first author was partly supported by Research Programme MSM6840770038 of the Czech Ministry of Education and by Project 103/10/0628 of the Grant Agency of the Czech Republic  ... 
doi:10.1007/978-1-4614-0769-0_20 fatcat:h6pch2tgxne4zaqgq5djlkjtfq

Projection methods in conic optimization [article]

Didier Henrion, Jérôme Malick (INRIA Rhône-Alpes / LJK Laboratoire Jean Kuntzmann)
2011 arXiv   pre-print
There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace.  ...  This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization.  ...  Acknowledgements The work of the first author was partly supported by Research Programme MSM6840770038 of the Czech Ministry of Education and by Project 103/10/0628 of the Grant Agency of the Czech Republic  ... 
arXiv:1103.1511v1 fatcat:qnyxxpiy5ffbznqn6x7zvjsvui

Proximal Distance Algorithms: Theory and Examples [article]

Kevin L. Keys and Hua Zhou and Kenneth Lange
2018 arXiv   pre-print
For convex problems, we prove global convergence.  ...  Since many explicit projections and proximal maps are known, it is straightforward to derive and implement novel optimization algorithms in this setting.  ...  Acknowledgments We thank Joong-Ho Won for many insightful discussions. In partiular, he pointed out the utility of the least squares criterion (6) .  ... 
arXiv:1604.05694v3 fatcat:tplargdqwjh6hjwgprej64n36u

Optimizing radar waveforms using generalized alternating projections

Dylan Eustice, Charles Baylis, Casey Latham, Robert J. Marks, Lawrence Cohen
2015 2015 Texas Symposium on Wireless and Microwave Circuits and Systems (WMCS)  
An algorithm is proposed which uses a generalized method of alternating projections to synthesize waveforms with desired ambiguity function properties.  ...  This work provides a computationally intelligent methodology to dynamically optimize detection in radar applications and a foundation for future work in joint circuit optimization for spectral compliance  ...  The authors would like to thank Keysight Technologies for costfree loan of the Advanced Design System simulation software.  ... 
doi:10.1109/wmcas.2015.7233196 fatcat:radsao36ybc6foi4twkbtkuerq

Affine Buildings and Tropical Convexity [article]

Michael Joswig, Bernd Sturmfels, Josephine Yu
2007 arXiv   pre-print
Our results include a convex hull algorithm for the Bruhat--Tits building of SL_d(K) and techniques for computing with apartments and membranes.  ...  While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to  ...  Hence any convex set within some apartment of B d arises as the output of an algorithm for Computational Problem B. Such algorithms are our topic in Section 6.  ... 
arXiv:0706.1918v1 fatcat:buwv765yk5erdmc6gkhjmdsd4u
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