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A Linear Programming Relaxation for Binary Tomography with Smoothness Priors

S. Weber, C. Schnorr, J. Hornegger
2003 Electronic Notes in Discrete Mathematics  
À ¹ f AE Ç Ä P ¼ AE Ä { È È ½ u » AE Â · À b ¹ { W É Ê y À Ä ¬ Å Ë À e ¹ ¥ Ì ® · ¬ º q ¾ Á ¦ ¹ f ½ ¼ Í | Î ¿ p · È Á Ã ½ p · ¬ Ä # À ® Ï Ë À e ¹ { Ð { À e Ä { AE AE Á Ã ½ Ð § Ñ i Ò z Ó W Ô x À · ¬ È Ä  ...  ¹ { Ð { À e Ä Ë ¿ Å a è w â Ä # À ä { ¹ ( Ý é Ä { ½ Ù ¹ f AE ë ê Á ¦ · ¬ È Ã Ù ¼ è ã W ¹ f AE Â Á Ã ½ Ä # ¾ l ¹ { À e Á Ã Ä { È á Ë ¾ Á Ã AE Á ¦ Ø ¬ Ä # ¾ Á ¦ ¹ f ½ ì è 6 Ò z Ó Ï é · ¬ È Ã Ä ¥ Õ 6 Ä #  ... 
doi:10.1016/s1571-0653(04)00490-1 fatcat:2wmj5xc7uzdhpihw6hx4oy7gky

Discrete tomography by convex–concave regularization and D.C. programming

T. Schüle, C. Schnörr, S. Weber, J. Hornegger
2005 Discrete Applied Mathematics  
A quadratic objective functional over binary variables comprising the squared projection error and a prior penalizing non-homogeneous regions, is supplemented with a concave functional enforcing binary  ...  Our approach is applicable to quite general objective functions over binary variables with constraints and thus applicable to a wide range of problems within and beyond the field of discrete tomography  ...  [11] introduced the following linear integer programming problem for binary tomography: and suggested a range of greedy approaches within a general framework for local search.  ... 
doi:10.1016/j.dam.2005.02.028 fatcat:2ebeb7f6gjatfcl2ex3kyeit2i

A Linear Programming Approach to Limited Angle 3D Reconstruction from DSA Projections [chapter]

S. Weber, T. Schüle, C. Schnörr, J. Hornegger
2003 Informatik aktuell  
a smoothness prior.  ...  Objectives: We investigate the feasibility of binary-valued 3D tomographic reconstruction using only a small number of projections acquired over a limited range of angles.  ...  Volumegraphics (http://www.volumegraphics.com) kindly provided us with their volume rendering software. All 3D models in this paper were rendered with VGStudioMax 1.1.  ... 
doi:10.1007/978-3-642-18993-7_9 fatcat:vaj6qn7pcnefxcmisyhwlzb5pq

Prior Learning and Convex-Concave Regularization of Binary Tomography

Stefan Weber, Thomas Schüle, Christoph Schnörr
2005 Electronic Notes in Discrete Mathematics  
A convex reconstruction functional, comprising the projections equations and a smoothness prior, was complemented with a concave penalty term enforcing binary solutions.  ...  We show that the difference-of-convex-functions DC-programming framework is flexible enough to cope with this more general model class.  ...  Smoothness priors are convenient from the computational viewpoint because they result in convex functionals having relaxed the binary constraint.  ... 
doi:10.1016/j.endm.2005.05.071 fatcat:d67n54u745ghfdf3mkp5gfs4gu

TomoGC: Binary Tomography by Constrained GraphCuts [chapter]

Jörg Hendrik Kappes, Stefania Petra, Christoph Schnörr, Matthias Zisler
2015 Lecture Notes in Computer Science  
We present an iterative reconstruction algorithm for binary tomography, called TomoGC, that solves the reconstruction problem based on a constrained graphical model by a sequence of graphcuts.  ...  A comprehensive numerical evaluation demonstrates that the proposed method can reconstruct objects from a small number of projections more accurate and also faster than competitive methods.  ...  The relaxed linear program can then have nonbinary solutions.  ... 
doi:10.1007/978-3-319-24947-6_21 fatcat:h4yf3qrzxrdkresc3sg7owv6tu

Discrete Tomography by Continuous Multilabeling Subject to Projection Constraints [chapter]

Matthias Zisler, Stefania Petra, Claudius Schnörr, Christoph Schnörr
2016 Lecture Notes in Computer Science  
We present a non-convex variational approach to non-binary discrete tomography which combines non-local projection constraints with a continuous convex relaxation of the multilabeling problem.  ...  A competitive numerical evaluation using standard test-datasets demonstrates a significantly improved reconstruction quality for noisy measurements from a small number of projections.  ...  [29, 21] proposed to combine a quadratic program with a non-convex penalty which gradually enforces binary constraints. More recently, Kappes et al.  ... 
doi:10.1007/978-3-319-45886-1_21 fatcat:i6nzn4fvljcwvfqftz5nm56mnq

Adaptive Reconstruction of Discrete-Valued Objects from few Projections

Thomas Schüle, Stefan Weber, Christoph Schnörr
2005 Electronic Notes in Discrete Mathematics  
Recently, we proposed an algorithm for binary tomography based on DC ( difference of convex functions ) programming [13, 15] .  ...  The proposed algorithm remains practicable for multi-valued reconstructions, and even with a large number of discrete values.  ...  Binary Objects Reconstruction Using a linear transformation for the projection process of transmission tomography, the reconstruction problem becomes an inverse problem for Ax = b (1) with unknown x  ... 
doi:10.1016/j.endm.2005.05.074 fatcat:uf4sz54zinf67ccqqujtkefh3e

Binary Tomography by Iterating Linear Programs from Noisy Projections [chapter]

Stefan Weber, Thomas Schüle, Joachim Hornegger, Christoph Schnörr
2004 Lecture Notes in Computer Science  
In this paper we improve the behavior of a reconstruction algorithm for binary tomography in the presence of noise.  ...  The objective function contains a smoothness prior that favors spatially homogeneous solutions and a concave functional gradually enforcing binary solutions.  ...  [7] introduced the following linear integer programming problem for binary tomography: max x∈{0,1} n e, x , e := (1, . . . , 1) , Ax ≤ b , (3) and suggested a range of greedy approaches within a general  ... 
doi:10.1007/978-3-540-30503-3_3 fatcat:dbg2rfeld5f7tjdzh6ir36bn3m

On image reconstruction algorithms for binary electromagnetic geotomography

Rafal Zdunek
2008 Theoretical Computer Science  
The assumption for a binary representation of the image to be reconstructed substantially relaxes image reconstruction problems related to ill-posedness that comes from an intrinsic limitation of an angular  ...  We test two algorithms for binary tomography, where the penalty term is based on the Markov Random Field (MRF) model.  ...  The author is also grateful to the anonymous reviewers for their valuable comments and suggestions.  ... 
doi:10.1016/j.tcs.2008.06.007 fatcat:gzecuujwbbh57exaptlmypbs7y

Efficient Message Passing for 0-1 ILPs with Binary Decision Diagrams [article]

Jan-Hendrik Lange, Paul Swoboda
2021 arXiv   pre-print
We present a message passing method for 0-1 integer linear programs.  ...  Our algorithm is based on a decomposition of the original problem into subproblems that are represented as binary decision diagrams.  ...  Implementation with BDDs We have described above a generic DBCA procedure for optimizing a Lagrangean relaxation of 0-1 integer linear programs and a primal search heuristic for finding solutions given  ... 
arXiv:2009.00481v2 fatcat:aq3kyrwrkbe2dfpjy4fekhpdvq

Image Reconstruction by Multilabel Propagation [chapter]

Matthias Zisler, Freddie Åström, Stefania Petra, Christoph Schnörr
2017 Lecture Notes in Computer Science  
The proposed objective function is efficiently minimized via DC programming which amounts to solving a sequence of convex programs, with guaranteed convergence to a critical point.  ...  Each convex program is solved by a generalized primal dual algorithm. This entails the evaluation of a proximal mapping, evaluated efficiently by a fixed point iteration.  ...  We compare our model to state-of-the-art approaches for non-binary discrete tomography in limited angles scenarios.  ... 
doi:10.1007/978-3-319-58771-4_20 fatcat:inyieqhgxnfqbelqrehezz6x6q

Binary Tomography with Deblurring [chapter]

Stefan Weber, Thomas Schüle, Attila Kuba, Christoph Schnörr
2006 Lecture Notes in Computer Science  
We study two scenarios of limited-angle binary tomography with data distorted with an unknown convolution: Either the projection data are taken from a blurred object, or the projection data themselves  ...  To this end, a recently introduced Difference-of-Convex-Functions programming approach to limited-angle binary tomographic reconstruction is complemented with Expectation-Maximization iteration.  ...  Problem Statement Binary Tomography and Reconstruction by DC-Programming We consider the reconstruction problem of transmission tomography for binary objects.  ... 
doi:10.1007/11774938_30 fatcat:n3oxcpnl4zh2hirmxlfjnnmnim

CoShaRP: A Convex Program for Single-shot Tomographic Shape Sensing [article]

Ajinkya Kadu, Tristan van Leeuwen, K. Joost Batenburg
2020 arXiv   pre-print
Hence, the shape prior transforms a linear ill-posed image estimation problem to a non-linear problem of estimating the roto-translations of the shapes.  ...  Moreover, it is more challenging than conventional tomography where a sufficiently large number of projection angles forms the measurements, allowing for a simple inversion process.  ...  Acknowledgments The authors thank Nick Luiken for stimulating discussions. This work was supported by the Dutch Research Council (grant no. OCENW.XS.039).  ... 
arXiv:2012.04551v2 fatcat:apimporhznfw5atjyziwibw2a4

Non-convex image reconstruction via Expectation Propagation [article]

Anna Paola Muntoni and Rafael Díaz Hernández Rojas and Alfredo Braunstein and Andrea Pagnani and Isaac Pérez Castillo
2018 arXiv   pre-print
This maximization can be performed with standard local optimization tools when the function is concave, but it is generally intractable for realistic priors, which are non-concave.  ...  We show, by means of extensive simulations, that, compared to state-of-the-art algorithms for this task, Expectation Propagation paired with very simple but non log-concave priors, is often able to reconstruct  ...  ), Quadratic Programming (QP) (for 2 smoothness) and, for binary reconstruction only, the BP algorithm.  ... 
arXiv:1809.03958v1 fatcat:oyyn5qalyfbh3djw4dtotamocu

Limited-view binary tomography reconstruction assisted by shape centroid

Tibor Lukić, Péter Balázs
2021 The Visual Computer  
In this paper, the binary tomographic reconstruction problem for very limited projection data availability is considered.  ...  Being this inverse problem highly ill-posed, we propose a new reconstruction model that uses a shape centroid-based regularization term, i.e., we assume that the center of gravity of the object of interest  ...  This research was supported by the project "Integrated program for training new generation of scientists in the fields of computer science," No. EFOP-3.6.3-VEKOP-16-2017-00002.  ... 
doi:10.1007/s00371-020-02044-8 pmid:33456100 pmcid:PMC7802814 fatcat:7g4qunfwi5di3eks7lxytgjeem
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