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| caption2 = Example of application of the Chambolle-Pock algorithm to image reconstruction.
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In [[mathematics]], the '''Chambolle-Pock algorithm''' is an [[algorithm]] used to solve [[convex optimization]] problems. It was introduced by Antonin Chambolle and Thomas Pock<ref name=":0">{{Cite journal |last1=Chambolle |first1=Antonin |last2=Pock |first2=Thomas |date=2011-05-01 |title=A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging |url=https://doi.org/10.1007/s10851-010-0251-1 |journal=Journal of Mathematical Imaging and Vision |language=en |volume=40 |issue=1 |pages=120–145 |doi=10.1007/s10851-010-0251-1 |s2cid=207175707 |issn=1573-7683}}</ref> in 2011 and has since become a widely used method in various fields, including [[Digital image processing|image processing]],<ref name=":1" /><ref name=":2" /><ref name=":3" /> [[computer vision]],<ref>{{Cite
The Chambolle-Pock algorithm is specifically designed to efficiently solve convex optimization problems that involve the minimization of a non-smooth [[Loss function|cost function]] composed of a data fidelity term and a regularization term.<ref name=":0" /> This is a typical configuration that commonly arises in ill-posed imaging [[Inverse problem|inverse problems]] such as [[image reconstruction]],<ref name=":1">{{Cite journal |last1=Sidky |first1=Emil Y |last2=Jørgensen |first2=Jakob H |last3=Pan |first3=Xiaochuan |date=2012-05-21 |title=Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle–Pock algorithm |journal=Physics in Medicine and Biology |volume=57 |issue=10 |pages=3065–3091 |doi=10.1088/0031-9155/57/10/3065 |issn=0031-9155 |pmc=3370658 |pmid=22538474|arxiv=1111.5632 |bibcode=2012PMB....57.3065S }}</ref> [[Noise reduction|denoising]]<ref name=":2">{{Cite journal |last1=Fang |first1=Faming |last2=Li |first2=Fang |last3=Zeng |first3=Tieyong |date=2014-03-13 |title=Single Image Dehazing and Denoising: A Fast Variational Approach |url=http://epubs.siam.org/doi/10.1137/130919696 |journal=SIAM Journal on Imaging Sciences |language=en |volume=7 |issue=2 |pages=969–996 |doi=10.1137/130919696 |issn=1936-4954}}</ref> and [[inpainting]].<ref name=":3">{{Cite journal |last1=Allag |first1=A. |last2=Benammar |first2=A. |last3=Drai |first3=R. |last4=Boutkedjirt |first4=T. |date=2019-07-01 |title=Tomographic Image Reconstruction in the Case of Limited Number of X-Ray Projections Using Sinogram Inpainting |url=https://doi.org/10.1134/S1061830919070027 |journal=Russian Journal of Nondestructive Testing |language=en |volume=55 |issue=7 |pages=542–548 |doi=10.1134/S1061830919070027 |s2cid=203437503 |issn=1608-3385}}</ref>
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'''end do'''
{{algorithm-end}}
Moreover, the convergence of the algorithm slows down when <math>L</math>, the norm of the operator <math>K</math>, cannot be estimated easily or might be very large. Choosing proper [[Preconditioner|preconditioners]] <math>T</math> and <math>\Sigma</math>, modifying the proximal operator with the introduction of the [[Inner product space#Euclidean vector space|induced norm]] through the operators <math>T</math> and <math>\Sigma</math>, the convergence of the proposed preconditioned algorithm will be ensured.<ref>{{Cite
==Application==
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