| caption2 = Application of the Chambolle-Pock algorithm to the test image with noise.
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A typical application of this algorithm is in the image [[Noise reduction|denoising]] framework, based on total variation.<ref name=":2" />. It operates on the concept that signals containing excessive and potentially erroneous details exhibit a high total variation, which represents the integral of the absolute value gradient of the image.<ref name=":2" /> By adhering to this principle, the process aims to decrease the total variation of the signal while maintaining its similarity to the original signal, effectively eliminating unwanted details while preserving crucial features like edges. In the classical bi-dimensional discrete setting,<ref>{{Cite journal |last=Chambolle |first=Antonin |date=2004-01-01 |title=An Algorithm for Total Variation Minimization and Applications |url=https://doi.org/10.1023/B:JMIV.0000011325.36760.1e |journal=Journal of Mathematical Imaging and Vision |language=en |volume=20 |issue=1 |pages=89–97 |doi=10.1023/B:JMIV.0000011325.36760.1e |s2cid=207622122 |issn=1573-7683 |s2cid=207622122}}</ref>, consider <math>\mathcal{X} = \mathbb{R}^{NM}</math>, where an element <math> u\in\mathcal{X} </math> represents an image with the pixels values collocated in a Cartesian grid <math>N\times M</math>.<ref name=":0" />
Define the inner product on <math> \mathcal{X} </math> as<ref name=":0" />
