Nonlinear total variation based noise removal algorithms.

*(English)*Zbl 0780.49028Summary: A constrained optimization type of numerical algorithm for removing noise from images is presented. The total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lagrange multipliers. The solution is obtained using the gradient-projection method. This amounts to solving a time dependent partial differential equation on a manifold determined by the constraints. As \(t\to\infty\) the solution converges to a steady state which is the denoised image. The numerical algorithm is simple and relatively fast. The results appear to be state-of-the-art for very noisy images. The method is noninvasive, yielding sharp edges in the image. The technique could be interpreted as a first step of moving each level set of the image normal to itself with velocity equal to the curvature of the level set divided by the magnitude of the gradient of the image, and a second step which projects the image back onto the constraint set.

##### MSC:

49N70 | Differential games and control |

49N75 | Pursuit and evasion games |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

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\textit{L. I. Rudin} et al., Physica D 60, No. 1--4, 259--268 (1992; Zbl 0780.49028)

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