total variation norm

Total variation norm is commonly used to make the visual data more local smooth. 

1)  definition

For matrices, the TV norm is defined as 

where 

 

For multidimensional data, the TV norm can be given by

2) algorithm (primal-dual gradient algorithm)

 basis:

 

the general inpainting problem with TV norm model:

 

 the dual expression of TV norm can be represened as

   

Then the inpainting problem (2) can be tackled by solving the following minmax problem.

 

 

 

 

 

 reference 

 1] Zhu M, Chan T. An efficient primal-dual hybrid gradient algorithm for total variation image restoration[J]. UCLA CAM Report, 2008, 34.

 2] Ye X, Zhou H. FAST TOTAL VARIATION WAVELET INPAINTING VIA APPROXIMATED PRIMAL-DUAL HYBRID GRADIENT ALGORITHM[J]. Inverse Problems & Imaging, 2013, 7(3).

3]  Zhang B, Zhu Z, Xu C. A primal-dual multiplier method for total variation image restoration[J]. Applied Numerical Mathematics, 2019.

来源：https://www.cnblogs.com/jianiliunier/p/11392838.html