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What is the notion of total variation and how is total variation norm calculated in an image?

More exactly, I want to calculate and understand the meaning of $ \left \|X \right \|_{TV} $ if $X$ is an image.

If I optimize: $$ min_{X} \left \|X \right \|_{TV} + {other . terms} $$ how will $X$ look like and what properties would be exaggerated in image, $X$ ?

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The Total Variation of an image $ I $ can be calculated in one of 3 methods (See The Meaning of the Terms Isotropic and Anisotropic in the Total Variation Framework):

  • Anisotropic TV - $ \operatorname{TV} \left( I \right) = \sum_{x} \| \nabla I (x) \|_1 $.
  • Isotropic TV - $ \operatorname{TV} \left( I \right) = \sum_{x} \| \nabla I (x) \|_2 $.
  • Isotropic Squared TV - $ \operatorname{TV} \left( I \right) = \sum_{x} \| \nabla I (x) \|_2^2 $.

In practice, all yield almost the same result.

The Total Variation is a measure of how close the image is to a Piece Wise Constant function. Namely, How sparse are the changes in the gradient of the image.

You may read more at Total Variation Denoising.

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