What Does the Total Variation Norm Mean in the Context of Image Processing

Question

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, $X$ ?

Answer 1

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.