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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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