I will divide my answer into 3 sections. ## The Distribution of the Derivative of Images Take a real world image, any image. Apply the derivative operator on it (Namely apply the kernel $ \left[ 1, -1 \right] $ on it. Display the histogram of the filtered image. I took this image: [![enter image description here][1]][1] The histogram I got is this: [![enter image description here][2]][2] This distribution is very similar to [Laplace Distribution][3]. Clearly, this distribution is "Sparse" namely most of the values are 0 (Or close) and very few are different. In real world we assume most of the values which are small yet not zero are actually due to the noise. ## Total Variation Optimization Problem Look at the Optimization Function: $$ \hat{x} = \arg \min_{x} {\| Hx -y \|}^{2} + \lambda {\| G x \|}_{1} $$ Where $ H $ is the Blur Operator and $ G $ is the Derivative Operator. Now, you can look at it in the Sparse sense or you can look at it as the MAP solution given a Laplace prior for the Gradient. As written above, it fits very well to real world images. ## Sparse Optimization Look at the Optimization Function: $$ \hat{x} = \arg \min_{x} {\| Hx -y \|}^{2} + \lambda {\| G x \|}_{0} $$ Where $ H $ is the Blur Operator and $ G $ is the Derivative Operator. This optimization problem clearly promote Sparse solution (With respect to the Derivative). Yet, this is a very hard problem to solve. Hence it was shown that under some circumstances the solution of the problem with $ {\ell}_{1} $ coincide with this solution. Moreover, by looking at the Unit Sphere of different norms (And the Pseudo Norm $ {\ell}_{o} $, or more correctly Cardinality Function) you can see why the lower the norm the Sparse Solution it promotes. All in all you have many point of view on the same problem. [1]: https://i.sstatic.net/hLPA3.png [2]: https://i.sstatic.net/qBbwY.png [3]: https://en.wikipedia.org/wiki/Laplace_distribution