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 o f different norms (And the Pseudo Norm $ {\ell}_{o} $) 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