# The Differences between Super Resolution and Denoising and Deblurring

In the field of computer vision/image processing, what are the differences between super-resolution and de-noising/de-blurring? Thanks.

All 3 of them fall into the category of Inverse Problems in the Image Processing world.

Lets assume a Linear Model and then we will be able to show all 3 of them as parameters of the same framework. Then the differences will be clear and one could generalize it into Non Linear settings as well.

Have a look on the following model:

$$y \approx A x + w$$

Where $x$ is the image we want to infer from the given image $y$, $A$ is the degradation operator applied on $x$ and $w$ is white additive noise.

1. Denoising
For denoising only noise is added hence $A = I$ and we left with estimating $x$ from a noisy measurements.
2. Deblurring
In that case $A$ is a matrix form of some Low Pass Filter (Circulant Square Matrix) which applies a blur on the image. We try to infer $x$ from a blurred and noisy version of it given by $y$.
3. Super Resolution
In this case take $A$ from (2) and remove few rows of it. Namely we get only some of the data in $x$ and the data is both blurred and noisy and we try to infer the whole data of $x$ given $y$.

Usually the problems above aren't well defined which means they require some kind of regularization.

Have a look on the following problem:

$$\arg \min_{x} \frac{1}{2} {\left\| A x - y \right\|}_{2}^{2} + \lambda r \left( x \right)$$

This is the Linear Inverse problem where you can adapt to any of the cases above. The $r \left( \cdot \right)$ is the regularization function which means we have some model on the data. For instance for Images many use the Total Variation model which means the Image is Piece Wise Smooth (Sparse Derivatives).