# Recommended Order of Performing Denoising, Deblurring and Super Resolution on an Image

In the normal workflow of image processing, there are multiple operations, increase image contrast, image denoising, image deblur and image super-resolution. What are the sequence/order of performing these different operations. Are there any overlap among these operations?

• Do you have enough information to validate an answer? – Laurent Duval Oct 3 at 20:27

If your image is modeled as an image which is noisy, blurry and heavily decimated the optimal thing to do is estimate the image given that model.

The model is well defined in @Laurent Duval's answer.
I'd remark that in most real world cases the blurring is spatially variant hence it can't be modeled by convolution (Well, it is a generalized convolution).

In practice it is too hard (Was?) to solve such things and like in most cases we take the greedy approach: Step by Step.

Since most algorithms are SNR dependent, it makes sense first to handle the noise by Denoising. Pay attention that if we assume the image is blurred we have spatially correlated noise which is more tricky to deal with.

Then I'd handle the Super Resolution / Deblurring.
Since those are closely related I'd certainly solve them in one step.
Yet again, in case it is important to do them separately, start with Deblurring then Super Resolution.

Contrast, noise, blurring and subsampling can be formulated with together in a single optimization framework. This can be dealt with, provided that loss and penalty functions are tractable, in many fashions, in an iterative fashion. So they overlap. A global model is:

$$y = D(\phi(h\ast x)+w)$$

with:

• $$y$$: observed signal
• $$x$$: initial discrete signal
• $$h$$: impulse response of convolution filter
• $$\phi$$: non-linear function (e.g. saturation)
• $$w$$: white noise
• $$D$$: decimation

It is possible to start from the least to the most linear operation, as linear yields the most constrained/structured algorithms. For one recent global attempt, with sparse assumptions: Signal Reconstruction from Sub-sampled and Nonlinearly Distorted Observations:

For dealing with sparse models, a large number of continuous approximations of the 0 penalization have been proposed. However, the most accurate ones lead to non-convex optimization problems. In this paper, by observing that many such approximations are piecewise rational functions, we show that the original optimization problem can be recast as a multivariate polynomial problem. The latter is then globally solved by using recent optimization methods which consist of building a hierarchy of convex problems. Finally, experimental results illustrate that our method always provides a global optimum of the initial problem for standard 0 approximations. This is in contrast with existing local algorithms whose results depend on the initialization.