A simple solution for image restoration = optimizing the least squares criterion.

Calculate the image  which minimizes the function J(X) :
  J(X)=|| Y-HX || ²  

This inverse problem is ill-posed (in the sense of Hadamard) :

the solution is not unique ;

it may be unstable.

Reconstruction  noise amplification

Small variations of  the observed image Y  high variations of the reconstructed solution X.
 
 

Example (synthetic image) :

Original image	Corrupted image	Reconstructed solution

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Calculate the image which minimizes the energy U(X) :

 
 =  data-dependent term (Y=data)

   =  regularization term, which penalizes noisy solutions

,  = hyperparameters of the model

gradients of X = differences between neighbour pixels 

= Phi-function, it takes into account the constraints imposed on the solution