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Reconstruction - Regularization
Least squares
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

    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

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    André Jalobeanu - 24 Aug 1998