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A simple solution for image restoration = optimizing the least squares criterion.
Calculate the image which minimizes the function J(X) :Regularization
J(X)=|| Y-HX || 2
This inverse problem is ill-posed (in the sense of Hadamard) :Reconstruction noise amplification
the solution is not unique ; it may be unstable.
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