The underlying principle used to restore an image is known generally as a reconstruction or inverse problem. This usually involves finding a method of inverting certain classes of integral equations. In the case of a deconvolution problem, these equations are of the type where is the kernel associated with the operator . This is basically an ill-posed problem (Tikhonov and Arsenin 1977) because it does not fulfill the three Hadamard conditions of existence, uniqueness, and stability of the solution. The last condition of stability causes the main problems because if is not continuous, the solutions are unstable with respect to slight variations of initial data . This means in practice that a slight error in the data may lead to a very large error in the solution.

When the problem is discretized, its transposition into finite dimension certainly eliminates the question of continuity of , but the difficulties of the underlying infinite problem result again in numerical instabilities due to the ill-conditioned nature of the matrix of the system. As a result, the noise is enhanced and the solutions may be singular. Let be some variation of the data . Denoting by the corresponding variation of , one has , where , and being the largest and the smallest eigenvalues of the imaging operator. The condition number provides then a measure of the difficulty about the resolution of equation , and hence a measure of the robustness of the problem under consideration.

To circumvent the ill-conditionedness nature of this problem, one is led to postulate that the properties of the solution are not entirely contained in the equation to be solved. One is then led to introduce a priori information on the solution which takes a regularizing part in the deconvolution process. This information is the basis for classes of linear or non-linear methods which are known as deterministic or Bayesian ones.

Fri Apr 15 16:01:02 EDT 1994