RESUMEN
We address the problem of space-invariant image restoration when the blurring operator is not known exactly, a situation that arises regularly in practice. To account for this uncertainty, we model the point-spread function as the sum of a known deterministic component and an unknown random one. Such an approach has been studied before, but the problem of estimating the parameters of the restoration filter to our knowledge has not been addressed systematically. We propose an approach based on a Gaussian statistical assumption and derive an iterative, expectation-maximization algorithm that simultaneously restores the image and estimates the required filter parameters. We obtain two versions of the algorithm based on two different models for the statistics of the image. The computations are performed in the discrete Fourier transform domain; thus they are computationally efficient even for large images. We examine the convergence properties of the resulting estimators and evaluate their performance experimentally.
Asunto(s)
Procesamiento de Imagen Asistido por Computador , Modelos Teóricos , Algoritmos , Análisis de Fourier , Humanos , Dispersión de RadiaciónRESUMEN
In this paper, we examine the restoration problem when the point-spread function (PSF) of the degradation system is partially known. For this problem, the PSF is assumed to be the sum of a known deterministic and an unknown random component. This problem has been examined before; however, in most previous works the problem of estimating the parameters that define the restoration filters was not addressed. In this paper, two iterative algorithms that simultaneously restore the image and estimate the parameters of the restoration filter are proposed using evidence analysis (EA) within the hierarchical Bayesian framework. We show that the restoration step of the first of these algorithms is in effect almost identical to the regularized constrained total least-squares (RCTLS) filter, while the restoration step of the second is identical to the linear minimum mean square-error (LMMSE) filter for this problem. Therefore, in this paper we provide a solution to the parameter estimation problem of the RCTLS filter. We further provide an alternative approach to the expectation-maximization (EM) framework to derive a parameter estimation algorithm for the LMMSE filter. These iterative algorithms are derived in the discrete Fourier transform (DFT) domain; therefore, they are computationally efficient even for large images. Numerical experiments are presented that test and compare the proposed algorithms.
RESUMEN
In this paper, the problem of restoring an image distorted by a linear space-invariant (LSI) point-spread function (PSF) that is not exactly known is formulated as the solution of a perturbed set of linear equations. The regularized constrained total least-squares (RCTLS) method is used to solve this set of equations. Using the diagonalization properties of the discrete Fourier transform (DFT) for circulant matrices, the RCTLS estimate is computed in the DFT domain. This significantly reduces the computational cost of this approach and makes its implementation possible even for large images. An error analysis of the RCTLS estimate, based on the mean-squared-error (MSE) criterion, is performed to verify its superiority over the constrained total least-squares (CTLS) estimate. Numerical experiments for different errors in the PSF are performed to test the RCTLS estimator. Objective and visual comparisons are presented with the linear minimum mean-squared-error (LMMSE) and the regularized least-squares (RLS) estimator. Our experiments show that the RCTLS estimator reduces significantly ringing artifacts around edges as compared to the two other approaches.