Image Estimation and Uncertainty Quantification

Abstract

Recorded images are usually contaminated by blur and noise. The restoration of such altered images is an ill-posed problem. Even if the blur is known, the unknown noise leads to uncertainty in the restored image. The naive restoration approach fails since it contains a lot of noise at high frequencies that destroys the computed restored image. To remedy this problem, this work focuses on the computation of the restored image by using spectral filters that give weight to components of the image that are not so contaminated by noise. We use different filtering methods such as the Truncated Tikhonov, Truncated SVD, and new methods that we created here and we seek to find a near optimal choice of the filter parameter which will give the best approximation of the original image. We define and compute the Picard Parameter when the problem satisfies the Discrete Picard Condition, and with that we estimate the noise properties. Also, we develop a new method to compute the near optimal solution by using statistical analysis which also gives us a way to estimate the error of the solution, a way to quantify uncertainty.