Image Estimation and Uncertainty Quantification

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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.