Infimal convolution approaches for image recovery

Abstract

The quality of image captured by acquisition devices has increased drastically over the years largely due to a revolution in imaging sensor capability. But, image acquisition under low illumination continues to be a bottleneck for imaging devices such as optical microscopes leading to blurred and noisy images. A potential solution to this limitation is a computational approach known as image restoration. An image restoration algorithm recovers an estimate of the original image from a noisy blurred observation while assuming a knowledge of the image degradation model. The restoration problem is even more challenging when it comes to a spatio-temporal image as a good restoration scheme needs to be mindful of presence of motion in the measured image. This means that in spatio-temporal image restoration problem, the algorithm should ensure temporal regularity of restored image in addition to spatial regularity. Regularization based image restoration attempts to pose image restoration problem as a regularized optimization problem given the measured image. We develop methods using the concept of infimal convolution from convex analysis to design effective and efficient restoration schemes for images and spatio-temporal images. In our first work, we propose a family of derivative based regularization which we call generalized unitary invariant regularization and it belongs to a class of infimal convolution based functions. We also design an algorithmic scheme to optimize the resultant optimization problem. We demonstrate the quality of proposed algorithm and restoration scheme through multiple experiments on simulated data. iIn our second work, we develop a method for restoration of spatio-temporal images measured from TIRF uorescence microscopes where a sequence of noisy blurred images are observed over time. We once again develop an in mal convolution based approach to design a novel spatio-temporal regularizer that is tailor made for above class of spatiotemporal images. The proposed regularization is designed to ensure both spatial and temporal regularity of restored signal. The resultant regularization functional is de ned as an optimization problem where the cost is a weighted sum of two constituent functions where the two functions play the role of promoting spatial and temporal regularity respectively. We also design an algorithm to optimize the resultant restoration problem using this regularization. We demonstrate the quality of the proposed algorithm by testing the restoration quality against spatio-temporal measurements collected from TIRF fluorescence microscopes. In the third and final work we develop an algorithm to solve the problem of estimating the relative weights in spatio-temporal regularization functional designed based on infimal convolution formulation. We propose a renewed optimization model where the spatio-temporal signal is estimated together with the better quality image estimate by incorporating the weights as part of the optimization problem. We also design an iterative scheme to optimize the resultant joint optimization model. We demonstrate the effectiveness of this scheme against other joint optimization schemes for spatio-temporal signal estimation