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Abstract

The goal of this book was to introduce the reader to the recent advances in the field of uncertainty quantification and error propagation for computer vision, image processing, and image analysis that are based on partial differential equations (PDEs). Our original motivation was the observation that more and more often, measured data in the form of images are being used as part of the simulation pipeline, and that error propagation and uncertainty quantification—starting from images but then moving through the pipeline—are important to understanding the ending result of this scientific or engineering process. To analyze the image processing pipeline, we have presented concepts that enable error propagation to be formulated with a set of basic operations: the idea of stochastic images and corresponding (numerical PDE) operations on those images. From this perspective, one approach commonly used within the engineering literature is to employ Monte Carlo sampling techniques to attempt to quantify the impact of errors and variability on an engineering pipeline. Within the uncertainty quantification world, alternative approaches such as the class of generalized polynomial chaos (gPC) methods, which make assumptions about the ability to approximate the process of interest through polynomial approximations, have gained traction as a way of accelerating the convergence of the error and uncertainty quantification process in a computationally tractable way. In this book, we have shown the results of our exploration of the use of the gPC methodology for image processing and computer vision problems. We have relied on the fact that all the prerequisites of the gPC framework (e.g., finite variance, smoothness in the stochastic space, etc.) are intrinsically satisfied in the image processing context. In particular, we have shown how the generalized polynomial chaos (gPC) approach and its corresponding rules for computation (e.g., sums, products, projections on the gPC space, etc.) lead to straightforward generalizations to image processing and computer vision techniques. We have shown that the gPC methodology, when combined with image processing and computer vision techniques, provides a powerful approach to error propagation and uncertainty quantification, and to accessing system sensitivity.

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