Testing Statistical Hypotheses

TL;DR: In this paper, the authors discuss the power of systematic modeling in solving imaging problems which arise in industry, such as nondestructive testing, as well as the role of regularization.

Abstract: This coherent and well-balanced book is an important contribution to the literature on mathematical image analysis. The author has a clear purpose in mind: demonstrate the power of systematic modeling in solving imaging problems which arise in industry, such as nondestructive testing. As a result, there are few books which are this “mathematical” and this “practical” at the same time. It is this systematic effort to illustrate advanced modeling that makes this book distinct and at least as useful as many books on machine vision which are less theoretical. Conversely, whereas other books cover many of the same mathematical topics, in particular stochastic inverse problems, high-dimensional Bayesian inference, regularization, and Markov random fields, none of them to my knowledge is as well-grounded in real applications. Within the world of imaging, this is a book about so-called “mid-level” vision. Except for some material on tomographic reconstruction, it is not about image formation (i.e., sensors to images), nor is it about image understanding (i.e., images to words, as in semantic scene interpretation). Rather, it is primarily about extracting structures (e.g., curves) from images and “inverting” transformations from images to images (e.g., denoising and deconvolution) with a special emphasis on industrial applications. The Introduction is engaging and concrete. In fact, the book starts off with a rich variety of examples in order to illustrate generic difficulties, such as geometric and photometric distortions and random effects, as well as the basic mathematical concepts, especially probabilistic image models and the role of regularization. The argument for rigorous modeling is clear and convincing. Parts I and II are about spline models and Markov models, respectively; Part III, “Modeling in Action,” develops applications in detail. All the major themes, regularization, inference, parameter estimation, optimization, and stability, are illustrated in Part I in the context of one-dimensional splines. Part II is the mathematical core. Since the 1980s, it has become commonplace to treat problems such as denoising, deconvolution, and motion estimation as stochastic inverse problems within the framework of Bayesian inference. The common objective is to estimate the state of a “hidden variable” x (e.g., a noise-free picture) based on an observed variable y, usually a single image, for example, a noisy picture, a pair of images, or perhaps a sequence of images. The observations are related to x by a data or forward model (e.g., additive white noise), which is derived from appropriate physical and statistical arguments. In a Bayesian framework, the data model is the conditional distribution p(y|x) of the data given the hidden variable. Once coupled with a “prior model” p(x), namely a data-independent distribution over the hidden variable, inference proceeds by analyzing the resulting “posterior distribution” p(x|y). In particular, the model of p(x|y), or maximum a posteriori estimator, is often the final target. Regularization is achieved by carefully crafting p(x) to account for a priori information about expected geometrical patterns or statistical regularities. In this book a “model” is then a joint distribution p(x, y) over all the variables of interest, or simply an expression for the resulting posterior distribution p(x|y). In most cases, these are Gibbs distributions, and modeling boils down to specifying a “posterior energy” U(x|y), where p(x|y) ∝ exp{−U(x|y)} and the energy U(x|y) is of the form U(x, θ1) + U(y|x, θ2) where θ1 and θ2 are unknown parameters. Usually, the estimate x̂(y) is the value which minimizes this energy, equivalently, maximizes the posterior probability. All of this is discussed in Part II, as well as stochastic algorithms, which are used for estimating (θ1, θ2) and for inferring x̂. In most cases, these models are not intended to capture reality in the sense that samples from p(x) look “real,” for example, like actual radiographic images. Instead, the purpose is to encode those elements of prior knowledge which are essential to overcome the ill-conditioned nature of the inverse problem. Indeed, the connections between Bayesian inference and regularization theory is treated at length, as is the one between discrete and variational methods. Many of the models are complex, the result of “long and laborious modeling procedures.” In Part III the author demonstrates this process in detail, utilizing the methods from both earlier parts but emphasizing discrete models and combinatorial optimization. Again, this is genuinely instructive, an especially attractive aspect of the book which sets it apart from others which cover the Bayesian methodology. The driving application is nondestructive testing. The author borrows freely from his considerable industrial experience. Many general types of problems are analyzed including denoising, deblurring, scatter analysis, and the estimation of the sensitivity function. Scatter and sensitivity analysis involve the most intensive and high-dimensional modeling due to intricate degradation mechanisms. In particular, scatter is more difficult to model than noise and blur, taking into account both the absorption and scatter of photons. The sensitivity problem is handled using the techniques on splines developed in Part I. Two examples treated in more detail are detecting a family of regular curves of unknown number and length (which represent filaments appearing in highly degraded images) and reconstructing an image from radiographic projections, which is ill-posed due to the small number of projections and treated using Markov regularization. In this case, due to the absence of information from the data, the prior model plays an especially pivotal role, involving a parametric model of the shape of the object to be reconstructed. Other examples involve a two-dimensional (2D) version of electrophoresis, focal analysis for 2D seismic migration, denoising NMR images, nondestructive testing from X-rays of a metallic assembly, deblurring X-rays, the detection defects in materials, often in the form of filaments and very difficult to distinguish from artifacts, and of course, determining the internal structure of a 3D object from projections. Such problems arise in various areas of manufacturing (e.g., designing printed circuits), monitoring sensitive structures (e.g., nuclear reactors), and in medical imaging (e.g., both transmission and emission tomography). Finally, this book is easily accessible to graduate students in applied mathematics, engineering, and computer science with a standard background in linear algebra, analysis, and probability theory. No measure theory is necessary nor any previous exposure to image processing or pattern recognition. All the ideas are derived from first principles.