Mathematics and Computation in Imaging Science and Information Processing

The following sections are included:

• Contents

• Foreword

• Preface

Subdivision surfaces have become a standard geometric modeling tool for a variety of applications. This survey is an introduction to subdivision algorithms for arbitrary meshes and related mathematical theory; we review the most important subdivision schemes, the theory of smoothness of subdivision surfaces, and known facts about approximation properties of subdivision bases.

In these lectures we review a few high order accurate numerical methods for solving time dependent Hamilton-Jacobi equations. We will start with a brief introduction of the Hamilton-Jacobi equations, the appearance of singularities as discontinuities in the derivatives of their solutions hence the necessity to introduce the concept of viscosity solutions, and first order monotone numerical schemes on structured and unstructured meshes to approximate such viscosity solutions, which can be proven convergent with error estimates. We then move on to discuss high order accurate methods which are based on the first order monotone schemes as building blocks. We describe the Essentially Non-Oscillatory (ENO) and Weighted Essentially Non-Oscillatory (WENO) schemes for structured meshes, and WENO schemes and Discontinuous Galerkin (DG) schemes for unstructured meshes.

To recover a sharp image from its blurry observation is the problem known as image deblurring. It frequently arises in imaging sciences and technologies, including optical, medical, and astronomical applications, and is crucial for allowing to detect important features and patterns such as those of a distant planet or some microscopic tissue.

Mathematically, image deblurring is intimately connected to backward diffusion processes (e.g., inverting the heat equation), which are notoriously unstable. As inverse problem solvers, deblurring models therefore crucially depend upon proper regularizers or conditioners that help secure stability, often at the necessary cost of losing certain high-frequency details in the original images. Such regularization techniques can ensure the existence, uniqueness, or stability of deblurred images.

The present work follows closely the general framework described in our recent monograph [18], but also contains more updated views and approaches to image deblurring, including, e.g., more discussion on stochastic signals, the Bayesian/Tikhonov approach to Wiener filtering, and the iterated-shrinkage algorithm of Daubechies et al. [30,31] for wavelet-based deblurring. The work thus contributes to the development of generic, systematic, and unified frameworks in contemporary image processing.

This paper reviews the fundamentals of the data hiding problem. Data hiding can be viewed as a game between two teams (embedder/decoder vs attacker), and optimal data-hiding and attack strategies may be developed in this context. This paper presents a framework for developing such strategies as well as practical codes. The theory is applied to image watermarking examples.

In the last few years, we have seen many new and powerful steganography and steganalysis techniques reported in the literature. In the following tutorial we go over some general concepts and ideas that apply to steganography and steganalysis. We review and discuss the notions of steganographic security and capacity. Some of the more recent image steganography and steganalysis techniques are analyzed with this perspective, and their contributions are highlighted.

Association rules are “if-then rules” with two measures which quantify the support and confidence of the rule for a given data set. Having their origin in market basket analysis, association rules are now one of the most popular tools in data mining. This popularity is to a large part due to the availability of efficient algorithms. The first and arguably most influential algorithm for efficient association rule discovery is Apriori.

In the following we will review basic concepts of association rule discovery including support, confidence, the apriori property, constraints and parallel algorithms. The core consists of a review of the most important algorithms for association rule discovery. Some familiarity with concepts like predicates, probability, expectation and random variables is assumed.