Course outline:

As interesting mathematical techniques used in image processing are in itself -- the goal is to solve application problems related to image data effectively and efficiently.
The aim of this introductory course is to give an introduction to the topic and to provide an application perspective, without abandoning the reference to mathematics.

• Introduction
  • What is an image?
  • The reasonable value of images in natural sciences
  • The universe of image sciences
  • Image-based research and knowledge discovery
  • Mathematical tasks in image sciences: overview and classification
• Overview on image acquisition and reconstruction techniques
  • Imaging Modalities - from subatomic to cosmic length scale
  • Tomographic techniques
  • Holographic techniques
  • Diffraction limited techniques & super-resolution techniques
  • Current trends in imaging and math-related research topics
• X-ray and electron tomography - an exemplary modality and its success story:
  • Principles and practical realization
  • Related image processing and analysis tasks
  • Applications in medicine, material science, nano-technology and semi-conductor industry
  • Applications in biological research: from molecular structure elucidation to depiction of biological functional units in action
  • Math-related research topics
• Introduction to image segmentation
  • From visual perception to the image segmentation task
  • Attempts to grasp the image segmentation task mathematically
  • Practically used image segmentation techniques; strengths and deficiencies, mathematical classification
  • Spatiotemporal image segmentation and tracking
  • Current trends and math-related research topics
• Geometrical shapes from images
  • Mathematical representations of shapes
  • Geometry reconstruction from images
  • Shape spaces and statistical shape models (SSM)
  • Application: SSM as priors in image segmentation and in tomographic reconstruction
  • Application: Simulation based surgery planning
• Computer generated images
  • Computer Graphics in 45 minutes: mathematical models, major techniques
  • Data Visualization in 45 minutes: concepts, major techniques
  • Challenges and math-related research topics

Course outline:

This lecture series focusses on the application of computational harmonic analysis to imaging science. The key idea is to sparsely represent an image by a specifically chosen representation system from this area, and use this representation for regularization of diverse inverse problems in imaging science such as inpainting.

We will start our tour with an introduction to wavelet theory, which studies properties of wavelet systems as maybe one of the most prominent systems from computational harmonic analysis, also discussing the success of wavelets for image compression. Since however, many imaging problems are governed by anisotropic structures such as sharp edges, the need arose to design representation systems which deliver sparse representations for such structures.

This will lead us to shearlet systems (see also http://www.shearlab.org/), which are the most widely used anisotropic representation systems today. Main reasons for this are their optimal sparse approximation properties within a model situation in combination with their unified treatment of the continuum and digital realm, leading to faithful implementations. An additional advantage is the availability of stable compactly supported systems for high spatial localization. After an introduction, we will discuss in detail the theoretical foundation of shearlets.

In a last part, we will focus on the application of shearlet systems to diverse inverse problems from imaging science. Exemplary inverse problems which we will solve using shearlets for regularization are inpainting, magnetic resonance imaging, and the inverse scattering problem. Each application will be studied both theoretically and numerically.

Course outline:

Numerous image processing tasks are defined as the solution of an optimization problem. The optimization problem (often called an objective) accounts for the model of the recording process and for the expected or desired features of the sought-after image. Usual approaches to construct an objective are Bayesian statistics, PDE's, calculus of variations, and regularisation. In spite of their philosophical differences, they lead to quite similar objective functionals. Essentially, they amount to a weighted combination of a data-fidelity term and a (possibly adaptive) regularization term. Challenging theories has established bridges between disparate methods based on optimization, diffusion and frame representations. These results gave rise various practical ramifications of the objective functionals used in imaging sciences. However, the usual approaches to formulate an objective yield solutions whose features are hard to control.

This short course presents a systematic approach to the problem of the choice of pertinent objective functionals for image processing. To this end, the focus is on the properties of the optimal solutions of an objective as an implicit function of both the data and the shape of the objective itself. This point of view leads to an intrinsic relationship between modelling and conception of relevant optimization problems. It provides a framework to unify the theory on optimization-based methods and to address rigorously the problem of the choice of objective functionals for image processing.

The goals of this course are the following:

• to understand the practical issues governing the proper choice of an objective for image processing;
• to show how to conceive objective functionals in such a way that their minimizers exhibit some desired or expected properties;
• to provide a systematic way to compare existing objectives for image processing.

This course is based on a series of analytical results which characterize some salient properties exhibited by the optimal solutions of different families of objectives. Points of interest are, e.g., the recovery of edges and of regions that are homogeneous or sparse over some manifolds, the processing of textures, under different observations systems and noises. These are shown to be determined by some attributes of the objective functional relevant to its shape -- (non)smoothness or (non)convexity, or more specific attributes. Numerical examples are used to illustrate the theory and stability results are provided. Applications where specific objectives are conceived using the mathematical results on optimal solutions are presented.

By way of conclusion, open questions ranging from concepts to practical imaging problems are discussed.

Course outline:

This short course discusses a class of algorithmic framework, called Coordinate Update (CU), which is useful for solving problems involving large-scale datasets and/or high-dimensional variables. CU algorithms decompose a problem into simple, scalable subproblems. These simple subproblems are formed by fixing (treating as constants) all but a small block of coordinates. CU algorithms can solve problems with linear and nonlinear mappings, smooth and nonsmooth functions, as well as convex and nonconvex problems. In addition, they are easy to parallelize. CU algorithms (such as the Gauss-Seidel and alternating projection/minimization algorithms) have existed since the early development of numerical analysis and optimization. However, their key developments since the 1980s are largely ignored by graduate textbooks. Recently, modern applications in image processing, signal processing, and large-scale computational statistics have yielded new problems well suited for CU methods.

The great performance of coordinate update algorithms depends on smart coordinate selection, solving simple subproblems, as well as efficient (sequential or parallel) implementations. This short course addresses these points from both theoretical and practical perspectives.

Theoretically, the short course covers the basic algorithms of the CU method, demonstrates how to derive CU algorithms for a set of imaging problems, and reviews their convergence guarantees. Topics such as operator splitting, primal-dual decomposition, and async-parallel computing will be also discussed.

Practically, the short course provides the students with hands-on experience of implementing single-threaded CU applications in Matlab and multithreaded ones in C++. The basics of parallel programming will be reviewed. Finally, async-parallel computing on multiple-core platforms is demonstrated with the ARock software package, an asynchronous coding framework for coordinate update written in C++.

Course outline:

This lecture series addresses the important task of image segmentation, i.e., the automated detection of (in some sense) homogeneous image features. It will pursue two different paradigms:

• Edge detector based segmentation;
• Segmentation based on the Mumford-Shah model.

Each of these models results in a shape optimization problems with the contour of the unknown (and to be segmented) features as the unknowns. In addition, the Mumford-Shah model simultaneously also tries to recover the image. The lecture series will therefore also address the mathematical foundations:

• Variational models for segmentation;
• Shape sensitivity calculus (in a nutshell);
• Shape gradient and shape Newton schemes;
• Level set methods for numerical realization.

For the specific case of piecewise constant a priori information, so-called

• topological sensitivities

will be discussed. They allow for a user-free initialization of the, e.g., Mumford-Shah based segmentation.
Finally, all these techniques are combined and utilized when simultaneously recovering coil sensitivities in magnetic resonance imaging. It is known from the Biot-Savart law that image contrast decays with an increasing distance to the coil. This usually complicates image segmentation due to reduced data quality. Accordingly, the lecture series will end with a

• variational approach to simultaneous recovery, segmentation and coil sensitivity estimation.

Course outline:

In the lecture series, we discuss imaging from the perspective of inverse problems, especially variational regularization. Formally, in this approach, the imaging task is formulated as a suitable optimization problem, where the objective function is a weighted combination of the data fidelity term and one or multiple penalty terms. The penalty describes the desired qualitative properties. In recent years, variational regularization has witnessed significant progress in modeling, theory, numerics and applications. We plan to cover some of these advances. The lecture consists of the following two parts, each with approximately three lectures:

1. Bayesian inversion
The Bayesian framework provides a principled yet versatile framework for developing image reconstruction algorithms, including variational type methods. In this part, we will discuss fundamentals of the Bayesian framework and computational tools, including statistical modeling (noise statistics, prior model, maximum a posterior estimator), Monte Carlo simulation, approximate inference methods (variational Bayesian and expectation propagation).

2. Variational regularization
Variational regularization in imaging often leads to complicated optimization problems. In this second part, we shall discuss important theoretical and practical issues for general (not necessary convex) penalties, e.g., existence, consistency, discretization, parameter choice and fast solvers. Throughout the lectures, the ideas will be illustrated with examples from linear and nonlinear imaging problems, and many ongoing research questions will also be discussed.

Course Outline:

The course addresses the important problem of image registration (aka co-registration, fusion, matching, morphing, warping). Given two or more perspectives of a scene, the objective is to automatically determine correspondences between geometrical points in the scene. In particular, in medical imaging this problem is ineluctable in applications such as the fusion of information acquired from different imaging devices like PET and CT, motion correction for image formation or longitudinal studies of the development of diseases.
This lecture series focuses on variational image registration, where the desired solution is characterized as a minimizer of an application-specific energy functional. In this course, important specification of this class of functionals (data-fit and regularization) and numerical approaches to compute solutions will be discussed.

1. Introduction
Brief outline, introduction and overview on the topics of the lectures, references
2. Applications
Based on a variety of medical applications, the richness and the challenges of image registration are demonstrated
3. Medical images
A brief introduction to various imaging modalities (X-ray, CT, MR, US, SPECT, PET) is given and the particularities of medical images are highlighted
4. Mathematical models for images
A mathematical model for images is presented. This lecture covers interpolation and discretization, grids, scale, resolution, derivatives
5. Transformation models
Euler and Lagrange models for image deformations are considered
6. Image distance measures
Important options such as feature based (landmarks, moments, localizer, etc.) surface or point based (point clouds, level sets, etc.) , volumetric measures (norms, correlation, gradient fields), and information theory based measures (mutual information) are introduced and discussed
7. Regularization
The necessity of regularization is explained and various options of regularization such as diffusion, linear elasticity, curvature and hyper-elasticity are discussed
8. Numerical Methods
The optimize-then-discretize and the discretize-then-optimize approaches are explained and discussed. Different concepts such as a hierarchy of discretization and convexification are outlined
9. Constrained image registration
If time permits, an overview on constrained image registration is given, focusing on point, volumetric, and local rigidity constraints

Course outline:

This lecture addresses state-of-the-art continuous convex and non-convex optimization methods for image processing, computer vision, and machine learning. We will discuss the whole design chain starting from the continuous variational model, its discretization and efficient optimization.

A short outline of the course is as follows:

• Typical optimization problems in imaging
• Basic notions of convexity
• Gradient methods
• Saddle-point methods
• Non-convex optimization
• Applications

From a theoretical point of view we will mainly address the convergence properties of the optimization algorithms and the main theoretical properties of the variational models.

From a practical point of view, we will focus of important applications including:

• Image restoration
• Image deconvolution
• Image segmentation
• Optical Flow
• MRI reconstruction
• Curvature minimization
• Learning of sparse models
• Classification

For some of these models (as time permits) we will develop and implement (in Octave) suitable optimization algorithms.