“Image Processing” 

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

A graduate course in analysis and one in linear algebra plus some programing experience are strongly recommended.

The main activity of the Graduate Summer School will be a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

COURSE TITLES AND DESCRIPTIONS:

The 2010 Summer Session in “Image Processing” will consist of nine graduate level lecture series. On any day during the summer session, three lectures will be offered. Graduate students are asked to attend the lectures as well as two daily problem sessions associated with the lecture and led by a graduate TA.

Richard Baraniuk, Rice University

Compressive Sensing: Sparsity-Based Signal Acquisition and Processing

Sensors, imaging systems, and communication networks are under increasing pressure to accommodate ever larger and higher-dimensional data sets; ever faster capture, sampling, and processing rates; ever lower power consumption; communication over ever more difficult channels; and radically new sensing modalities. The foundation of today’s digital data acquisition systems is the Shannon/Nyquist sampling theorem, which asserts that to avoid losing information when digitizing a signal or image, one must sample at least two times faster than the signal’s bandwidth, at the so-called Nyquist rate. Unfortunately, the physical limitations of current sensing systems combined with inherently high Nyquist rates impose a performance brick wall to a large class of important and emerging applications.

This lecture will overview some of the recent progress on compressive sensing, a new approach to data acquisition in which analog signals are digitized not via uniform sampling but via measurements using more general, even random, test functions. In stark contrast with conventional wisdom, the new theory asserts that one can combine “sub-Nyquist-rate sampling” with digital computational power for efficient and accurate signal acquisition. The implications of compressive sensing are promising for many applications and enable the design of new kinds of analog-to-digital converters; radio receivers, communication systems, and networks; cameras and imaging systems, and sensor networks.

Antonin Chambolle, École Polytechnique

Total-Variation based image reconstruction.

In the introduction we will recall the reason for which the Total Variation (TV) was introduced as a powerful tool for image recovery. The focus of the first lectures will be mostly on theoretical aspects. The definition and essential properties of the TV will be detailed, variational problems involving the related perimeter functional will also be considered. Then, we will study the “Rudin-Osher-Fatemi” problem (from a convex analysis point of view, the proximal operator associated to the TV). We will try to analyse some interesting properties of the solutions, including regularity issues.

A second part of the lectures will address algorithmic issues and describe the standard and less standard numerical methods for solving efficiently TV-like problems. In a last lecture, we will discuss original extensions which involve TV-like functionals.

Michael Elad, Israel Institute of Technology

Sparse & Redundant Representations – From Theory to Applications in Image Processing

Modeling natural image content is key in image processing. Armed with a proper model, one can handle various tasks such as denoising, restoration, separation, interpolation and extrapolation, compression, sampling, analysis and synthesis, detection, recognition, and more. Indeed, a careful study of the image processing literature reveals that there is an evolution of such models and their use in applications.

This short-course is all about one such model, which I call Sparse-Land for brevity. This specific model is intriguing and fascinating because of the beauty of its theoretical foundations, the superior performance it leads to in various applications, its universality and flexibility in serving various data sources, and its unified view, which makes all the above processing tasks clear and simple. In this course we shall starts with the mathematical foundations of this model, and then turn to present several image processing applications, where it is shown to lead to state-of-the-art results.

Anna Gilbert, University of Michigan

A survey of sparse approximation

The past 10 years have seen a confluence of research in sparse approximation amongst computer science, mathematics, and electrical engineering. Sparse approximation encompasses a large number of mathematical, algorithmic, and signal processing problems which all attempt to balance the size of a (linear) representation of data and the fidelity of that representation. I will discuss several of the basic algorithmic problems and their solutions, including connections to streaming algorithms and compressive sensing.

Yann LeCun, New York University

Learning Image Feature

Image recognition systems have traditionally consisted of a hard-wired feature extractor, followed by a trainable supervised classifier. One could argue that the next challenge of computer vision, machine learning, and image processing, is to devise methods that can automatically learn the feature extractor from labeled and unlabeled data. a particularly relevant task for image recognition is the so-called “Deep Learning” problem, whose object is to learn hierarchies of features with multiple levels of abstraction, and suitable invariances.

I will discuss the foundations of unsupervised learning in the probabilistic and non-probabilistic (energy-based) frameworks, and will present several feature-learning models, including sparse coding, sparse auto-encoders, denoising auto-encoders, and Restricted Boltzmann Machines. I will also discuss supervised and semi-supervised methods for deep learning, including metric learning criteria. I will then discuss specific model architectures for image recognition, based on stacks on non-linear filter banks. A number of applications to object dectection, object recognition, and vision-based navigation for mobile robots will be described, and some will be demonstrated live.

Guillermo Sapiro, University of Minnesota

Dictionary learning for efficient signal representation

The goal of this class is to introduce participants to the area of dictionary learning for sparse representations. We will cover fundamental theory and computational algorithms as well as applications in signal processing. Topics we plan to cover include:

• Sparse coding (OMP, LASSO, group LASSO, LARS, soft thresholding, etc).
• Metric learning and k-flats.
• Dictionary learning (MOD, K-SVD, on-line learning, etc).
• Data as dictionaries and sparse graphs.

Zuowei Shen, National University of Singapore

Wavelet and Wavelet Frames in Imaging Science

From the beginning of sciences, visual observations have played major roles. With the rapid progress in video and computer technology, computers have become powerful enough to process image data. As a result, image processing techniques are now applied to virtually all natural sciences and technical disciplines.

Mathematical analysis makes image processing algorithms predictable, accurate and, in some cases, optimal. New mathematical methods often result in novel approaches that can solve previously intractable problems or that are much faster or more accurate than previous approaches. The speed up that can be gained by fast algorithm is considerable. Fast algorithms make many image processing techniques applicable and reduce the hardware cost considerably.

Wavelet and Wavelet frame based methods are relatively new mathematical tools that allow us to quickly manipulate images, for example, high-resolution image reconstructions in some applications, or image compressions in other applications. The wavelet or wavelet frame algorithms decompose and arrange an image data into strata reflecting their relative importance. This allows a rapid access to good coarse resolution of the image while retaining the flexibility for increasingly fine representations. It leads to algorithms that give sparse and accurate representations of image and medical image for efficient computation, analysis, storage, restorations and communication. In this series talk, I will illustrate how the wavelet theory is developed and applied to various applications in image processing which includes deblurring, denoising, inpainting and image decomposition.

Joseph M. Teran, UCLA

Numerical Methods for Elasticity Problems in Biomechanics

Elasticity plays a fundamental role in many biomechanics and computer graphics related problems. I will talk about numerical methods used for simulating elastic phenomena in soft tissues, skeletal muscles as well as techniques inspired by imaging for determining elastic constitutive models. Specifically, I will discuss common difficulties encountered and my recent algorithm developments to address them. I will put specific emphasis on applications in computer graphics based special effects for virtual characters. Topics include robust resolution of: severe nonlinearity, near incompressibility, extremely large deformation, collision/contact and solid/fluid coupling. I will also discuss the impact of multi-core accelerated computation for such problems and the potentially revolutionary applications it will admit.

Ross Whitaker, University of Utah

Statistical Models and Methods in Image Analysis

These lectures cover a series of topics on the application of statistical models to several important problems in image analysis. The focus is on several types of models from statistics and machine learning, including parametric and nonparametric techniques, and strategies for developing these models from image data. We will also discuss the applications of these models including estimates of measurement error and prior information. Applications will include statistics of image neighborhoods for feature detection, denoising and segmentation, models of solid objects for statistical shape analysis, and nonlinear manifold learning for analyzing characteristics of large image databases.