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PREP Workshop: The Mathematics of Images March 14, 2005 - March 18, 2005
Registration Deadline: March 01, 2005 over 19 years ago
To apply for Funding you must register by: December 14, 2004 over 19 years ago
Parent Program:
Organizers Kathryn Leonard , David Mumford

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This workshop is aimed at faculty who wish to learn about this exciting field and would like to enrich a variety of undergraduate courses with new examples and applications. The workshop is being held in collaboration with the Mathematical Association of America as part of the MAA's Professional Enhancement Program (PREP). See the PREP website for information about registration and participant support. Note that the application deadline is February 1, 2005. Present day mathematical challenges in image analysis span a wide range of mathematical territory, including harmonic analysis (the development of new mathematical tools for decomposing image data into elementary units), variational calculus and PDE’s (segmenting images into regions representing distinct objects), geometry (understanding the geometry within an image as well as understanding the geometry of spaces arising from images), statistical learning theory (learning from empirical data about the underlying basic components of image data) and the search for new stochastic models, e.g. of 'shape' and of grammars. There are even important technical challenges; recently, the problem of image 'inpainting' (filling-in missing information in images) has stimulated demand for existence and solution theory for certain high-order nonlinear PDE.s. In this workshop three themes will be emphasized: the need for statistics in making inferences about the patterns and objects present in images; the use of partial differential equations in image enhancement algorithms; and the problems of describing ‘shape’ and using shape for object recognition. As lead-in to the program, participants will be required to learn and work with Matlab, and will be provided with images and specific problems to play with, in order to get some experience with the basic tools of the subject. There will be approximately 18 lectures during the five days of the workshop. Each lecture will be a modified form of a lecture that could be delivered in an undergraduate classroom. In some lectures we will rely more on the background knowledge of the participants and provide material which can be adapted to an undergraduate lecture. Interspersed with these lectures will be discussions. Participants will be divided into small discussion groups of three or four, based on their interests. These groups will meet periodically to work on problems, and to discuss how various aspects of image analysis fit into courses in the undergraduate curriculum and into undergraduate research projects related to their own work. On the last day of the workshop, each participant will present these ideas to the rest of the group. More information at: http://www.maa.org/prep/2005/ Outline of PREP workshop Revised 2/18/05 Revised Schedule (2/18/05) Changes are in blue Monday 9:30-11:30 D.Mumford - What is the field of vision and what is its mathematical side? Outline of topics to be presented. Round table introductions and forming working groups 11:30-1:30 Lunch provided at MSRI sixth floor, time for discussion (catered lunch served at 12:00) 1:30-3:00 K.Leonard - Images as functions I: Edges and segmentation, filters, Fourier expansions and wavelets. Part lecture, part workshop with MatLab. Introducing two vision competitions, one with texture, one with shape. 3:00-3:30 Tea Break (sixth floor MSRI) 3:30-5:00 R.Romano - The geometry of imaging: multiple cameras, inferring 3D structure with stereoscopic images 5:00-7:30 Dinner Break 7:30-9:00 Computer Lab at MSRI open Tuesday 9:30-11:00 D.Mumford - Statistics as a tool for classifying images. Features and histograms, 3 fundamental tools in statistical pattern recognition (for use in the competition) 11:30-1:30 Lunch provided at MSRI sixth floor, time for discussion (catered lunch served at 12:00) 1:30-3:00 J.Carter - Images as functions II: evolving an image with _‘heat-type’ PDE’s and applications to image enhancement: Part lecture, part workshop with MatLab 3:00-3:30 Tea Break (sixth floor MSRI) 3:30-5:00 K.Leonard - The geometry inside images. The set of shapes as a metric space. 5:00-7:30 Group Dinner (catered or restaurant TBD) 7:30-9:00 Computer Lab at MSRI open Wednesday 9:30-11:00 TBA - Graph theory and linear algebra as tools for image analysis. Use of the eigenvectors of the discrete graph Laplacian to segment an image AFTERNOON EXCURSION to the San Francisco EXPLORATORIUM 11:00-12:30 Travel to SF 12:30-1:30 Pick up your lunch at the Exploratorium food court 1:00-?? Play! Thursday 9:30-11:00 M.Harrison - More statistical methods, Bayes’s rule, likelihood ratios, inference 11:30-1:30 Lunch provided at MSRI sixth floor, time for discussion (catered lunch served at 12:00) 1:30-3:00 TBA - Shrink wrapping shapes, the variational approach known as ‘snakes’. 3:00-3:30 Tea Break (sixth floor MSRI) 3:30-5:30 Free time to work on course material and the competition. 5:30-7:30 Dinner Break 7:30-9:00 Computer Lab at MSRI open Friday 9:30-11:00 D.Donoho - Information theory, signals and images, coding and counting bits 11:30-1:30 Lunch provided at MSRI sixth floor, time for discussion (catered lunch served at 12:00) 1:30-3:00 D.Mumford/K.Leonard - Idea of a diffeomorphism, ‘shortest path’ from one image to another, application to stereo 3D reconstruction and to morphing 3:30-5:30 General discussion of using images and matlab in courses, announce winners of competition Registration: Please register on-line at the MAA website. PREP workshop on the Mathematics of Images, March 14-18 David Mumford and Kathryn Leonard Expectations and Goals The primary goal of the PREP workshop is to introduce ways in which the exciting field of Image Analysis fits into a myriad of undergraduate courses. In pursuit of that goal, we also hope to build familiarity with the programming environment Matlab, and to give a flavor of current areas of research in Image Analysis. The workshop will be organized around an image recognition challenge. Each day will consist of a few lectures and a hands-on activity period. The lectures are intended to explain techniques of image analysis and to point out specific mathematics used in image analysis that naturally fit into particular undergraduate classes. The activity periods are intended to demonstrate computational activities to be included in undergraduate class-work, as well as to build skills that will help participants compete in the challenge. Before the workshop, we expect participants to familiarize themselves with basic Matlab commands (see Matlab Guide below), to solve a few illustrative Matlab problems, and to read over some introductory texts on Image Processing (see Suggested Readings). During the workshop, participants should anticipate devoting a reasonable amount of time to the image recognition challenge and related Matlab projects. The computer lab will be available after hours to facilitate that work. On the afternoon of final day of the workshop, participants will discuss what they have learned and ways to incorporate that knowledge into undergraduate classes. At the conclusion of the discussion, we will announce the winners of the recognition challenge and analyze the outcomes of the various approaches to the challenge. We hope to produce an “Image Analysis Activity Book” based on the activities of the workshop and the discussion on the final day. Matlab Guide A wonderful tutorial for basic Matlab commands is: http://mathcs.holycross.edu/~spl/MATLAB. A site that explains how to load and save images is: http://amath.colorado.edu/courses/4720/2000Spr/Labs/Worksheets/Matlab_tutorial/matlabimpr.html Some other commands that we recommend exploring (for instance, through the Matlab Help site) are: quit rand use of logical operators to produce binary strings. find sort more (on, off) If you are interested in playing around with image processing techniques, browsing the image processing toolbox (if you have access to it) is a great way to begin to understand the various ways images can be analyzed and manipulated. Googling “matlab tutorial” will return many sites, some of which will be more helpful than others. For those of you used to other languages, the biggest surprise is going to be that you can almost always avoid loops, using the built-in parallelism of matrix operations. For speed, you must in fact do this. Thus you can sum 100,000 terms of the harmonic series without a for loop as sum(1./(1:100000)) Some MatLab problems: 1. How about calculating pi? Take some standard series from calculus and sum as above. How many decimal places can you get before limitations of time or space stop you? Another way is to use random numbers: rand(1,n) will give you a row vector of n random numbers between 0 and 1. Use 2n random numbers like this to get n random points in the unit square and count how many fall in the positive quadrant of the unit circle! 2. A nice dataset to play with is “topo”, which you get via “load topo”. It is a 180 by 360 matrix of the mean altitude (or depth) of the land (or sea) at the patch of the earth given by that latitude and longitude. It’s fun to do “imagesc(topo)” (you can play with the color map too) or “surf(topo)” (sometimes surf(topo, ‘EdgeColor’, ‘none’) is better to suppress the grid lines). But here’s a multi-variable calculus problem: calculate the volume of the land above sea level and the volume of the water below sea level. What is their ratio? (Hint: ‘.*’ should be used to bring in the jacobian factor.) Another fun thing is to do ‘oceans = topo < 0’ and ‘hist(oceans)’. 3. We have attached 3 pdf’s made with MatLab like this: we took images but considered them as functions of 2 variables and plotted their graphs. Thus white areas are elevated, black depressed. The 1st 2 are our faces and the 3rd is a mystery object. Which face is whose? Can you guess anything about our facial features? What is the mystery object? This is not a MatLab question but is meant to start you thinking of images as functions. Suggested Readings The best book on computer vision may be: Computer Vision: A Modern Approach by David A. Forsyth, Jean Ponce An excellent introduction to the geometry of imaging is http://www.robots.ox.ac.uk/~vgg/hzbook/hzbook2/HZintroduction.pdf
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC