David Mumford (Brown University), Jitendra Malik (University of California, Berkeley), Donald Geman (John Hopkins University) and David Donoho (Stanford University)

The field of image analysis is one of the newest and most active sources of inspiration for applied mathematics. While the scientific study of vision dates from the work of Helmholtz, the unfolding of its mathematical side started some 20 years ago: wavelets were invented at about this time in vision and other applications; ideas from statistical mechanics and variational calculus were related to the fundamental problem of segmenting images, i.e. finding objects, distinguishing foreground from background; and the formulation of image analysis as a problem in Bayesian statistical inference was introduced. Soon after, a series of quite novel non-linear PDE's were discovered which unified many image analysis problems.

Present day mathematical challenges in image analysis span a wide range of mathematical territory, from harmonic analysis (the development of new mathematical tools for decomposing image data into elementary units), 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; very 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.

We expect that the push and pull between image analysis and applied mathematics will remain a strong factor in the foreseeable future. From its inception, image analysis has always been one of the most interdisciplinary of fields, and so this program is open to all mathematicians, statisticians, engineers, computer scientists and life scientists interested in image analysis.

As a symbol of the many challenges remaining to be faced in this area, consider the problem of segmenting an image into objects, parts and foreground/background in the same way that humans do it. Although this has been one of main goals in image analysis for 20 years, the best current algorithms, as illustrated in the figure above from Jitendra Malik's group, produce very plausible segmentations but still lack the ability to extract and label regions in a way compatible with human image understanding.