In the past decade, a number of independent developments in mathematical analysis, in computer vision, in pattern recognition, and in statistical analysis have independently developed tools and theories which can now be seen to be closely related, as parts of an emerging area which should be called Multiscale Geometric Analysis. The purpose of this meeting is to crystallize this emerging field and to stimulate cross-disciplinary exchanges which will accelerate its formation and development.
The goal of this program is to detect, organize, represent, and manipulate data which nominally span a high-dimensional space but which contain important features which are approximately concentrated on lower-dimensional subsets (curves, sheets, etc.). This is a problem that comes up in image analysis where edges are important features, in volumetric 3-d imaging, where filaments and tubes are important, and in high-dimensional statistical data analysis where the data cloud concentrates near special low-dimensional subsets. The tools of MGA range from multiscale approximation of data in dyadic cubes by k-dimensional flats, as in Jones’ traveling salesman theorem, to multiscale radon transformation, as in beamlet analysis, to special space-frequency tilings, as in curvelet analysis. Computational tools and mathematical theories are under development, and some initial results are very impressive. For example, MGA provides:
(i) in mathematical analysis, a complete understanding of when a pointset actually lies on a lower-dimensional rectifiable set,
(ii) in approximation theory, data compression of objects with edges which achieves performance levels far beyond existing wavelet coders, and which are in fact asymptotically optimal; and
(iii) in statistical signal processing, detectors for faint curvilinear structures in noisy images which are orders of magnitude more sensitive than previously proposed methods based e.g. on edge detectors, and again are provably asymptotically optimal.
There are exciting emerging applications of these ideas in particle physics data analysis, in computer vision, and, most recently, in the analysis of massive digital astronomical catalogs. We speculate that there are many other applications to be developed, for example in materials science and medical imaging.
We believe that holding a meeting of this kind at IPAM can lead to a new synthesis of ideas and numerous valuable collaborations and initiatives.