August 18, 2002
Publications in Mathematics and Shape by David Mumford, Ralph Teixeira, et al. :
BibTeX references.
Conglin Lu, Yan Cao, David Mumford
Vol. 13, No. 1/2, March/June 2002, pp. 65-81.
In many areas of computer vision, such as multiscale analysis and shape description, an image or surface is smoothed by a nonlinear parabolic partial differential equation to eliminate noise and to reveal the large global features. An ideal flow, or smoothing process, should not create new features. In this paper we describe in detail the effect of a number of flows on surfaces on the parabolic curves, the ridge curves, and umbilic points. In particular we look at the mean curvature flow and the two principal curvature flows. Our calculations show that two principal curvature flows never create parabolic and ridge curves of the same type as the flow, but no flow is found capable of simultaneously smoothing out all features. In fact, we find that the principal curvature flows in some cases create a highly degenerate type of umbilic. We illustrate the effect of these flows by an example of a 3-D face evolving under principal curvature flows.
Ralph Costa Teixeira
Vol. 13, No. 1/2, March/June 2002, pp. 135-155.
What happens to the medial axis of a curve that evolves through MCM (mean curvature motion)? We explore some theoretical results regarding properties of both medial axes and curvature motions. Specifically, we present a set of conditions on the local validity of a medial axis transform and a differential equation for the change of smooth parts of the medial axis when its generating curve evolves under MCM. Finally, we also introduce a differential equation that describes the evolution of the distance transform of a curve under MCM. A companion article will use singularity theory to classify all generic changes in the medial axis of a curve evolving through MCM.
P. W. Hallinan, G. Gordon, A.L. Yuille, P. Giblin, D. Mumford
A.K.Peters, 1999. 270 pages.
The human face is perhaps the most familiar and easily recognized object in the world, yet both its three-dimensional shape and its two-dimensional images are complex and hard to characterize. This book develops the vocabulary of ridges and parabolic curves, of illumination eigenfaces and elastic warpings for describing the perceptually salient features of a face and its images. The book also explores the underlying mathematics and applies these mathematical techniques to the computer vision problem of face recognition, using both optical and range images.
PhD thesis, Harvard University, Cambridge, Massachusetts, June 1998, 139 pages.
Advisor: David Mumford
in "Proc. 1st European Congress of Mathematics", Birkhauser-Boston, 1994,
and in revised form in "Perception as Bayesian Inference",
edited by D.Knill and W.Richards, Cambridge Univ. Press, pp. 25-62,
1996.
in "Proc. Geometric Methods in Computer Vision Conference",
Soc. Photo-optical & Ind. Engineers (SPIE), vol. 1570, 1991, pp.
2-10.
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