Professor Robert Hardt
W.L. MOODY PROFESSOR OF MATHEMATICS, RICE UNIVERSITY
Some New Uses of Functions with Finite Total Variation
1) Estimate the length of a continuous curve using drafting dividers.
2) Enhance an image by sharpening edges and smoothing roughness.
3) Find a spanning surface of least area in a very singular space X.
These 3 problems can all be attacked using the notion of the Total Variation (TV) of a function.
For 1) the usual formula for the length of a curve f :[a, b] -+ RN is the definition of TV( f).
For 2) one considers a two-variable function g :[a, b] x [c, d] -+ [0, 1] giving the grayscale intensity of an image in the rectangle [a, b] x [c, d]. Assuming TV(g), suitably defined, is finite, edges and roughness are described using different parts of the derivative of g. The models for image enhancement that we will discuss involve interesting POE's and many open questions.
For 3) we consider functions h whose values are finite sums of point masses in X. Assuming X has a distance function, we find a geometrically reasonable notion of the distance between 2 such sums. Then functions with TV(h) < oo essentially determine the surfaces and give a surprising amount of regularity. Analysis in singular spaces has had wide applications from algebraic geometry to data analysis.