Next: MORE ON GAUSSIAN CURVATURE Up: Claerbout & Fomel: Gaussian Previous: What is the L1

# Gaussian curvature

I proposed that we find out the differential equation that describes the bending of paper and use it as a regularization. The idea is to encourage a Busch-like behavior. As with L1, I would like to have a linear operator to preserve the uniqueness of the solution. Uniqueness gives reliability. My exerience has taught me that if a method has multiple isolated minima, I will descend into the wrong one. If the paper-bending operator is nonlinear, I could linearize it.

Bill Symes suggested the Gaussian curvature. My favorite search engine (google.com) quickly gave me several references. Indeed a sheet of paper does seem to have a Gaussian curvature of zero. The Gaussian curvature of a 2-D function vanishes wherever the the function is locally one dimensional. The Gaussian curvature is the product of the principal curvatures. The Gaussian curvature is
 (10)
For small dips, the numerator is the important part. The numerator is the determinant of the Hessian,
 (11)

We might regularize a collection of data points by minimizing this determinant. I have begun looking for references that may have previously investigated this very basic idea. Unfortunately, the function is nonlinear. We can linearize it. Replacing h by and dropping terms in h2 we get
 (12)
The most important question is: what is ? How do we initialize it, and how can we safely update it? A way to initialize is to approximate the initial data by a best fitting one-dimensional parabola. One way to stablize is to smooth it in patches.

I am reminded of "LOMOPLAN", an earlier idea I had to fit a best plane wave, then use it to define a linear operator to use as a weighting function in estimation. The idea is that a sedimentary section consists of a single local plane wave. Perhaps that two-stage least squares process is akin to linearizing the Gaussian curvature.

Next: MORE ON GAUSSIAN CURVATURE Up: Claerbout & Fomel: Gaussian Previous: What is the L1
Stanford Exploration Project
4/27/2000