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

$$h_{xx}h_{yy}-h_{xy}^2 \over 1 + h_x^2 +h_y^2$$ (10)

For small dips, the numerator is the important part. The numerator is the determinant of the Hessian,

$$\det \left\vert \begin{array} {cc} h_{xx} & h_{xy} \\ h_{yx} & h_{yy} \end{array} \right\vert$$ (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 $\bar h +h$ and dropping terms in h² we get

$$0 \quad\approx\quad (\bar h_{xx}\bar h_{yy}-\bar h_{xy}^2) + \bar h_{xx}h_{yy} + h_{xx}\bar h_{yy} - 2\bar h_{xy} h_{xy}$$ (12)

The most important question is: what is $\bar h(x,y)$? How do we initialize it, and how can we safely update it? A way to initialize $\bar h(x,y)$is to approximate the initial data by a best fitting one-dimensional parabola. One way to stablize $\bar h(x,y)$ 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.