Leonard has a brief powerpoint presentation on using Singular Value Decomposition to represent terrain as a sum of vector product surfaces.

DARPA_GIS.ppt

He observes that if you take the SVD of a matrix that represents a height field, then the height field is decomposed into a sum of rank-1 matrices, each given by the outer product of corresponding eigenvectors. Successive eigenvectors encode lower energy modes, so that if you keep only the first k modes for an n by n terrain, you represent it with only 2nk+k numbers, yet still get a guarantee on the quality of approximation. Linear operators like differentiation will distribute across the sums and vector products, so you can compute slope on the decomposed representation. (Although you lose the guarantee on the quality of approximation of the derivative, how good an approximation do you actually get?)

-- JackSnoeyink - 02 Feb 2005

One simple idea: Compute singular value decomposition to give a representation of terrain as a sum of vector products (orthonormal vectors multiplied to give matrices, then added with decreasing weights from the SVD).

The derivative surface is a also a sum of vector products (of difference vectors, but these are no longer normal -- no longer unit length).

Find out the lenghts of difference vectors for a terrain, and take the most significant difference vectors and use them to reconstruct the difference surface.

• How well do you reconstruct the difference surface by taking the first k vectors of the SVD of the difference surface vs. the most significant vectors from the SVD of the height?
  • How do you measure "how well"?
  • What is "most significant"?

-- JackSnoeyink - 04 Feb 2005
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