Stereographic Projection for general surfaces

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SUMMARY

The discussion centers on the application of stereographic projection in generating reflectance maps for illuminated surfaces, particularly in the context of computer vision. The participants reference the book "Computer Vision" by Michael Brady and H. G. Barrow, which discusses stereographic projection for spheres and its generalization to other surfaces. The Gaussian sphere is identified as a key concept in understanding this generalization, although the specifics remain unclear to the participants. The conversation highlights the need for a clearer explanation of how stereographic projection can be applied to various surface types beyond the sphere.

PREREQUISITES
  • Understanding of stereographic projection principles
  • Familiarity with Gaussian spheres in mathematical contexts
  • Knowledge of reflectance mapping techniques in computer vision
  • Basic concepts of parameterization in geometry
NEXT STEPS
  • Research the mathematical foundations of stereographic projection
  • Explore the role of Gaussian spheres in surface parameterization
  • Study reflectance mapping methods in computer vision applications
  • Investigate generalizations of stereographic projection to non-spherical surfaces
USEFUL FOR

Mathematicians, computer vision researchers, and anyone interested in the application of stereographic projection to various surface types in visual computing.

Deltinu
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Stereographic Projection for "general" surfaces

First off, sorry if this is in the wrong forum. I came across this while studying computer vision, but it's of a somewhat mathematical nature. Please move it if it's in the wrong place.

In the book I'm reading*, stereographic projection is used to generate a reflectance map of an illuminated surface. A similar approach can be found here: http://books.google.com/books?id=XEOsAAAAIAAJ&pg=PA117&source=gbs_toc_r&cad=0_0#PPA154,M1 (that link should get you to page 154 of a book called "Computer Vision" by Michael Brady and H. G. Barrow)

My problem is this: the stereograpic projection is put forward for a sphere, then (apparently) generalised to more general surfaces. However this generalisation isn't made explicit. What is it? Am I missing something really simple? EDIT: From further reading, I've come across the Gaussian sphere, which seems to play a role here...but the details are still a little hazy to me.

Many thanks in advance!

*"Artificial Intelligence" 2 ed. by Patrick Henry Wilson, picked up at a local charity shop for £2.99 :-)
 
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Um, it looks like the book you linked to was just using the projection to parameterize the sphere, and thus the set of surface orientations. Of course, this parameterization will induce local coordinate systems for most places on suitably well-behaved surfaces (is this the ultimate goal? Google wouldn't let me read very far). What did the other book say that makes you think that you need to generalize directly.