Help with Photogrammetry / Projective Geometry

[McC]

I figure this falls under the 'Logic' and 'Science' banners, so apologies if it belongs elsewhere.

I'm trying to understand the mathematical underpinnings of applied photogrammetry, projective geometry, and image-based geometric reconstruction. My end goal is to take a collection of photos with arbitrary, unknown camera positions, identify the pixel coordinates of major features in common across these images, and then input these coordinates into a formula (almost certainly involving matrices) to generate close-to-accurate 3D coordinates.

Based on this PDF (which is just one of many I've read and saved today), there does seem to be an equation that does what I want: , where W is a matrix, M is a matrix, and S is a matrix. f is the number of images and n is the number of features to be tracked. Each element of W is defined by u, which is a 2x1 matrix representing a pixel coordinate pair in the image (x,y).

Any of the math brains out there able to help me go from W to S? Basically, I want to restructure the equation so that it's . I know W, I want to find S, but I have no idea how to construct M.

[Kuroneko]

Unless your photos were taken with an orthographic fisheye lens, this simply isn't going to work. However, when I looked up the references in your pdf, I came across
-- Conrad Poelman and Takeo Kanade, "A Paraperspective Factorization Method for Shape and Motion Recovery," tech. report CMU-CS-93-219, Computer Science Department, Carnegie Mellon University, December, 1993.
I've not read it, but its abstract implies that it is much closer to what you want.

[McC]

I found another PDF that outlines a method for doing this in detail using just two images. I'm a little rusty with my matrix math, though, so it'll take me some time to parse through.