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Understanding Structure from Motion: Projection and Affine Models

In today's lecture, we explore the concept of Structure from Motion (SfM), focusing on how to recover spatial point locations using corresponding images. We'll discuss the projection matrices of cameras and the challenges of working with correspondences alone. Starting from simplified imaging models, we will investigate the projection equation and its implications, including affine models and the effects of weak perspective. Finally, we will touch on the use of Singular Value Decomposition (SVD) to aid in solving the structure and motion recovery problem.

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Understanding Structure from Motion: Projection and Affine Models

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  1. Lecture 20: Structure from Motion

  2. Announcements • Proposals due today • or Wednesday if you need an extra day or two • I will schedule project meetings next week

  3. Today • We've talked about finding corresponding points in images • If we know the projection matrices of the cameras, we can recover the locations of points in space • What if we only know the correspondences? • Known as “Structure from Motion”

  4. Today • Today we will be talking about solving the problem under a simplifying assumption • To understand the assumption, we'll first talk about a simplified model of imaging

  5. The equation of projection We know: so (Image from Slides by Forsyth)

  6. The equation of projection Makes things hard! We know: so (Image from Slides by Forsyth)

  7. Weak perspective • Issue • perspective effects, but not over the scale of individual objects • collect points into a group at about the same depth, then divide each point by the depth of its group • Adv: easy • Disadv: wrong Effectively dividing by a constant z (Image from Slides by Forsyth)

  8. Affine Model • The projection equation can be written as • No division! • Okay approximation when variation in depth is small relative to the overall depth of the object 3D Coordinate

  9. Basic problem • Given n fixed points observed by m affine cameras we can say that for each point • For large enough m and n this is solvable • Up to an ambiguity • If M and P are a solution, so is 2x4 matrix Invertible 3x3 matrix

  10. Affine Structure and Motion from Two Images • Projection equations

  11. Leads to condition • Take advantage of affine ambiguity (see text), we can rewrite this as

  12. Which is • One equation per set of correspondences • Can solve with 4 sets of corresponding (u,v) and (u',v') • Given new correspondence, solve

  13. What if I have multiple images? • Basic Equations • If I stack the m instances across cameras

  14. Since I'm tracking multiple points • Can stack these into a matrix (from Forsyth and Ponce)

  15. Side Trip: SVD (from Forsyth and Ponce)

  16. More properties of SVD

  17. Back to Recovering Structure and Motion • D is a product of a 2mx3 matrix and 3xn matrix • Rank 3 • So, using SVD

  18. Using the SVD • If • We claim that

  19. Results (Tomasi and Kanade '92) • Input

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