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Lecture 20: Structure from Motion

Lecture 20: Structure from Motion. Announcements. Proposals due today or Wednesday if you need an extra day or two I will schedule project meetings next week. Today. We've talked about finding corresponding points in images

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Lecture 20: Structure from Motion

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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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