1 / 32

Multi-linear Systems and Invariant Theory in the Context of Computer Vision and Graphics

Multi-linear Systems and Invariant Theory in the Context of Computer Vision and Graphics Class 3: Infinitesimal Motion CS329 Stanford University. Amnon Shashua. Material We Will Cover Today. Infinitesimal Motion Model. Infinitesimal Planar Homography (8-parameter flow).

timon-velasquez
Download Presentation

Multi-linear Systems and Invariant Theory in the Context of Computer Vision and Graphics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Multi-linear Systems and Invariant Theory in the Context of Computer Vision and Graphics Class 3: Infinitesimal Motion CS329 Stanford University Amnon Shashua

  2. Material We Will Cover Today • Infinitesimal Motion Model • Infinitesimal Planar Homography (8-parameter flow) • Factorization Principle for Motion/Structure Recovery • Direct Estimation

  3. Infinitesimal Motion Model Rodriguez Formula:

  4. Infinitesimal Motion Model

  5. Reminder: Assume:

  6. Infinitesimal Motion Model Let

  7. Infinitesimal Motion Model

  8. Infinitesimal Planar Motion (the 8-parameter flow)

  9. Infinitesimal Planar Motion (the 8-parameter flow)

  10. Infinitesimal Planar Motion (the 8-parameter flow) Note: unlike the discrete case, there is no scale factor

  11. Reconstruction of Structure/Motion (factorization principle) Note: 2 interchanges 1 interchanges

  12. Reconstruction of Structure/Motion (factorization principle)

  13. Reconstruction of Structure/Motion (factorization principle) Let be the “flow” of point i at image j (image 0 is ref frame)

  14. Reconstruction of Structure/Motion (factorization principle) Given W, find S,M (using SVD) Let for some Goal: find such that using the “structural” constraints on S

  15. Reconstruction of Structure/Motion (factorization principle) Goal: find such that using the “structural” constraints on S Columns 1-3 of S are known, thus columns 1-3 of A can be determined. Columns 4-6 of A contain 18 unknowns: eliminate Z and one obtains 5 constraints

  16. Reconstruction of Structure/Motion (factorization principle) Goal: find such that using the “structural” constraints on S Let because

  17. Reconstruction of Structure/Motion (factorization principle) because Each point provides 5 constraints, thus we need 4 points and 7 views

  18. Direct Estimation The grey values of images 1,2 Goal: find u,v per pixel

  19. Direct Estimation Assume: We are assuming that (u,v) can be found by correlation principle (minimizing the sum of square differences).

  20. Direct Estimation Taylor expansion:

  21. Direct Estimation gradient of image 2 image 1 minus image 2

  22. Direct Estimation “aperture problem”

  23. Direct Estimation Estimating parametric flow: Every pixel contributes one linear equation for the 8 unknowns

  24. Direct Estimation Estimating 3-frame Motion: Combine with:

  25. Direct Estimation Let

  26. Direct Estimation image 1 to image 2 image 1 to image 3 Each pixel contributes a linear equation to the 15 unknown parameters

  27. Direct Estimation: Factorization Let be the “flow” of point i at image j (image 0 is ref frame)

  28. Direct Estimation: Factorization

  29. Direct Estimation: Factorization Recall:

  30. Direct Estimation: Factorization

  31. Direct Estimation: Factorization Rank=6 Rank=6 Enforcing rank=6 constraint on the measurement matrix removes errors in a least-squares sense.

  32. Direct Estimation: Factorization Once U,V are recovered, one can solve for S,M as before.

More Related