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Single-view metrology

Single-view metrology. Magritte, Personal Values , 1952. Many slides from S. Seitz, D. Hoiem. Review: Camera projection matrix. camera coordinate system. world coordinate system. i ntrinsic parameters. extrinsic parameters. Camera parameters. Intrinsic parameters

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Single-view metrology

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  1. Single-view metrology Magritte, Personal Values, 1952 Many slides from S. Seitz, D. Hoiem

  2. Review: Camera projection matrix camera coordinate system world coordinate system intrinsic parameters extrinsic parameters

  3. Camera parameters • Intrinsic parameters • Principal point coordinates • Focal length • Pixel magnification factors • Skew (non-rectangular pixels) • Radial distortion

  4. Camera parameters • Intrinsic parameters • Principal point coordinates • Focal length • Pixel magnification factors • Skew (non-rectangular pixels) • Radial distortion • Extrinsic parameters • Rotation and translation relative to world coordinate system

  5. Camera calibration

  6. Xi xi Camera calibration • Given n points with known 3D coordinates Xi and known image projections xi, estimate the camera parameters

  7. Camera calibration: Linear method Two linearly independent equations

  8. Camera calibration: Linear method • P has 11 degrees of freedom • One 2D/3D correspondence gives us two linearly independent equations • Homogeneous least squares: find p minimizing ||Ap||2 • Solution given by eigenvector of ATA with smallest eigenvalue • 6 correspondences needed for a minimal solution

  9. Vertical vanishing point (at infinity) Vanishing line Vanishing point Vanishing point Camera calibration • What if world coordinates of reference 3D points are not known? • We can use scene features such as vanishing points Slide from Efros, Photo from Criminisi

  10. vanishing point v Recall: Vanishing points image plane camera center line in the scene • All lines having the same direction share the same vanishing point

  11. Computing vanishing points • X∞ is a point at infinity, vis its projection: v = PX∞ • The vanishing point depends only on line direction • All lines having direction D intersect at X∞ v X0 Xt

  12. Calibration from vanishing points • Let us align the world coordinate system with three orthogonal vanishing directions in the scene: v2 v1 . v3

  13. Calibration from vanishing points • p1= P(1,0,0,0)T – the vanishing point in the x direction • Similarly, p2 and p3 are the vanishing points in the y and z directions • p4= P(0,0,0,1)T– projection of the origin of the world coordinate system • Problem: we can only know the four columns up to independent scale factors, additional constraints needed to solve for them

  14. Calibration from vanishing points • Let us align the world coordinate system with three orthogonal vanishing directions in the scene: • Each pair of vanishing points gives us a constraint on the focal length and principal point

  15. Calibration from vanishing points Can solve for focal length, principal point Cannot recover focal length, principal point is the third vanishing point

  16. Rotation from vanishing points • Thus, • Get λ by using the constraint ||ri||2=1.

  17. Calibration from vanishing points: Summary • Solve for K (focal length, principal point) using three orthogonal vanishing points • Get rotation directly from vanishing points once calibration matrix is known • Advantages • No need for calibration chart, 2D-3D correspondences • Could be completely automatic • Disadvantages • Only applies to certain kinds of scenes • Inaccuracies in computation of vanishing points • Problems due to infinite vanishing points

  18. Making measurements from a single image http://en.wikipedia.org/wiki/Ames_room

  19. Recall: Measuring height 5.3 5 Camera height 4 3.3 3 2.8 2 1

  20. Measuring height without a ruler Z O ground plane • Compute Z from image measurements • Need more than vanishing points to do this

  21. The cross-ratio • A projective invariant: quantity that does not change under projective transformations (including perspective projection)

  22. The cross-ratio • A projective invariant: quantity that does not change under projective transformations (including perspective projection) • What are invariants for other types of transformations (similarity, affine)? • The cross-ratio of four points: P4 P3 P2 P1

  23. scene cross ratio C image cross ratio Measuring height  T (top of object) t r R (reference point) H b R B (bottom of object) vZ ground plane

  24. Measuring height without a ruler

  25. t v H image cross ratio vz r vanishing line (horizon) t0 vx vy H R b0 b

  26. 2D lines in homogeneous coordinates • Line equation: ax + by + c = 0 • Line passing through two points: • Intersection of two lines: • What is the intersection of two parallel lines?

  27. t v H image cross ratio vz r vanishing line (horizon) t0 vx vy H R b0 b

  28. 1 4 2 3 4 3 2 1 Measurements on planes Approach: unwarp then measure What kind of warp is this?

  29. Image rectification p′ p • To unwarp (rectify) an image • solve for homographyH given p and p′ • how many points are necessary to solve for H?

  30. Image rectification: example PierodellaFrancesca, Flagellation, ca. 1455

  31. Application: 3D modeling from a single image J. Vermeer,Music Lesson, 1662 A. Criminisi, M. Kemp, and A. Zisserman, Bringing Pictorial Space to Life: computer techniques for the analysis of paintings, Proc. Computers and the History of Art, 2002 http://research.microsoft.com/en-us/um/people/antcrim/ACriminisi_3D_Museum.wmv

  32. Application: 3D modeling from a single image D. Hoiem, A.A. Efros, and M. Hebert, "Automatic Photo Pop-up", SIGGRAPH 2005. http://www.cs.illinois.edu/homes/dhoiem/projects/popup/popup_movie_450_250.mp4

  33. Application: Image editing Inserting synthetic objects into images: http://vimeo.com/28962540 K. Karsch and V. Hedau and D. Forsyth and D. Hoiem, “Rendering Synthetic Objects into Legacy Photographs,” SIGGRAPH Asia2011

  34. Application: Object recognition D. Hoiem, A.A. Efros, and M. Hebert, "Putting Objects in Perspective", CVPR 2006

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