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Computer Vision

Computer Vision. Spring 2010 15-385,-685 Instructor: S. Narasimhan WH 5409 T-R 10:30am – 11:50am Lecture #10. Shape from Shading Lecture #10. Image Intensity and 3D Geometry. Shading as a cue for shape reconstruction What is the relation between intensity and shape? Reflectance Map.

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Computer Vision

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  1. Computer Vision Spring 2010 15-385,-685 Instructor: S. Narasimhan WH 5409 T-R 10:30am – 11:50am Lecture #10

  2. Shape from Shading Lecture #10

  3. Image Intensity and 3D Geometry • Shading as a cue for shape reconstruction • What is the relation between intensity and shape? • Reflectance Map

  4. : source brightness : surface albedo (reflectance) : constant (optical system) Image irradiance: Let then Reflectance Map - RECAP • Relates image irradiance I(x,y) to surface orientation (p,q) for given source direction and surface reflectance • Lambertian case:

  5. Reflectance Map (Lambertian) Iso-brightness contour cone of constant Reflectance Map - RECAP • Lambertian case

  6. Reflectance Map - RECAP iso-brightness contour • Lambertian case Note: is maximum when

  7. Shape from a Single Image? • Given a single image of an object with known surface reflectance taken under a known light source, can we recover the shape of the object? • Given R(p,q) ( (pS,qS) and surface reflectance) can we determine (p,q) uniquely for each image point? NO

  8. Solution • Take more images • Photometric stereo (previous class) • Add more constraints • Shape-from-shading (this class)

  9. Photometric Stereo

  10. Solution • Take more images • Photometric stereo (previous class) • Add more constraints • Shape-from-shading (this class)

  11. Human Perception

  12. Human Perception

  13. Examples of the classic bump/dent stimuli used to test lighting assumptions when judging shape from shading, with shading orientations (a) 0° and (b) 180° from the vertical. Thomas R et al. J Vis 2010;10:6

  14. Human Perception • Our brain often perceives shape from shading. • Mostly, it makes many assumptions to do so. • For example: • Light is coming from above (sun). • Biased by occluding contours. by V. Ramachandran

  15. See Ramachandran’s work on Shape from Shading by Humans http://psy.ucsd.edu/chip/ramabio.html

  16. (f,g)-space Problem (p,q) can be infinite when Redefine reflectance map as Stereographic Projection (p,q)-space (gradient space)

  17. and are known The values on the occluding boundary can be used as the boundary condition for shape-from-shading Occluding Boundaries

  18. Minimize Image Irradiance Constraint • Image irradiance should match the reflectance map (minimize errors in image irradiance in the image)

  19. : surface orientation under stereographic projection Smoothness Constraint • Used to constrain shape-from-shading • Relates orientations (f,g) of neighboring surface points Minimize (penalize rapid changes in surface orientation f and g over the image)

  20. Minimize Shape-from-Shading weight • Find surface orientations (f,g) at all image points that minimize smoothness constraint image irradiance error

  21. Of course you can consider more neighbors (smoother results) Find and that minimize Numerical Shape-from-Shading • Smoothness error at image point (i,j) • Image irradiance error at image point (i,j) (Ikeuchi & Horn 89)

  22. Find and that minimize If and minimize , then where and are 4-neighbors average around image point (k,l) Numerical Shape-from-Shading (Ikeuchi & Horn 89)

  23. Update rule Numerical Shape-from-Shading • Use known values on the occluding boundary to constrain the solution (boundary conditions) • Compare with for convergence test • As the solution converges, increase to remove the smoothness constraint (Ikeuchi & Horn 89)

  24. Results

  25. Results

  26. Next Part • Geometry • Image projection • Motion and Tracking • Stereo • Range image sensors

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