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Learn how to estimate the 3D shape of an object from shading in an image using photometric stereo. This technique assumes Lambertian reflection and requires multiple images with different lighting conditions. Discover how to compute surface normals and solve for the surface shape using integrability constraints.

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Announcements

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Presentation Transcript

  1. Announcements • Project 3b due Friday

  2. Last time • Shape from shading • Photometric Stereo

  3. Lambertian reflection is light source intensity Lets assume that can achieve this by dividing each pixel in the image by image intensity at a single point Lighting direction (same for all points) Albedo at a point Surface normal at a point

  4. Shape from shading Input: Output: - Single Image - 3D shape of the object in the image Z(x,y) I(x,y)

  5. Shape from shading Suppose • You can directly measure angle between normal and light source • Not quite enough information to compute surface shape • But can be if you add some additional info, for example • Assume normals along the sihouette are known • Constraints on neighboring normals—“integrability” • Smoothness

  6. Surface Normal A surface A point on the surface: Tangent directions

  7. Shape from shading But both unknowns come from an integrable surface: Z(x,y) thus we can use the integrability constraint:

  8. Photometric stereo Input: Output: - Several Images - 3D shape of the * same object object in the images * different lightings - Albedo * same pose - Lighting

  9. L3 L2 L1 Photometric stereo N V Can write this as a matrix equation:

  10. Solving the equations

  11. Least squares solution: Solve for N, kd as before What’s the size of LTL? More than three lights Get better results by using more lights

  12. Example Recovered albedo Recovered normal field Forsyth & Ponce, Sec. 5.4

  13. Color images The case of RGB images • get three sets of equations, one per color channel: • Simple solution: first solve for N using one channel or grayscale • Then substitute known N into above equations to get kd s

  14. Where do we get the lighting directions?

  15. Capture lighting variation Illuminate subject from many incident directions From Ravi Ramamoorthi

  16. Example images: From Ravi Ramamoorthi

  17. Computing light source directions Trick: place a chrome sphere in the scene • the location of the highlight tells you where the light source is

  18. Recall the rule for specular reflection For a perfect mirror, light is reflected about N • We see a highlight when V = R • then L is given as follows:

  19. Computing the light source direction Can compute  (and hence N) from this figure Now just reflect V about N to obtain L Chrome sphere that has a highlight at position h in the image N h H rN  c C sphere in 3D image plane

  20. Depth from normals What we have What we want Forsyth & Ponce, Sec. 5.4

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