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Computer Vision

Computer Vision. Spring 2006 15-385,-685 Instructor: S. Narasimhan Wean 5403 T-R 3:00pm – 4:20pm Lecture #12. Midterm – March 9. Syllabus – until and including Lightness and Retinex Closed book, closed notes exam in class. Time: 3:00pm – 4:20pm Midterm review class next Tuesday (March 7)

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Computer Vision

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  1. Computer Vision Spring 2006 15-385,-685 Instructor: S. Narasimhan Wean 5403 T-R 3:00pm – 4:20pm Lecture #12

  2. Midterm – March 9 Syllabus – until and including Lightness and Retinex Closed book, closed notes exam in class. Time: 3:00pm – 4:20pm Midterm review class next Tuesday (March 7) (Email me by March 6 specific questions) If you have read the notes and readings, attended all classes, done assignments well, it should be a walk in the park

  3. Mechanisms of Reflection source incident direction surface reflection body reflection surface • Surface Reflection: • Specular Reflection • Glossy Appearance • Highlights • Dominant for Metals • Body Reflection: • Diffuse Reflection • Matte Appearance • Non-Homogeneous Medium • Clay, paper, etc Image Intensity = Body Reflection + Surface Reflection

  4. Example Surfaces Surface Reflection: Specular Reflection Glossy Appearance Highlights Dominant for Metals Body Reflection: Diffuse Reflection Matte Appearance Non-Homogeneous Medium Clay, paper, etc Many materials exhibit both Reflections:

  5. Diffuse Reflection and Lambertian BRDF source intensity I incident direction normal viewing direction surface element • Surface appears equally bright from ALL directions! (independent of ) albedo • Lambertian BRDF is simply a constant : • Surface Radiance : source intensity • Commonly used in Vision and Graphics!

  6. Diffuse Reflection and Lambertian BRDF

  7. White-out: Snow and Overcast Skies CAN’T perceive the shape of the snow covered terrain! CAN perceive shape in regions lit by the street lamp!! WHY?

  8. Diffuse Reflection from Uniform Sky • Assume Lambertian Surface with Albedo = 1 (no absorption) • Assume Sky radiance is constant • Substituting in above Equation: Radiance of any patch is the same as Sky radiance !! (white-out condition)

  9. Specular Reflection and Mirror BRDF source intensity I specular/mirror direction incident direction normal viewing direction surface element • Valid for very smooth surfaces. • All incident light energy reflected in a SINGLE direction (only when = ). • Mirror BRDF is simply a double-delta function : specular albedo • Surface Radiance :

  10. Combing Specular and Diffuse: Dichromatic Reflection Observed Image Color = a x Body Color + b x Specular Reflection Color Klinker-Shafer-Kanade 1988 R Color of Source (Specular reflection) Does not specify any specific model for Diffuse/specular reflection G Color of Surface (Diffuse/Body Reflection) B

  11. Diffuse and Specular Reflection diffuse specular diffuse+specular

  12. Photometric Stereo Lecture #12

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

  14. Equation of plane or Let Surface normal Surface Normal surface normal

  15. Normal vector Source vector plane is called the Gradient Space (pq plane) • Every point on it corresponds to a particular surface orientation Gradient Space

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

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

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

  19. Diffuse peak Specular peak Reflectance Map • Glossy surfaces (Torrance-Sparrow reflectance model) diffuse term specular term

  20. 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

  21. Solution • Take more images • Photometric stereo • Add more constraints • Shape-from-shading (next class)

  22. Photometric Stereo

  23. Photometric Stereo Lambertian case: Image irradiance: • We can write this in matrix form:

  24. inverse Solving the Equations

  25. Least squares solution: Moore-Penrose pseudo inverse • Solve for as before More than Three Light Sources • Get better results by using more lights

  26. Color Images • The case of RGB images • get three sets of equations, one per color channel: • Simple solution: first solve for using one channel • Then substitute known into above equations to get • Or combine three channels and solve for

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

  28. Specular Reflection - Recap • For a perfect mirror, light is reflected about N • We see a highlight when • Then is given as follows:

  29. Computing the Light Source Direction Chrome sphere that has a highlight at position h in the image • Can compute N by studying this figure • Hints: • use this equation: • can measure c, h, and r in the image N h H rN c C sphere in 3D image plane

  30. V2 V1 N Depth from Normals • Get a similar equation for V2 • Each normal gives us two linear constraints on z • compute z values by solving a matrix equation

  31. Limitations • Big problems • Doesn’t work for shiny things, semi-translucent things • Shadows, inter-reflections • Smaller problems • Camera and lights have to be distant • Calibration requirements • measure light source directions, intensities • camera response function

  32. Trick for Handling Shadows • Weight each equation by the pixel brightness: • Gives weighted least-squares matrix equation: • Solve for as before

  33. Original Images

  34. Results - Shape Shallow reconstruction (effect of interreflections) Accurate reconstruction (after removing interreflections)

  35. Results - Albedo No Shading Information

  36. Original Images

  37. Results - Shape

  38. Results - Albedo

  39. Results • Estimate light source directions • Compute surface normals • Compute albedo values • Estimate depth from surface normals • Relight the object (with original texture and uniform albedo)

  40. Next Class • Shape from Shading • Reading: Horn, Chapter 11.

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