Computer Vision

1. Computer Vision Spring 2010 15-385,-685 Instructor: S. Narasimhan PH A18B T-R 10:30am – 11:50am Lecture #13

2. 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

3. 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:

4. Diffuse Reflection and Lambertian BRDF

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. 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?

7. 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 :

8. 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

9. Diffuse and Specular Reflection diffuse specular diffuse+specular

10. Photometric Stereo Lecture #9

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

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

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

14. : 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:

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

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

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

18. 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

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

20. Photometric Stereo

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

22. inverse Solving the Equations

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

24. 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

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

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

27. 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

28. 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

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

30. Results: Lambertian Sphere Input Images Needles are projections of surface normals on image plane Estimated Surface Normals Estimated Albedo

31. Lambertain Mask

32. Results – Albedo and Surface Normal

33. Results: Lambertian Toy I.2 Input Images Estimated Surface Normals Estimated Albedo

34. 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

35. Results – Shape of Mask

36. 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)

37. Original Images

38. Results - Albedo No Shading Information

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

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