Analysis of Lighting Effects

1. Analysis of Lighting Effects Outline: • The problem • Lighting models • Shape from shading • Photometric stereo • Harmonic analysis of lighting

5. Applications Modeling the effect of lighting can be used for • Recognition – particularly face recognition • Shape reconstruction • Motion estimation • Re-rendering • …

6. Lighting is Complex • Lighting can come from any direction and at any strength • Infinite degree of freedom

7. Issues in Lighting • Single light source (point, extended) vs. multiple light sources • Far light vs. near light • Matt surfaces vs. specular surfaces • Cast shadows • Inter-reflections

8. Lighting • From a source – travels in straight lines • Energy decreases with r2 (r – distance from source) • When light rays reach an object • Part of the energy is absorbed • Part is reflected (possibly different amounts in different directions) • Part may continue traveling through the object, if object is transparent / translucent

9. Specular Reflectance • When a surface is smooth light reflects in the opposite direction of the surface normal

10. Specular Reflectance • When a surface is slightly rough the reflected light will fall off around the specular direction

11. Lambertian Reflectance • When the surface is very rough light may be reflected equally in all directions

12. Lambertian Reflectance • When the surface is very rough light may be reflected equally in all directions

13. Lambertian Reflectance

14. Lambert Law q or

15. BRDF • A general description of how opaque objects reflect light is given by the Bidirectional Reflectance Distribution Function (BRDF) • BRDF specifies for a unit of incoming light in a direction (θi,Φi) how much light will be reflected in a direction (θe,Φe) . BRDF is a function of 4 variables f(θi,Φi;θe,Φe). • (0,0) denotes the direction of the surface normal. • Most surfaces are isotropic, i.e., reflectance in any direction depends on the relative direction with respect to the incoming direction (leaving 3 parameters)

16. Why BRDF is Needed? Light from front Light from back

17. Most Existing Algorithms • Assume a single, distant point source • All normals visible to the source (θ<90°) • Plus, maybe, ambient light (constant lighting from all directions)

18. Shape from Shading • Input: a single image • Output: 3D shape • Problem is ill-posed, many different shapes can give rise to same image • Common assumptions: • Lighting is known • Reflectance properties are completely known – For Lambertian surfaces albedo is known (usually uniform) • First solutions: Horn, 1977

19. convex concave

20. convex concave

21. HVS Assumes Light from Above

22. HVS Assumes Light from Above

23. HVS Assumes Light from Above

24. HVS Assumes Light from Above

25. Lambertian Shape from Shading (SFS) • Image irradiance equation • Image intensity depends on surface orientation • It also depends on lighting and albedo, but in SFS those assumed to be known

26. Surface Normal • A surface z(x,y) • A point on the surface: (x,y,z(x,y))T • Tangent directions tx=(1,0,p)T, ty=(0,1,q)T with p=zx, q=zy

27. Lambertian SFS • We obtain • Proportionality – because albedo is known up to scale • For each point one differential equation in two unknowns, p and q • But both come from an integrable surface z(x,y) • Thus, py= qx (zxy=zyx). • Therefore, one differential equation in one unknowns (Horn, 1977)

28. Lambertian SFS

29. SFS with Fast Marching • Suppose lighting coincides with viewing direction l=(0,0,1)T, then • Therefore • For general l we can rotate the camera

30. Distance Transform • is called Eikonal equation • Consider d(x) s.t. |dx|=1 • Assume x0=0 d x x0

31. Distance Transform • is called Eikonal equation • Consider d(x) s.t. |dx|=1 • Assume both x0=0 and x1=0 • Minimum at every point (shortest distance) d x x1 x0

32. SFS with Fast Marching • - Some places are more difficult to walk than others • Solution to Eikonal equations –using a variation of Dijkstra’s algorithm • Initial condition: we need to know z at extrema • Starting from lowest points, we propagate a wave front, where we gradually compute new values of z from old ones (Kimmel and Sethian, 2001)

33. Results

34. Photometric Stereo (Woodham 1980) • Fewer assumptions are needed if we have several images of the same object under different lightings • In this case we can solve for both lighting, albedo, and shape • This can be done by Factorization • Recall that • Ignore the case θ>90°

35. Photometric Stereo - Factorization (Hayakawa, 1994) Goal: given M, find L and S What should rank(M) be?

36. Photometric Stereo - Factorization • Use SVD to find a rank 3 approximation • Define • So • Factorization is not unique, since , A invertible To reduce ambiguity we impose integrability

37. Reducing Ambiguity (Belhumeur, Kriegman, Yuille, 1999) • Assume • We want to enforce integrability • Notice that • Denote by the three rows of A, then • From which we obtain

38. Reducing Ambiguity • Transforming a surface linearly maintains integrability • It can be shown that this is the only transformation that maintains integrability • Such transformations are called “generalized bas relief transformations” (GBR) • Thus, by imposing integrability the surface is reconstructed up to GBR

39. Relief Sculptures

40. Summary • Lighting effects are complex • Algorithms for SFS and photometric stereo for Lambertian object illuminated by a single light source • Harmonic analysis extends this to multiple light sources • Handling specularities, shadows, and inter-reflections is difficult