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

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

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

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

Lighting From a source – travels in straight lines Energy decreases with r 2 (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

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

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

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

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

Lambertian Reflectance

Lambert Law or 

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 (θ i,Φ i ) how much light will be reflected in a direction (θ e,Φ e ). BRDF is a function of 4 variables f(θ i,Φ i ;θ e,Φ e ). (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)

Why BRDF is Needed? Light from front Light from back

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)

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

convex concave

convex concave

HVS Assumes Light from Above

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

Surface Normal A surface z(x,y) A point on the surface: (x,y,z(x,y)) T Tangent directions t x =(1,0,p) T, t y =(0,1,q) T with p=z x, q=z y

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, p y = q x (z xy =z yx ). Therefore, one differential equation in one unknowns (Horn, 1977)

Lambertian SFS

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

Distance Transform is called Eikonal equation Consider d(x) s.t. |d x |=1 Assume x 0 =0 x0x0 d x

Distance Transform is called Eikonal equation Consider d(x) s.t. |d x |=1 Assume both x 0 =0 and x 1 =0 Minimum at every point (shortest distance) x0x0 d x x1x1

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)

Results

Photometric Stereo 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° (Woodham 1980)

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

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

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

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

Relief Sculptures

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