Lighting affects appearance

What is the Question ? (based on work of Basri and Jacobs, ICCV 2001) Given an object described by its normal at each surface point and its albedo (we will focus on Lambertian surfaces) 1.What is the dimension of the space of images that this object can generate given any set of lighting conditions ? 2. How to generate a basis for this space ?

# # # # #1 ParrotPhoneFaceBall (Epstein, Hallinan and Yuille; see also Hallinan; Belhumeur and Kriegman) Dimension: Empirical Study

Domain Lambertian No cast shadows (“convex” objects) Lights are distant n l 

   Lambert Law k(  max ( cos , 0) Lighting to Reflectance: Intuition

Three point-light sources, l  ), Illuminating a sphere and its reflection r  ). Profiles of l  and r  Lighting to Reflectance: Intuition

Images... Lighting Reflectance where...

Spherical Harmonics (S.H.) Orthonormal basis,, for functions on the sphere. n’th order harmonics have 2n+1 components. Rotation = phase shift (same n, different m). In space coordinates: polynomials of degree n.

S.H. analog to convolution theorem Funk-Hecke theorem: “Convolution” in function domain is multiplication in spherical harmonic domain.filter. k

Harmonic Transform of Kernel

Amplitudes of Kernel n

Energy of Lambertian Kernel in low order harmonics k  is a low pass filter

Reflectance Functions Near Low-dimensional Linear Subspace Yields 9D linear subspace.

Forming Harmonic Images Z Y X XZ YZ XY

How accurate is approximation? Point light source 9D space captures 99.2% of energy

How accurate is approximation? Worst case. 9D space captures 98% of energy DC component as big as any other. 1st and 2nd harmonics of light could have zero energy

How Accurate Is Approximation? Accuracy depends on lighting. For point source: 9D space captures 99.2% of energy For any lighting: 9D space captures >98% of energy.

Accuracy of Approximation of Images Normals present to varying amounts. Albedo makes some pixels more important. Worst case approximation arbitrarily bad. “Average” case approximation should be good.

Summary Convex, Lambertian objects: 9D linear space captures >98% of reflectance. Explains previous empirical results. For lighting, justifies low-dim methods.

Models Query Find Pose Compare Vector: I Matrix: B Harmonic Images Recognition

Experiments (Basri&Jacobs) 3-D Models of 42 faces acquired with scanner. 30 query images for each of 10 faces (300 images). Pose automatically computed using manually selected features (Blicher and Roy). Best lighting found for each model; best fitting model wins.

Results 9D Linear Method: 90% correct. 9D Non-negative light: 88% correct. Ongoing work: Most errors seem due to pose problems. With better poses, results seem near 100%.

“kernel” can be far from low-pass. Ongoing work: Specularity

Specularity (2) Example: Phong model Product of 3 terms Not a convolution Solution from Atomic Spectroscopy (Wigner))