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Lambertian Reflectance and Linear Subspaces

Lambertian Reflectance and Linear Subspaces . Ronen Basri David Jacobs Weizmann NEC. Lighting affects appearance. How Complicated is Lighting?. Lighting => infinite DOFs. Set of possible images infinite dimensional (Belhumeur and Kriegman).

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Lambertian Reflectance and Linear Subspaces

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  1. Lambertian Reflectance and Linear Subspaces Ronen Basri David Jacobs Weizmann NEC

  2. Lighting affects appearance

  3. How Complicated is Lighting? • Lighting => infinite DOFs. Set of possible images infinite dimensional (Belhumeur and Kriegman) • But, in many cases, lighting => 9 DOFs.

  4. Our Results • Convex, Lambertian objects: 9D linear space captures >98% of reflectance. • Explains previous empirical results. (Epstein, Hallinan and Yuille; Hallinan; Belhumeur and Kriegman) • For lighting, justifies low-dim methods. • Simple, analytic form. => New recognition algorithms.

  5. Domain Domain Lambertian No cast shadows Lights far and isotropic n l q llmax (cosq, 0)

  6. Reflectance Lighting Images ...

  7. Lighting to Reflectance: Intuition

  8. + + + (See D’Zmura, ‘91; Ramamoorthi and Hanrahan ‘00)

  9. Spherical Harmonics • Orthonormal basis for functions on the sphere • Funk-Hecke convolution theorem • Rotation = Phase Shift • n’th order harmonic has 2n+1 components.

  10. Amplitudes of Kernel n

  11. Reflectance functions near low-dimensional linear subspace Yields 9D linear subspace.

  12. 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.

  13. Forming Harmonic images l lZ lX lY lXY lXZ lYZ

  14. 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.

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

  16. Comparison Methods • Linear: • Non-negative light: (See Georghides, Belhumeur and Kriegman) • Non-negative light, first order approximation:

  17. Previous Linear Methods • Shashua. With no shadows, i=lln with B =[lX,lY,lZ]. • First harmonic, no DC • Koenderink & van Doorn heuristically suggest using l too.

  18. PCA on many images Amano, Hiura, Yamaguti, and Inokuchi; Atick and Redlich; Bakry, Abo-Elsoud, and Kamel; Belhumeur, Hespanha, and Kriegman; Bhatnagar, Shaw, and Williams; Black and Jepson; Brennan and Principe; Campbell and Flynn; Casasent, Sipe and Talukder; Chan, Nasrabadi and Torrieri; Chung, Kee and Kim; Cootes, Taylor, Cooper and Graham; Covell; Cui and Weng; Daily and Cottrell; Demir, Akarun, and Alpaydin; Duta, Jain and Dubuisson-Jolly; Hallinan; Han and Tewfik; Jebara and Pentland; Kagesawa, Ueno, Kasushi, and Kashiwagi; King and Xu; Kalocsai, Zhao, and Elagin; Lee, Jung, Kwon and Hong; Liu and Wechsler; Menser and Muller; Moghaddam; Moon and Philips; Murase and Nayar; Nishino, Sato, and Ikeuchi; Novak, and Owirka; Nishino, Sato, and Ikeuchi; Ohta, Kohtaro and Ikeuchi; Ong and Gong; Penev and Atick; Penev and Sirivitch; Lorente and Torres; Pentland, Moghaddam, and Starner; Ramanathan, Sum, and Soon; Reiter and Matas; Romdhani, Gong and Psarrou; Shan, Gao, Chen, and Ma; Shen, Fu, Xu, Hsu, Chang, and Meng; Sirivitch and Kirby; Song, Chang, and Shaowei; Torres, Reutter, and Lorente; Turk and Pentland; Watta, Gandhi, and Lakshmanan; Weng and Chen; Yuela, Dai, and Feng; Yuille, Snow, Epstein, and Belhumeur; Zhao, Chellappa, and Krishnaswamy; Zhao and Yang.

  19. Comparison to PCA • Space built analytically • Size and accuracy known • More efficient time, When pose unknown, rendering and PCA done at run time.

  20. Experiments • 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.

  21. 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%.

  22. Summary • We characterize images object produces. • Useful for recognition with 3D model. • Also tells us how to generalize from images.

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