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Lambertian Reflectance and Linear Subspaces

This paper explores the use of linear subspaces to capture the reflectance of convex, Lambertian objects, explaining previous empirical results and justifying low-dimensional methods for lighting. The accuracy of the approximation is discussed, and comparison methods with matrix and vector queries are presented. Experimental results demonstrate the effectiveness of the proposed approach for recognition with 3D models.

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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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