A General BRDF Representation Based on Tensor Decomposition

1. A General BRDF RepresentationBased on TensorDecomposition Ahmet Bilgili1, Aydın Öztürk2 and Murat Kurt1 1 International Computer Institute, Ege University, TURKEY 2 Department of Computer Engineering, Yasar University, TURKEY

2. Our Goal • Given a set of precise reflectance measurements from real surfaces is it possible to represent these measurements compactlyand accurately? • The proposed method should alsolend itself to developing anefficient and simple importance sampling algorithm. isotropic anisotropic

3. Previous Work – Analytical Models Analytical BRDF Models [CT81] [EBJ*06] Emprical BRDF Models Anisotropic BRDF Models Lineer BRDF Models Physically based BRDF Models Phong [Pho75] Blinn-Phong [Bli77] Ward [Ward92] Lafortune et al. [LFTG97] Ward-Duer [Due05] Torrance-Sparrow [TS67] Cook-Torrance [CT81] He et al. [HTSG91] Oren-Nayar [ON94] Kajiya [Kaj85] Poulin-Fournier [PF90] Ward [War92] Lafortune et al. [LFTG97] Ashikhmin-Shirley [AS00] Ward-Duer [Due05] Edwards et al. [EBJ*06] • Westin et al. [WAT92] • Koenderink et al. [KvDS96] • Schröder and Sweldens [SS95] • Lalonde and Fournier[LF97] • Stark et al. [SAS05] • Öztürk et al. [OKBG08]

4. Previous Work – Data-Driven Models Data-Driven BRDF Models [MPBM03] [LRR04] Measurement based BRDF Models Factorization based BRDF Models Matusik et al. [MPBM03] Romerio et al. [RVZ08] Kautz and McCool [KM99] McCool et al. [MAA01] Lawrence et al. [LRR04] [MAA01]

5. Previous Work – Importance Sampling Importance Sampling 400 samples/pixel 400 samples/pixel [EBJ*06] [LRR04] Factorization based BRDF Models Analytical BRDF Models General BRDF Sampling Methods Lawrence et al. [LRR04] Lawrence et al. [LRR05] Montes et al. [MUGL08] Phong [Pho75] Blinn-Phong [Bli77] Ward [War92] Lafortune [LFTG97] Ashikhmin-Shirley [AS00] Ward-Duer [Due05] Edwards et al. [EBJ∗06]

6. Previous Work – Tensor Factorization Computer Graphics [SZC∗07] [VT04] Data Compression BRDF Data Representation • [WWS*05] Sun et al. [SZC∗07] Vasilescu and Terzopulos [VT04] Wang et al. [WWS*05] BTF Data Representation • Original [VT04] • [WWS*05]

7. Key Idea 1D Vector I X P X K 1D Vector J X Q J Y g A Scalar P X Q x R Tucker I T Z 3D Tensor Data I X J X K 1D Vector K X R Project 3D Tensor data into products of 1D functions and a core tensor: P = Q = R =1

8. Our BRDF Representation • Our BRDF model is based on halfway vector representation. • We used logarithmic transformation of measured BRDF data (non-negativity). • Our Tucker approximation for a 4D BRDF data: • To improve the accuracy of the approximation wepropose applying the Tucker factorization recursively (error modeling approach).

9. Error Modeling Approach Tucker Tucker Tucker The final logBRDF values:

10. Importance Sampling • If the BRDF data is properly normalized,it can be viewed as sampled frequencies ofa multi-variate probability distribution [ÖKB10]. • Thenstandard statistical methods can be used to generate incidentvectors for a given outgoing direction. Normalizing coefficient of Normalizing coefficient of

11. Importance Sampling • We experimentally analyzed Tucker factors of both isotropic and anisotropic measured BRDF data set [MPBM03, NDM05]. • Based on the empirical properties explained, the Tucker factorization can be used to reduce the 4D sampling problem into a 2D case.

12. Importance Sampling- Tucker Factors

13. Importance Sampling – Isotropic

14. Importance Sampling – Anisotropic

15. Results- Isotropic & Anisotropic 32.073 46.369 41.349 37.878 38.886 33.123 36.637 blue-fabric, blue-metallic-paint, nickel, yellow-matte-plastic, grease-covered-steel red-velvet, yellow-satin

16. Results- Comparison on Isotropic Materials • 100 isotropic materials from MIT MERL database. • 6 well-known BRDF models are used in comparison. • Our proposed model gives the highestPSNR values in 66 cases and performing well for the remaining 34 materials.

17. Results- Alum-bronze Reference Image Ashikhmin-Shirley, 34.370 Cook-Torrance, 30.862 Edwards et al., 27.982 Lawrence et al., 32.629 Ward, 25.475 Ward-Duer, 26.146 Our model, 37.866

18. Results- Alum-bronze-Difference Images Reference Image Ashikhmin-Shirley, 34.370 Cook-Torrance, 30.862 Edwards et al., 27.982 Lawrence et al., 32.629 Ward, 25.475 Ward-Duer, 26.146 Our model, 37.866

19. Results- Nylon Reference Image Ashikhmin-Shirley, 30.720 Cook-Torrance, 30.934 Edwards et al., 30.830 Lawrence et al., 23.720 Ward, 29.802 Ward-Duer, 30.105 Our model, 38.025

20. Results- Nylon-Difference Images Reference Image Ashikhmin-Shirley, 30.720 Cook-Torrance, 30.934 Edwards et al., 30.830 Lawrence et al., 23.720 Ward, 29.802 Ward-Duer, 30.105 Our model, 38.025

21. Results- Silver-metallic-paint Reference Image Ashikhmin-Shirley, 29.282 Cook-Torrance, 28.901 Edwards et al., 32.361 Lawrence et al., 33.190 Ward, 25.373 Ward-Duer, 28.910 Our model, 40.191

22. Results- Silver-metallic-paint-Difference Images Reference Image Ashikhmin-Shirley, 29.282 Cook-Torrance, 28.901 Edwards et al., 32.361 Lawrence et al., 33.190 Ward, 25.373 Ward-Duer, 28.910 Our model, 40.191

23. Results- Comparison on Princeton Scene Reference Image Ashikhmin-Shirley, 33.656 Cook-Torrance, 30.240 Edwards et al., 25.604 Lawrence et al., 33.403 Ward, 22.916 Ward-Duer, 31.126 Our model, 35.274

24. Results- Importance Sampling Constant Environment Grace Environment

25. Results- Importance Sampling Comparison on Princeton Scene Ashikhmin-Shirley sampling, 256 samples/pixel, Time: 1067.392 sec Edwards et al. sampling, 256 samples/pixel, Time: 1109.015 sec Lawrence et al. sampling, 256 samples/pixel, Time: 1161.327 sec Our factored sampling, 256 samples/pixel, Time: 1261.461 sec

26. Results- Comparison on Rendering Times & Storage Needs Storage Needs Rendering times (in seconds)

27. Conclusions • Introduced a factored representation of the BRDF that is general, accurate, compact and amenable to importance sampling: • Correct parameterization of incoming direction. • Decomposition into small set of one-dimensional factored forms. • Importance sampling with numerical inversion.

28. Future Works • Factored forms for • Higher dimensional data: SvBRDFs, BTF, BSSRDF.. • Implementation of our factored BRDF representation in real-time global illumination algorithms.

29. Thank You Thank You http://ube.ege.edu.tr/~kurt/