Skip to main content
SpringerLink
Log in

An objective criterion for stopping light–surface interaction. Numerical validation and quality assessment

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

In most algorithms of global illumination, light–surface interaction terminates declaring that result at some point is close enough to some reference ground truth data. The underlying principle of such criterion is to minimize the processing time without compromising the (subjective) visual perception of the resulting image. We introduce an objective-driven condition for stopping the simulation of light transport. It is inspired by the physical meaning of light propagation. Besides, it takes into account that computations are performed in finite precision. Its main feature is the definition of the threshold establishing the maximum number of pixels that are completed in finite precision. Its value is computed at run time depending on the brightness of the image. As a proof of concept of the validity of this approach, we employ the stopping condition in a light tracing algorithm, propagating light that is generated by the light source. We assess the quality of the computed image by measuring the Peak Signal-to-Noise Ratio and the Structured Similarity Index error metrics on the standard scene of the Cornell Box. Numerical validation is performed by comparing results with the output of the NVIDIA\(^{\circledR }\) Iray render whose stopping condition is based on Russian roulette and on the elapsed time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Notes

  1. We assume L be a continuous function of x and \(\varTheta _{\mathrm{out}}\).

  2. This result agrees to the physical dispersion of the radiance L.

  3. We use the symbol \(|\cdot |\) to denote the cardinality of a given set.

  4. We use the notation fl(w), for denoting the machine number of \(\mathcal {F}\) corresponding to \(w\in \mathfrak {R}\).

  5. To seek efficient algorithms for implementing the ray casting function r is one of the topics that still are under investigation (see the most recent papers [5, 35]).

  6. We are mainly interested in extrapolating the growth factor of \(w_L(a_{\mathrm{max}})\) as a function of \(a_{\mathrm{max}}(S)\in [0,1)\) rather than the fitting accuracy. So, we determine the polynomial as the polynomial of degree 2 of best fit (in the least square sense).

References

  1. Amann, J., Weber, B., Wthrich, C. A.: Using image quality assessment to test rendering algorithms, Journal of WSCG. In: International Conference on Computer Graphics, Visualization and Computer Vision, vol. 20, ISSN 1213-6972, Union Agency, (2012)

  2. Bala, K., Dutre, P., Bekaert, P.: Advanced Global Illumination, 2nd edn. A K Peters/CRC Press, Natick (2006)

    Google Scholar 

  3. Cohen, M.F., Chen, S.E., Wallace, J.R., Greenberg, D.P.: A progressive refinement approach to fast radiosity image generation. SIGGRAPH ’88’ Comput. Graph. 22, 75 (1988)

    Article  Google Scholar 

  4. Cohen, M., Wallace, J.: Radiosity and Realistic Image Synthesis. Academic Press Prof, Cambridge (1993)

    MATH  Google Scholar 

  5. Cozzi, P., Stoner, F.: GPU ray casting of virtual globes. In: ACM SIGGRAPH 2010, 2010, Los Angeles, California, pp. 128:1–128:1, ACM, (2010)

  6. Dalquist, G., Björck, Åke: Numerical Methods. Prentice-Hall, Upper Saddle River (1974)

    Google Scholar 

  7. D’Amore, L., Marcellino, L., Mele, V., Romano, D.: Deconvolution of 3D fluorescence microscopy images using graphics processing units. In: Lecture Notes in Computer Science, Proceedings of the 9th International Conference on Parallel Processing and Applied Mathematics—Volume 7203, Part I, pp. 690-699, Springer-Verlag Berlin, Heidelberg, (2012)

  8. D’Amore, L., Campagna, R., Galletti, A., Marcellino, L., Murli, A.: A smoothing spline that approximates Laplace transform functions only known on measurements on the real axis. Inverse Probl. 28, 025007 (2012). doi:10.1088/0266-5611/28/2/025007

  9. D’Amore, L., Laccetti, G., Romano, D., Scotti, G., Murli, A.: Towards a parallel component in a GPU-CUDA environment: a case study with the L-BFGS Harwell routine. Int. J. Comput. Math. 92(1), 59–76 (2015)

    Article  MATH  Google Scholar 

  10. D’Amore, L., Murli, A.: Regularization of a Fourier series method for the Laplace transform inversion with real data. Inverse Probl. 18, 1185–1205 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Demmel, J.: The probability that a numerical analysis problem is difficult. Math. Comput. 50(182), 449–480 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dutré, P., Willems, Y.D.: Importance-driven Monte Carlo light tracing. In: Photorealistic Rendering Techniques, Part III, Focus on Computer Graphics, pp. 188–197, (1995)

  13. Goral, C.M., Torrance, K.E., Greenberg, D.P., Battaile, B.: Modeling the interaction of light between diffuse surfaces. In: Proceedings of the 11th Annual Conference on Computer Graphics and Interactive Techniques H. Christiansen, Ed. SIGGRAPH ’84. ACM Press, New York, NY, pp. 213–222, (1984)

  14. Hamby, D.M.: A Review of techniques for parameter sensitivity analysis of environmental models. Environ. Monit. Assess. 32, 135–154 (1994)

    Article  Google Scholar 

  15. He, Xiao D., Torrance, Kenneth E.: Francois X. Sillion, and Donald P. Greenberg. A Comprehensive Physical Model for Light Reflection. In: Computer Graphics, 25(4), Proceedings, Annual Conference Series, 1991, ACM SIGGRAPH, pp. 175–186

  16. He, X.D., Torrance, K.E., Sillion, F.X., Greenberg, D.P.: A comprehensive physical model for light reflection. In: SIGGRAPH’91 Conference Proceedings, 1991, LasVegas, United States. ACM Press, (1991)

  17. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, New Delhi (2002)

    Book  MATH  Google Scholar 

  18. Huges, J.F., Dam, A.V., McGuire, M., Sklar, D., Foley, J., Freiner, S., Akeley, K.: Computer Graphics, Principles and Practice, 3rd edn. Addison-Wesley, Boston (2014)

    Google Scholar 

  19. Kajiya, J. T.: The rendering equation. In: Proceedings of the 13th annual conference on Computer graphics and interactive techniques (SIGGRAPH ’86), pp. 143–150, ACM, New York, (1986)

  20. Kolmogorov, A. N., Fomin, S. V.: Elementi di teoria delle funzioni e di analisi funzionale, Ed. Mir, (1980)

  21. Krig, S.: Computer vision metrics—survey, taxonomy, and analysis. Springer-Verlag, ISBN: 978-1-4302-5929-9, (2010)

  22. Krommer, A.R., Ueberhuber, C.W.: Computational Integration. SIAM, New Delhi (1998)

    Book  MATH  Google Scholar 

  23. Laccetti, G., Lapegna, M., Mele, V., Romano, D.: A study on adaptive algorithms for numerical quadrature on heterogeneous GPU and multicore based systems, Chapter of book Parallel Processing and Applied Mathematics - PPAM 2013, Part I. (R. Wyrzykowski, J. Dongarra, et al., eds.), LNCS 8384, Springer, pp. 704–713 (2014)

  24. Laccetti, G., Lapegna, M., Mele, V., Romano, D., Murli, A.: A double adaptive algorithm for multidimensional integration on multicore based HPC systems. Int. J. Parallel Programm. 40(4), 397–409 (2012)

    Article  Google Scholar 

  25. Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Trans. Model. Comput. Simul. 8, 3 (1998)

    Article  MATH  Google Scholar 

  26. Muñoz-Rodrguez, J.A., Rodriguez-Vera, R.: Shape detection based on topography extraction from the width of the light line. Optik 111(10), 435–442 (2000)

    Google Scholar 

  27. Muñoz-Rodrguez, J.A., Rodriguez-Vera, R.: Image encryption based on a grating generated by a reflection intensity map. J. Mod. Opt. 52, 1385–1395 (2005)

    Article  Google Scholar 

  28. Muñoz-Rodrguez, J.A., Rodriguez-Vera, R.: Binocular imaging of a laser stripe and approximation networks for shape detection. Int. J. Imag. Syst. Technol. 17(2), 62–74 (2007)

    Article  Google Scholar 

  29. NVIDIA Advanced Rendering Center, NVIDIA Iray Whitepaper, Document Version 1.0, (2012)

  30. Oren, M., Nayar, S.K.: Generalization of Lambert’s reflectance model. Computer Graphics. In: Proceedings, Annual Conference Series, 1994, ACM SIGGRAPH, pp. 239–246. Orlando, Florida (1994)

  31. Szirmay-Kalos, L.: Monte-Carlo Methods in Global Illumination. Vienna University of Technology, Institute of Computer Graphics (1999)

  32. Veach, E.: Robust Monte Carlo Methods for light transport simulation, Ph.D. Thesis, (1997). Available at https://graphics.stanford.edu/papers/veach-thesis/thesis-bw.pdf

  33. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600 (2004)

  34. Wang, Z., Bovik, A.: Modern Image Quality Assessment, Synthesis Lectures on Image, Video, Multimedia Processing, 1st edn. Morgan & Claypool, San Rafael (2006)

    Google Scholar 

  35. Wei, F., JieQing, F.: Real-time rendering of algebraic B-spline surfaces via Bézier point insertion, Science China Information Sciences, Vol. 57, N. 1, Science China Press, pp.1-15, (2014)

Download references

Acknowledgements

We would like to express our very great appreciation to reviewers for their valuable and constructive suggestions.

Author information

Authors and Affiliations

  1. University of Naples Federico II, Naples, Italy

    Luisa D’Amore

  2. Institute of High Performance Computing and Networking of CNR, Naples, Italy

    Diego Romano

Authors
  1. Luisa D’Amore
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diego Romano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luisa D’Amore.

Appendix

Appendix

We prove Proposition 5.

Proof

Equation (2) can be written as the root finding problem, i.e

$$\begin{aligned} L(x, \varTheta )-\mathcal {H}[L(x,\varTheta )]= 0 \Leftrightarrow \mathcal {K}[L(x,\varTheta )]=0, \end{aligned}$$

where

$$\begin{aligned} \mathcal {K}: L(x,\varTheta ) \mapsto L(x, \varTheta ) - \mathcal {H}[L(x, \varTheta )]. \end{aligned}$$

Condition number of root finding problem is [6]

$$\begin{aligned} \mu _{\mathcal {K}}=\frac{1}{1-\Vert D_F(\mathcal {K})\Vert _{A \times \varOmega _x}}, \end{aligned}$$
(26)

where \(D_F\) denotes the Fréchet derivative [20]. As \(D_F(\mathcal {K}) = -\mathcal {H}\), from (26), it follows that

$$\begin{aligned} \mu _{\mathcal {K}}=\frac{1}{1-\Vert -\mathcal {H}\Vert _{A \times \varOmega _x}}= \frac{1}{1-a_{\mathrm{max}}(S)}. \end{aligned}$$

\(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

D’Amore, L., Romano, D. An objective criterion for stopping light–surface interaction. Numerical validation and quality assessment. J Math Imaging Vis 60, 18–32 (2018). https://doi.org/10.1007/s10851-017-0739-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-017-0739-z

Keywords

Mathematics Subject Classification

Search

Navigation