<figure><img src="image_71.png" /><figcaption>Figure 6.4.1: Photonic Crystals are periodical multi-layered systems where the index of refraction varies periodically in 1D or 2D or even 3D. Images from *Photonic Crystals, Molding the Flow of Light* [DGND08]. Observe that layered materials presented in Figure 6.1.4 are 1D and none-periodic systems.</figcaption></figure>

properties (e.g., roughness) and optical response has been extensively studied (e.g., [Ric51b, KoCF06]), further evaluation of recent BRDF models (e.g., [22]) is still required in order to, ideally, establish a bijection with, eventually, the manufacturing parameters of machining devices. This evaluation can only be done by accurately manufacturing control materials and then measuring them just as accurately. With the help of industrial partners (e.g., Saint-Gobain, Valeo) to manufacture the samples and colleagues from the electromagnetic community (e.g., Inria Project Atlantis), we will be able to establish up to what extent BRDF models based on diffraction are sufficiently accurate.

We have already established a first inverse method to design the appearance for a stack of thick layers (see, Bati et al. [19]). Thanks to our analytical results, our method is very efficient in terms of computation. Beyond thick-film systems, I also intend to look at more complex systems that generate interferences.

## Perturbed Thin-films without nano-particles

Such systems have industrial applications (cf. Figure 6.4.5 left) where they are considered as quasi-perfect when manufactured but they can also be found in biological realizations (cf. Figure 6.4.5 right) where they cannot be considered as perfect and where they can also be mixed with thick layers. Therefore, it is also necessary to be interested in the optical stability of such systems in order to establish which physical characteristics are sensitive (resp. invariant) and modify strongly (resp. weakly) the final optical response. Of course, this can be done by disrupting the different physical characteristics and comparing the variance of the different optical responses, obtained through numerical simulation.

To close the loop with a manufacturing process, members of the Manao team and I have started to collaborate with Mélanie Bay (former PhD Student at Cambridge University), who studied
<figure><img src="image_72.png" /><figcaption>Figure 6.4.2: The different types of 1D periodic systems. **(Left)** A Bragg Mirror system is structured which consists of alternating two optical materials with different indexes of refraction $η_1$ and $η_2$ and thicknesses ($αΔ$ and $(1-α)Δ$ with $α ∈ [0, 1]$). **(Right)** A clustered system is the concatenation of multiple Bragg mirrors with different thicknesses. When the thickness of a chirped system varies continuously, the system is said to be "clustered".</figcaption></figure>

<figure><img src="image_73.png" /><figcaption>Figure 6.4.3: Predicted appearance of a Bragg stack which physical properties are: $Δ = 700$nm, $N = 5$ and indexes of refraction are $n_1 = 1.4$ and $n_2 = 1.7 + 1i$. **Left:** Visualization of the different colour that can be achieved for each elevation angle (abscissa) and for each $α$ value (ordinate). **Right:** Rendered spectral images for different values of $α$. Thanks to Pierre Hubert-Briere for these images.</figcaption></figure>
<figure><img src="image_74.png" /><figcaption>Figure 6.4.4: Periodic systems with elliptical nano-particles. Image from [YLL$^{+}$18].</figcaption></figure>

(cf. [Bay21]) and fabricated multilayered-materials that are obtained through the structuring of hydroxypropyl cellulose (a bio-sourced and water-compatible polymer). For this specific type of systems, we aim at developing an efficient macroscopic BRDF and inverse it to establish a complete inverse design method for bio-inspired materials.

## Thin-films with nano-particles of arbitrary shapes

This type of system consists of a substrate on which a layer of nano-particles is deposited. The geometry of the nano-particles can be of any shape and the particles can be deposited in a periodic or semi-periodic way. An example of this type of system is presented in Figure 6.4.4. This type of structures has been seldomly studied in terms of BSDF appearance (only in terms of average color) despite the fact that the are promising in terms of expressiveness.

## Volumic Systems with 3D Correlation

In line with our work [30] and the work of Guo et al. [GJZ21] and in collaboration with K. Vynck, we intend to study the appearance of 3D correlated systems. This require to simulate the light transport inside a volume according to the RTE but with new phase function models and scattering and absorption properties. Contrary to the work of Guo et al. [GJZ21], we intend to study systems that are *not clustered* and that can even exhibit disordered hyperuniformity (e.g., [HS13]).

Finally, no matter the complexity of the system, my approach remains the same and requires:

(i) To solve the electromagnetic (EM) problem through simulation either in partnership with the Inria ATLANTIS team, or by using existing (EM) solvers such as Smuthi [ECT$^{+}$21] or RCWA [MG81] (e.g., $S^4$ [LF12]) according to the studied physical system. Typically, RCWA is meant for periodic or semi-periodic systems whereas Smuthi is for non-periodic systems.

(ii) To use the resulting tabulated solution from the EM solver to render a ground truth reference BSDF as well as a rendered image.
<figure><img src="image_75.png" /><figcaption>Figure 6.4.5: **Left:** Manufactured glasses from Saint Gobain with dozen of thin layers. Each layer thickness is about 10 nanometers and their combination produces colored interferences. Their realization requires a very fine control of the manufacturing process. **Center Left:** Natural example of thin layers that are not perfectly aligned and present irregularities: Graphium Sarpedon butterfly (images from [Sta14]). On the wings of the butterfly, unpigmented glass(**g**) or white (**w**) scales are present. **Center Right:** Electron microscopy scan of the part (**g**) revealing a complex and imperfect thin film structure. **Right:** Scattering measurement of the glass scale (**g**) showing strong angular dependence of the spectral reflectance. See also Figure 6.4.2 to illustrate the different 1D periodic structures.</figcaption></figure>

(iii) To develop a semi-analytical (or even closed-form if possible) model that is predictive enough to avoid the need of a full EM simulation every time an optical parameter is changed (e.g., thicknesses, absorption, ...)

(iv) To infer an inverse model from the direct model obtained in (iii). This step is most of the time non-trivial because there are no analytical solutions. However, we could benefit from Deep Learning approaches to guide us towards an initial solution. Since Deep Learning approaches require a lot of data to train the network, in our case, this translates to generate a BSDF for each set of parameters; the benefit of a predictive BSDF model developed in (iii) is thus paramount.

In the long term, I wish to develop models of ageing and degradation of materials (cf. [DRS08]) in order to predict the appearance of materials over time and also to virtually restore the **appearance of ancient objects**. For this purpose, the combination of experimental archaeology, optical measurements and modelling is promising. Experimental archaeology allows the reconstruction of the object as it was manufactured at the time and the comparison of the appearance measurements of the original object with the restored object. These measurements can then be used to develop appearance models for ageing or restoration. Furthermore, the same models can then be applied to objects of the same period in order to restore them virtually, thus avoiding long measurement campaigns. Our recent work [31] is a first step in that direction.

In an **industrial context**, with the help of collaborations with material science experts, comparing measurements of artificially accelerated aged and new materials will allow the development of more predictive models than those usually developed in computer graphics (see [MG08] for an overview).
<figure><img src="image_76.png" /></figure>
LIST OF EXTERNAL REFERENCES

[ACGO18] Rachel A. Albert, Dorian Yao Chan, Dan B. Goldman, and James F. O'Brien.
Approximate svBRDF Estimation From Mobile Phone Video.
In Wenzel Jakob and Toshiya Hachisuka, editors, *Eurographics Symposium on Rendering - Experimental Ideas & Implementations*. The Eurographics Association, 2018.

[APS00] Michael Ashikhmin, Simon Premoze, and Peter Shirley.
A microfacet-based BRDF generator.
In *Proc. ACM SIGGRAPH '00*, pages 65–74, 2000.

[AWL15] Miika Aittala, Tim Weyrich, and Jaakko Lehtinen.
Two-shot SVBRDF capture for stationary materials.
*ACM Trans. Graph. (Proc. SIGGRAPH)*, 34(4):110:1–110:13, July 2015.

[BAEDR08] Aner Ben-Artzi, Kevin Egan, Fredo Durand, and Ravi Ramamoorthi.
A precomputed polynomial representation for interactive BRDF editing with global illumination.
*ACM Trans. Graph.*, 27 (2):13 (1–13), May 2008.

[Bat21] Mégane Bati.
*Design inverse de l'apparence de matériaux multicouches*.
PhD thesis, 2021.
2021BORD0396.

[Bay21] Mélanie Marthe Bay.
*The interplay between order and disorder in cholesteric hydroxypropyl cellulose films*.
PhD thesis, University of Cambridge. Department of Chemistry, 2021.

[BJ19] Benedikt Bitterli and Wojciech Jarosz.
Selectively metropolised monte carlo light transport simulation.
*ACM Trans. Graph.*, 38(6), nov 2019.

[Bli77] James F. Blinn.
Models of light reflection for computer synthesized pictures.
In *ACM SIGGRAPH '77*, pages 192–198, 1977.

[BLPW14] Adam Brady, Jason Lawrence, Pieter Peers, and Westley Weimer.
Genbrdf: Discovering new analytic brdfs with genetic programming.
*ACM Trans. Graph.*, 33(4), jul 2014.

[BNM15] Samuel D. Butler, Stephen E. Nauyoks, and Michael A. Marciniak.
Experimental analysis of bidirectional reflectance distribution function cross section
conversion term in direction cosine space.
*Opt. Lett.*, 40(11):2445–2448, Jun 2015.

[BOK11] Ahmet Bilgili, Aydn Öztürk, and Murat Kurt.
A General BRDF Representation Based on Tensor Decomposition.
*Computer Graphics Forum*, 2011.

[BP20] James C. Bieron and Pieter Peers.
An adaptive brdf fitting metric.
*Computer Graphics Forum*, 39(4), July 2020.

[BS87] P. Beckmann and A. Spizzichino.
*The scattering of electromagnetic waves from rough surfaces*.
Artech House, 2 edition, 1987.

[BSH12] Mahdi M. Bagher, Cyril Soler, and Nicolas Holzschuch.
Accurate fitting of measured reflectances using a Shifted Gamma micro-facet distri-
bution.
*Computer Graphics Forum*, 31(4):1509–1518, June 2012.

[BSN16] Mahdi M. Bagher, John Snyder, and Derek Nowrouzezahrai.
A non-parametric factor microfacet model for isotropic brdfs.
*ACM Trans. Graph.*, 35(5), July 2016.

[Bur12] Brent Burley.
Physically-based shading at Disney.
In Stephen Hill and Stephen McAuley, editors, *Siggraph course: Practical Physically
Based Shading in Film and Game Production*. ACM, 2012.

[BWP<sup>+</sup>20] Benedikt Bitterli, Chris Wyman, Matt Pharr, Peter Shirley, Aaron Lefohn, and Wojciech Jarosz.
Spatiotemporal reservoir resampling for real-time ray tracing with dynamic direct lighting.
*ACM Trans. Graph.*, 39(4), aug 2020.
[CCFM22] O. Clausen, Y. Chen, A. Fuhrmann, and R. Marroquim.
Investigation and simulation of diffraction on rough surfaces.
*Computer Graphics Forum*, n/a(n/a), 2022.

[CDPS09] Massimiliano Corsini, Matteo Dellepiane, Federico Ponchio, and Roberto Scopigno.
Image-to-geometry registration: A Mutual Information method exploiting
illumination-related geometric properties.
*Computer Graphics Forum*, 28(7):1755–1764, oct 2009.

[Cha60] S. Chandrasekhar.
*Radiative Transfer*.
Dover Books on Intermediate and Advanced Mathematics. Dover Publications, 1960.

[CJZ77] E. L. Church, H. A. Jenkinson, and J. M. Zavada.
Measurement of the Finish of Diamond-Turned Metal Surfaces By Differential Light
Scattering.
*Optical Engineering*, 16(4):164360, 1977.

[CJZ79] E. L. Church, H. A. Jenkinson, and J. M. Zavada.
Relationship between Surface Scattering and Microtopographic Features.
*Optical Engineering*, 18(2):182125, 1979.

[CKS+17] Chakravarty R. Alla Chaitanya, Anton S. Kaplanyan, Christoph Schied, Marco Salvi, Aaron Lefohn, Derek Nowrouzezahrai, and Timo Aila.
Interactive reconstruction of monte carlo image sequences using a recurrent denoising autoencoder.
*ACM Trans. Graph.*, 36(4), July 2017.

[CSCC14] David Coleman, Ioan Sucan, Sachin Chitta, and Nikolaus Correll.
Reducing the barrier to entry of complex robotic software: a moveit! case study, 2014.

[CT82] R. L. Cook and Kenneth E. Torrance.
A reflectance model for computer graphics.
*ACM Trans. Graph.*, 1(1):7–24, 1982.

[CT86] E. L. Church and P. Z. Takacs.
Statistical and signal processing concepts in surface metrology.
In *Proc. SPIE 0645, Optical Manufacturing, Testing and Aspheric Optics*, volume
0645, pages 107–115, 1986.

[dGM07] Luca di Gaspero and Eric Moyer.
Quadprog++.
https://sourceforge.net/projects/quadprog/, 2007.
[dGM16] Luca di Gaspero and Eric Moyer.
Quadprog++.

https://github.com/liuq/QuadProgpp, 2016.

[DGND08] Joannopoulos John D., Johnson Steven G., Winn Joshua N., and Meade Robert D.
*Photonic Crystals, Molding the Flow of Light*.

Princeton University Press, 2008.

[DRS08] Julie Dorsey, Holly Rushmeier, and Franois Sillion.
*Digital Modeling of Material Appearance*.

Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 2008.

[DSHL10] Holger Dammertz, Daniel Sewtz, Johannes Hanika, and Hendrik P. A. Lensch.
Edge-avoiding À-trous wavelet transform for fast global illumination filtering.

In *Proceedings of the Conference on High Performance Graphics*, HPG '10, page 67–75, Goslar, DEU, 2010. Eurographics Association.

[DWT+10] Yue Dong, Jiaping Wang, Xin Tong, John Snyder, Yanxiang Lan, Moshe Ben-Ezra,
and Baining Guo.
Manifold bootstrapping for SVBRDF capture.

*ACM Trans. Graph.*, 29:98 (1–10), July 2010.

[ECT+21] Amos Egel, Krzysztof M. Czajkowski, Dominik Theobald, Konstantin Ladutenko,
Alexey S. Kuznetsov, and Lorenzo Pattelli.

Smuthi: A python package for the simulation of light scattering by multiple particles
near or between planar interfaces.

*Journal of Quantitative Spectroscopy and Radiative Transfer*, 273:107846, 2021.

[EG04] Tanos Mikhael Elfouhaily and Charles-Antoine Guérin.

A critical survey of approximate scattering wave theories from random rough surfaces.

*Waves in Random Media*, 14(4):R1–R40, 2004.

[FHF+17] Luca Fascione, Johannes Hanika, Marcos Fajardo, Per Christensen, Brent Burley,
and Brian Green.

Path tracing in production - part 1: Production renderers.

In *ACM SIGGRAPH 2017 Courses*, SIGGRAPH '17, pages 13:1–13:39, 2017.

[FJM+20] J. R. Frisvad, S. A. Jensen, J. S. Madsen, A. Correia, L. Yang, S. K. S. Gregersen,
Y. Meuret, and P.-E. Hansen.

Survey of models for acquiring the optical properties of translucent materials.

*Computer Graphics Forum*, 39(2):729–755, 2020.
[FVH<sup>+</sup>13] J. Filip, R. Vávra, M. Haindl, P. id, M. Krupika, and V. Havran.
Brdf slices: Accurate adaptive anisotropic appearance acquisition.
In *2013 IEEE Conference on Computer Vision and Pattern Recognition*, pages 1468–1473, June 2013.

[Gal19] Alain Galvan.
Machine learning denoising. a review of different techniques for denoising monte-carlo images with machine learning.
https://alain.xyz/blog/machine-learning-denoising, Nov. 2019.

[GGG<sup>+</sup>16] D. Guarnera, G. C. Guarnera, A. Ghosh, C. Denk, and M. Glencross.
BRDF representation and acquisition.
*Computer Graphics Forum*, 35(2):625–650, may 2016.

[GGSK19] P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek.
Variance-aware multiple importance sampling.
*ACM Trans. Graph.* (*SIGGRAPH Asia 2019*), 38(6), 2019.

[GJZ21] Yu Guo, Adrian Jarabo, and Shuang Zhao.
Beyond mie theory: Systematic computation of bulk scattering parameters based on microphysical wave optics.
*ACM Trans. Graph.*, 40(6), dec 2021.

[GKPB04] Pascal Gautron, Jaroslav Křivánek, Sumanta N. Pattanaik, and Kadi Bouatouch.
A novel hemispherical basis for accurate and efficient rendering.
In Henrik Wann Jensen and Alexander Keller, editors, *Proc. Eurographics Symposium on Rendering 2004*, pages 321–330. Eurographics Association, 2004.

[GKPS12] I. Georgiev, J. Křivánek, S. Popov, and P. Slusallek.
Importance caching for complex illumination.
In *Computer Graphics Forum*, volume 31, pages 701–710. Wiley Online Library, 2012.

[Gre09] Eugène Grenon.
Observer l'extrêmement petit: au microscope, c'est possible, 2009.

[Gue20] G. Guennebaud.
Eigen version of the Quadprog++ library.
https://github.com/ggael/QuadProg, 2020.

[Hal07] Johan Hallenberg.
Robot Tool Center Point Calibration using Computer Vision Master's Thesis in Computer Vision by.
Master's thesis, Linköping University, 2007.

[Har75] James E. Harvey.
*Light-Scattering Characteristics of Optical Surface.*
PhD thesis, University of Arizona, 1975.
Adviser: R. V. Shack.

[Hei14] Eric Heitz.
Understanding the Masking-Shadowing Function in Microfacet-Based BRDFs.
*Journal of Computer Graphics Techniques*, 3(2):32–91, June 2014.

[Hes95] Tim Hesterberg.
Weighted Average Importance Sampling and Defensive Mixture Distributions.
*Technometrics*, 37(2):185–194, 1995.

[HEV⁺¹⁶a] S. Herholz, O. Elek, J. Vorba, H. Lensch, and J. Křivánek.
Product importance sampling for light transport path guiding.
*In Computer Graphics Forum*, volume 35, pages 67–77. Wiley Online Library, 2016.

[HEV⁺¹⁶b] Sebastian Herholz, Oskar Elek, Jiří Vorba, Hendrik Lensch, and Jaroslav Křivánek.
Product importance sampling for light transport path guiding.
*Computer Graphics Forum*, 35(4):67–77, 2016.

[HHN⁺¹⁷] Vlastimil Havran, Jan Hošek, Šárka Němcová, Jiří Čáp, and Jiří Bittner.
Lightdrum—Portable light stage for accurate BTF measurement on site.
*Sensors (Switzerland)*, 17(3), 2017.

[HJBS79] B. J. Hoenders, E. Jakeman, H. P. Baltes, and B. Steinle.
K correlations and facet models in diffuse scattering.
*Optica Acta: International Journal of Optics*, 26(10):1307–1319, 1979.

[HLZ10] Michael Holroyd, Jason Lawrence, and Todd Zickler.
A coaxial optical scanner for synchronous acquisition of 3D geometry and surface reflectance.
*ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010*, 29(4):1, jul 2010.

[HP03] Jefferson Y. Han and Ken Perlin.
Measuring bidirectional texture reflectance with a kaleidoscope.
*ACM Trans. Graph.*, 22(3):741–748, July 2003.

[HS13] Jakub Haberko and Frank Scheffold.
Fabrication of mesoscale polymeric templates for three-dimensional disordered photonic materials.
*Opt. Express*, 21(1):1057–1065, Jan 2013.
[HSCD12] James E. Harvey, Sven Schröder, Narak Choi, and Angela Duparré.
Total integrated scatter from surfaces with arbitrary roughness, correlation widths,
and incident angles.
*Optical Engineering*, 51(1):013402–1–013402–11, 2012.

[HST10] Kaiming He, Jian Sun, and Xiaoou Tang.
Guided image filtering.
In Kostas Daniilidis, Petros Maragos, and Nikos Paragios, editors, *Computer Vision
– ECCV 2010*, pages 1–14, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg.

[HTSG91] Xiao D. He, Kenneth E. Torrance, François X. Sillion, and Donald P. Greenberg.
A comprehensive physical model for light reflection.
In *Proceedings of the 18th Annual Conference on Computer Graphics and Inter-
active Techniques*, SIGGRAPH '91, page 175–186, New York, NY, USA, 1991.
Association for Computing Machinery.

[Int19] Intel open image denoise. high performance denoising library for ray-tracing.
https://openimagedenoise.github.io/, 2019.

[JEK⁺¹⁹] Wojciech Jarosz, Afnan Enayet, Andrew Kensler, Charlie Kilpatrick, and Per Chris-
tensen.
Orthogonal array sampling for Monte Carlo rendering.
*Computer Graphics Forum (Proceedings of EGSR)*, 38(4):135–147, July 2019.

[Jen01] Henrik Wann Jensen.
*Realistic Image Synthesis Using Photon Mapping*, page 208.
A.K. Series / CRC Press, 2001.

[JMLH01] H. W. Jensen, S. R. Marschner, M. Levoy, and P. Hanrahan.
A practical model for subsurface light transport.
In *Proceedings of the 28th Annual Conference on Computer Graphics and Inter-
active Techniques*, SIGGRAPH '01, page 511–518. Association for Computing
Machinery, 2001.

[JS06] Nocedal J. and Wright S.
*Numerical Optimization*.
Springer-Verlag, New York Inc., July 2006.

[Kaj86] James T. Kajiya.
The rendering equation.
*SIGGRAPH Comput. Graph.*, 20(4):143–150, August 1986.
[KBS15]
Nima Khademi Kalantari, Steve Bako, and Pradeep Sen.
A machine learning approach for filtering monte carlo noise.
*ACM Trans. Graph.*, 34(4), July 2015.

[KDS96]
Jan Koenderink, Andrea Doorn, and Marigo Stavridi.
BRDF expressed in terms of surface scattering modes.
In *Proc. Eur. Conf. on Comp. Vision '96*, pages 28–39, 1996.

[KNRS13]
Johannes Kohler, Tobias Noll, Gerd Reis, and Didier Stricker.
A full-spherical device for simultaneous geometry and reflectance acquisition.
In *Proceedings of IEEE Workshop on Applications of Computer Vision*, pages 355–362, 2013.

[KoCF06]
A. Krywonos and University of Central Florida.
Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction, 2006.

[Kry06]
Andrey Krywonos.
*Predicting Surface Scatter using a Linear Systems Formulation of Non-Paraxial Scalar Diffraction.*
PhD thesis, University of Central Florida, 2006.
Adviser: J. E. Harvey.

[KVG+19]
I. Kondapaneni, P. Vévoda, P. Grittmann, T. Skřivan, P. Slusallek, and J. Křivánek.
Optimal multiple importance sampling.
*ACM Trans. Graph.*, 38(4), 2019.

[KW09]
Malvin H. Kalos and Paula A. Whitlock.
*Monte Carlo Methods.*
John Wiley & Sons, Ltd, 2009.

[LBGO15]
Ronan Le Breton, Guillaume Ged, and Gaël Obein.
Out of plane brdf measurement at lne-cnam using "condor" our primary goniospectrophotometer.
In *28th CIE SESSION, Manchester, United Kingdom*, volume 1, pages 1401–1407.
CIE 216:2015, 2015.

[Len89]
Reimar K. Lenz.
A New Technique for Fully Autonomous and Efficient 3D Robotics Hand/Eye Calibration.
*IEEE Transactions on Robotics and Automation*, 5(3):345–358, 1989.
[LF12]
Victor Liu and Shanhui Fan.
$S^4$: A free electromagnetic solver for layered periodic structures.
*Computer Physics Communications*, 183(10):2233–2244, 2012.

[LFTG97]
Eric P. F. Lafortune, Sing-Choong Foo, Kenneth E. Torrance, and Donald P. Green-
berg.
Non-linear approximation of reflectance functions.
In *ACM SIGGRAPH '97*, pages 117–126, 1997.

[LKB<sup>+</sup>22]
Daqi Lin, Markus Kettunen, Benedikt Bitterli, Jacopo Pantaleoni, Cem Yuksel, and
Chris Wyman.
Generalized resampled importance sampling: Foundations of restir.
*ACM Trans. Graph.*, 41(4), jul 2022.

[LKYU12]
Joakim Löw, Joel Kronander, Anders Ynnerman, and Jonas Unger.
Brdf models for accurate and efficient rendering of glossy surfaces.
*ACM Trans. Graph.*, 31(1), February 2012.

[LLCK19]
Edward Liu, Ignacio Llamas, Juan Cañada, and Patrick Kelly.
Cinematic rendering in ue4 with real-time ray tracing and denoising.
In Eric Haines and Tomas Akenine-Möller, editor, *Ray Tracing Gems*, chapter 19,
pages 289–319. Apress, 2019.

[Lou04]
M. I. A. Lourakis.
levmar: Levenberg-marquardt nonlinear least squares algorithms in c/c++, 2004.

[LRR04]
Jason Lawrence, Szymon Rusinkiewicz, and Ravi Ramamoorthi.
Efficient brdf importance sampling using a factored representation.
In *ACM SIGGRAPH 2004 Papers*, SIGGRAPH '04, page 496–505, New York, NY,
USA, 2004. Association for Computing Machinery.

[Lu14]
Heqi Lu.
*Importance Sampling of Realistic Light Sources*.
Thesis, Université de Bordeaux, February 2014.

[Luc20]
Antoine Lucat.
*Acquisition opto-numérique de vêtements asiatiques anciens*.
PhD thesis, Université de Bordeaux, 2020.
2020BORD0012.

[MAA01]
Michael D. McCool, Jason Ang, and Anis Ahmad.
Homomorphic factorization of brdfs for high-performance rendering.
In *Proceedings of the 28th Annual Conference on Computer Graphics and Inter-
active Techniques*, SIGGRAPH '01, page 171–178, New York, NY, USA, 2001.
Association for Computing Machinery.

[Mar98] Stephen R Marschner.
*Inverse rendering for computer graphics*.
PhD thesis, Cornell University, 1998.

[Mat22] Mathworks.
Matlab implementation of constrained nonlinear optimization algorithms, 2022.

[McA02] David K. McAllister.
A generalized surface appearance representation for computer graphics.
*thesis, Chapel Hill*, page 129, 2002.

[Mes88] J.B. Messerschmitt.
Ueber diffuse reflexion.
*Annalen der Physik*, page 867, 1888.

[MG81] M. G. Moharam and T. K. Gaylord.
Rigorous coupled-wave analysis of planar-grating diffraction.
*J. Opt. Soc. Am.*, 71(7):811–818, Jul 1981.

[MG08] S. Mérillou and D. Ghazanfarpour.
Technical section: A survey of aging and weathering phenomena in computer graph-
ics.
*Comput. Graph.*, 32(2):159–174, April 2008.

[MMBJ17] Michael Mara, Morgan McGuire, Benedikt Bitterli, and Wojciech Jarosz.
An efficient denoising algorithm for global illumination.
In *Proceedings of High Performance Graphics*, New York, NY, USA, July 2017. ACM.

[MMS<sup>+</sup>04] Gero Müller, Jan Meseth, Mirko Sattler, Ralf Sarlette, and Reinhard Klein.
Acquisition, synthesis and rendering of bidirectional texture functions.
In Christophe Schlick and Werner Purgathofer, editors, *Eurographics 2004, State of the Art Reports*, pages 69–94. INRIA and Eurographics Association, September 2004.

[MPBM03] Wojciech Matusik, Hanspeter Pfister, Matt Brand, and Leonard McMillan.
A data-driven reflectance model.
*ACM Trans. Graph.*, 22(3):759–769, July 2003.

[MTM96] Michael I. Mishchenko, Larry D. Travis, and Daniel W. Mackowski.
T-matrix computations of light scattering by nonspherical particles: A review.
*J. Quant. Spectrosc. Radiat. Transf.*, 55(5):535–575, 1996.

[NB11]
Colette Naud and André Bergeron.
La lumière et l'éclairage, 2011.

[NDM05]
A. Ngan, F. Durand, and W. Matusik.
Experimental Analysis of BRDF Models.
In *Proc. Eurographics Symposium on Rendering*, pages 117–226, 2005.

[NKRS15]
Tobias Nöll, Johannes Köhler, Gerd Reis, and Didier Stricker.
Fully automatic, omnidirectional acquisition of geometry and appearance in the
context of cultural heritage preservation.
*Journal on Computing and Cultural Heritage*, 8:1–28, 02 2015.

[NRH⁺77]
F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis.
*Geometrical Considerations and Nomenclature for Reflectance*.
National Bureau of Standards, 1977.

[NVi18]
NVidia.
Nvidia rtx platform.
https://developer.nvidia.com/rtx, 2018.

[OLK⁺21]
Y. Ouyang, S. Liu, M. Kettunen, M. Pharr, and J. Pantaleoni.
Restir gi: Path resampling for real-time path tracing.
*Computer Graphics Forum*, 40(8):17–29, 2021.

[Opt19]
Nvidia optix ai-accelerated denoiser.
https://developer.nvidia.com/optix-denoiser, 2019.

[PBD⁺10]
Steven G. Parker, James Bigler, Andreas Dietrich, Heiko Friedrich, Jared Hoberock, David Luebke, David McAllister, Morgan McGuire, Keith Morley, Austin Robison, and Martin Stich.
Optix: A general purpose ray tracing engine.
*ACM Trans. Graph.*, 29(4), jul 2010.

[PBPP11]
Anthony Pajot, Loïc Barthe, Mathias Paulin, and Pierre Poulin.
Representativity for robust and adaptive multiple importance sampling.
*IEEE Trans. Visualization and Computer Graphics*, 17(8):1108–1121, 2011.

[PG10]
J.M. Palmer and B.G. Grant.
*The Art of Radiometry*.
Press Monographs. SPIE Press, 2010.
[PH10] Matt Pharr and Greg Humphreys.
*Physically Based Rendering: From Theory to Implementation.*
Morgan Kaufmann, 2 edition, 2010.

[Pho75] Bui Tuong Phong.
Illumination for computer generated pictures.
*Commun. ACM*, 18(6):311–317, June 1975.

[PJH16] Matt Pharr, Wenzel Jakob, and Greg Humphreys.
*Physically Based Rendering.*
Morgan Kaufmann, November 2016.

[Ray07] Lord Rayleigh.
On the dynamical theory of gratings.
*Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character*, 79(532):399–416, 1907.

[Rei08] Vincent Reillon.
*Characterization and modelling of the optical and colorimetric properties of lustred ceramics.*
Thesis, Université Pierre et Marie Curie - Paris VI, October 2008.

[Ric51a] Stephen O. Rice.
Reflection of electromagnetic waves from slightly rough surfaces.
*Communications on Pure and Applied Mathematics*, 4(2-3):351–378, 1951.

[Ric51b] Stephen O. Rice.
Reflection of electromagnetic waves from slightly rough surfaces.
*Communications on Pure and Applied Mathematics*, 4(2-3):351–378, 1951.

[Ric72] William Hadley Richardson.
Bayesian-based iterative method of image restoration*.
*J. Opt. Soc. Am.*, 62(1):55–59, Jan 1972.

[Rus98] Szymon Rusinkiewicz.
A new change of variables for efficient BRDF representation.
In *Proc. EGWR '98*, pages 11–22, 1998.

[RVZ08] Fabiano Romeiro, Yuriy Vasilyev, and Todd Zickler.
Passive reflectometry.
In *Proc. Eur. Conf. on Comp. Vision*, pages 859–872, 2008.

[RWS$^{+}$11] Peiran Ren, Jiaping Wang, John Snyder, Xin Tong, and Baining Guo.
Pocket reflectometry.

In *ACM SIGGRAPH 2011 Papers*, SIGGRAPH '11, pages 45:1–45:10, New York, NY, USA, 2011. ACM.

[SAWG91] François X. Sillion, James R. Arvo, Stephen H. Westin, and Donald P. Greenberg. A global illumination solution for general reflectance distributions. In *Proc. ACM SIGGRAPH '91*, pages 187–196, 1991.

[SBN15] Cyril Soler, Mahdi M. Bagher, and Derek Nowrouzezahrai. Efficient and accurate spherical kernel integrals using isotropic decomposition. *ACM Trans. Graph.*, 34(5), November 2015.

[SCV07] O. Salazar Celis, A. Cuyt, and B. Verdonk. Rational approximation of vertical segments. *Numerical Algorithms*, 45:375–388, 2007.

[SDC⁺¹¹] Sven Schröder, Angela Duparré, Luisa Coriand, Andreas Tünnermann, Dayana H. Penalver, and James E. Harvey. Modeling of light scattering in different regimes of surface roughness. *Opt. Express*, 19(10):9820–9835, May 2011.

[SHSK19] M. Sbert, V. Havran, and L. Szirmay-Kalos. Optimal deterministic mixture sampling. In *Eurographics (Short Papers)*, pages 73–76, 2019.

[Smi67] B. Smith. Geometrical shadowing of a random rough surface. *IEEE Transactions on Antennas and Propagation*, 15(5), September 1967.

[Sol21] Cyril Soler. private communication, February 2021.

[SP94] François X. Sillion and Claude Puech. *Radiosity and Global Illumination*, volume 11. Morgan Kaufmann Publishers, 1994.

[SR09] Olga Sorkine and Michael Rabinovich. Least-squares rigid motion using svd. Technical Report February, 2009.

[SSK03] Mirko Sattler, Ralf Sarlette, and Reinhard Klein. Efficient and realistic visualization of cloth.
In *Proceedings of the 14th Eurographics Workshop on Rendering*, EGRW '03, pages 167–177, Aire-la-Ville, Switzerland, Switzerland, 2003. Eurographics Association.

[SSW<sup>+</sup>14] Christopher Schwartz, Ralf Sarlette, Michael Weinmann, Martin Rump, and Reinhard Klein.
Design and implementation of practical bidirectional texture function measurement devices focusing on the developments at the university of bonn.
*Sensors*, 14(5), April 2014.

[SSWK13a] Christopher Schwartz, Ralf Sarlette, Michael Weinmann, and Reinhard Klein.
Dome ii: A parallelized btf acquisition system.
In *Holly Rushmeier and Reinhard Klein, editors, Eurographics Workshop on Material Appearance Modeling: Issues and Acquisition*, pages 25–31. Eurographics Association, June 2013.

[SSWK13b] Christopher Schwartz, Ralf Sarlette, Michael Weinmann, and Reinhard Klein.
DOME II: a parallelized BTF acquisition system.
*Workshop on Material Appearance Modeling (2013)*, pages 25–31, 2013.

[Sta99] Jos Stam.
Diffraction shaders.
In *SIGGRAPH '99*. ACM, 1999.

[Sta14] Doekele Stavenga.
Thin film and multilayer optics cause structural colors of many insects and birds.
*Materials Today: Proceedings*, 1, 12 2014.

[Sto95] John C. Stover.
*Optical Scattering: Measurement and Analysis*.
SPIE Optical Engineering Press, 1995.

[SWR<sup>+</sup>11] Christopher Schwartz, Michael Weinmann, Roland Ruiters, Arno Zinke, Ralf Sarlette, and Reinhard Klein.
Capturing shape and reflectance of food.
In *SIGGRAPH Asia 2011 Sketches*, SA '11, pages 28:1–28:2, New York, NY, USA, December 2011. ACM.

[SWRK11] Christopher Schwartz, Michael Weinmann, Roland Ruiters, and Reinhard Klein.
Integrated High-Quality Acquisition of Geometry and Appearance for Cultural Heritage.
In *The 12th International Symposium on Virtual Reality, Archaeology and Cultural Heritage VAST 2011*, pages 25–32, 2011.
[TCE05]
Justin F. Talbot, David Cline, and Parris Egbert.
Importance resampling for global illumination.
In *Proceedings of the Sixteenth Eurographics Conference on Rendering Techniques*, EGSR '05, page 139–146, Goslar, DEU, 2005. Eurographics Association.

[Tha03]
Fritz Thaler.
Die diffuse reflexion des lichtes an matten oberflächen.
*Annalen der Physik*, 316(8):996–1019, 1903.

[TS66]
K. E. Torrance and E. M. Sparrow.
Off-Specular Peaks in the Directional Distribution of Reflected Thermal Radiation.
*Journal of Heat Transfer*, 88(2):223–230, 05 1966.

[TU⁺²⁰]
Tanaboon Tongbuasirilai, Jonas Unger, Joel Kronander, and Murat Kurt.
Compact and intuitive data-driven brdf models.
*The Visual Computer*, 36(4):855–872, Apr 2020.

[Vea97]
Eric Veach.
*Robust Monte Carlo methods for light transport simulation*, volume 1610.
Stanford University PhD thesis, 1997.

[Vea98a]
Eric Veach.
*Robust Monte Carlo Methods for Light Transport Simulation*.
PhD thesis, Stanford University, Stanford, CA, USA, 1998.
AAI9837162.

[Vea98b]
Eric Veach.
*Robust Monte Carlo Methods for Light Transport Simulation*.
PhD thesis, Stanford University, Stanford, CA, USA, 1998.
AAI9837162.

[VH98]
C. L. Vernold and J. E. Harvey.
A modified beckmann-kirchhoff scattering theory.
*Proc. SPIE*, 3426, Scattering and Surface Roughness II:51–56, 1998.

[WAT92]
Stephen Westin, James Arvo, and Kenneth Torrance.
Predicting reflectance functions from complex surfaces.
In *Proc. ACM SIGGRAPH '92*, pages 255–264, 1992.

[WDR11]
Hongzhi Wu, Julie Dorsey, and Holly Rushmeier.
Physically-based interactive bi-scale material design.
In *Proceedings of the 2011 SIGGRAPH Asia Conference*, SA '11, New York, NY, USA, 2011. Association for Computing Machinery.
[WK15] Michael Weinmann and Reinhard Kleiny.
Advances in geometry and reflectance acquisition (Course Notes).
In *SIGGRAPH Asia 2015 Courses, SA 2015*. Association for Computing Machinery,
Inc, nov 2015.

[WMLT07] B. Walter, S. Marschner, H. Li, and K. E. Torrance.
Microfacet models for refraction through rough surfaces.
In *Eurographics Symposium on Rendering*, pages 195–206, 2007.

[WND⁺14] A. Wilkie, S. Nawaz, M. Droske, A. Weidlich, and J. Hanika.
Hero wavelength spectral sampling.
In *Proceedings of the 25th Eurographics Symposium on Rendering*, EGSR '14,
pages 123–131, Aire-la-Ville, Switzerland, Switzerland, 2014. Eurographics
Association.

[WW07] A. Weidlich and A. Wilkie.
Arbitrarily layered micro-facet surfaces.
In *Proc. of the 5th International Conference on Computer Graphics and Interactive
Techniques in Australia and Southeast Asia*, GRAPHITE '07, page 171–178.
ACM, 2007.

[WW09] A. Weidlich and A. Wilkie.
Exploring the potential of layered brdf models.
In *ACM SIGGRAPH ASIA 2009 Courses*, SIGGRAPH ASIA '09, pages 7:1–7:58.
ACM, 2009.

[Yee66] Kane S. Yee.
Numerical solution of initial boundary value problems involving maxwell's equations
in isotropic media.
*IEEE Trans. Antennas and Propagation*, pages 302–307, 1966.

[YHW⁺18] Ling-Qi Yan, Miloš Hašan, Bruce Walter, Steve Marschner, and Ravi Ramamoorthi.
Rendering specular microgeometry with wave optics.
*ACM Trans. Graph.*, 37(4), jul 2018.

[YLL⁺18] Bo Yang, Wenwei Liu, Zhancheng Li, Hua Cheng, Shuqi Chen, and Jianguo Tian.
Polarization-sensitive structural colors with hue-and-saturation tuning based on
all-dielectric nanopixels.
*Advanced Optical Materials*, 6(4):1701009, 2018.

[ZCD⁺16] Zhiming Zhou, Guojun Chen, Yue Dong, David Wipf, Yong Yu, John Snyder, and Xin
Tong.
Sparse-as-possible svbrdf acquisition.

*ACM Trans. Graph.*, 35(6):189:1–189:12, November 2016.

[ZERB05] Todd Zickler, Sebastian Enrique, Ravi Ramamoorthi, and Peter Belhumeur.
Reflectance sharing: Image-based rendering from a sparse set of images.
In *Proc. Eurographics Symposium on Rendering*, pages 253–265, 2005.

[Zha00] Zhengyou Zhang.
A flexible new technique for camera calibration.
Technical Report 11, Microsoft Research, 2000.

[ZJL+15] M. Zwicker, W. Jarosz, J. Lehtinen, B. Moon, R. Ramamoorthi, F. Rousselle, P. Sen, C. Soler, and S.-E. Yoon.
Recent advances in adaptive sampling and reconstruction for monte carlo rendering.
*Comput. Graph. Forum*, 34(2):667–681, May 2015.

[ZTZ05] O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu.
*The Finite Element Method: Its Basis and Fundamentals, Sixth Edition*.
Butterworth-Heinemann, 6 edition, May 2005.
<figure><img src="image_77.png" /></figure>
BIBLIOGRAPHY

[1] Ramon Hegedus, Antoine Lucat, Justine Redon, and Romain Pacanowski.
Isotropic BRDF Measurements with Quantified Uncertainties.
In *EUROGRAPHICS WORKSHOP ON MATERIAL APPEARANCE MODELING*, Dublin,
Ireland, June 2016.

[2] Antoine Lucat, Ramon Hegedus, and Romain Pacanowski.
Diffraction Removal in an Image-based BRDF Measurement Setup.
In *EI 2018 - Electronic Imaging Material Appearance 2018*, pages 1–6, Burlingame,
California, United States, January 2018.

[3] Antoine Lucat, Ramon Hegedus, and Romain Pacanowski.
Diffraction effects detection for HDR image-based measurements.
*Optics Express*, 25(22):2921–2929, October 2017.

[4] Antoine Lucat, Ramon Hegedus, and Romain Pacanowski.
Diffraction Prediction in HDR measurements.
In *EUROGRAPHICS WORKSHOP ON MATERIAL APPEARANCE MODELING*, Helsinki,
Finland, 2017.

[5] Romain Pacanowski, Xavier Granier, Christophe Schlick, and Poulin Pierre.
Sketch and Paint-based Interface for Highlight Modeling.
In *Fifth Eurographics conference on Sketch-Based Interfaces and Modeling*, Annecy, France,
June 2008.

[6] Romain Vergne, Romain Pacanowski, Pascal Barla, Xavier Granier, and Christophe Schlick.
Light Warping for Enhanced Surface Depiction.
*ACM Transactions on Graphics*, 28(3):25:1–25:8, July 2009.

[7] Romain Vergne, Romain Pacanowski, Pascal Barla, Xavier Granier, and Christophe Schlick.
Improving Shape Depiction under Arbitrary Rendering.
*IEEE Transactions on Visualization and Computer Graphics*, 17(8):1071–1081, June 2011.

[8] Boris Raymond, Pascal Barla, Romain Pacanowski, and Xavier Granier.
Understanding Anisotropic Highlights for Interactive Appearance Editing.
PRISM2: The science of light & shade, October 2013.
Poster.

[9] Romain Pacanowski, Oliver Salazar-Celis, Christophe Schlick, Xavier Granier, Poulin Pierre, and Cuyt Annie.
Rational BRDF.
*IEEE Transactions on Visualization and Computer Graphics*, 18(11):1824–1835, March 2012.

[10] Laurent Belcour, Romain Pacanowski, Marion Delahaie, Aude Laville-Geay, and Laure Eupherte.
BRDF Measurements and Analysis of Retroreflective Materials.
*Journal of the Optical Society of America*, 31(12):2561–2572, December 2014.

[11] Kevin Vynck, Romain Pacanowski, Pascal Barla, Xavier Granier, and Philippe Lalanne.
Engineering nanostructured surfaces with correlated disorder for visual appearance design.
In *Workshop on "Correlated Disorder and hyperuniformity in soft matter and photonics"*,
Paris, France, October 2018.

[12] Heqi Lu, Romain Pacanowski, and Xavier Granier.
Position-Dependent Importance Sampling of Light Field Luminaires.
*IEEE Transactions on Visualization and Computer Graphics*, 21(2):241–251, 2015.

[13] Romain Pacanowski, Xavier Granier, Christophe Schlick, and Poulin Pierre.
Volumetric Vector-based Representation for Indirect Illumination Caching.
*Journal of Computer Science and Technology (JCST)*, 25(5):925–932, September 2010.

[14] Heqi Lu, Romain Pacanowski, and Xavier Granier.
Real-Time Importance Sampling of Dynamic Environment Maps.
In *Eurographics 2013 - Short Papers*, pages 65–68, Girona, Spain, May 2013. Eurographics.

[15] Heqi Lu, Romain Pacanowski, and Xavier Granier.
Second-Order Approximation for Variance Reduction in Multiple Importance Sampling.
*Computer Graphics Forum*, 32(7):131–136, October 2013.

[16] David Murray, Sofiane Benzait, Romain Pacanowski, and Xavier Granier.
On Learning the Best Balancing Strategy.
In *Eurographics 2020 - Annual Conference of the European Association for Computer Graphics*, volume xx, pages 1–4, Norrköping / Virtual, Sweden, May 2020.

[17] Arthur Dufay, Pascal Lecocq, Romain Pacanowski, Jean-Eudes Marvie, and Xavier Granier.
Cache-friendly micro-jittered sampling.
In *SIGGRAPH 2016*, Anaheim, United States, July 2016.
[18] Mégane Bati, Romain Pacanowski, and Pascal Barla.
Comparative study of layered material models.
In H. Rushmeier and R. Klein, editors, *Workshop on Material Appearance Modeling*,
Strasbourg, France, July 2019.

[19] Mégane Bati, Pascal Barla, and Romain Pacanowski.
An Inverse Method for the Exploration of Layered Material Appearance.
*ACM Transactions on Graphics*, August 2021.

[20] Laurent Belcour, Oliver Salazar-Celis, and Romain Pacanowski.
An Addendum to Rational Function Fitting.
Technical report, p., December 2013.
This report is an addendum to the Rational BRDF publication available in hal-00678885.

[21] Laurent Belcour, Pascal Barla, and Romain Pacanowski.
ALTA: A BRDF analysis library.
In Holly Rushmeier and Reinhard Klein, editors, *EGSR Workshop on Material Appearance Modeling*, Workshop on Material Appearance Modeling 2014, Lyon, France, June 2014.
Eurographics.

[22] Nicolas Holzschuch and Romain Pacanowski.
A Two-Scale Microfacet Reflectance Model Combining Reflection and Diffraction.
*ACM Transactions on Graphics*, 36(4):12, July 2017.
Article 66.

[23] Pascal Barla, Laurent Belcour, and Romain Pacanowski.
In Praise of an Alternative BRDF Parametrization.
In *Workshop on Material Appearance Modeling*, Proceedings of the Workshop on Material Appearance Modeling 2015, Darmstadt, Germany, June 2015.

[24] Romain Pacanowski.
I3D Presentation of TVCG Paper Rational BRDF.
In *ACM Siggraph Symposium on Interactive 3D Graphics and Games*, Orlando, United States, March 2013.

[25] Nicolas Holzschuch and Romain Pacanowski.
A physically accurate reflectance model combining reflection and diffraction.
Research Report RR-8807, INRIA, November 2015.

[26] Nicolas Holzschuch and Romain Pacanowski.
The Effects of Digital Cameras Optics and Electronics for Material Acquisition.
In H. Rushmeier and R. Klein, editors, *Workshop on Material Appearance Modeling*, pages 1–4, Helsinki, Finland, June 2017.
[27] Arthur Dufay, David Murray, Romain Pacanowski, et al.
The Malia Rendering Framework.
https://pacanows.gitlabpages.inria.fr/MRF/main.md.html, 2019.

[28] Mégane Bati, Romain Pacanowski, and Pascal Barla.
Numerical Analysis of Layered Materials Models.
Research report, Institut d'Optique Graduate School ; Université de Bordeaux ; CNRS ;
INRIA, June 2019.

[29] David Murray, Alban Fichet, and Romain Pacanowski.
Efficient Spectral Rendering on the GPU for Predictive Rendering.
In *Ray Tracing Gems II*, pages 673 – 698. Springer, 2021.

[30] Kevin Vynck, Romain Pacanowski, Adrian Agreda, Arthur Dufay, Xavier Granier, and Philippe Lalanne.
The visual appearances of disordered optical metasurfaces.
*Nature Materials*, 21:1035–1041, May 2022.

[31] Aurélie Mounier, Markus Schlicht, Maud Mulliez, Romain Pacanowski, Antoine Lucat, and Pascal Mora.
In search of the lost polychromy of English medieval alabaster panels in the Southwest of France.
*Color Research and Application*, February 2020.