DOCSLIB.ORG

Efficient Freeform Lens Optimization for Computational Caustic Displays

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Efficient Freeform Lens Optimization for Computational Caustic Displays Efficient freeform lens optimization for computational caustic displays Gerwin Damberg1;3 and Wolfgang Heidrich2;1;4 1Department of Computer Science, University of British Columbia 2Visual Computing Center, King Abdullah University of Science and Technology [email protected] [email protected] www.cs.ubc.ca/ gdamberg/ Abstract: Phase-only light modulation shows great promise for many imaging applications, including future projection displays. While images can be formed efficiently by avoiding per-pixel attenuation of light most projection efforts utilizing phase-only modulators are based on holographic principles utilizing interference of coherent laser light and a Fourier lens. Limitations of this type of an approach include scaling to higher power as well as visible artifacts such as speckle and image noise. We propose an alternative approach: operating the spatial phase modulator with broadband illumination by treating it as a programmable freeform lens. We describe a simple optimization approach for generating phase modulation patterns or freeform lenses that, when illuminated by a colli- mated, broadband light source, will project a pre-defined caustic image on a designated image plane. The optimization procedure is based on a simple geometric optics image formation model and can be implemented compu- tationally efficient. We perform simulations and show early experimental results that suggest that the implementation on a phase-only modulator can create structured light fields suitable, for example, for efficient illumination of a spatial light modulator (SLM) within a traditional projector. In an alternative application, the algorithm provides a fast way to compute geometries for static, freeform lens manufacturing. © 2015 Optical Society of America OCIS codes: (080.4225) Nonspherical lens design; (120.2040) Displays; (100.3190) Inverse problems; (110.1758) Computational imaging. References and links 1. L. Lesem, P. Hirsch, and J. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM Journal of Research and Development 13, 150–155 (1969). 2. P. R. Haugen, H. Bartelt, and S. K. Case, “Image formation by multifacet holograms,” Applied Optics 22, 2822– 2829 (1983). 3. G. Damberg, H. Seetzen, G. Ward, W. Heidrich, and L. Whitehead, “3.2: High dynamic range projection sys- tems,” in “SID Symposium Digest of Technical Papers,” , vol. 38 (Wiley Online Library, 2007), vol. 38, pp. 4–7. 4. M. Berry, “Oriental magic mirrors and the laplacian image,” European journal of physics 27, 109 (2006). 5. M. Papas, W. Jarosz, W. Jakob, S. Rusinkiewicz, W. Matusik, and T. Weyrich, “Goal-based caustics,” in “Com- puter Graphics Forum,” , vol. 30 (Wiley Online Library, 2011), vol. 30, pp. 503–511. 6. T. Kiser, M. Eigensatz, M. M. Nguyen, P. Bompas, and M. Pauly, Architectural CausticsControlling Light with Geometry (Springer, 2013). 7. Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Transactions on Graphics (TOG) 33, 74 (2014). 8. Y. Yue, K. Iwasaki, B.-Y. Chen, Y. Dobashi, and T. Nishita, “Pixel art with refracted light by rearrangeable sticks,” in “Computer Graphics Forum,” , vol. 31 (Wiley Online Library, 2012), vol. 31, pp. 575–582. 9. Y. Ohno, “Color rendering and luminous efficacy of white led spectra,” in “Optical Science and Technology, the SPIE 49th Annual Meeting,” (International Society for Optics and Photonics, 2004), pp. 88–98. 1. Introduction In this work we propose to use phase only spatial light modulation combined with broadband illumination for image formation. We achieve this by treating the spatial phase modulator as a programmable freeform lens, and devising a simple and computationally efficient optimization procedure to derive a lens surface or modulation pattern that will form a caustic representing a predefined target image when illuminated by a collimated, broadband light source. Our research draws from a number of different research fields, including: Holographic Displays. Early holographic image formation models [1] have been adapted to create digital holograms [2]. Most of the common approaches require coherent light which has several disadvantages. Coherent light can result in high resolution artifacts, including screen speckle and diffraction on structures such as the discrete pixel grid of a SLM. On the other hand, using broadband light sources can eliminate screen speckle, and low-frequency modulation patterns can significantly reduce diffraction artifacts. Remaining diffraction is averaged out by the broadband nature of the illumination, resulting in a small amount of blur that can be modeled and compensated for [3]. Freeform Lenses. Recently, there has been strong interest in freeform lens design, both for general lighting applications and also to generate images from caustics [4]. In the latter application, we can distinguish between discrete optimization methods that work on a pixelated version of the problem (e.g. [5]), and those that optimize for continuous surfaces without obvious pixel structures (e.g. [6, 7, 8]). The current state of the art [8] defines an optimization problem on the gradients of the lens surface, which then have to be integrated up into a height field. In addition to low computational performance, this leads to a tension between satisfying a data term (the target caustic image) and maintaining the integrability of the gradient field. In our work we derive a simple and efficient formulation in which we optimize directly for the phase function (i.e. the shape of the wavefront in the lens plane) without the need for a subsequent integration step. This is made possible by a new parameterization of the problem that allows us to express the optimization directly in the lens plane rather than the image plane. 2. Freeform Lensing 2.1. Phase Modulation Image Formation To derive the image formation model for a phase modulator, we consider the geometry shown in Fig. 1: a lens plane and an image plane (screen) are parallel to each other at focal distance f . Collimated light is incident at the lens plane from the normal direction, but a phase modulator in the lens plane distorts the phase of the light, resulting in a curved phase function p(x) which corresponds to a local deflection of the light rays. lens image plane plane f φ u-x x=(x,y) u=(u,v) phase function image p(x) i(u) Fig. 1. Geometry for image formation model: Phase modulation in lens plane at focal dis- tance f from image plane resulting in curvature of the wavefront (phase function p(x)). Light deflection. With the paraxial approximation sinf ≈ f we obtain the following equation for the mapping between x on the lens plane and u on the image plane: u(x) ≈ x + f · ∇p(x): (1) Intensity modulation. Using the above geometric mapping, we derive the intensity change associated with this distortion as follows. Let dx be a differential area on the lens plane, and let du = m(x) · dx be the differential area of the corresponding region in the image plane, where m(:) is a spatially varying magnification factor. The intensity on the image plane is then given as dx 1 i(u(x)) = i = i ; (2) du 0 m(x) 0 where i0 is the intensity of the collimated light incident at the lens plane. In the following we set i0 = 1 for simplicity of notation. lens plane image plane ε x+(0, ) u(x+(0,ε)) dx m⋅dx u(x+(ε,0)) x x+(ε,0) u(x) intensity i0 intensity i0/m Fig. 2. Intensity change due to distortion of a differential area dx. The magnification factor m(:) can be expressed in terms of the derivatives of the mapping between the lens and image planes (also compare Fig. 2): ¶ ¶ m(x) = u(x) × u(x) ≈ 1 + f · ∇2 p(x): (3) ¶x ¶y This yields the following expression for the intensity distribution in the image plane: 1 i(x + f · ∇p(x)) = : (4) 1 + f · ∇2 p(x) In other words, the magnification m, and therefore the intensity i(u) on the image plane can be directly computed from the Laplacian of the scalar phase function in the lens plane. 2.2. Optimization Problem While it is possible to directly turn the image formation mode from Equation 4 into an op- timization problem, we found that we can achieve better convergence by first linearizing the equation with a first-order Taylor approximation, which yields i(x + f · ∇p(x)) ≈ 1 − f · ∇2 p(x); (5) where the left hand side can be interpreted as a warped image ip(x) = i(x + f · ∇p(x)) for which the target intensity i(u) in the image plane has been warped backwards onto the lens plane using the geometric distortion u(x) produced by a known phase function p(x). With this parameterization, the continuous least-square optimization problem for determining the desired phase function becomes Z 2 2 pˆ(x) = argminp(x) ip(x) − 1 + f · ∇ p(x) dx: (6) x This problem can be solved by iterating between updates to the phase function and updates to the warped image, as shown in Algorithm 1: Algorithm 1 Freeform lens optimization // Initialization 0 ip(x) = i(u) while not converged do // phase update 2 (k) R (k−1) 2 p (x)=argminp(x) x ip (x) − 1 + f · ∇ p(x) dx // image warp (k) k ip (x)=i(x + f · ∇p (x)) end while After discretization of i(:) and p(:) into pixels, the phase update corresponds to solving a linear least squares problem with a discrete Laplace operator as the system matrix. We can solve this positive semi-definite system using a number of different algorithms, including Conjugate Gradient, BICGSTAB and Quasi Minimal Residual (QMR). The image warp corresponds to a texture mapping operation and can be implemented on a GPU. We implement a non-optimized prototype of the algorithm in the Matlab programming environment using QMR as the least squares solver.
Load more

Recommended publications
  • Classical and Modern Diffraction Theory
    Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Classical and Modern Diffraction Theory Edited by Kamill Klem-Musatov Henning C. Hoeber Tijmen Jan Moser Michael A. Pelissier SEG Geophysics Reprint Series No. 29 Sergey Fomel, managing editor Evgeny Landa, volume editor Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Society of Exploration Geophysicists 8801 S. Yale, Ste. 500 Tulsa, OK 74137-3575 U.S.A. # 2016 by Society of Exploration Geophysicists All rights reserved. This book or parts hereof may not be reproduced in any form without permission in writing from the publisher. Published 2016 Printed in the United States of America ISBN 978-1-931830-00-6 (Series) ISBN 978-1-56080-322-5 (Volume) Library of Congress Control Number: 2015951229 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Dedication We dedicate this volume to the memory Dr. Kamill Klem-Musatov. In reading this volume, you will find that the history of diffraction We worked with Kamill over a period of several years to compile theory was filled with many controversies and feuds as new theories this volume. This volume was virtually ready for publication when came to displace or revise previous ones. Kamill Klem-Musatov’s Kamill passed away. He is greatly missed. new theory also met opposition; he paid a great personal price in Kamill’s role in Classical and Modern Diffraction Theory goes putting forth his theory for the seismic diffraction forward problem.
    [Show full text]
  • Image-Space Caustics and Curvatures
    Image-space Caustics and Curvatures Xuan Yu Feng Li Jingyi Yu Department of Computer and Information Sciences University of Delaware Newark, DE 19716, USA fxuan,feng,[email protected] Abstract Caustics are important visual phenomena, as well as challenging global illumination effects in computer graph- ics. Physically caustics can be interpreted from one of two perspectives: in terms of photons gathered on scene geom- etry, or in terms of a pair of caustic surfaces. These caustic surfaces are swept by the foci of light rays. In this paper, we develop a novel algorithm to approximate caustic sur- faces of sampled rays. Our approach locally parameterizes rays by their intersections with a pair of parallel planes. We show neighboring ray triplets are constrained to pass simul- taneously through two slits, which rule the caustic surfaces. We derive a ray characteristic equation to compute the two slits, and hence, the caustic surfaces. Using the characteris- tic equation, we develop a GPU-based algorithm to render the caustics. Our approach produces sharp and clear caus- tics using much fewer ray samples than the photon mapping method and it also maintains high spatial and temporal co- herency. Finally, we present a normal-ray surface repre- Figure 1. We use our caustic-surface-based sentation that locally parameterizes the normals about a algorithm to render the refraction caustics surface point as rays. Computing the normal ray caustic cast by a crystal bunny of 69473 triangles. On surfaces leads to a novel real-time discrete shape operator. an NVidia GeForce7800, our method renders at 115 fps at an image resolution of 512x512.
    [Show full text]
  • Caustic Architecture Article
    Tricks'of'the'light' An#extended#version#of#an#article#in#New$Scientist,#30#January#2013# # Two#men#enter#the#darkened#stage,#apparently#carrying#a#thick#slab#of#badly# made#glass,#like#the#stuff#in#the#windows#of#old#houses#that#turns#the#world# outside#wobbly.#They#hold#it#up,#and#computer#scientist#Mark#Pauly#shines#a# torch#at#it.#The#crowd#in#this#Parisian#auditorium#gasps#and#then#breaks#into# spontaneous#applause.# For#there#on#the#screen#behind,#conjured#out#of#this#piece#of#nearFfeatureless# material#–#not#glass,#in#fact,#but#transparent#acrylic#plastic#(Perspex)#–#is#a# projected#image#of#Alan#Turing,#the#computer#pioneer#whose#centenary#is# celebrated#this#year.#Every#thread#of#his#thick#tweed#jacket#is#picked#out#in#light# and#shadow.#But#where#is#the#image#coming#from?#It#can#only#be#the#transparent# slab,#but#there#seems#to#be#nothing#there#to#produce#it,#nothing#but#a#slightly# uneven#surface.# # An#image#of#Alan#Turing#conjured#from#light#passing#through#a#slab#of#Perspex#at#the# Advances#in#Architectural#Geometry#conference#in#Paris,#September#2012.# This#image#is#made#from#rays#refracted,#folded#and#focused#by#the#slightly# uneven#surface#of#the#acrylic#block.#It’s#similar#to#the#filigree#of#bright#bands#seen# on#the#bottom#of#a#swimming#pool#in#the#sunlight,#called#a#caustic#and#caused#by# the#way#the#wavy#surface#refracts#and#focuses#light.#Caustics#are#familiar#enough,# but#they#never#looked#like#this#before.#Those#made#by#sunlight#shining#through# an#empty#glass#are#a#random#mass#of#cusps#and#squiggles.#Pauly,#a#specialist#in#
    [Show full text]
  • Real-Time Caustics
    EUROGRAPHICS 2003 / P. Brunet and D. Fellner Volume 22 (2003), Number 3 (Guest Editors) Real-Time Caustics M. Wand and W. Straßer WSI/GRIS, University of Tübingen Abstract We present a new algorithm to render caustics. The algorithm discretizes the specular surfaces into sample points. Each of the sample points is treated as a pinhole camera that projects an image of the incoming light onto the diffuse receiver surfaces. Anti-aliasing is performed by considering the local surface curvature at the sample points to filter the projected images. The algorithm can be implemented using programmable texture mapping hardware. It allows to render caustics in fully dynamic scenes in real-time on current PC hardware. Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture / Image Generation – Display Algo- rithms; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism 1. Introduction tion step has to be performed that is often even more ex- pensive. Thus, the technique is usually not very efficient. A real-time simulation of the interaction of light with Although the implementation techniques for raytracing complex, dynamically changing scenery is still one of the queries have made impressive advances in the last few major challenges in computer graphics. In this paper, we years29, raytracing based algorithms still need a consider- look at a special global illumination problem, rendering of able amount of computational power (such as a cluster of caustics. Caustics occur if light is reflected (or refracted) at several high end CPUs) to calculate global illumination one or more specular surfaces, focused into ray bundles of solutions in real time30.
    [Show full text]
  • Adaptive Spectral Mapping for Real-Time Dispersive Refraction By
    Adaptive Spectral Mapping for Real-Time Dispersive Refraction by Damon Blanchette A Thesis Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the Degree of Master of Science in Computer Science by ___________________________________ January 2012 APPROVED: ___________________________________ Professor Emmanuel Agu, Thesis Adviser ___________________________________ Professor Matthew Ward, Thesis Reader ___________________________________ Professor Craig Wills, Head of Department Abstract Spectral rendering, or the synthesis of images by taking into account the wavelengths of light, allows effects otherwise impossible with other methods. One of these effects is dispersion, the phenomenon that creates a rainbow when white light shines through a prism. Spectral rendering has previously remained in the realm of off-line rendering (with a few exceptions) due to the extensive computation required to keep track of individual light wavelengths. Caustics, the focusing and de-focusing of light through a refractive medium, can be interpreted as a special case of dispersion where all the wavelengths travel together. This thesis extends Adaptive Caustic Mapping, a previously proposed caustics mapping algorithm, to handle spectral dispersion. Because ACM can display caustics in real-time, it is quite amenable to be extended to handle the more general case of dispersion. A method is presented that runs in screen-space and is fast enough to display plausible dispersion phenomena in real-time at interactive frame rates. i Acknowledgments I would like to thank my adviser, Professor Emmanuel Agu, for his guidance, laughs, and answering my hundreds of questions over the year it took to complete this thesis. The beautiful video card helped, too.
    [Show full text]
  • Advanced Computer Graphics CS 563: Adaptive Caustic Maps Using Deferred Shading
    Advanced Computer Graphics CS 563: Adaptive Caustic Maps Using Deferred Shading Frederik Clinckemaillie Computer Science Dept. Worcester Polytechnic Institute (WPI) ItIntrod ucti on: CtiCaustics Reflective Caustics Refractive Caustics IdiIntroduction: CiCaustic MiMapping Much faster than path tracing algorithms Two‐pass process Similar to photon mapping Creates a caustic intensity map CtiCaustic MiMapping Three Step Process 1) Photon Emission 2) Rearrangement into Caustic Map 3) Caustic Map Projection CtiCaustic MiMapping Phot on EiiEmission Rasterizes from light view to generate grid of photons Creates photon buffer 2D image storing final photon hit points In conjunction with shadow maps Allows quick lookups to determine Indirect lighting from caustics CtiCaustic Map CtiCreation Controls lighting quality and cost Splatting photons into caustic map becomes bo ttlenec k Crisp noise‐free images require millions of phthotons Not feasible in interactive time Hierarchical caustic maps: discard unimportant parts of photon buffer to improve speed Uses multi‐resolution caustic map to reduce splatting costs Problems Poor photon sampling due to rasterization leads to under and over sampling Proper sampling resolution cannot be determined Millions of photons are required for high‐quality caustics. Processing each is too expensive Photon sampling location can change between frames, leading to coherency problems Photon sampling is not dynamic DfDeferre d Shadi ng Good sampling rates cannot be computed Ideally, number of photons is determined adtildaptively Hierarchical Caustic Map(HCM) Has a maximum number of photons, not all are processed Pho ton EiiEmission and CtiCaustic Map GtiGeneration should be coupled DfDeferre d Shadi ng Postpones final illumination computations until visible fragments are identified HCMs generates grid of phthotons and only processes relevant ones DfDeferre d Sha ding never generates ilirrelevan t photons.
    [Show full text]
  • Controlling Caustics
    Controlling Caustics Mark Pauly1, Michael Eigensatz, Philippe Bompas, Florian Rist2, Raimund Krenmuller2 1 EPFL 2 TU Wien Keywords 1 = Caustic 2 = Refraction 3 = Refl ection 4 = Design 5 = Milling 6 = Slumping Abstract Caustics are captivating light patterns created by materials focusing or diverting Figure 1 Left: A typical caustic created by a curved glass object appears chaotic and random. Right: Our light by refraction or refl ection. We know method creates glass objects that cast controlled caustics by optimization of the glass surface geometry alone. caustics as random side effects, appearing, The portrait of Alan Turing emerges from a beam of uniform light that is refracted by the circular glass piece. for example, at the bottom of a swimming pool, or generated by many glass objects, like drinking glasses or bottles. In this paper we show that it is possible to control caustic patterns to form almost any desired shape by optimizing the geometry of the refl ective or refractive surface generating the caustic. A seemingly fl at glass window, for example, can produce the image of a person as a caustic pattern on the fl oor, generated solely by the sunlight entering through that window. We demonstrate how this surprising result offers a new perspective on light control and the use of caustics as an inspiring design element in architecture, product design and beyond. Several produced samples illustrate that physical realizations of such optimized geometry are feasible. 1 Introduction The interaction of light with glass plays an important role for the perception and functionality of many glass products. One of the most fascinating light phenomena with glass objects are caustics: light gets focused and diverted when passing through the glass, creating an intriguing pattern of varying light intensity.
    [Show full text]
  • Geometric Optics 1 7.1 Overview
    Contents III OPTICS ii 7 Geometric Optics 1 7.1 Overview...................................... 1 7.2 Waves in a Homogeneous Medium . 2 7.2.1 Monochromatic, Plane Waves; Dispersion Relation . ........ 2 7.2.2 WavePackets ............................... 4 7.3 Waves in an Inhomogeneous, Time-Varying Medium: The Eikonal Approxi- mationandGeometricOptics . .. .. .. 7 7.3.1 Geometric Optics for a Prototypical Wave Equation . ....... 8 7.3.2 Connection of Geometric Optics to Quantum Theory . ..... 11 7.3.3 GeometricOpticsforaGeneralWave . .. 15 7.3.4 Examples of Geometric-Optics Wave Propagation . ...... 17 7.3.5 Relation to Wave Packets; Breakdown of the Eikonal Approximation andGeometricOptics .......................... 19 7.3.6 Fermat’sPrinciple ............................ 19 7.4 ParaxialOptics .................................. 23 7.4.1 Axisymmetric, Paraxial Systems; Lenses, Mirrors, Telescope, Micro- scopeandOpticalCavity. 25 7.4.2 Converging Magnetic Lens for Charged Particle Beam . ....... 29 7.5 Catastrophe Optics — Multiple Images; Formation of Caustics and their Prop- erties........................................ 31 7.6 T2 Gravitational Lenses; Their Multiple Images and Caustics . ...... 39 7.6.1 T2 Refractive-Index Model of Gravitational Lensing . 39 7.6.2 T2 LensingbyaPointMass . .. .. 40 7.6.3 T2 LensingofaQuasarbyaGalaxy . 42 7.7 Polarization .................................... 46 7.7.1 Polarization Vector and its Geometric-Optics PropagationLaw. 47 7.7.2 T2 GeometricPhase .......................... 48 i Part III OPTICS ii Optics Version 1207.1.K.pdf, 28 October 2012 Prior to the twentieth century’s quantum mechanics and opening of the electromagnetic spectrum observationally, the study of optics was concerned solely with visible light. Reflection and refraction of light were first described by the Greeks and further studied by medieval scholastics such as Roger Bacon (thirteenth century), who explained the rain- bow and used refraction in the design of crude magnifying lenses and spectacles.
    [Show full text]
  • Rendering Techniques for Transparent Objects
    173 Rendering Techniques for Transparent Objects Mikio Shinya* Takafumi Saito Tokiichiro Takahashi NTT Human Interface Laboratories 1-2356, Take, Yokosuka-shi, Kanagawa, Japan Abstract situations, however, simple application of ray tracing is not Three techniques are proposed for more realistic adequate to render photo-realistic images of transparent objects, because other important phenomena occur due to image synthesis of transparent objects. The tech­ refraction and/or reflection. For example: niques simulate important phenomena due to refrac­ tion and/or reflection, such as focuses and disper­ • caustics and focuses; Refraction and reflection sion. For efficient simulation of focuses and caus­ causes light concentration, resulting in a complex pat­ tics, a method called grid-pencil tracing is proposed, tern of caustics and fo cuses. This phenomenon is of­ based on the ideas of pencil tracing and backward ten observed in water scenes (swimming pools, sea­ ray tracing. An anti-aliasing filter is designed for side, and so on) and can even be produced by a glass grid-pencil tracing in area-source illumination envi­ of water. ronments. The theory of pencil tracing is also ap­ • dispersion; Due to dispersion, a rainbow spectrum plied to dispersion simulation. An extension of the is often produced by transparent objects, such as prisms, theory provides a basis for dispersi ve pencil tracing, gemstones, and cut-glass. where both spatial coherency and 'wavelength co­ herency' is effectively used for fast image synthesis • extremely bright spots; Reflection of primary of dispersive objects. light sources are manifested as extremely bright spots. Furthermore, a linear filtering technique is ap­ The glint off a knife or shimmering light reflected off plied to effectively express extremely bright spots a body of water, for example.
    [Show full text]
  • Caustics Mapping: an Image-Space Technique for Real-Time Caustics Musawir Shah, Sumanta Pattanaik
    SCHOOL OF ENGINEERING AND COMPUTER SCIENCE, UNIVERSITY OF CENTRAL FLORIDA, CS TR 50-07 1 Caustics Mapping: An Image-space Technique for Real-time Caustics Musawir Shah, Sumanta Pattanaik Abstract— In this paper, we present a simple and prac- tical technique for real-time rendering of caustics from reflective and refractive objects. Our algorithm, concep- tually similar to shadow mapping, consists of two main parts: creation of a caustic map texture, and utilization of the map to render caustics onto non-shiny surfaces. Our approach avoids performing any expensive geometric tests, such as ray-object intersection, and involves no pre-computation; both of which are common features in previous work. The algorithm is well suited for the standard rasterization pipeline and runs entirely on the graphics hardware. I. INTRODUCTION AUSTICS are complex patterns of shimmering C light that can be seen on surfaces in presence of reflective or refractive objects, for example those formed on the floor of a swimming pool in sunlight. Caustics occur when light rays from a source, such as the sun, get refracted, or reflected, and converge at a single point on a non-shiny surface. This creates the non-uniform dis- Fig. 1. Image rendered using the caustics mapping algorithm. This result was obtained using double surface refraction (both for the tribution of bright and dark areas. Caustics are a highly appearance of the bunny as well as for the caustics) at the rate of desirable physical phenomenon in computer graphics due 31 fps. to their immersive visual appeal. Some very attractive results have been produced using off-line high quality rendering systems; however, real-time caustics remain open to more practical solutions.
    [Show full text]
  • Reparameterizing Discontinuous Integrands for Differentiable Rendering Guillaume Loubet, Nicolas Holzschuch, Wenzel Jakob
    Reparameterizing Discontinuous Integrands for Differentiable Rendering Guillaume Loubet, Nicolas Holzschuch, Wenzel Jakob To cite this version: Guillaume Loubet, Nicolas Holzschuch, Wenzel Jakob. Reparameterizing Discontinuous Integrands for Differentiable Rendering. ACM Transactions on Graphics, Association for Computing Machinery, 2019, 38 (6), pp.228:1 - 228:14. 10.1145/3355089.3356510. hal-02497191 HAL Id: hal-02497191 https://hal.inria.fr/hal-02497191 Submitted on 3 Mar 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Reparameterizing Discontinuous Integrands for Differentiable Rendering GUILLAUME LOUBET, École Polytechnique Fédérale de Lausanne (EPFL) NICOLAS HOLZSCHUCH, Inria, Univ. Grenoble-Alpes, CNRS, LJK WENZEL JAKOB, École Polytechnique Fédérale de Lausanne (EPFL) (a) (b) (c) (d) A scene with complex geometry and visibility (1.8M triangles) Gradients with respect to scene parameters that affect visibility Fig. 1. The solution of inverse rendering problems using gradient-based optimization requires estimates of pixel derivatives with respect to arbitrary scene parameters. We focus on the problem of computing such derivatives for parameters that affect visibility, such as the position and shape of scene geometry (a, c) and light sources (b, d). Our renderer re-parameterizes integrals so that their gradients can be estimated using standard Monte Carlo integration and automatic differentiation—even when visibility changes would normally make the integrands non-differentiable.
    [Show full text]
  • Computational Optimal Transport
    Full text available at: http://dx.doi.org/10.1561/2200000073 Computational Optimal Transport with Applications to Data Sciences Full text available at: http://dx.doi.org/10.1561/2200000073 Other titles in Foundations and Trends R in Machine Learning Non-convex Optimization for Machine Learningy Prateek Jain and Purushottam Ka ISBN: 978-1-68083-368-3 Kernel Mean Embedding of Distributions: A Review and Beyond Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur and Bernhard Scholkopf ISBN: 978-1-68083-288-4 Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions Andrzej Cichocki, Anh-Huy Phan, Qibin Zhao, Namgil Lee, Ivan Oseledets, Masashi Sugiyama and Danilo P. Mandic ISBN: 978-1-68083-222-8 Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives Andrzej Cichocki, Anh-Huy Phan, Qibin Zhao, Namgil Lee, Ivan Oseledets, Masashi Sugiyama and Danilo P. Mandic ISBN: 978-1-68083-276-1 Patterns of Scalable Bayesian Inference Elaine Angelino, Matthew James Johnson and Ryan P. Adams ISBN: 978-1-68083-218-1 Generalized Low Rank Models Madeleine Udell, Corinne Horn, Reza Zadeh and Stephen Boyd ISBN: 978-1-68083-140-5 Full text available at: http://dx.doi.org/10.1561/2200000073 Computational Optimal Transport with Applications to Data Sciences Gabriel Peyré CNRS and DMA, ENS Marco Cuturi Google and CREST/ENSAE Boston — Delft Full text available at: http://dx.doi.org/10.1561/2200000073 R Foundations and Trends in Machine Learning Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 United States Tel.
    [Show full text]