Looking against the light: How perception of translucency … · 2018-12-08 · Looking against the light: How perception of translucency depends on lighting direction Bei Xiao #

Looking against the light How perception of translucencydepends on lighting direction

Bei Xiao $Department of Computer Science American University

Washington DC USA

Bruce Walter $Department of Computer Science Cornell University Ithaca

NY USA

Ioannis Gkioulekas $Harvard School of Engineering and Applied Sciences

Cambridge MA USA

Todd Zickler $Harvard School of Engineering and Applied Sciences

Cambridge MA USA

Edward AdelsonDepartment of Brain and Cognitive Sciences

Massachusetts Institute of Technology Cambridge MA USA

Kavita Bala $

Computer Science and Artificial Intelligence LaboratoryMassachusetts Institute of Technology Cambridge MA USA

Department of Computer Science Cornell UniversityIthaca NY USA

Translucency is an important aspect of materialappearance To some extent humans are able toestimate translucency in a consistent way acrossdifferent shapes and lighting conditions ie to exhibittranslucency constancy However Fleming and Bulthoff(2005) have shown that that there can be large failuresof constancy with lighting direction playing an importantrole In this paper we explore the interaction of shapeillumination and degree of translucency constancy moredeeply by including in our analysis the variations intranslucent appearance that are induced by the shape ofthe scattering phase function This is an aspect oftranslucency that has been largely neglected We usedappearance matching to measure how perceivedtranslucency depends on both lighting and phasefunction The stimuli were rendered scenes thatcontained a figurine and the lighting direction wasrepresented by spherical harmonic basis functionObservers adjusted the density of a figurine under onelighting condition to match the material property of atarget figurine under another lighting condition Acrossthe trials we varied both the lighting direction and thephase function of the target The phase functions weresampled from a 2D space proposed by Gkioulekas et al

(2013) to span an important range of translucentappearance We find the degree of translucencyconstancy depends strongly on the phase functionrsquoslocation in the same 2D space suggesting that the spacecaptures useful information about different types oftranslucency We also find that the geometry of an objectis important We compare the case of a torus which hasa simple smooth shape with that of the figurine whichhas more complex geometric features The complexshape shows a greater range of apparent translucenciesand a higher degree of constancy failure In summaryhumans show significant failures of translucencyconstancy across changes in lighting direction but theeffect depends both on the shape complexity and thetranslucency phase function

Introduction

Many natural materials we encounter every day aretranslucent including the food we eat the liquids wedrink and our skin The translucent appearance of

Citation Xiao B Walter B Gkioulekas I Zickler T Adelson E amp Bala K (2014) Looking against the light how perception oftranslucency depends on lighting direction Journal of Vision 14(3)17 1ndash22 httpwwwjournalofvisionorgcontent14317doi10116714317

Journal of Vision (2014) 14(3)17 1ndash22 1httpwwwjournalofvisionorgcontent14317

doi 10 1167 14 3 17 ISSN 1534-7362 2014 ARVOReceived October 7 2013 published March 13 2014

Downloaded From httpsjovarvojournalsorgpdfaccessashxurl=datajournalsjov932817 on 12082018

these materials is caused by internal volumetricscattering Figure 1A shows photographs of a cherubunder two different lighting directions One distinctivefeature of the material is that it is permeable to lightWe see part of the green background through the thinparts of the cherub such as its wings This is becausesome light penetrates into the cherub that allowed thebackground to be seen through the object

Humans are skilled at discriminating subtle differ-ences in translucency such as when they discriminatemilk from cream wax from soap and human fleshfrom a doll Figure 1B shows photographs undernatural illumination of the same lsquolsquofrog princersquorsquo shapemade of two different materials soap and wax the

translucent appearance of the two materials is verydifferent Since in the natural world translucentmaterials are the norm not the exception (Donner etal 2009) it makes sense that the human visual systemis well engineered to analyze them However very littleis known about how the visual system recognizes anddiscriminates translucent materials from images

Creating controlled inputs to study translucencywith real objects is a challenge as it is hard tomanufacture objects with specific scattering parame-ters Further until very recently rendering realistictranslucent objects was not feasible using computersimulation thus preventing the use of such stimuli forcontrolled studies Therefore most previous work in

Figure 1 Translucent materials (A) Studio photographs of a cherub figurine made with polyurethane lit from two lighting directions

from behind and from the side The one on the left looks more translucent (B) Photographs of two different translucent materials

under the same natural illumination The same lsquolsquofrog princersquorsquo shape was made of two different translucent materials soap and wax

Observers can discriminate the subtle differences between the two materials

Journal of Vision (2014) 14(3)17 1ndash22 Xiao et al 2

translucency perception has focused on stimuli withsimple shapes unnatural illumination and a limited setof materials These works hypothesized that image cuessuch as specular highlights blur image contrast andcolor variation could influence the perception oftranslucency However to understand translucencyperception in the real world we need to expand on eachof these three parameters We need to consider complexshapes realistic lighting and a richer set of materials

It is commonly observed that lighting has dramaticeffects on translucent objects They tend to appearmore translucent under backlighting For examplepeople often hold up a translucent object against a lightwhile studying it This change in lighting direction alsocauses many of the measureable image properties (suchas image contrast location of specular highlights castshadows luminance histograms etc) to changeStudying the effect of lighting direction on translucencycan be very informative of the images cues used by thevisual system In addition the lighting effect variesdepending on the 3D shapes and material properties ofobjects For example the translucent appearance of ajade bracelet might be more sensitive to a change oflighting direction than a bar of soap Thereforestudying lighting on translucency perception with acontrolled 3D shape and material properties is impor-tant

Changing lighting direction also influences theperception of 3D shape of an object which indirectlyinfluences the perception of material properties (Ger-ardin Kourtzi amp Mamassian 2010 Koenderink amp vanDoorn 2001 Olkkonen amp Brainard 2011 Todd 2004Wijntjes amp Pont 2010) A previous study on the effectof lighting direction on translucency used a simpleshape a torus (Fleming amp Bulthoff 2005) Byobserving translucent objects in real life we findlighting direction influences objects that have avariation of thin and thick geometry such as thecherub shown in Figure 1A in a different way from anobject with a simple geometry such as a torus (Xiao etal 2012) Gkioulekas et al (2013) showed thatchanging the shape of the spherical scattering distri-bution (which is called phase function details areprovided in the section lsquolsquoBackground and relatedworkrsquorsquo) affected translucent appearance This effect issmall for thick geometries because the appearance isdominated by high-order scattering However thephase function can impact appearance in a perceptuallyimportant way near thin geometric structures

The present study aims to systematically understandhow lighting direction affects the perception oftranslucency of objects using natural shape realisticlighting and advanced rendering methods Our resultshows that lighting direction has a different effect fordifferent phase functions especially for 3D shapes thatcontain thin geometric structures

Background and related work

We begin by introducing the physical process ofsubsurface scattering that is responsible for theappearance of the translucent objects the phasefunction models and current rendering methods fortranslucency We will then review previous studies onmaterial perception and constancy with an emphasison the effects of lighting environment and 3D shape

Translucent materials

Subsurface scattering Figure 2A illustrates how lightinteracts with translucent objects and the parameterscontrolling subsurface scattering When light hits thesurface of an opaque object it bounces back from thepoint that is illuminated When light hits a translucentobject only some of it bounces off from the surfaceThe rest penetrates into the object refracts andscatters multiple times within the body of the mediumuntil it re-emerges outside from a different location Incomputer graphics this process can be modeled usingthe radiative transfer equation (RTE) (Ishimaru 1978)This approach models the material using three wave-length dependent functions the scattering coefficient rsthe absorption coefficient ra and the phase function pFor light at a single wavelength rs and ra describe howrapidly the light is scattered and absorbed in themedium When light is scattered the phase functionwhich is a probability distribution over directionsdescribes the angular distribution of scattered lightThe sum of the scattering and absorption coefficients iscalled the extinction coefficient rtfrac14rsthorn ra also referredto as the density of the material and the ratio rsrt iscalled the albedo The density controls the degree oftranslucency of the medium A more translucentmedium has a low density while a less translucentmedium has a high density When light interacts withthe medium the albedo controls what fraction of thislight is scattered instead of being absorbed Thescattered light then continues in new directionscontrolled by the phase function In this paper we usethe density the albedo and the phase function todescribe the scattering behaviorPhase function model In this work we focus on theeffect of phase function and study how the combinationof phase function and lighting direction affect perceivedtranslucency We are motivated by the followingreasons First the phase function can impact appear-ance in perceptually important ways near thin geo-metric structures where light undergoes only a handfulof scattering events before exiting towards the observerFigure 2B illustrates how two different phase functionsaffect the appearance of translucent images especiallynear thin edges Second since lighting direction oftenaffects the appearance of thin edges (see Figure 2B)

this leads us to ask whether there is an interactionbetween phase function and lighting direction on theeffects of translucent appearance for complex-shapedobjects that have both thin and thick parts Thereforeto thoroughly study the effects of lighting direction webelieve it is necessary to explore the perceptual space ofthe phase function

A common analytical form of the phase functionused in graphics is due to Henyey and Greenstein(1941) It is controlled by a single parameter and canrepresent both predominantly forward and predomi-nantly backward scattering To model more complex

types of scattering a linear combination of the twoHenyey and Greenstein (HG) lobes one forward andone backward can be used but there are still materialsfor which this is inadequate It was shown in(Gkioulekas et al 2013) that the appearance ofmaterials such as microcrystalline wax and white jadecannot be reproduced using HG phase functions alone

Gkioulekas et al (2013) extended the phase functionfamily to consider lobes from another single-parameterfamily the von Mises-Fisher (vMF) family Thisextension allowed us to have all the linear combinationsof two types of lobes from both HG and vMF

Figure 2 Illustration of volumetric scattering and the phase function (A) Illustration of subsurface scattering Left Translucent

appearance is caused by scattering of light inside the volume of an object and described by the radiative transfer equation Middle A

closer look of subsurface scattering Light travels through a medium The extinction coefficient controls the distance before a volume

event occurs Then at each volume event light may get absorbed with some probability determined by the absorption coefficient

Otherwise light is scattered into some new direction within the volume Right The phase function describes the angular distribution

of the scattered light The green circle represents an isotropic phase function meaning one that scatters equally in all directions (B)

The effect of the phase function on the final appearance of an object depends strongly on its geometry The lsquolsquoLucyrsquorsquo model is rendered

with two different phase functions represented by orange and purple frames The shape of the phase function is displayed below the

images The green circle which represents an isotropic phase function is added to both In the thick object parts such as the body

the difference in the two images is small However in the thin object parts such as the wing the image differences help us

distinguish between the two materials

representing a much larger space of phase functionsand hence a much more diverse set of materials Thisextension increased the physical parameter space butvery different phase functions could produce the samevisual appearance This suggested the perceptual spaceof translucency was much smaller than this expandedphysical parameter space That study established alower-dimensional embedding of the phase functionthat captures most of the variance in perception

To achieve this goal Gkioulekas et al (2013)combined psychophysics with computational analysisThis involved three steps First we densely sampled thephysical parameter space of phase functions tocompute thousands of images and employed anonmetric multidimensional scaling methods (WillsAgarwal Kriegman amp Belongie 2009) with variousimage distance measures to find a two-dimensionalembedding of the space (Figure 3B) The two-dimen-sional embedding is consistent across different distancemetrics shapes and lighting Second we ran apsychophysical study where observers performedpaired-comparisons between images We found thatthese perceptual distances were consistent with a two-dimensional embedding that was similar to that foundcomputationally thereby affirming the perceptualrelevance of that embedding Thirdly we performedstatistical analysis of the computationally obtained andperceptually consistent two-dimensional embedding toinvestigate how its dimensions related to phase functionshape We identified two linearly independent axes thatwere described as simple analytic expressions ofgeneralized first and second moments of the phasefunctions We also obtained a learned distance metricthat could be used to compute an efficient approxi-mation to the perceptual distance between any twotabulated phase functions (For details of the param-eterization of the 2D space please see Gkioulekas et al2013) The visualization of the 2D phase function wasprovided in Figure A1 in the Appendix

Figure 3A shows how phase function can affect theappearance of translucent objects while the otherscattering parameters are kept the same The left imageshows the rendered lsquolsquoLucyrsquorsquo model (httpgraphicsstanfordedudata3Dscanrep) using a forward vMFand isotropic phase function and the right imageshows a lsquolsquoLucyrsquorsquo rendered with a mixture of a forwardHenyey and Greenstein and a backward von Mises-Fisher phase function Both objects are side-lit ThelsquolsquoLucyrsquorsquo on the left looks more diffusive (lsquolsquomarble-likersquorsquo)and the lsquolsquoLucyrsquorsquo on the right looks glassy (lsquolsquojade-likersquorsquo)Figure 3B shows a 2D computational embedding ofphase functions and the positions of the two phasefunctions used to render images in Figure 3A in the 2Dspace The possible differences in image brightness arecaused by the physics of the scattering of the two-phasefunctions Moving from left to right on the embedding

results in phase functions with larger variance implyingthat more light gets scattered towards side-directionsand therefore less light exits from the front for thecamera to observe This was discussed extensively in thediscussion section in the paper by Gkioulekas et al2013Rendering translucency Solving the radiative transferequations is computationally expensive In computergraphics approximations of the RTE have been used toavoid the cost of simulation Multiple scatteringdefined as light that has scattered many times withinthe medium often varies smoothly across the surfaceand is independent of the viewing direction Thepioneering work by Jensen (2001) proposed the dipolediffusion model an approximate solution based on thediffusion equation This model is fast and widely usedand it has created a large body of research in dipolesolutions and multiple scattering The dipole approx-imation however is based on a semi-infinite planarmodel and can be quite inaccurate in thin or highlycurved regions This approximation is also onlysuitable for thick objects whose appearance is domi-nated by high-order scattering In this work we insteadused volumetric Monte Carlo path tracing to directlysolve for the radiative transfer without the need forsuch approximations While more expensive thisapproach is guaranteed to produce the correct solutionunder all lighting material and shape conditionsPerception of translucent materials There has been along history of research on transparent materials(Helmholtz 1867 Koffka 1935 Metelli 1974) Buttraditional research on the perception of transparentmaterials has focused on scenes with overlapping thintransparent layers (Adelson 1993 Anderson 1997Anderson 2003 Anderson amp Khang 2010 Beck 1978Beck amp Ivry 1988 Beck Prazdny amp Ivry 1984 Chenamp DrsquoZmura 1998 Kersten 1991 Kersten BulthoffSchwartz amp Kurtz 1992 Khang amp Zaidi 2002Koenderink amp van Doorn 2001 Sayim amp Cavanagh2011 Watanabe amp Cavanagh 1992 Watanabe ampCavanagh 1993) Metelli developed an episcotistermodel to explain perception of transparent objects thatare infinitely thin neutral density filters However thismodel cannot explain the effects of subsurface scatter-ing where light travels within the body of the objectFurthermore the image cues in translucent layers aredifferent than those observed in transparent layers

There has been little research on translucencyperception of three-dimensional objects with a fewnotable exceptions that we discuss now Fleming andBulthoff (2005) was the first study to use syntheticimages of 3D objects to investigate the perception oftranslucency Their study examined the effects ofseveral low-level image cues such as specular high-lights saturation in color edge blurring imagecontrast and luminance histogram on the perception of

translucency Fleming and Bulthoff also studied theeffect of direction of illumination direction on trans-lucency perception They used a simple torus-shapedobject and varied the illumination direction by rotatinga point light source Their images were rendered usingan isotropic phase function and the dipole approxi-mation method (Jensen 2001) They asked observers toadjust the scattering coefficient of a torus to match thetranslucency of a target torus under a different lightingdirection They found that observers were bad atdiscounting the effect of lighting source direction and

objects tend to appear more translucent when illumi-nated from the back In this paper we use a similarmatching paradigm and extend their work by usingstate-of-the-art rendering to capture the translucentappearances of complex-shaped objects under differentlighting directions

Motoyoshi (2010) found that manipulating thecontrast and blur of the nonspecular image componentof translucent objects could alter the translucentappearance Nagai et al (2013) used a psychophysicalreverse correlation method to extract spatial regions

Figure 3 Phase functions and translucent appearance (A) lsquolsquoLucyrsquorsquo rendered with two different phase functions but with the same

density and albedo (B) Two-dimensional computational embedding of phase functions constructed using a multi-dimensional scaling

method (Gkioulekas et al 2013) The purple dot marks the phase function used to render the left lsquolsquoLucyrsquorsquo and the orange dot marks

the phase function that used to render the right lsquolsquoLucyrsquorsquo in A The gray dots represent full set of images used in the computational

embeddings (for details please see (Gkioulekas et al 2013 figure 7)

related to translucency perception from renderedimages of objects They show that the global root meansquare contrast within an entire rendered image wasnot related to perceptual translucency but the localmean luminance of specific image regions within theimage correlates well with perceived translucency

Recent work by Fleming Jakel and Maloney (2011)proposes image cues for thick transparent objects thathave irregular shapes They discovered that imagedistortions that occur when a textured background isvisible through a refractive object are a key cue that thevisual system can use to estimate an objectrsquos intrinsicmaterial properties

Material constancy under variation of lighting direction

Humans are good and fast at perceiving materialproperties of objects from images (Sharan Rosenholtzamp Adelson 2009) Similar to color constancy the visualsystem must be able to discount variation of lighting andviewpoint to achieve a stable representation of materialproperties (lsquolsquomaterial constancyrsquorsquo) There have beenmany studies on material constancy under variations ofillumination geometry but most have focused on surfacegloss and roughness (for reviews see Anderson 2011Maloney amp Brainard 2010) In a pioneering studyFleming Dror and Adelson (2003) illustrated theimportance of real-world illumination on materialperception Obein Knoblauch amp Vieot (2004) examinedhow gloss perception is affected by illuminationdirection and find that gloss difference scales obtainedunder two different illuminations are very similarimplying that humans can achieve lsquolsquogloss constancyrsquorsquoHo Landy and Maloney (2006) and Ho Maloney andLandy (2007) studied how surface roughness perceptionwas affected by scene illumination and viewpoints andfound that observers exhibited some constancy tovariation of viewpoint Doerschner Boyaci and Malo-ney (2010) studied how perceived gloss could betransferred from one light field to another Despite all ofthe progress very little is known about how illuminationdirection affects translucency perception

3D shape and material perception

It was previously found that the perception of 3Dshape interacted with the perception of materialproperties though most results pertain to the percep-tion of surface gloss Fleming Torralba and Adelson(2004) Norman Todd and Orban (2004) and Toddand Norman (2003) showed that specular reflectanceaided shape estimation Muryy Welchman Blake andFleming (2013) further demonstrated that disparity ofspecular highlights was important in shape estimationVangorp Laurijssen and Dutre (2007) showed theimportance of 3D shape in the estimation of surface

reflectance from images However a recent brainimaging study suggested that surface cues that con-tribute to object shape are processed separately fromsurface cues that are linked to an objectrsquos materialproperties (Cant Large McCall amp Goodale 2008)

In this paper we expand the study of the perceptionof translucency to objects with natural and complexshapes and with an expanded set of material param-eters from those that have been previously studiedUsing complex shapes spherical harmonic basisfunctions to represent light physically accurate ren-dering techniques and an expanded space of phasefunctions to render rich materials we find that lightingdirection has a strong effect on translucent appearancefor some phase functions but not others This effect canbe predicted from the position of the phase function ona 2D perceptual phase function space established inGkioulekas et al (2013)

Method

We conduct an appearance-matching task to mea-sure how perceived translucency depends on bothlighting direction and phase function We use thelsquolsquoLucyrsquorsquo model from the Stanford 3D scanning repos-itory as the 3D shape and spherical harmonicillumination of varying dominant direction

Stimuli

Rendering

Figure 4 shows an example trial The scene containsa lsquolsquoLucyrsquorsquo angel which is roughly 7 cm tall standing ona dark gray floor The interior of the angel is modeledas a volumetric homogeneous scattering medium withan index of refraction of 15 As mentioned beforethese media can be characterized by three parametersdensity albedo and phase function In our experi-ments observers adjust the density of the medium usinga logarithmic (base frac14 2) scale We used an albedo of099 in our experiments and 12 different phasefunctions chosen from Gkioulekas et al (2013)

The surface of the lsquolsquoLucyrsquorsquo is modeled as a roughrefractive surface (Walter Marschner Li amp Torrance2007) with Beckmann roughness of 01 This choiceaffects the highlights and refraction We use therougher appearance of the surface so that the lsquolsquoLucyrsquorsquoappears to be more natural

Spherical harmonic basis lighting

Our goal is to create lighting while controlling itsdirection front back and side-lit The illumination has

Figure 4 Experimental interface We use a matching task to measure perceived translucency under different lighting conditions

During each trial the observer adjusts the density of a simulated lsquolsquoLucyrsquorsquo angel (match image on the right) using a slider to match the

material properties of an angel of a particular density (target image on the left) simulated under another lighting direction After each

trial the observer is asked to answer whether they are happy with the match or not The observer is instructed that this is a

subjective task The image has the size of 481 middot 421 pixels The viewing distance is 50 cm The visual angle of the image is 598 middot 538

its maximum value in one direction and decreasesmonotonically to zero in the opposite direction Moreprecisely its value is proportional to eth1thorn coshTHORN3 where his the angle to the maximal direction of the illuminationpattern This simulates a relatively large area of lightingwhile also providing a strong sense of lighting direc-tionality (such as side- or front-lit) We can represent thispattern using 16 spherical harmonic coefficients and thisrepresentation also allows arbitrarily rotating theillumination from a single volumetric simulation In ourexperiments we chose to use only a small set ofpreselected illumination directions The lighting direc-tion is controlled by the polar and the azimuthal anglesso that the maximum brightness can occur at anydirection we want (to get front-lit side-lit back-litconditions etc) For more details about the sphericalharmonic representation of lighting please see Ram-amoorthi and Hanrahan (2001)

2D embedding of phase functions

In this experiment we uniformly sample 12 phasefunctions from the perceptually uniform two-dimen-sional embedding established in our previous study(Gkioulekas et al 2013) which we then use in all of ourexperiments Figure 5 shows example images renderedusing the 12 phase functions and their correspondingpositions in the 2D embedding Doing the sampling thisway means that the set of phase functions we select isrepresentative of all of the perceptually importantappearance effects phase functions can produce Previ-ous studies used phase functions lying only on a one-dimensional slice concentrated to the leftmost part ofthis appearance space Consequently they neglectedmany perceptual effects that can be critical foraccurately reproducing the appearance of differentclasses of translucent materials such as jade As ourresults demonstrate it is necessary to study the full spaceof phase functions as the magnitude of the effects weobserve varies considerably depending on the phasefunction in the appearance space The details of the fullembedding of the phase functions can be seen in figures7 8 and 11 in Gkioulekas et al 2013 The parametersof the 12 phase functions are provided in the Appendixat the end of the paper

Procedure

An asymmetric matching task is used to measurechanges in perceived translucency caused by variationsof lighting direction and phase function parametersFigure 4 shows the interface of the experiment Theimage on the left is the target image The one on theright is the match image The target image is renderedunder a specific lighting direction and with a specific

phase function at each trial The match image isrendered with the same phase function as the target andunder the lsquolsquosidersquorsquo lighting direction (azimuthal angle frac140) The observerrsquos task is to use the slider to adjust thetranslucent appearance of the lsquolsquoLucyrsquorsquo on the right side(match image) so that it appears to be made of the samematerial as the lsquolsquoLucyrsquorsquo on the left side (target image)Before the experiment the observer confirms thatchanging from 0 to 10 in logarithmic density scale (leftto right in Figure 4 on the sliding bar) makes thelsquolsquoLucyrsquorsquo change from appearing transparent (lsquolsquoglassyrsquorsquo)to opaque (lsquolsquomarblersquorsquo) During the trial the observercan adjust the apparent translucency of the matchlsquolsquoLucyrsquorsquo through 40 values in its densities in a rangefrom 0 (log) to 10 (log) with a step of 025 (log) Theobserver has unlimited viewing time After the observerfinishes the adjustment he or she is asked to press aradio button to indicate whether the match issatisfactory Then the observer can move to the nexttrial in the sequence by pressing the lsquolsquoNextrsquorsquo button onthe graphical user interface

Experimental conditions

Figure 5 shows target images of the lsquolsquoLucyrsquorsquo renderedwith 12 phase functions and three different lightingdirections at densityfrac14 4(log) In the experiment thedensity of the target lsquolsquoLucyrsquorsquo can be one of the fourvalues 3 4 5 and 6 in the logarithmic density scales

The lighting directions of both target and matchimages are controlled by changing the polar and theazimuthal angles To vary the lighting direction only inone dimension at each trial the polar angle is fixed ateither 908 (horizon lighting) or 308 (overhead lighting)The azimuthal angle can be one of three conditionsfront lighting (2708) side-back lighting (308) andback lighting (908) The lighting conditions were chosenbased on pilot studies There are a total of 288conditions (12 phase functions six lighting conditionsand four densities)

Figure 6 shows example images rendered with polarangle 908 and 308 The match lsquolsquoLucyrsquorsquo has the samepolar angle as the target lsquolsquoLucyrsquorsquo But the lightingdirection of the match is always at zero (side-lit) Wechose the azimuthal angle to be zero to avoid havingthe same direction as one of the target lightingdirections The complete sets of target images and theparameter files used in the experiment are provided inthe supplementary website1

Display

Observers perform the experiments in a dark roomand the images are displayed on a 13-inch MacBook

Pro laptop monitor (Apple Inc Cupertino CAdynamic range 601 gamma 22 maximum lumi-nance 340 cdm2) The images are tone-mappedlinearly with clipping and gamma-corrected subse-quently We scale the values of an image by anexposure value Then we convert the images to 8-bitinteger scaled by 255 Values which scale to 10 afterexposure will be mapped to an 8-bit value of 255

causing darkening and loss of contrast in the corre-

sponding regions for this reason we chose exposures

where this effect was minimal Because images with

polar angle 308 are significantly brighter than images

with polar angle 908 (because much of the lighting is

near or below the horizon or ground plane in the 908

condition) we use different exposure values (075 for

Figure 5 Examples of experiment stimuli (polar anglefrac14 908 and densityfrac14 4 (log) for three lighting directions azimuthal anglefrac14308

908 2708 The phase functions numbers correspond to the numbers of the phase functions sampled from the 2D embedding shown in

Figure 3B (To see the original version of the figure please go to the Supplementary Material website)

polarfrac14 308 and 085 for polar frac14 908) for them so thatthe ground plane looks similar in brightness

Observers

Six naıve observers (four females and two males)with normal or corrected-to-normal visual acuitycompleted the task The average age of the observers is25 years

Results

The stimuli in our experiment vary along threedimensions density phase function and lightingdirection We now present our results that answer thefollowing three questions (1) How well can observersdiscriminate material properties (defined by densitieswith fixed-phase functions) under different lightingconditions (2) How does lighting direction affecttranslucent appearance (3) How does the phasefunction interact with the lighting direction fortranslucent appearance First we look at materialdiscrimination independent of lighting direction to seewhether changing the density in the scattering param-eters correlates with translucency perception Secondwe examine whether lighting direction affects translu-cency perception for example whether the stimulirendered under front lighting look more opaque thanthe stimuli rendered under back lighting Finally welook at how the effect of lighting direction ontranslucent appearance depends on phase functionshape

How well can observers discriminatetranslucent materials under different lightingconditions

Figure 7A plots the mean data across observers forthe 12 phase functions when the polar angle is set to be908 The x-axis represents the density of the targetlsquolsquoLucyrsquorsquo varying from 1 to 10 in log scale (from hightranslucency to low translucency) The y-axis representsthe mean density across the observersrsquo matches Thediagonal line represents perfect matching Since thematching experiment is asymmetric meaning the matchlsquolsquoLucyrsquorsquo is always rendered with a different lightingdirection (Side-lit Azimuthal anglefrac1408) than the targetlsquolsquoLucyrsquorsquo (Azimuthal angle frac14308 908 and 2708) it isnot expected that the data would lie along the diagonalIf the data lie above the diagonal line it means theperceived density is higher than the target This is thecase for most of the green line in the graphs (front

lighting conditions) If the data lie below the diagonalline it means the perceived density is lower than thetarget This is the case for some of the red and blue linesin the graphs (side- and backlighting conditionsrespectively)

First we ask whether changing the density aphysical parameter alters the perception of translucentappearance Figure 7A shows that for all conditions asthe target density increases (this corresponds to lesstranslucent appearance of the material) observersrsquomatches increase in a monotonic and linear fashion(most data lies along a line except a few cases for back-lit conditions) As the density increases the lsquolsquoLucyrsquorsquo isjudged to appear less translucent We confirmed this byfitting the data with linear functions and plot histo-grams of the slopes of the fitted lines (Figures 7C and8C) We find the slopes are centered on 10 whichsuggests the monotonic relationship between thedensity parameter change and the changes in observersrsquomatches

How does lighting direction affect translucentappearance

If lighting direction had no effect on perceivedtranslucency the colored lines representing matches forthe three lighting conditions in Figure 7 would beoverlapping But Figure 7 shows that the green line issignificantly above the red and the blue lines for somephase function conditions (9 10 11 12 7 6 5) Thismeans that observers perceived the front-lit lsquolsquoLucyrsquorsquo toappear less translucent than the side-lit and back-litlsquolsquoLucyrsquorsquo which is consistent with previous findings forthe torus stimuli (Fleming amp Bulthoff 2005)

Also the effect of lighting direction is not symmetricThe gap between the green lines and the diagonal linesis much bigger than the gap between the red or bluelines and the diagonal lines This shows that the

Figure 6 lsquolsquoLucyrsquorsquo with two different polar angle conditions polar

angle frac14 308 and polar angle frac14 908 for a particular phase

function and density (Phase functionfrac14 7 and densityfrac14 4 (log)

azimuthal angle frac14 908

Figure 7 Results from the lsquolsquoLucyrsquorsquo matching task of polar anglefrac14 908 (A) Mean subject matches plot against target densities for 12

phase functions (polar angle frac14 908) The p values are computed using ANOVA with repeated measures representing significance

between the front lighting and the back lighting conditions (B) Mapping the effect of lighting of each condition on the 2D embedding

of phase functions The purple circles represent phase functions for which lighting direction has no significant effect The orange

circles represent phase functions for which lighting direction has significant effect (C) The histograms of the slopes of linear fit for

each scatter plots shown in (A) The mean of the slopes is 101

observers perceive the front-lit target stimuli as

considerably less translucent than the match stimuli

but the back-lit stimuli only slightly more translucent

than the match stimuli Figure 7 also shows that

lighting direction has little or no significant effect for

phase function conditions 1 2 3 and 4 We used an

ANOVA test with repeated measures (within observers)

to examine the difference between front- and back-

lighting conditions The p value of the main effect

(lighting direction) is displayed on the plot for each

phase function condition

How does the phase function interact with thelighting direction for translucent appearance

The data shows that lighting direction affects somephase function more than others We now want toconnect the results in the current experiment with the2D embedding of phase functions we established inearlier work (Gkioulekas et al 2013) We want to askwhether the effect of lighting direction on differentphase functions can be predicted from their corre-sponding positions in the 2D embedding For eachphase function condition we compute the p value with

Figure 8 Results from the lsquolsquoLucyrsquorsquo matching task polar anglefrac14 308 (A) Mean subject matches plotted against target densities for 12

phase functions (B) Mapping the effect of lighting of each condition on the 2D embedding of phase functions The symbols have the

same meaning as those in Figure 7 (C) The histograms of the slopes of linear fit for each scatter plots shown in (A) The mean of the

slope is 102

a within-observer ANOVA on the difference betweenthe front- and the back-lighting conditions

In Figure 7B the phase function that has asignificant p value is marked with an orange dot ontop of each panel and the phase function withnonsignificant p value is marked with a purple dot oneach panel We then label the corresponding testphase functions in the 2D embedding of phasefunctions We can see in Figure 7B that the purpledots lie on the leftmost column of the phase functionspace (single-lobe Henyey-Greenstein or isotropicphase functions) whereas all the orange dots lie onthe right-hand side of the phase function space(double lobe combining HG and von Mises-Fisherfunctions) This shows that the strength of the effectof the lighting direction on translucent materials canbe predicted by the position of the correspondingphase function on the 2D space The lightingdirection has a strong effect on phase functions thathave two lobes with either HG thornHG or HG thorn vMFphase functions (see figure 11 in Gkioulekas et al2013) for the locations of different types of phasefunction on the 2D embedding) In fact the strength

of lighting direction becomes stronger as we movefrom the left to the right on the 2D space Note thatphase functions in the right part of the embedding canyield relatively sharp (glassy) appearances whereasthe leftmost column phase functions yield diffusiveappearances

Figure 8 shows the results from the conditions wherethe polar angle is set to be 308 The data is plotted in thesame fashion as in Figure 7 for the same group ofobservers The results for polar angle 308 are qualita-tively the same as for 908 except that unlike before theeffect of lighting direction for phase function 8 is notsignificant (ANOVA test with repeated measure pfrac1401) Note that the effect of lighting direction for thisphase function is also relatively small in Figure 7

Observersrsquo confidence

At the end of each trial during the experiment theobserver is asked to indicate whether they are lsquolsquohappyrsquorsquowith the match This indicates observersrsquo confidence onthe matches Figure 9 plots the percentage of trials in

Figure 9 Confidence ratings averaged across all observers The bars represent the percentage of trials aggregated across all observers

deemed lsquolsquosatisfactoryrsquorsquo in their matches for each phase function condition of both azimuthally angles The vertical bar represents each

phase function with the same number scheme as in Figures 7 and 8 The purple colors represent phase functions that have no

significant lighting effects while brown colors represent phase functions that have significant lighting effects

which all observers indicated they were lsquolsquosatisfiedrsquorsquoversus the total number of trials for each phasefunction condition On average the percentage is about78 suggesting observers are overall confident in theirmatches The plot also shows that observers haveslightly higher confidence for the conditions that haveno significant effects of lighting direction on thematches than the conditions that have significanteffects of lighting direction (mean confidence rating ofthe purple conditions is 084 mean confidence rating ofthe brown-yellow conditions is 073)

Discussion

In this paper we explore the interaction of shapeillumination and level of translucency More impor-tant we also consider the type of translucency asspecified by the scattering phase function We useappearance matching to assess the degree of perceivedtranslucency We find that the degree of translucencyconstancy depends strongly on the phase functionrsquoslocation in the 2D perceptual space discovered inGkioulekas et al 2013 suggesting that the space

Figure 10 Examples of stimuli that subjects match to be perceptually equivalent in translucency Top row a front-lit target lsquolsquoLucyrsquorsquo and

the matched side-lit lsquolsquoLucyrsquorsquo rendered with the averaged density selected by the observers Bottom row a back-lit target lsquolsquoLucyrsquorsquo and

the matched side-lit lsquolsquoLucyrsquorsquo rendered with the averaged density set by the observer

captures genuinely useful information about differenttypes of translucency

Translucency discrimination and lightingdirection

From a material design point of view it is interestingto ask whether lighting direction affects materialdiscrimination Our data show that observers candiscriminate translucent appearance simulated withdifferent densities under all lighting directions All the

lines in Figures 7 and 8 have similar slopes to thediagonal indicating that the physical change in densitycan predict perceptual discrimination except for a fewcases in front-lighting conditions (Figures 7C and 8C)Fleming and Bulthoff (2005) show that when the torusis illuminated from the front the image does notchange very much as a function of the degree oftranslucency Stated another way the image informa-tion is not sufficiently discriminative using the imageinformation alone there is little basis for a visualsystem to distinguish between translucent and opaque

Figure 11 Both shape and lighting affects how phase function interacts with lighting direction in translucency perception Comparison

of the effect of lighting direction on object with thin geometric features lsquolsquoLucyrsquorsquo and lsquolsquoTorusrsquorsquo Top row lsquolsquoLucyrsquorsquo rendered with the same

area light source as used in the experiment Middle row lsquolsquoTorusrsquorsquo (with similar setting as Fleming amp Bulthoff 2005) rendered with a

point light source Bottom row lsquolsquoTorusrsquorsquo rendered with the same area light as lsquolsquoLucyrsquorsquo in the top row All the objects have the same

scattering parameters The chrome sphere included in the image shows the reflected image of the lighting environment

objects It is possible that this problem is idiosyncraticof the particular stimuli Fleming chose Our data showsthat even for the front-lit lsquolsquoLucyrsquorsquo the effect of densityon translucent appearance is strong and mostlymonotonically related to density changes (Figures 7and 8 green line) It is possible that the lsquolsquoLucyrsquorsquo hasmore features than the torus providing enough imagecues for the observers to distinguish between objectswith different degrees of translucency in front-litconditions

Which image regions are important

Since the lsquolsquoLucyrsquorsquo has a complex shape which partsof the image are the most important for perceptionFigure 10 compares the lsquolsquoLucyrsquorsquo rendered with theaverage matched value from all observers and thetarget lsquolsquoLucyrsquorsquo The two stimuli are perceptuallyequivalent in terms of translucency based on the dataThe top row in Figure 10 shows that on averageobservers matched a front-lit lsquolsquoLucyrsquorsquo with a muchlower density to a side-lit lsquolsquoLucyrsquorsquo with much higherdensity (almost double the density) This confirms theresults of Figure 7 that front lighting makes the objectappear less translucent than the side and backlightingThe bottom row in Figure 10 shows the averageobserversrsquo match of the object under the backlightingcondition The two lsquolsquoLucyrsquorsquo images look very differentin image contrast but the lsquolsquothinrsquorsquo parts of the objectsuch as the wings and hands of the lsquolsquoLucyrsquorsquo appear tobe almost transparent in both of the images Thissuggests that the thin geometric parts such as the edgescontribute most to the overall translucent appearanceof the lsquolsquoLucyrsquorsquo

This observation is consistent with a recent study byNagai et al (2013) which shows that certain regionscontribute more strongly to the translucent appearanceof rendered images than other regions Most of theregions extracted from the psychophysical reversecorrelation method used by the authors are thin partsnear the objectsrsquo edges or delicate structures andregions that have high contrast shading patterns (seefigure 7 in Nagai et al 2013) Fleming and Bulthoff(2005) also show similar results

Comparison with the findings in Fleming andBulthoff (2005)

The phase function in the top left corner of the 2Dspace (phase function 1) exhibits little effect of lightingdirection in translucent appearance in the currentstudy However Fleming and Bulthoff (2005) showedstrong lighting effects on translucent appearance Herewe speculate that the main causes of the difference in

the results could be the different geometry of the stimuliand the rendering methods Lucy provides far moredensity cues than the torus does especially for front-litconditions Also Fleming and Bulthoff (2005) useJensenrsquos BSSRDF approximation where we used amore accurate method in our study Jensen usessimilarity theory which potentially generates errorsfrom approximated refraction in single-scatter for side-lit conditions which is what Fleming and Bulthoff(2005) used for the target images The difference in themethods generates different translucent appearance Inaddition Fleming and Bulthoff (2005) use a point-lightsource and there is a strong cast shadow on the toruswhich affects translucent appearance We use arealights and there is not much shadow Figure 11demonstrates the effects of 3D shape and lighting onappearances

Importance of the complexity of the lightingenvironment

Most previous studies used a simple lightingenvironment such as a point light source (Fleming ampBulthoff 2005 Motoyoshi 2010 Nagai et al 2013)The bottom row in Figure 11 shows the same lsquolsquotorusrsquorsquoas in the middle row in Figure 11 but illuminated withthe area lights we use in our lsquolsquoLucyrsquorsquo experiment(generated by spherical harmonics up to the fourthorder) We observe that the effect of lighting ontranslucent appearance is less dramatic in the imagesshown in the bottom row than those in the middle rowin Figure 11 The back-lit lsquolsquotorusrsquorsquo appears lesstranslucent in the area lights than the backlit torusunder a point light source This observation indicatesthat the complexity of the lighting environment plays arole in how lighting affects translucency It is possiblethat more realistic lighting with a combination of bothlow and high (infinite) frequencies in the sphericalharmonics could result in less dramatic effects oflighting direction on translucent appearance

Implications for material constancy

Translucent objects interact with light in a complexway In this work we find that for some translucentmaterials lighting direction has a significant effect onappearance Thus for these conditions the visualsystem cannot completely discount the lighting direc-tion in material perception On the other hand thereare also conditions where observers can indeeddiscount lighting direction changes However previ-ous work has found humans are good at materialdiscrimination and recognition (Sharan et al 2009)Even for translucent objects in order to have a stable

percept of material properties the visual system hasto at least to an extent discount the changes causedby rotating the light source To draw an analogy withstudies of human color constancy constancy is goodin complex scenes where there are enough cues aboutthe illuminant such as nearby surfaces and specularhighlights on shinny surfaces But color constancy ispoor when the scene context is reduced (KraftMaloney amp Brainard 2002) For translucency wehypothesize that complex lighting and shape make itpossible for observers to discount lighting for simplematerials (isotropic phase functions) But observershave a hard time discounting the effects of lightingwhen the object has both the complex 3D shape and ismade of complex materials (for example having adouble-lobe phase functions with two types ofdistributions [eg vMF thornHG]) However thisspeculation of the relationship between complexity ofshape and of phase function in translucency constancyneeds more experimental support In addition most ofthe cues to the illuminant in our scenes are from theobject itself Though we believe there are sufficientcues to the illumination direction in the image (egcontrast between the wings and the body of the Lucy)the absence of cues to the illuminant could play a rolein poor constancy Examining how adding additionalcues to the illuminant would affect the results is aninteresting future direction

Conclusion and future work

In conclusion using a natural scene with a complex3D shape we find that lighting direction has a strongeffect on translucent appearance simulated with somephase functions but not others This effect can bepredicted by the position of the phase functions in a2D perceptual embedding established in a previouspublication (Gkioulekas et al 2013) The phasefunctions that are associated with the strong lightingdirection effect can simulate materials that appearglassy with sharp details On the other hand the phasefunctions that are associated with weak lightingdirection effect can simulate materials with a diffusiveappearance Our results suggest that more consider-ation should be given to the richness of materialproperties such as rich sets of phase function whenexamining the effects of lighting direction on translu-cency perception

We also find that the geometry of an object isimportant We compare the case of a torus which hasa simple smooth shape with that of a figurine whichhas more complex geometry with thick and thinsections and features at multiple scales The complexshape shows a greater range of apparent translucen-

cies which allows observers to discriminate differentdegrees of translucency But it also resulted in a higherdegree of constancy failure Although the complexfigure offers more cues to drive translucent appear-ance those cues do not necessarily increase theconstancy

The focus of this paper has been on the phasefunction but in order to fully understand how lightingaffects translucency perception and constancy weneed to explore other dimensions of scatteringparameters such as scattering coefficient albedo andtheir spectral variations Also in this work we usedsynthetic images In the real world there is higherdynamic range and observers have stereoscopic cueswhich might also help with translucency constancy Inaddition we only take an initial step towardsunderstanding the role of 3D shape in translucencyperception Future experiments should systematicallystudy how varying 3D shape affects translucencyperception using a rich set of material parametersFrom a computational view we would like to buildmodels that quantify scene complexities and predicthuman translucency discrimination and constancybased on scene parameters such as shape materialand lighting geometry Such models would be useful inmaterial design applications guiding what scenes oneshould use to exhibit certain translucent appearanceeffects and in user interfaces for rendering translucentmaterials

Keywords translucency material perception lightingphase functions 3D shape

Acknowledgments

We want to thank Cathy Tingle for the photographsof the cherubs in Figure 1 and other photographs thatinspired the experiment in this work We wish to thankShuang Zhao and Asher Dunn for their support ofrendering and useful discussions and Professor RolandFleming for sharing with us his rendering parametersand very useful discussions We also want to thank theStanford 3D Scanning Repository for providing theLucy This research is supported by NSF awards116173 (K B T Z E A) and 1161645 (K B) andAmazon Web Services in Education grant awards toI G and T Z

Commercial relationships noneCorresponding author Bei XiaoEmail beixiaomiteduAddress Department of Brain and Cognitive SciencesMassachusetts Institute of Technology CambridgeMA USA

Footnote

1Supplementary website httppeoplecsailmitedubeixiaoSupplementary_BX

References

Adelson E H (1993) Perceptual organization and thejudgment of brightness Science 262(5142) 2042ndash2044

Anderson B L (1997) A theory of illusory lightnessand transparency in monocular and binocularimages The role of contour junctions Perception26 419ndash454

Anderson B L (2003) The role of occlusion in theperception of depth lightness and opacity Psy-chological Review 110(4) 785

Anderson B L (2011) Visual perception of materialsand surfaces Current Biology 21(24) R978ndash983

Anderson B L amp Khang B G (2010) The role ofscission in the perception of color and opacityJournal of Vision 10(5)26 1ndash16 httpwwwjournalofvisionorgcontent10526 doi10116710526 [PubMed] [Article]

Beck J (1978) Additive and subtractive color mixturein color transparency Attention Perception ampPsychophysics 23(3) 265ndash267

Beck J amp Ivry R (1988) On the role of figuralorganization perceptual transparency Perception ampPsychophysics 44(6) 585ndash594

Beck J Prazdny K amp Ivry R (1984) The perceptionof transparency with achromatic colors Perceptionamp Psychophysics 35(5) 407ndash422

Cant J S Large M-E McCall L amp Goodale MA (2008) Independent processing of form colourand texture in object perception Perception 37 57ndash78

Chen V J amp DrsquoZmura M (1998) Test of conver-gence model for color transparency perceptionPerception 27(5) 595ndash608

Doerschner K Boyaci H amp Maloney L T (2010)Estimating the glossiness transfer function inducedby illumination change and testing its transitivityJournal of Vision 10(4)8 1ndash9 httpwwwjournalofvisionorgcontent1048 doi1011671048 [PubMed] [Article]

Donner C Lawrence J Ramamoorthi R Hachi-suka T Jensen H W amp Nayar S An empiricalBSSRDF model (2009) ACM Transactions on

Graphics (TOG) Proceedings of ACM SIGGRAPH28(3)

Fisher R (1953) Dispersion on a sphere Proceedingsof the Royal Society of London Series A Mathe-matical and Physical Sciences 217(1130) 295ndash305

Fleming R W amp Bulthoff H (2005) Low-level imagecues in the perception of translucent materialsACM Transactions on Applied Perception (TAP)2(3) 346ndash382

Fleming R W Dror R O amp Adelson E H (2003)Real-world illumination and the perception ofsurface reflectance properties Journal of Vision3(5)3 347ndash368 httpwwwjournalofvisionorgcontent353 doi101167353 [PubMed] [Article]

Fleming R W Jakel F amp Maloney L T (2011)Visual perception of thick transparent materialsPsychological Science 22(6) 812ndash820

Fleming R W Torralba A amp Adelson E H (2004)Specular reflections and the perception of shapeJournal of Vision 4(9)10 798ndash820 httpwwwjournalofvisionorgcontent4910 doi1011674910 [PubMed] [Article]

Gerardin P Kourtzi Z amp Mamassian P (2010)Prior knowledge of illumination for 3D perceptionin the human brain Proceedings of the NationalAcademy of Sciences 107(37) 16309ndash16314

Gkioulekas I Xiao B Zhao S Adelson E HZickler T amp Bala K (2013) Understanding therole of phase function in translucent appearanceACM Transaction of Graphics (TOG) 32 5

Helmholtz H V (1867) LXIII On Integrals of thehydrodynamical equations which express vortex-motion The London Edinburgh and Dublin Phil-osophical Magazine and Journal of Science 33(226)485ndash512

Henyey L G amp Greenstein J L (1941) Diffuseradiation in the galaxy The Astrophysical Journal93 70ndash83

Ho Y-X Landy M S amp Maloney L T (2006)How direction of illumination affects visuallyperceived surface roughness Journal of Vision 6(5)8 634ndash648 httpwwwjournalofvisionorgcontent658 doi101167658 [PubMed] [Article]

Ho Y-X Maloney L T amp Landy M S (2007) Theeffect of viewpoint on perceived visual roughnessJournal of Vision 7(1)1 1ndash16 httpwwwjournalofvisionorgcontent711 doi101167711 [PubMed] [Article]

Ishimaru A (1978) Wave propagation and scatteringin random media and rough surfaces Proceedingsof IEEE 79(10) 1359ndash1368

Jensen H W (2001) A practical model of sub-surface

transport Proceedings of SIGGRAPH 2001 511ndash518

Kersten D (1991) Transparency and the cooperativecomputation of scene attributes ComputationalModels of Visual Processing 209ndash228

Kersten D Bulthoff H H Schwartz B L amp KurtzK J (1992) Interaction between transparency andstructure from motion Neural Computation 4(4)573ndash589

Khang B-G amp Zaidi Q (2002) Accuracy of colorscission for spectral transparencies Journal ofVision 2(6)3 451ndash466 httpwwwjournalofvisionorgcontent263 doi101167263 [PubMed] [Article]

Koenderink J J amp van Doorn A J (2001) Shadingin the case of translucent objects In B E Rogowitzamp T N Pappas (Eds) Proceedings of SPIE (pp312ndash320) Bellingham WA SPIE

Koffka K (1935) Principles of Gestalt psychologyLondon Lund Humphries

Kraft J M Maloney S I amp Brainard D H (2002)Surface-illuminant ambiguity and color constancyEffects of scene complexity and depth cuesPerception 31(2) 247ndash263

Maloney L T amp Brainard D H (2010) Color andmaterial perception Achievements and challengesJournal of Vision 10(9)19 1ndash6 httpwwwjournalofvisionorgcontent10919 doi10116710919 [PubMed] [Article]

Metelli F (1974) The perception of transparencyScientific American 230 90ndash98

Motoyoshi I (2010) Highlight-shading relationship asa cue for the perception of translucent andtransparent materials Journal of Vision 10(9)6 1ndash11 httpwwwjournalofvisionorgcontent1096doi1011671096 [PubMed] [Article]

Muryy A A Welchman A E Blake A amp FlemingR W (2013) Specular reflections and the estima-tion of shape from binocular disparity Proceedingsof the National Academy of Sciences USA 110(6)2413ndash2418

Nagai T Ono Y Tani Y Koid K Kitazaki M ampNakauchi S (2013) Image regions contributing toperceptual translucency A psychophysical reverse-correlation study i-Perception 6 407ndash429

Norman J F Todd J T amp Orban G A (2004)Perception of three-dimensional shape from spec-ular highlights deformations of shading and othertypes of visual information Psychological Science15(8) 565ndash570

Obein G Knoblauch K amp Vieot F (2004)Difference scaling of gloss Nonlinearity binocu-

larity and constancy Journal of Vision 4(9)4 711ndash720 httpwwwjournalofvisionorgcontent494doi101167494 [PubMed] [Article]

Olkkonen M amp Brainard D H (2011) Joint effectsof illumination geometry and object shape in theperception of surface reflectance i-Perception 2(9)1014ndash1034

Ramamoorthi R amp Hanrahan P (2001) A signal-processing framework for inverse rendering Pro-ceedings of SIGGRAPH 2001 117ndash128

Sayim B amp Cavanagh P (2011) The art oftransparency i-Perception 2(7) 679

Sharan L Rosenholtz R amp Adelson E (2009)Material perception What can you see in a briefglance Journal of Vision 9(8) 784 httpwwwjournalofvisionorgcontent98784 doi10116798784 [Abstract]

Todd J T (2004) The visual perception of 3D shapeTrends in Cognitive Sciences 8(3) 115ndash121

Todd J T amp Norman J F (2003) The visualperception of 3-D shape from multiple cues Areobservers capable of perceiving metric structurePerception amp Psychophysics 65(1) 31ndash47

Vangorp P Laurijssen J amp Dutre P (2007) Theinfluence of shape on the perception of materialreflectance ACM Transactions on Graphics 26(3)77

Walter B Marschner S R Li H amp Torrance K E(2007) Microfacet models for refraction throughrough surfaces Rendering Techniques 2007 (Pro-ceedings of the Eurographics Symposium on Ren-dering)

Watanabe T amp Cavanagh P (1992) Depth captureand transparency of regions bounded by illusoryand chromatic contours Vision Research 32(3)527ndash532

Watanabe T amp Cavanagh P (1993) Transparentsurfaces defined by implicit X junctions VisionResearch 33(16) 2339ndash2346

Wijntjes M W A amp Pont S C (2010) Illusory glosson Lambertian surfaces Journal of Vision 10(9)131ndash12 httpwwwjournalofvisionorgcontent10913 doi10116710913 [PubMed] [Article]

Wills J Agarwal S Kriegman D amp Belongie S(2009) Toward a perceptual space for gloss ACMTransactions on Graphics (TOG) 28(4) 105

Xiao B Gkioulekas I Dunn A Zhao S AdelsonE Zickler T amp Bala K (2012) Effects of shapeand color on the perception of translucencyJournal of Vision 12(9) 948 httpwwwjournalofvisionorgcontent129948 doi101167129948 [Abstract]

Appendix

A Phase function models and the parameters

The phase functions were defined mathematicallyeither by a Henyey-Greenstein function (HG) or a vonMises Fisher function (vMF) or a combination ofboth Henyey and Greenstein (1941) describe the phasefunction as the following

PHGethhTHORN frac141

4p1 g2

eth1thorn g2 2gcoshTHORN32

eth1THORN

In Gkioulekas et al (2013) we expanded the phasefunction by adding a lobe that is shaped according tothe von Mises-Fisher distribution on the sphere ofdirections (Fisher 1953) Its probability density func-tion is given as

pvMFethhTHORN frac14j

2psinhjexpethjcoshTHORN eth2THORN

h [0 p] and has a single shape parameter j

jfrac14 2C(1 MC) where C is the average cosine

C frac14 2pZ p

hfrac140

coshpethhTHORNsinhdh

MC frac14 2pZ p

hfrac140

ethcoshTHORN2pethhTHORNdh

We represent the phase function as linear mixtures ofthe two distributions We selected g from 0 603605 608 609 6095 for the vMF lobes we foundthrough experimentation that j 100 producedunrealistic results and selected j 6 165 610625 675 6100 We also chose values of the mixingweight of the first lobe from 02 03 04 05 06 0708 09 099 with the corresponding weight on thesecond lobe being one minus this value SupplementaryTable 1 shows the parameters of the 12 phase functionsused in this study

B 2D embedding of phase function and thestimuli used in this study

Supplementary Figure 1 shows the computationalembedding of the phase functions discovered inGkioulekas et al (2013) and the stimuli images used inthis paper arranged as in the embedding We see thatmoving from top to bottom results in more diffusionan effect similar to that achieved by increasing thedensity of the scattering process Moving from left toright results in greater surface detail and more lsquolsquoglassyrsquorsquoappearance By selecting different points in theembedding we can achieve different trade-offs betweendiffusion and sharpness

Figure A1 HG (shows in blue) and vMF (shows in red) phase functions for increasing value of average cosine C shown as increase in

saturation vMF lobes are wider and not as strongly forward-scattering as HG lobe This figure is copied from figure 3 in Gkioulekas et

al (2013)

Phase function

Forward lobe Backward lobeWeight

j g j g w

1 1 05 08

2 25 0 05

3 08 5 09

4 08 5 099

5 08 09 06

6 10 75 07

7 25 25 08

8 25 095 09

9 100 095 06

10 095 100 07

11 75 100 08

12 100 75 09

Supplementary Table 1 Parameters of the 12 phase functions used in the study Each phase function can have two lobes the forwardlobe and the backward lobe For each lobe it can be either defined by HG or vMF functions The weight (w) is the weight of the linearsum of the two lobes that is always applied to the first lobe (which is the forward lobe)

Figure A2 Two-dimensional computational embedding and the experimental images (A) Two-dimensional embedding of images

rendered with phase functions sampled from the physical parameter space produced in cubic root metric Gray dots represent the

753 rendered images (for details see Gkioulekas et al 2013) The numbers label the stimuli used BA subset of the stimulus images

arranged in a grid according to their positions in the computational embedding The numbers are in correspondence with those

shown in (A) The lsquolsquoLucysrsquorsquo were rendered side-lit and have the same parameters except for the phase functions [azimuthal anglefrac14 08

polar angle frac14 908 and density frac14 4 (log)]

Introduction

Method

Results

Discussion

Adelson1

Anderson1

Anderson2

Anderson3

Anderson4

Doerschner1

Donner1

Fisher1

Fleming1

Fleming2

Fleming3

Fleming4

Gerardin1

Gkioulekas1

Helmholtz1

Henyey1

Ishimaru1

Jensen1

Kersten1

Kersten2

Khang1

Koenderink1

Koffka1

Kraft1

Maloney1

Metelli1

Motoyoshi1

Muryy1

Nagai1

Norman1

Obein1

Olkkonen1

Ramamoorthi1

Sayim1

Sharan1

Vangorp1

Walter1

Watanabe1

Watanabe2

Wijntjes1

Wills1