Closeup pictures in the center used spatially arying te xture to interpolate between re57347ectance models for each point on the teapot Abstract present generati model for isotropic bidirectional re 57347ectance distrib ution functions BRDFs based o ID: 27034 Download Pdf

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Closeup pictures in the center used spatially arying te xture to interpolate between re57347ectance models for each point on the teapot Abstract present generati model for isotropic bidirectional re 57347ectance distrib ution functions BRDFs based o

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Data-Driven Reectance Mo del ojciech Matusik Hanspeter Pster Matt Brand Leonard McMillan Figur 1: Renditions of materials generated using our model: steel teapot with greasy ngerprints (left), teapot with rust forming (right). Closeup pictures in the center used spatially arying te xture to interpolate between reectance models for each point on the teapot. Abstract present generati model for isotropic bidirectional re- ectance distrib ution functions (BRDFs) based on acquired re- ectance data. Instead of using analytical

reectance models, we represent each BRDF as dense set of measurements. This al- lo ws us to interpolate and xtrapolate in the space of acquired BRDFs to create ne BRDFs. treat each acquired BRDF as single high-dimensional ector tak en from space of all possi- ble BRDFs. apply both linear (subspace) and non-linear (mani- fold) dimensionality reduction tools in an ef fort to disco er lo wer dimensional representation that characterizes our measurements. let users dene perceptually meaningful parametrization direc- tions to na vig ate in the reduced-dimension BRDF space. On the lo

w-dimensional manifold, mo ement along these directions pro- duces no el ut alid BRDFs. eyw ords: Light Reection Models, Photometric Measurements, Reectance, BRDF Image-based Modeling Intro duction fundamental problem of computer graphics rendering is model- ing ho light is reected from surf aces. class of functions called MIT Cambridge, MA. Email: ojciech@graphics.lcs.mit.edu MERL, Cambridge, MA. Email: [pster ,brand]@merl.com UNC, Chapel Hill, NC. Email: mcmillan@cs.unc.edu Bidirectional Reectance Distrib ution Functions (BRDFs) charac- terizes the

process where light transport occurs at an idealized sur ace point. raditionally ph ysically inspired analytic reection models [Cook and orrance 1982] [He et al. 1991] [He et al. 1992] pro vide the BRDFs used in computer graphics. These BRDF models are only approximations of reectance of real materials. Furthermore, most analytic reection models are limited to describing only par ticular subclasses of materials gi en model can represent only the phenomena for which it is designed. Signicant ef forts ha been xpended on impro ving these models by incorporating the

rel- ant aspects of the underlying ph ysics. Man of these models are based on material parameters that in principle could be measured, ut in practice are dif cult to acquire. An alternati to directly measuring model parameters is to ac- quire actual samples from BRDF using some ersion of gonio- spectro-reectometer [Marschner et al. 2000] [Cornell [CUReT [ST ARR [Dana 2001] [W ard 1992] and then t the measured data to selected analytic model using arious optimization techniques [W ard 1992] [Y et al. 1999] [Lafortune et al. 1997] [Lensch et al. 2001]. There are se eral

shortcomings to this measure-and-t ap- proach. First, BRDF represented by the analytic function with the computed parameters is only an approximation of real reectance; measured alues of the BRDF are usually not xactly equal to the alues of the analytic model. The measure-and-t approach is of- ten justied by assuming that there is inherent noise in the mea- surement process and that the tting process lters out these errors. This point of vie ho we er ignores more signicant modeling er rors due to approximations made in the analytic surf

ace reection model. Man of the salient and distincti aspects of an objects reection properties might lie within the range of these modeling errors. Second, the choice of the error function er which the op- timization should be performed is not ob vious. or xample, er ror based on the Euclidean distance is poor metric since it tends to eremphasize the importance of the specular peaks (these are usually much higher than the rest) and ignore the of f-specular al- ues. Finally there is no guarantee that the optimization process will yield the best model. Since most BRDF models are

highly non-linear the optimization frame orks used in the tting process rely hea vily on initial guesses of the models parameters. The qual- ity of these initial guesses can ha dramatic impact on the nal

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parameter alues of the model. Another approach is to acquire dense measurements of the BRDF and use these measurements directly in the rendering pro- cess. This approach preserv es those subtleties of the measured data that are lost in data-tting approach. Ho we er it is time- consuming since it requires reectance measurements for all de- sired

materials in the scene. Furthermore, we end up with collec- tion of measured BRDFs and not with parameterized reectance model. An change to the material property ould require nding real material with the desired property and acquiring its reectance. suggest an alternati sampling-based approach for model- ing surf ace reectance. capitalize on the act that it is feasible to rapidly acquire accurate reectance measurements using image- based techniques. acquire BRDFs for lar ge representati set of materials. Materials in our collection include metals,

paints, ab- rics, minerals, synthetics, or anic materials, and others. intro- duce ne approach to BRDF modeling, an approach that is data dri en it interpolates/e xtrapolates ne BRDFs from the represen- tati BRDF data. Our approach has the adv antage that the pro- duced BRDFs look ery realistic since the are based on the mea- sured BRDFs. Furthermore, we pro vide set of intuiti parameters that allo users to change the properties of the output BRDF also let users specify their wn parameters by labeling fe rep- resentati BRDFs. belie that this ay of specifying model parameters mak es our model

much easier to use and control than the analytic models in which the meaning of parameters is often non-intuiti [Pellacini et al. 2000]. In our model, we do not ant to store all acquired BRDFs x- plicitly This leads us to the analysis of the space of all possible BRDFs for common materials in the orld. BRDF for these ma- terials is not an arbitrary function, and we seek representation for all possible functions corresponding to ph ysical BRDFs. treat each of our acquired BRDFs as single high-dimensional ector where each measurement is an element of this ector Then we ap- ply both linear and

non-linear dimensionality reduction tools to ob- tain lo dimensional manifold that characterizes the set of BRDFs we measured. In the process we also obtain mapping between the embedding manifold and the original BRDF space. Therefore we can al ays compute the corresponding BRDF for each point on the manifold. An interesting side ef fect of our approach is that it suggests an inherent dimensionality for the space of all isotropic BRDFs. summarize, the main contrib utions of this paper are: introduce no el model for an isotropic BRDF that is based on measured reectance for lar ge set of

materials. introduce set of perceptually-based parameters for this model. also let users specify their wn parameters. analyze both linear and non-linear dimensionality of the space of isotropic BRDFs. In our model the parameter alues are pre-dened for man typical materials the materials we ha measured. Using our model we can also generate dif cult to represent ef fects such as rust, oxidation, or dust. Previous rk The alue of ph ysically accurate reectance models has long been understood within the computer graphics community [Blinn 1977]. The ailability of BRDF models

based on the actual ph ysics of light transport and alidated by empirical measurements were signicant catalyst in this realization [T orrance et al. 1966] [T ro w- bridge and Reitz 1975]. Ph ysical accurac as an impetus behind the de elopment of man subsequent computer graphics reection models [Cook and orrance 1982][He et al. 1991]. An interest- ing transition occurred with [W ard 1992], when ard de eloped BRDF model that, while not strictly ph ysically based, as capa- ble of describing most signicant reection phenomena. He went to great ef fort to ensure that

his model obe yed the most basic of ph ysi- cal la ws (reciprocity and ener gy conserv ation), and signicantly he t his model' parameters to actual material measurements. More recently the ailability of lo w-cost digital cameras has rekindled interest in BRDF acquisition and modeling. One particularly ambi- tious undertaking is the CUReT BRDF database [Dana et al. 1999] [CUReT ]. The CUReT database represents approximately 200 re- ectance measurements er arying incident and reected angles for planar patch of 60 dif ferent materials. ith uniform material sample

this amounts to relati ely sparsely sampled BRDF Such sparsely sampled BRDF is not directly useful as table-based BRDF function; thus, it as necessary to t an analytic function in order to get useful model. Marschner [Marschner et al. 2000] constructed another signi- cant BRDF measurement system. His system, although limited to only isotropic BRDF measurements, as both ast and rob ust. In particular his system took unique adv antage of reciprocity bilateral symmetry and multiple simultaneous measurements to achie un- precedented le erage from each reection measurement.

This of- fers signicant adv antage. It lters measurement noise due to minute ariations er the surf ace, errors due to spatial ariations in photosite response within the image sensor and ariations in il- lumination intensity In the ace of such statistical eraging, one is hard pressed to attrib ute the ine vitable residual errors that occur when model tting to additional systematic noise, rather than ail- ings of the analytic model. The inherent dimensionality of BRDF combined with the de- sire to sample it at high resolutions in order to model specular inci- dent, and

retroreection ef fects, leads to an unwieldy sampling and storage problem. Man researchers ha addressed this specic problem by searching for more appropriate basis for represent- ing BRDFs. Spherical harmonics [W estin et al. 1992] and spheri- cal elets [Schr oder and Sweldens 1995] are natural choices for representing the angular parameters of the BRDF Other ef cient representations include elets [Lalonde and ournier 1997], Zer nick polynomials [K oenderink et al. 1996],and separable approx- imations obtained using singular alue decomposition [Kautz and McCool 1999] or

purely positi matrix actorization [McCool et al. 2001]. Furthermore, recent image-based approaches to BRDF modeling [Lensch et al. 2001] ha demonstrated the po wer of us- ing linear combinations of compact reectance function basis sets for modeling spatially arying BDRFs. Such linear decomposi- tions lead to an interesting question: can the true space of potential BRDFs be described as linear combination of basis functions? Clearly actorizations of the sort used to compress BRDFs are lin- ear allo wing for arbitrary mixtures of their basis ectors to t gi en set of data. If this

decomposition approach were in act alid, it ould imply that linear combinations of actual BRDFs might be used to model original and ph ysically plausible reection models. Exploring the ramications of this ypothesis is one of our moti a- tions for de eloping sample-based generati model. Data Acquisition In order to acquire suf cient number of adequately sampled BRDFs, it as necessary to uild measurement de vice. Our mod- eling approach placed tw requirements on the acquired data: rst, that each BRDF be sampled densely enough that it could be used di- rectly as

table-based model, and second, that the space of BRDFs be sampled adequately so as to span the range of models that we hope to generate. Accordingly we ha uilt BRDF measurement de vice suitable for rapidly acquiring high-quality BRDFs for wide

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Figur 2: photograph of our high-speed BRDF measurement antry range of dif ferent materials (see Figure ). The image-based BRDF measurement de vice described by [Marschner et al. 2000] inspired our design. Our acquisition system requires spherically homogenous sam- ple of the material. The system is placed in completely isolated room

painted in black matte. It consists of the follo wing compo- nents: QImaging Retig 1300 (a 10 bit, and 1300x1030 res- olution Fire wire camera), Kaidan MDT -19 (a precise computer controlled turntable), and Hamamatsu SQ Xenon lamp (a lamp with stable light emission output and continuous and relati ely constant radiation spectrum er the visible light range). The lamp is mounted on an arm to the turntable. The light orbits the mea- surement sample placed at the center of rotation; the camera is sta- tionary Our camera is geometrically calibrated using the technique described in [Zhang 1998]. The

position of the light source is de- termined using contact digitizer (F AR Arm). use the same digitizer to determine the position of the center of the material sam- ple. The radius of the sample is measured with calipers. The light source mo es in increments of approximately 0.5 from the point xactly opposite the camera (the sample is in between the camera and the light source) to the point xactly in front of the camera. tak total of 330 high dynamic range pictures to co er the re- quired half circle. This process tak es about hours. or each high dynamic range picture we tak total of 18 10-bit

photographs. The xposure time ranges from 40 microseconds to 20 seconds. use the act that our CCD camera has ery linear response curv to deri the high dynamic measurement. or each pix el in the image we t line to the xposure time vs. radiance alues. The slope of the line is used as the radiance estimate. The correlation of this line is higher than 0.998. Each acquired image of the sample sphere represents man BRDF samples. Essentially each pix el of the sphere is treated as separate BRDF measurement. In order to compute the specic BRDF alue for gi en pix el we perform the follo

wing steps. First, we intersect the ray dened by the pix el with the sphere to deter mine point Then, we compute the normal at point on the sphere, the ector and the distance to the light source, and the ector to the camera pix el. Ne xt, we compute the irradiance at point due the light source (taking into account distance to the light source and foreshortening). Finally we compute the BRDF alue as the ratio of the high dynamic range radiance to the irradiance. Data Rep resentation found that specular peaks were dif cult to represent using the natural coordinate system in ou ).

Ev en when binning BRDF at dense grid (e ery spacing for each dimension), it is not possible to reproduce original images (the specular highlight becomes an al shape, oriented at dif ferent directions). use dif ferent coordinate system, described in [Rusinkie wicz 1998] and sho wn in Figure 3. This coordinate frame is based on the an- gles with respect to the half-angle (half-v ector between incoming and outgoing directions). This coordinate frame allo ws us to ary the sampling density near the specular highlight. Specically we ary (angle between the normal and the half-v ector),

assigning smaller bins for alues near specular reection and lar ger bins for angles ar ay from the specular reection. Figur 3: The standard coordinate frame is sho wn on the left. Rusinkie wicz' coordinate system is sho wn on the right. still discretize into 90 bins and into 360 bins. This results in total of 90 90 360 2,916,000 bins for each color component. halv this number to 1,458,000 by enforcing the reciprocity constraint: (1) ith this constraint we need only to discretize into 180 bins. Figur 4: log images of sphere (alumina oxide). real image is sho wn on the left.

synthesized image using tab ulated BRDF data is sho wn on the right. Our measurement process gi es us typically 20-80 million BRDF samples for each material. reduce the noise in our mea- surements by remo ving the outliers in each bin (lo west and highest 25% of the alues), and we erage the remaining measurements. This statistical smoothing is intended to remo systematic noise as well as compensate for small ariations in material properties er the sample. As nal alidation we render synthesized ersion of our sample sphere and compare it to the corresponding acquired high dynamic range

image. conduct this inspection for all in- put light congurations. Pictures for typical acquired material are in sho wn in Figure 4. The rendered images reproduce the input images ery well. ha used our de vice to acquire BRDF mea- surements of more than 130 dif ferent materials, including metals, plastics, painted surf aces, and cloth. Figure depicts some of the materials that were sampled. ha remo ed from further anal- ysis some materials that xhibited signicant subsurf ace scattering, anisotrop or non-homogenity

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Figur 5: Pictures of 100 of our acquired

materials. Data Analysis These sampled BRDFs can be used directly by renderer Se v- eral xamples of that are sho wn in Figure 6, where teapot is ren- dered under natural illumination using the ra acquired data. Our ultimate goal, ho we er is to construct an empirical BRDF model that can be used to generate no el, yet plausible, reectance func- tions directly from this database. be gin with the follo wing as- sumption: if we treat each of our BRDF samples as high dimen- sional ector in an abstract BRDF space, we xpect that all ph ysical BRDFs lie upon lo wer dimensional manifold within

this space indicati of their inherent dimensionality This is common as- sumption used by others [Cula and Dana 2001] and it is consis- tent with the relati ely small number of parameters seen in analytic BRDF models. Therefore, we breakdo wn the task of constructing an empirical BRDF model into tw phases: disco ering this lo wer dimensional model, and dening an interpolation scheme within this lo wer -dimensional subspace. 5.1 Linea Analysis In the case where the ph ysical BRDF manifold lies on linear sub- space, the analysis tools for both manifold disco ery and interpo- lation are

well kno wn. In this case, Principal Component Analysis (PCA)[Bishop 1995] ef fecti ely determines set of basis ectors that span the desired subspace, and linear combinations of sam- ples can be used for interpolation. Linear manifold approaches ha pro en xtremely ef fecti in some problem domains, such as ace synthesis [Blanz and etter 1999] and radiance interpolation [Chen et al. 2002]. Potential linear manifolds are generally sug- gested when there is noticeable plateau in the magnitudes of the sorted eigen alues. When this plateau occurs on the eigen alue, we can model the data as

k-dimensional linear subspace with residual error bounded by the square root of the sum of the squares of the remaining eigen alues. be an our analysis of the BRDF samples by searching for linear embedding manifold (a yperplane). The three color chan- nels of each BRDF sample were assembled into column ector and concatenated to form 4,374,000 by 104 measurement ector matrix Figur 6: Rendered teapots using BRDFs from our database: nick el, hematite, gold paint, and pink abric. Figur 7: Plot of the eigen alues resulting from PCA of the data set. perform the analysis in the log space (we apply

the natural log arithm to each element of ector ). There are se eral reasons for this normalization. First, there is huge dif ference (on the order of fe magnitudes) between the specular and non-specular alues of the BRDF If used in the original space, the analysis tools ould associate more importance to noise in the specular alues than the actual non-specular components. The linear analysis ould depre- ciate importance of these non-specular alues (the non-specular al- ues are perceptually important). Our operation is also justied by the act that the human visual system is sensiti to

ratios rather than absolute radiance alues. Singular alue decomposition as then applied to (a 104x104 matrix). The singular alues in this case are the squares of the desired eigen alue magnitudes. plot of these eigen alues is sho wn in Figure 7. also sho in Figure the reconstruction of typical material using rst 1, 5, 10, 20, 30, 45, 60, and all principal components. see that good reconstruction is usually obtained using the rst 30-40 components. While there is considerable all of in the sequential alues seen in this plot, the plateau is reached around 45 eigen alue (the re-

construction error is about 1% at that point). This dimension of the embedding subspace is considerably higher than our intuition ould suggest, based on the typical number of parameters used in analytic BRDF models. eried that the 45-dimensional space dened by the rst principal components reconstructs all our mea- sured BRDFs well. Ho we er it spans space that is bigger than the space of all possible BRDFs. are able to nd the points in this

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Figur 8: Reconstruction of BRDF from principal components in the order of increasing number of components

mean, 5, 10, 20, 30, 45, 60, and all. subspace that do not correspond to an ph ysical materials. In other ords, using linear combinations of the components, we can obtain the data samples that do not look lik BRDFs. illustrate this point in Figure 9. Moreo er in order to span the whole space, we ould need to ha at least 45 parameterization directions in order to reach all specied BRDFs. This suggests that the space of all possible BRDFs lies on lo wer -dimensional manifold that is non- linearly embedded in the 45D linear space. In the ne xt section, we apply recently de eloped

nonlinear dimensionality reduction tech- niques to disco er this lo wer dimensional manifold. 5.2 Nonlinea dimensionalit reduction Nonlinear dimensionality reducers (NLDR) compute lo w-distortion embeddings of high-dimensional data in lo w-dimensional tar get spaces. The nonlinearity usually obtains from the act that only local relationships in the ambient space are preserv ed while long- distance relationships are presumed to be corrupted by the curv ature of the manifold in the ambient space. First-generation NLDRs such as nonmetric MDS [Kruskal and ish 1978], IsoMap [T enenbaum et al.

2000], and LLE [Ro weis and Saul 2000] generalize PCA to gi lo w-dimensional embeddings of the data, ut of fer no map- ping of the data points. Recently tw second-generation methods ha been announced that of fer continuous mappings between the embedding an the original (ambient) space: Automatic Alignment [T eh and Ro weis 2003] combines LLE with set of pre-estimated local dimensionality reducers–each of which is presumed to be t- ted to relati ely at subset of the manifold–and solv es for mix- ture of these projections that globally attens the data while min- imizing

barycentric distortion in each point neighborhood. Chart- ing [Brand 2003] solv es for ernel-based mixture of projections that minimizes Euclidean distortion of local neighborhoods; it in- cludes solution for the local dimensionality reducers needed by automatic alignment. chose to use charting because it is x- plicitly designed to ork well with small numbers of samples and to suppress measurement noise, tw conditions that tend to break methods for dimensionality reduction from local relationships. Figure 10 gi es the main geometric intuition behind charting. First one solv es for set of

at “pancak e Gaussians that smoothly Figur 10: simple charting xample. Points sampled from unkno wn manifold (gray curv e) are projected onto three subspaces (red, green, and blue lines) and assigned probability (indicated by size) according to their distance from the point where the chart touches the manifold. minimal-distortion mer ger of these charts gi es attening of the manifold in lo wer dimensional space, where the mapped locations of points are the probability-weighted combinations of their chart-specic locations. co er the data manifold, in the sense that

adjoining Gaussians ha similar orientation. The dominant ax es of each Gaussian specify subspace. Projecting the data into this subspace gi es “chart of one part of the manifold. chart preserv es local structure where it touches the manifold and suppresses measurement noise that dis- places samples of the manifold. data point has location and probability in ery chart. Due to curv ature of the manifold, chart gi es ery distorted picture of ara ay points; these points are assigned ery lo probability The pancak Gaussians are solv ed under criterion that opti- mizes the charts for the ensuing

“connection. The connection is an af ne mer ger of all charts in the tar get space–ef fecti ely at- tening of the manifold that minimally distorts all charts and max- imizes agreement between erlapping charts of the locations of points to which the assign high probability The connection gi es

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Figur 9: Nonlinear spaces generate alid BRDFs where linear spaces ail. Original BRDF corresponding to point on 45 dimensional yperplane (left). Ph ysically implausible reectance (hole in the middle of the specular highlight) corresponding to mo ving ay from point

on the 45 dimensional linear subspace (center). Ph ysically plausible reectance corresponding to mo ving equally ar ay from point on the 15 dimensional non-linear manifold (right). Figur 11: Data reconstruction error as function of the dimen- sionality of the global chart. The sharp drop in this error curv indicates that 10-dimensional chart is suf cient for the BRDF data. In act, that chart has better reconstruction error than 25- dimensional PCA. mappings between the ambient and tar get spaces, which are simply mixtures of af ne projections, weighted by the

probability that point “belongs to each chart. The dimensionality-reducing map- ping from the ambient to the tar get space ef fecti ely imposes lo w- dimensional coordinate system on the samples, while the in erse mapping gi es smoothly curving lo w-dimensional surf ace in the ambient space, ef fecti ely reconstructing the original manifold. or charting, one must specify set of chart centers, width parameter for the Gaussians, and tar get dimensionality used the def ault settings: one chart centered on each data point and half the erage distance between each point and its closest neighbor Note

that locating chart on point does not cause the manifold to pass through that point–only near it. See [Brand 2003] for additional details. As with PCA, the data-reconstruction error of charted data set gi es an indication of the true dimensionality of the manifold. Fig- ure 11 sho ws that our BRDF data probably lies on 10D manifold. The reconstruction error does not decline monotonically because each dimensionality may merit dif ferent attening. or xample, if the data were sampled from truncated cone, the best 1D chart ould simply be height along the cone, while the best 2D chart ould

atten the cone into an annulus. Each attening ould sup- press the noise in dif ferent directions, some more fortuitous than others. While the 10D manifold xhibits good reconstruction of the orig- inal data, our goal is to synthesize no el BRDFs. ith that in mind we chose to ork on 15D manifold because interpolations on it pass en closer to the data density (with error comparable to 45D PCA reconstruction). Moreo er this dimensionality is roughly con- sistent with pre vious isotropic BRDF models [W ard 1992], [Lafor tune et al. 1997], and [K oenderink et al. 1996], which ha at

least 10 de grees of freedom. charted manifold of BRDF data mak es it possible to treat the space of BRDFs as if it were linear and to identify meaningful ax es of ariation in this embedding space. An interpolating or x- trapolating line in this space is nonlinear curv in the original BRDF space that passes closer to the data density that the equi a- lent straight line ould (on erage), simply because it stays on the manifold where straight line does not. This translates directly to superior BRDF synthesis, as will be demonstrated belo Mo del Construction In order to use our sample-based

reectance model it is necessary to de elop intuiti user interf aces for specifying and xploring ne materials. in estig ated methods for characterizing material traits by analogies deri ed from the xisting samples. belie that such methods pro vide the best and most intuiti user interf ace [Pellacini et al. 2000]. Our model is uilt from actual ph ysical measurements and it re- produces these measurements. Therefore, we ha dened model parameters for lar ge collection of materials materials we ha measured. belie that the most useful scheme of na vig ation is when users can choose

as starting point some type of the material similar to the one the desire. In our case the can pick an of the measured materials. Then, the ould change the reectance prop- erties of this material according to one of the follo wing schemes (these na vig ation schemes are applicable for both linear and non- linear manifold models). The simplest method is to choose another BRDF and mo in this direction. Although of limited use, this method orks well for perceptually similar materials. more useful approach is to dene directions corresponding to desired trait (the parameterization

direction is 45D ector for linear space and 15D ector for nonlinear space). pick some arbitrary point on the manifold and then mo in the direction de- ned by the ector by adding it to the current position to increase the trait, or subtracting it to decrease the trait. can backproject the current point onto the original BRDF space to check the corre- sponding BRDF Ne xt, we describe arious procedures for identi-

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fying trait ectors. Our modeling approach requires the user to specify suf cient set of traits. This specication can be as simple as binary

classi- cation (i.e., noting whether each acquired BRDF has the specied trait.) also allo the user to lea BRDF unspecied in cases where the trait is hard to determine or simply does not apply Usu- ally the more samples we specify for each class the more precise the direction is. There are man dif ferent ays to dene the parameterization di- rections based on the classication. ha xamined and alu- ated fe (A) Mean dif ference [Blanz and etter 1999]: In this approach we compute the erage of each BRDF in each comple- mentary pair of clusters associated with

trait (i.e., those samples with, and those without) in the embedding space. Then the ec- tor between these complement erages in the embedding space is the parameterization direction. This direction ector is then applied (added or subtracted) to the current point in the embedding space. (B) Support ector machines [V apnik 1995]: Support ector ma- chines determine the yperplane which separates the data points in the rst material class from the data points in the second class. The partitioning yperplane has maximum distance to the closest points (called support ectors) in both material

classes. The parameteriza- tion direction is the normal to this yperplane. The yperplane is dened in 15D space for non-linear analysis and 45D for the linear space. This method also tells us on which side of the yperplane the current point is, and ho ar the point is from the plane. (C) Fisher' linear discriminant [Duda and Hart 1973]: Each material class cor responds to some distrib ution of high-dimensional data (15D for non-linear analysis and 45D for linear analysis). Fisher' linear dis- criminant denes projection of these distrib utions on the axis such that the distrib

utions projected on this axis are the most separable (the projection maximizes the distance between the means of the tw classes while minimizing the ariance of each class). In prac- tice, support ector machines performed the best on our data set and Fisher' linear discriminant performed the poorest. Since we ant our model to preserv the basic principles of ph ysics, we ha to disallo mo ements on the manifold that do not adhere to these principles. consider the three follo wing princi- ples: Reciprocity: As mentioned before, reciprocity in our model is met by def ault since we store only half

of the BRDF ector Non-ne ati vity: allo the user to mo only in the space so that all the alues in the backprojected ector are positi e. Ener gy conserv ation: unit of light ener gy is applied at some incoming light direction. If the sum of ener gy in all outgoing directions is less than one (we assume that the surf ace does not emit ener gy by itself) then the ener gy is conserv ed. This has to be true for all incoming light directions in order for BRDF to follo ener gy conserv ation. enforce this and do not allo the users to produce BRDFs for which the sum of ener gy for an incoming direction

is greater than one. Results Once the BRDFs are acquired and alidated, as described in section 4, we performed both linear and non-linear dimensionality reduc- tion as described section 5. then set out to construct perceptual BRDF model using the techniques outlined in section 6. This sec- tion presents the results from typical model construction session. test subject as ask ed to characterize each of the BRDFs from our database using 16 dif ferent traits. These included ed- ness gr eenness blueness specularness dif fuseness glossiness Figur 12: Dif fuseness trait vs specularness trait. Observ

that the dif fuseness and specularness traits xhibit weak in erse correla- tion. The green, blue, and red ectors denote projections of the BRDF interpolations sho wn in the second, third, and fourth ro ws of Figure 16 respecti ely Figur 13: Metallic-lik trait vs specularness trait. Observ that the metallic-lik and specularness traits xhibit weak correlation. The green, blue, and red ectors denote projections of the BRDF interpolations sho wn in the second, third, and fourth ro ws of Fig- ure 16 respecti ely metallic-lik plastic-lik oughness silverness gold-lik fabric- lik acrylic-lik gr

easiness dustiness rubber -lik In sense, these parameters are arbitrary since the classication is completely based on the subject' interpretation. could ha chosen traits without ph ysical connotations, such as ugly or pleasing Alterna- ti ely the traits could ha been based on actual measurable quan- tities, such as conducti vity and mean surf ace ariation. Our test subject characterized each BRDF as one of three choices: 1) pos- sessing the particular trait, 2) not possessing the trait, or 3) unclear then used the subject' characterizations to uild model in both the linear and

non-linear embedding spaces using Support ector Machines. The results from this trait-based analysis are sho wn as projec- tions onto the deri ed trait ectors in Figures 12, 13, and 14. These projections are computed in the linear embedding space gi en by

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our non-linear model. Observ that the metallic and specular char acteristics are weakly correlated, the specular and dif fuse traits are weakly in erse-correlated, and the glossy and dif fuse traits are in erse-correlated. This is what we ould xpect. Note that we do not mak attempts to model independent traits in either our

trait selection or trait ector deri ations. Therefore, we xpect that addi- tion of particular trait to an xisting BRDF may ef fect other traits. This lack of parameter independence is tradeof that we accept in order to establish perceptually meaningful parameters in our mod- eling approach. Despite the act that the parameterization ectors are not orthogonal, the did span the whole 15D non-linear embed- ding space and pro vide an intuiti set of “dials for users to design materials. Figur 14: Glossiness trait vs dif fuseness trait. Observ that the glossiness and dif fuseness traits xhibit an in

erse correlation. The green, blue, and red ectors denote projections of the BRDF inter polations sho wn in the second, third, and fourth ro ws of Figure 16 respecti ely Once trait ectors are established, we can add and subtract them from our data points in our embedding space. In Figure 15 we demonstrate four xamples of arying user -specied traits using the linear model. The rst ro sho ws teapot rendered using our Gold- aint BRDF on the ar left, and the ef fect of adding the edness trait in successi steps to the right. The second ro starts from our SpecularGold BRDF (left) with

successi additions of the silverness trait. The third ro adds the gold-lik trait to the Blue- GlossyP aint BRDF (left). Finally the fourth ro sho ws the addition of the specularness trait to the Blac kMattePlastic BRDF It is our xperience that the linear model gi es reasonable BRDFs if only small displacements are permitted. If the displacement is too lar ge, ph ysically in alid BRDFs result (as illustrated in Figure 9). then applied the same trait classications and Support ec- tor Machine calculations to the embedding space of our non-linear model. Figure 16 demonstrates xamples using

this approach. The rst ro of Figure 16 sho ws our Copper BRDF on the left, with successi additions of the oughness trait. The second ro be gins with our Gr eenAcrylic BRDF and sho ws the addition of the blueness trait. The trajectory of this path is also illustrated in Fig- ures 12, 13, and 14. Notice that color -change specication is not particularly correlated with an of traits used for these projec- tions. Thus, we ould xpect relati ely small mo ements and no preferred direction. The third ro on the other hand, represents the addition of the metallic trait to the

ioletAcryllic BRDF model, whose path is also illustrated in the projections in Figures 12, 13, and 14. The path trajectory of this xample conforms to our xpec- tations, and its magnitude is lar ge in these visualizations since the metallic trait is correlated to the glossiness and specularness traits used as axis. The fourth ro starts with ellowDif fuseP aint BRDF and sho ws the addition of the glossiness trait, which is depicted as the red path in Figures 12, 13, and 14. The direction of this path is as we ould xpect, and it has lar ge magnitude due to the act the the ellowDif fuseP aint BRDF

is located ar ay from the glossy xamples in the projections sho wn. Ov erall the non-linear basis set results in more rob ust model than our linear basis set, in that we were able to mo lar ge dis- tances within the non-linear embedding space and still generate ph ysically plausible BRDFs with the xpected appearance. Our modeling approach also allo ws us to associate approximate trait ectors with ph ysical processes. This can be done in one of tw ays, by tting least-squares line to path of specied BRDFs in the embedding space, or by computing local piece wise dif ference ectors

between xamples. As an xample, we ha modeled metal oxidation. measured the reectance changes as metal as xposed to an acidic en vironment. It changed from highly spec- ular polished material to black matte material. The acquired four BRDFs determine path in the embedding space. The intermediate stages are interpolated in the embedding space and backprojected to the sample space (Figure 17). Figure illustrates another pro- cess rust formation. used spatially arying te xture to select rust le els for each point on the teapot. are currently measuring more processes lik this such as copper

patination and other types of rust formation. uture rk and Conclusions In this paper we ha introduced ne approach for modeling isotropic BRDFs. Our model generates ne surf ace reectance models by forming combinations from set of densely-sampled, acquired BRDFs. are hopeful that data-dri en reectance mod- eling approaches, lik ours, can greatly xpand the range of material models used in computer graphics rendering. In order to de elop an ef fecti and ef cient interpolation scheme we choose to rst analyze the inherent dimensionality of our data set. this end we

applied both in linear subspace and non- linear manifold analysis. The results of this analysis are suggesti of the erall structure of BRDFs. Specically we found that the linear subspace model lent itself to the creation of ph ysically im- plausible BRDFs, and lar ge number of dimensions (around 45) were required to adequately represent our measurements. Nonethe- less, we still found the linear subspace model to be useful for in- terpolation er small distances. The nonlinear model, on the other hand, as much more compact in its dimensionality (around 14 dimensions for the same accurac

as the 45-dimension linear sub- space model), and more rob ust in its ability to interpolate plausi- ble BRDFs er long distances. Ho we er we caution ag ainst er generalizing from our results. are comfortable in saying that our modeling approach ef fecti ely represents our data set, ut our sample size is still relati ely small to dra conclusions re arding the fundamental nature of isotropic BRDFs. Ho we er we are op- timistic that techniques lik ours can be used to greatly xpand our kno wledge in these areas. also ha demonstrated methods for dening intuiti pa- rameters for na vig ating

within BRDF models. These techniques can easily be customized for range of industrial and artistic appli- cations. Furthermore, the can be personalized for indi vidual use or made objecti by incorporating ph ysical measurements. The adv antages of our data-dri en BRDF model include high de gree of realism, perceptually meaningful parameterization, rel- ati ease of modeling for comple surf ace materials, and speed of aluation. The main disadv antage of the model is its size. belie that the model we propose can easily be incorpo- rated into xisting rendering systems. also hope to xtend our

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Figur 15: Na vig ation in the linear space. Each ro corresponds to changing one parameter of the model. The rst ro sho ws an increase in the edness trait applied to the GoldP aint BRDF The second ro illustrates an increase in the silverness trait applied to the SpecularGold BRDF Ro three applies the gold-lik trait to the BlueGlossyP aint BRDF The fourth ro sho ws an increase in the specularness trait applied to the Blac kMattePlastic BRDF ork in sample-based reectance modeling to include anisotrop (4D BRDF), macro-scale surf ace ariations typically described by

BTFs, and subsurf ace scattering ef fects (BSSRDF). Another ob vi- ous xtension ould be to use this model in solving in erse render ing problems. Ackno wledgments ould thank Ste en Gortler and Julie Dorse for helpful discus- sions; Fredo Durand for help with writing the shader; Henrik ann Jensen for the Dali renderer; Joe Marks for his continuing support; the anon ymous re vie wers for their constructi comments; and Jen- nifer Roderick Pster for proofreading the paper arts of this ork were supported by NSF CAREER grant 9875859. References 1995. Neur al Networks for attern Reco gnition

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Figur 16: Na vig ation on the non-linear manifold. Each ro corresponds to changing one parameter of the model. The rst ro sho ws an increase in the oughness trait applied to the Copper BRDF The second ro illustrates an increase in the blueness trait applied to the Gr eenAcrylic BRDF Ro three applies the metallic-lik trait to the ioletAcryllic BRDF The fourth and fth ro ws sho an increase in the glossiness trait applied to the ellowDif fuseP aint BRDF (the images

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Figur 17: Progression of the steel oxidation process. Our BRDF model also supports interpolation along ph ysically meaningful path. In this xample we start with completely polished steel sample (upper left) and progressi ely oxidize it. The nal black-oxidized sample is sho wn on the lo wer right. 1992. ast and accurate light reection model. Com- puter Gr aphics 26 Annual Conference Series, 253–254. 1999. Interacti rendering with arbi- trary BRDFs using

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