Compact and intuitive datadriven BRDF models
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Abstract
Measured materials are rapidly becoming a core component in the photorealistic image synthesis pipeline. The reason is that datadriven models can easily capture the underlying, fine details that represent the visual appearance of materials, which can be difficult or even impossible to model by hand. There are, however, a number of key challenges that need to be solved in order to enable efficient capture, representation and interaction with real materials. This paper presents two new datadriven BRDF models specifically designed for 1D separability. The proposed 3D and 2D BRDF representations can be factored into three or two 1D factors, respectively, while accurately representing the underlying BRDF data with only small approximation error. We evaluate the models using different parameterizations with different characteristics and show that both the BRDF data itself and the resulting renderings yield more accurate results in terms of both numerical errors and visual results compared to previous approaches. To demonstrate the benefit of the proposed factored models, we present a new Monte Carlo importance sampling scheme and give examples of how they can be used for efficient BRDF capture and intuitive editing of measured materials.
Keywords
Reflectance modeling Rendering Computer graphics1 Introduction
Over the last decade, we have seen the development of computer graphics algorithms and techniques which enable the quality and accuracy needed to make rendered images truly comparable to photographs of the same scene. The accuracy in the simulation of scattering at surfaces and computation of the light transport in a scene is determined by the way the material properties such as color and reflectance, modeled by the Bidirectional Reflectance Distribution Function (BRDF) [30], are measured and represented. This has led to the development of both accurate parametric models [3, 8, 11, 16, 24, 35, 47], and advanced datadriven methods [20, 26, 29] for describing, measuring and analyzing for virtually all classes of materials.
Recently, measured materials and appearance capture techniques have gained significant popularity, for an overview see [14, 23]. The advantage of datadriven approaches is that carefully measured BRDFs automatically bring the detailed appearance of realworld materials into rendering pipeline. There are, however, still a number of important challenges that need to be solved in order to make datadriven models flexible and easy to use in practice. First, although significant progress has been made, [1, 31], accurate BRDF and SvBRDF measurements are still challenging due to mechanical and computational complexity. Second, measured data are difficult to edit, analyze, and in other ways interact with. It is, therefore, necessary to further develop compact representations which allow for detailed representations of the reflectance distributions with minimal approximation errors.

A study of the separability of existing datadriven BRDF models.

Two new BRDF models specifically designed for efficient 3D and 2D representation with 1D separability.

A novel importance sampling strategy for datadriven BRDFs of PDV parameterization.

PDV basis measurement and intuitive BRDF editing from mixture of PDV factors.
2 Background
The analysis and new BRDF models presented in this paper build upon a large body of previous work in appearance measurement and modeling. The following subsections describe how our work relates to datadriven BRDF models, existing BRDF parameterizations, and previous approaches for BRDF factorization.
DataDriven BRDF Models The way in which light interacts with matter is, in computer graphics, described using the rendering equation [17]. In the rendering equation, the scattering at each surface point in a scene is modeled by the BRDF [30]. Accurate and efficient BRDF models are essential components in the rendering of complex synthetic scenes for photorealistic rendering. Measured BRDF data are commonly stored using the standard parameterization which expresses the BRDF directly in spherical coordinates, \((\omega _i, \omega _o)\). Reparameterization of BRDFs has been studied extensively for decades in order to find more efficient and compact representations. A good representation should ideally lead to reducing storage requirement (i.e., provides compactness), an intuitive representation for user editing purposes, and enable efficient importance sampling. Previous works have explored factorization methods in order to find compact representations directly in the standard parameterization [27, 44]. However, other parameterizations such as the HalfDiff parameterization [39] and the HalfOut parameterization [21] have also been proposed for obtaining more compact representations [7, 32, 38] and efficient importance sampling procedure [21]. However, previous works have not proposed both compact and intuitive representations of BRDF data, allowing for intuitive user edits, flexible and practical measurement setups, and efficient importance sampling.
In this work, we focus on reparameterization of isotropic BRDFs. Our work is inspired by the parametric ABC BRDF model [24], and we compare their projected deviation vector (PDV) parameterization to the more commonly used parameterizations used by Rusinkiewicz [39] and Lawrence et al. [21]. The main contributions of this paper are that the resulting BRDF parameterization enables us to represent measured BRDF data with as little as only two 1D factors for adequate representation, and we introduce a new representation of the deviation vector. In contrast to previous datadriven BRDF models, the proposed BRDF parameterization also induces a clear separability in both 2D and 3D representations. Moreover, we investigate factorization methods based on both 2D and 3D representations in order to show that the separability of our parameterization leads to very compact BRDF representations, well approximated with a rank1 tensor approximation with nonnegative factors intuitive for BRDF editing.
In many applications, it is useful to represent isotropic 3D BRDFs with 2D approximations. 2D BRDF representations have shown to be useful both for efficient storage, in acquisition systems and other computer vision problems [13, 37]. Romeiro et al. [37] proposed a simple transformation from 3D to 2D for measured BRDFs stored in the HalfDiff parameterization. In the HalfDiff parameterization, the parameters \((\theta _h, \theta _d, \phi _d)\) are reduced to \((\theta _h, \theta _d)\) by taking the average over \(\phi _d\). Stark et al. [43] presented mathematical relations of \((\omega _i, \omega _o)\) on Barycentric coordinates and proposed three 2D parameterizations. However, these parameterizations are based on unintuitive and complex mathematical functions which limit user intuition. Barla et al. [5] proposed a parameterization which is similar to the PDV; however, there is not, to our knowledge, any BRDF model support for their parameterization. Thus, it would limit the use of its parameterization and importance sampling strategy.
BRDF Factorization In previous work, it has been shown that measured BRDFs can be factored into basis representations to enable compact representations for efficient rendering and editing. Kautz and McCool [18] proposed the normalized decomposition (ND) method for interactive rendering. In their work, they used a modified HalfDiff parameterization to avoid numerical instability. McCool et al. [27] developed homomorphic factorization on the standard parameterization. The homomorphic factorization is applied on logarithmically transformed data to ensure that the factorized results do not contain negative values. Lawrence et al. [21] proposed the double factorization method on the HalfOut parameterization for material editing applications. The method uses nonnegative matrix Factorization (NMF) [22] on 4D BRDF data. They rearranged 4D BRDFs into 2D BRDFs so that the double factorization could be applied. However, accurate results required a high number of terms in the factorization. Bagher et al. [4] proposed factored BRDF models based on the microfacet BRDF model [11] by using specific weighting functions in optimization process. The resulting models are efficient representations; however, the importance sampling is still an issue on their models; thus, their models require complicated sampling method such as multiple importance sampling (MIS)[40, 41, 46] or precomputed sampling scheme [25]. BenArtzi et al.[6] proposed factored BRDF models in the form of linear combinations of wavelet bases to enable realtime BRDF editing and rendering. Their method supports both parametric and analytical BRDF models. An interesting venue for future work is to investigate how our models can be incorporated with their system for editing and rendering.
Tensor factorization methods have also been applied to BRDFs. Sun et al. [44] applied Tucker factorization on the standard parameterization for interactive relighting. Schwenk et al. [42] used CANDECOMP/PARAFAC Decomposition (CPD) method [9, 15] on the standard parameterization with an additional dimension which is for wavelength information. The method iteratively applies the rank1 approximation of the CPD method with repetition of residual factorization. They found that in standard spherical coordinates, the CPD method works better on diffuse and moderately glossy materials; while using the HalfDiff parameterization, CPD works well with glossy materials. A problem, however, is that the method needs at least four factor packs, i.e., the number of iterations on residual factorization, to accurately approximate BRDFs. Later, Ruiters and Klein [38] presented a way to improve optimization of tensor factorizations. They used the HalfDiff parameterization and applied the CPD method on BRDFs transformed to the \(\log \)domain using the square of the relative error as the error metric in the optimization. This error metric can provide better approximation compared to using the more common \(L_2\) error metric. However, the output needs up to eight components, i.e., rank8 approximation, to get accurate results. Bilgili et al. [7] employed the Tucker factorization on 4D BRDFs to represent both isotropic and anisotropic materials in the \(\log \)domain. In their work, they simplified the three color channels into one luminance channel and factorized the luminance data. To get RGB BRDF data, a linear regression model is fitted to each of color channels. It is reported that measured materials [26] need up to 1315 iterations of rank1 approximations to get accurate results. In contrast to previous work, we show our proposed models can accurately describe BRDFs as a single term rank1 nonnegative approximation, which allows for both compact and intuitive BRDF models.
3 BRDF parameterization
The first step in creating a separable BRDF representation is to select a suitable parameterization. In this section, we give an overview of the BRDF parameterizations introduced by Rusinkiewicz [39], Lawrence et al. [7, 21], and Löw et al. [24] and analyze their behavior with respect to separability and factorization. We show that they are all good choices for going from 3D to 2D and that the characteristics of the projected deviation vector parameterization[24] make it a particularly good choice for separable BRDF models. This analysis then forms the basis for the new models proposed in this paper.
The values of a BRDF typically exhibit a very high dynamic range [31]. To avoid computational problems in such as factorizations and basis representations, it is therefore common to transform the BRDF values to the \(\log \)domain. To represent BRDFs efficiently for factorization, we transform the data values using \(\rho _t = log(\rho + 1)\).
The HalfDiff parameterization has four parameters which are \((\theta _h, \phi _h, \theta _d, \phi _d)\). To represent isotropic BRDFs, \(\phi _h\) can be ignored because isotropic BRDFs are independent of \(\phi _h\) due to symmetry.
The HalfOutgoing vector parameterization (HalfOut) has been used for BRDF factorization in several previous studies [7, 21]. The HalfOut parameterization is formed by \((\omega _h, \omega _o)\), the half vector, and the outgoing vector. Figure 1b illustrates the vectors and their notations. When \(\omega _o\) is fixed, changing \(\omega _h\) results in moving \(\omega _i\). Here, the half vector is similar to the HalfDiff parameterization defined by the normalized sum of \(\omega _i\) and \(\omega _o\), given in Eq. 1.
3.1 Analysis of parameterizations
The different parameterizations described above can all accurately describe isotropic BRDFs using their inherent 3D representation. Since our goal is to reduce the dimensionality of the models required to accurately represent BRDF data, we analyze how well the 3D representation can be represented in 2D. To do this, we first, in Fig. 4, plot the projection of the 3D coordinates on the hemisphere onto the unit disk for regular points in the parameter spaces. The top row shows the HalfDiff parameterization, with parameters \((\theta _h, \theta _d, \phi _d)\), for a fixed \(\theta _h = 30^\circ , 70^\circ \). The plots show how \(\omega _o\) moves on the hemisphere and in the projection plane when \(\omega _d\) is varied. The middle row shows the HalfOut parameterization, with parameters \((\theta _o, \theta _h, \phi _h)\), and a fixed \(\theta _o = 30^\circ , 70^\circ \). The plots show how \(\omega _i\) changes as a function of \(\omega _h\). Finally, the bottom row shows the same plots for the PDV parameterization, with parameters \((\theta _r, d_p, \phi _p)\). By varying \(d_p\) and \(\phi _p\) with a fixed angle of \(\omega _r = 30^\circ , 70^\circ \), only \(\omega _o\) is changed and forms circles in the projection plane. In fact, for smooth surfaces Löw et al. [24] observed that these circles closely model the isocontours on the BRDF lobe. This means that the BRDF values along each circle are close to constant.
Further conclusions can be drawn from visualizing real BRDF data. Figure 5 illustrates an example from our investigation where the alumbronze BRDF is plotted in each of the parameterizations, respectively. The three leftmost columns demonstrate how the 2D slices in the 3D representation vary along the third dimension of each parameterization. Figure 5a–d illustrates the BRDF data in the HalfDiff parameterization along the \(\phi _d\), Fig. 5e–h illustrates the BRDF data in the HalfOut parameterization along the \(\phi _h\) dimension, and Fig. 5i–l illustrates the BRDF data in the PDV parameterization along the \(\phi _p\) dimension. The right column shows the corresponding 2D representation computed as the mean over all slices in the third parameter dimension for each parameterization. The plots illustrate that for all three parameterizations, the BRDF values are stable along the third dimension, which supports the conclusion that they are suitable choices for going from a 3D to a 2D representation. Visual inspection of the BRDFs in the MERL database also shows that data represented in the PDV parameterization is better aligned (horizontally and vertically) in the 2D plane compared to the HalfDiff and HalfOut parameterizations. As we show in the next section, this structure makes it possible to accurately factorize the 2D representation into a 1D + 1D factored BRDF model.
4 BRDF factorization
Accurate factorization requires both a representation which enables separability and a suitable factorization technique adhering to the requirements put by the application. The choice of parameterization, such as the PDV parameterization presented in the previous section, is thus a key for an efficient lowrank approximation as unstructured data or diagonally structured data results in high rank factorizations [33]. It is highly desirable that the BRDF data can be approximated using nonnegative rank1 approximations so that the data can be represented as nonnegative univariate functions which lead to intuitive representations for applications such as BRDF capture and editing.
For rank1 approximations, the three techniques are comparable in terms of approximation error. Figure 6 shows a comparison of Tucker decomposition and CPD applied to BRDF data from the MERL database [26] represented using the PDV, HalfOut, and HalfDiff parameterizations for 3D data described in the previous section. Figure 7 shows similar plots but compares NMF and CPD for factorization of 2D data as described in the previous section. Table 1 and 2 show the errors from Figures 6 and 7 in numerical form showing that the PDV parameterization exhibits the smallest mean error and low variance as compared to the HalfDiff and HalfOut parameterizations. Although the error varies between the different factorization models, the approximation accuracy is more or less the same for all three factorization techniques. Given advantages such as a lower computational complexity and efficient random access, we use the CPD method for both 3D and 2D data.
Rank1 iterative factorization In some cases, it is desirable to trade the single factor representation for achieving a higher accuracy in the factorization. This can be done using iterative factorization such that the residual is represented using a higher number of factors for the parameter dimensions, for an overview see [7, 19, 42]. In our models, we have the option of using an iterative method for residual factorization. Since the intuitiveness inherent to the single factor representation is lost, we do not enforce the nonnegative constraint for the residual factorization as this has the tendency to lead to a smaller error. Starting from the original tensor, we constraint the first factorization with the nonnegativity in order to preserve the physical meaning within the BRDFs. For the residual parts, we iteratively factorize the residuals without any constraint, allowing negative values.
5 New BRDF models
This section presents two new separable BRDF models, in 3D and in 2D, which are based on the above analysis of the PDV parameterization and factorization techniques. The separable models are designed to enable factorization into three or two 1D factors, respectively.
5.1 Importance sampling
The table shows mean, variance, minimum and maximum values of log relative errors of all 100 materials in the MERL database for each method, showing in Fig. 6
Method  Mean  Variance  Min  Max 

PDVTucker3D  \(\) 8.251643  0.263030  \(\) 9.763395  \(\) 7.662130 
PDVCPD3D  \(\) 8.213502  0.264648  \(\) 9.737277  \(\) 6.843860 
HOTucker3D  \(\) 7.586190  0.721300  \(\) 8.805157  \(\) 4.455247 
HOCPD3D  \(\) 7.586169  0.721289  \(\) 8.805092  \(\) 4.455237 
HDTucker3D  \(\) 7.676227  1.693427  \(\) 10.317398  \(\) 3.974776 
HDCPD3D  \(\) 7.676365  1.693840  \(\) 10.316965  \(\) 3.974837 
The table shows mean, variance, minimum and maximum values of log relative errors of all 100 materials in the MERL database for each method, showing in Fig. 7
Method  Mean  Variance  Min  Max 

PDV  NMF3D  \(\) 8.222539  0.236527  \(\) 9.622211  \(\) 7.661749 
PDV  CPD2D  \(\) 8.195418  0.227387  \(\) 9.619088  \(\) 7.384546 
HO  NMF3D  \(\) 7.497353  0.835841  \(\) 8.768603  \(\) 4.215825 
HO  CPD2D  \(\) 7.596027  1.001203  \(\) 9.405387  \(\) 4.215839 
HD  NMF3D  \(\) 7.541444  1.860140  \(\) 9.769958  \(\) 3.700108 
HD  CPD2D  \(\) 7.541442  1.860132  \(\) 9.769955  \(\) 3.700104 
It is unavoidable to sometimes choose \((d_p, \phi _p)\) pairs that result in invalid \(\omega _i\) directions, i.e., falls outside the domain of the parameters. However, when such a \((d_p, \phi _p)\) pair is chosen, it is straightforward to reject such samples without introducing inaccuracy using rejection sampling, for an overview see the book by Pharr and Humphreys [34].
Fitting time table of CPD3D for \(L = 1,5,10,15,20,25\) measured in seconds
Number of iterations  Min  Max  Mean  Variance 

1  5.44  8.48  6.89  0.87 
5  30.90  92.61  60.88  193.00 
10  65.10  201.75  128.74  1147.38 
15  112.08  339.16  212.41  2945.60 
20  157.91  464.54  297.84  5775.32 
25  205.70  597.35  385.42  9434.98 
We have found that removing \(sin(\theta _i)\) does not change the efficiency of the importance sampling significantly and that this approximation leads to high quality rendering results. The effect of the importance sampling is evaluated in the next section.
6 Results and evaluation
This section presents an evaluation of the new separable BRDF models described in Sect. 5, the different parameterizations described in Sect. 3 with respect to their ability to go from 3D to 2D representations, and the effect of the factorization techniques described in Sect. 4 in computing factored models using the different parameterizations. We also show examples of how the models presented in this paper can be used in applications such as storage, efficient BRDF capture and editing.
To measure the error introduced by the models, we use the relative root mean square error (RMS) over all the 100 BRDFs in the MERL database [26]. For each combination of BRDF model, parameterization and factorization, we compute the error over the entire hemisphere. In order to generate the same set of sample points and avoid unnecessary interpolation, we first use a random cosine sampling scheme in standard spherical coordinates [34] and then transform each sample into the parameter space of the parameterization where the error is measured. Both the PDV and HalfDiff models are represented at a resolution of \(90 \times 90 \times 360\) elements for the \((\theta _h, \theta _d, \phi _d)\) and \((\theta _r, d_p, \phi _p)\) parameter dimensions, respectively. We parameterized the HalfOut parameterization using \((\theta _o, \theta _h, \phi _h)\) and found that the \(\theta _h\) resolution needs to be higher than 90 for high fidelity renderings. Therefore, for the HalfOut, we used a resolution of \(45 \times 200 \times 360\) elements in order to ensure a fair comparison. The supplementary material includes both interactive examples for visual inspection of the proposed models and rendered examples of all BRDFs in the MERL database.
The CPD3D and CPD2D methods were implemented using the Nway Toolbox for MATLAB [2] with a nonnegativity constraint for the first iteration, \(L=1\). For the experiments, we used the MERL database (including all measured angles). For CPD3D, we used the PDV parameterization at a resolution of \(90 \times 90 \times 180\). All the rendering results were produced by using PBRT [34]. The fitting time comparison in Table 3 was carried out using an Intel(R) Xeon(R) W2123 3.60GHz computer with 32 GB memory.
The mean, minimum and maximum errors, and variance for three different parameterizations with \(L=10\) as illustrated in Fig. 9
Method  Mean  Variance  Min  Max 

PDV CPD3D L\(=\)10  \(\) 9.119674  0.432865  \(\) 10.903059  \(\) 7.764585 
HO CPD3D L\(=\)10  \(\) 9.018169  0.419751  \(\) 10.330703  \(\) 6.868430 
HD CPD3D L\(=\)10  \(\) 9.310537  0.854673  \(\) 11.282352  \(\) 7.033940 
The storage requirements for the BRDF models compared in Fig. 11
BRDF model  Bluemetallicpaint  Yellowmatteplastic  Nickel 

Measured  33.4 MB  33.4 MB  33.4 MB 
Lawrence et al.  139.0 KB  331.9 KB  96.5 KB 
Bilgili et al.  76.7 KB  73.2 KB  76.7 KB 
Our 3D Model, L = 10  87.1 KB  87.1 KB  87.1 KB 
Our 3D Model, L = 15  130.3 KB  130.3 KB  130.3 KB 
Our 3D Model, L = 20  173.5 KB  173.5 KB  173.5 KB 
Method  Mean  Variance  Min  Max 

Bagher et al. (Naive model)  \(\) 9.635750  0.681189  \(\) 11.509530  \(\) 7.829065 
PDVCPD3D L = 5  \(\) 9.052952  0.225250  \(\) 10.153436  \(\) 7.883952 
PDVCPD3D L = 15  \(\) 9.692110  0.377019  \(\) 11.469682  \(\) 8.071923 
PDVCPD3D L = 20  \(\) 9.843168  0.390666  \(\) 11.487065  \(\) 8.131693 
PDVCPD3D L = 25  \(\) 9.938137  0.418542  \(\) 11.600815  \(\) 8.146085 
Importance sampling Figure 13 compares the importance sampling technique developed for our factored models described in Sect. 5.1 to the techniques presented by Lawrence et al. [21], Edwards et al. [12], and Bilgili et al. [7]. The Figure shows renderings of the Princeton scene generated using the PBRT rendering system, described in the book by Pharr and Humphreys [34], with 256 path samples per pixel. The scene consists of Nickel, Yellowmatteplastic, and Bluemetallicpaint materials. The rendered images show that our importance sampling technique performs better than Lawrence et al. [21] and Edwards et al. [12] techniques numerically, but our technique gives numerically worse performance than Bilgili et al. [7] technique. A natural next step, however, would be to derive a more accurate approximation of the Jacobian which potentially would lead to even better results. The color shift on and near the teapot in Fig. 13c is due to Bilgili et al.’s representation inaccuracy. Because this representation factorizes intensity channel and then estimates color values linearly as a subprocedure by a few linear coefficients. Table 7 shows the rendering time for the images in Fig. 11 (in seconds). Even with our higher accurate models (\(L>15\)), our factored models have comparable computational cost compared to Edwards et al. [12], better computational cost compared to the technique presented by Bilgili et al. [7], but a higher computational cost compared to the technique presented by Lawrence et al. [21]. Note that our factored models compute probabilities on the fly and do not rely on precomputed lookup tables for importance sampling, but many previous BRDF representations (i.e., Bilgili et al. [7] and Lawrence et al. [21]) rely on precomputed lookup tables for importance sampling.
Limitations Our approach is dependent on the quantization of the \(D_p\) vector. In the experiments presented here, we computed the \(d_p\) step length based on the average of all materials in the MERL database. We have noticed that this results in unsuitable quantization for some materials, see specialwalnut224 in Fig. 10. This could be alleviated by computing different nonlinear quantizations for different material classes. Additionally, in order to make a comparison with Bagher et al. [4], we included all angles of the materials in the MERL database. Previous studies (e.g., Ngan et al. [28] and Löw et al. [24]) have pointed out that the measured values in the MERL database are increasingly unreliable toward grazing angles. It would therefore be interesting to evaluate the accuracy of the models within a restricted range of angles. Moreover, the Jacobian term used in the importance sampling is an approximation to the exact solution due to numerical instabilities. This could be improved by deriving a better approximation.
6.1 Applications
The rendering time (in seconds) for the renderings shown in Fig. 11 and a number of singlematerial spheres with material from the MERL database
BRDF model  Bluemetallicpaint  Yellowmatteplastic  Nickel  Princeton 

Edwards et al.  74.8  72.4  81.3  30.1 
Lawrence et al.  75.0  69.2  62.5  27.7 
Bilgili et al.  86.7  85.4  89.9  35.9 
Our factored L = 15  72.3  73.7  75.1  29.7 
Our factored L = 20  74.7  75.5  78.7  30.4 
Our factored L = 25  76.2  80.1  81.3  30.9 
BRDF reconstruction from factor measurements Using the factored 2D model described in Eq. 3, it is possible to capture a full isotropic BRDF by measuring the scaling of the lobe described by \(G_1(\theta _r)\) and the shape of the lobe described by \(G_2(d_p)\) in a planar slice of the BRDF. In the illustration of the 2D PDV parameterization in Fig. 5l, this would correspond to measure the 1D \(G_2(d_p)\) factor representing the vertical direction and the 1D \(G_1(\theta _r)\) factor representing the horizontal direction in the 2D matrix. As illustrated in Fig. 15 (left), the horizontal factor \(G_1(\theta _r)\) can be measured by moving a sensor and a calibrated light source in opposite directions with respect to the normal and capture the response in the mirror direction. The shape of the lobe described by \(G_2(d_p)\) can be measured by fixing the light source at a specific angle to the normal and move the sensor along the arc as illustrated in Fig. 15 (right). Please note that the \(G_2(d_p)\) factor needs to be normalized before it can be modulated with \(G_1(\theta _r)\) to form the full 2D matrix describing the BRDF. Figure 16 shows the reconstruction errors (red plot) obtained by simulating capture and reconstruction as described above for all BRDFs in the MERL database, and as a reference the error introduced by direct representation using our factored 3D model with \(L=1\) iteration. To simulate the capture of the \(G_2(d_p)\) factor, the light source was fixed at a \(70^\circ \) angle to the normal. Figure 17 shows example renderings of three reconstructed BRDFs.
7 Conclusions and future work
This paper presented a study of three different BRDF parameterizations and proposed two new factored BRDF models for 3D and 2D representations of isotropic BRDFs. The study showed that the HalfDiff [39], HalfOut [21], and PDV [24] parameterizations are all suitable for 2D representation of isotropic BRDFs with only small approximation errors. It also showed that the PDV parameterization structures the data in a way suitable for factorization. The evaluation demonstrated that the new models are flexible in terms of which parameterization is used and that they can represent measured BRDF with low numerical approximation errors, and produce visually plausible renderings. For efficient rendering, the paper also presented a new Monte Carlo importance sampling scheme based on the factored models.
The evaluation and the results point toward a number of interesting venues for future work. One interesting area is to further study how the choice of parameter quantization affects the accuracy of the factorization. Another important direction is to investigate how lowdimensional factored models can be adapted for accurate representation of anisotropic BRDFs. Finally, we will use the factored models to develop efficient BRDF capture systems and investigate how they can be incorporated in computer vision applications where BRDFs are needed to be characterized in the wild.
Notes
Acknowledgements
This work was supported by the Swedish Science Council through grant VR201505180, the strategic research environment ELLIIT, the Scientific and Technical Research Council of Turkey (Project No: 115E203), and the Scientific Research Projects Directorate of Ege University (Project No: 2015/BİL/043).
Compliance with Ethical Standards
Conflict of Interest
The authors declare that they have no competing interests.
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