Computers & Graphics 32 (2008) 149–158 Technical Section Linear approximation of Bidirectional Reflectance Distribution Functions Aydin Ozturk à , Murat Kurt, Ahmet Bilgili, Cengiz Gungor International Computer Institute, Ege University, Bornova, 35100 Izmir, Turkey Received 18 June 2007; received in revised form 19 November 2007; accepted 11 January 2008 Abstract Various empirical and theoretical models of the surface reflectance have been introduced so far. Most of these models are based on functions with non-linear parameters and therefore faces some computational difficulties involved in non-linear optimization processes. In this paper, we introduce a new approach for approximating Bidirectional Reflectance Distribution Functions (BRDF) by employing response surface methodology. The proposed model employs principal component transformations of the explanatory variables which are essentially functions of incoming and outgoing light directions. The resulting model is linear and can be used to represent both isotropic and anisotropic reflectance for diffuse and glossy materials. Considering some widely used reflection models including the Ward model, the Ashikhmin–Shirley model and the Lafortune model, we demonstrate empirically that satisfactory approximations can be made by means of the proposed general, simple and computationally efficient linear model. r 2008 Elsevier Ltd. All rights reserved. Keywords: BRDF representation; Reflection models; Rendering; Linear models; Principal components 1. Introduction Building a comprehensive reflection model that describes the interactions between light and materials is a funda- mental problem in computer graphics. To synthesize a realistic image of a scene, a complete description of reflectance is required for each surface in the scene. A class of functions called Bidirectional Reflectance Distribution Functions (BRDF) has been widely used to describe the surface reflectance [1]. Theoretically the BRDF is a function of a number of factors including incident light direction, reflected light direction, the wavelength and the surface position. Various models ranging from empirical models to physically based models for BRDFs have been developed to approximate surface reflectance. Each model can have better approx- imation over the others under certain conditions. Whatever model is selected for a certain application, one needs to determine its unknown parameters. A common approach for determining the model parameters is to estimate them from the experimental data based on the BRDF measurements. Mostly the least squares techniques are used for estimating the model parameters. However, most of the models are defined by some non-linear functions and estimating the underlying parameters is not straightforward. There are several shortcomings in model fitting when a BRDF is represented by a non-linear function [2]. One major problem is that non-linear least squares estimation requires employing some optimization algorithms. Depending on the number of lobes used in modeling the BRDF, the correspond- ing number of parameters to be estimated usually is large. For example, when Lafortune et al. [3] model is chosen to fit an isotropic material, there would be at least three and six non- linear parameters which should be estimated for one lobe and two lobe representations, respectively. Furthermore, optimiza- tion results closely depend on the choice of initial values of the non-linear parameters and a global minimum usually is not guaranteed. Also computational cost of estimating the non- linear parameters become high when large data set is used. ARTICLE IN PRESS www.elsevier.com/locate/cag 0097-8493/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cag.2008.01.004 à Corresponding author. Tel.: +90 232 388 72 28; fax: +90 232 388 72 30. E-mail addresses: aydin.ozturk@ege.edu.tr (A. Ozturk), murat.kurt@ege.edu.tr (M. Kurt), ahmetbilgili@gmail.com (A. Bilgili), cengiz.gungor@ege.edu.tr (C. Gungor).