Global optimization for estimating a multiple-lobe analytical BRDF
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Computer Vision and Image Understanding 115 (2011) 1679–1688
Contents lists available at ScienceDirect
Computer Vision and Image Understanding
journal homepage: /locate/cviu
Global optimization for estimating a multiple-lobe analytical BRDF
Chanki Yu, Yongduek Seo, Sang Wook Lee ⇑
Department of Media Technology, Sogang University, Seoul, Republic of Korea
1. Introduction Accurate description of surface reflection has been a topic of research in computer vision and graphics for decades. The Bidirectional Reflectance Distribution Function (BRDF) is a four-dimensional function that defines how light is reflected at a surface, and a number of BRDFs have been developed to model real-world material surfaces [18]. Analytical models have been popular for their compact representations, and some of those are developed for physical plausibility [1,4,5,10,16,19,23,24]. Since these models generally do not account for all the reflectance properties of all kinds of materials, approaches to representing reflectances on basis functions have recently been developed to describe reflectances from a wider range of material surfaces [2,20,21]. Despite all their comprehensiveness, however, the number of bases needs to be kept large to account for viewing and lighting variability and to maintain high frequency details. Once the analytical model is appropriately selected for a specific type of surface material and its parameters are accurately estimated, the model describes the reflection in a highly compact way. For estimating a BRDF, reflections are measured under various viewing and illumination angles, and the data is usually fitted to an analytical model using conventional (constrained) least-squares nonlinear minimization [13,17,24]. Many real-world surfaces
exhibit complex reflections that can only be adequately represented with multiple lobes if an analytical BRDF is used. Since highly nonlinear BRDFs that include multiple Gaussian-like functions can have a huge number of local minima, the nonlinear local minimization is prone to numerical instabilities and does not guarantee a global minimum. It has been noted that the fitting process becomes highly unstable for multiple lobes [17,27]. In traditional nonlinear optimization, the quality of the fit is dependent on good initial guesses. To make sure the optimization converges to a local minimum yielding a satisfactory result or hopefully the global minimum, the fitting quality of the result is visually/manually inspected and if necessary the optimization is restarted from a different set of initial guesses. However, even this supervised optimization does not necessarily guarantee the globally minimum solution. While global optimization has recently been an area of active research in geometric vision [9,11,12,26], little explicit effort has been made in photometric vision. To our knowledge, no previous work has given globally optimal solutions to the BRDF estimation problems under photometrically meaningful L1 or L2 cost functions. Our work shown in this paper focuses on the globally optimal parameter estimation of the Cook–Torrance model [5] with multiple specular lobes using a set of photometric measurements with known surface orientations and viewing/lighting directions. It is based on our recent work shown in [31]. The Cook–Torrance model has been developed based on the geometrical optics and considered one of the most physically plausible models. Nonlinearity arises from the surface roughness parameter in the Beckmann
⇑ Corresponding author.
E-mail addresses: ckyu@sogang.ac.kr (C. Yu), yndk@sogang.ac.kr (Y. Seo), slee@sogang.ac.kr (S.W. Lee). 1077-3142/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.cviu.2011.05.012
Article history: Received 16 April 2010 Accepted 6 May 2011 Available online 23 July 2011 Keywords: BRDF Analytical BRDF Parameter estimation Branch and bound Global optimization
a r t i c l e
i n f o
ห้องสมุดไป่ตู้a b s t r a c t
We present a global minimization framework for estimating the parameters of multiple-lobe analytical BRDF model using the techniques of convex programming and branch and bound. When an analytical BRDF is used to model a natural or artificial surface, reflections can often be represented accurately with multiple non-Lambertian lobes. A BRDF model with multiple lobes can be highly nonlinear and poses a challenge in data fitting for parameter estimation. Traditional local minimization suffers from local minima and requires a large number of initial conditions and supervision for successful results especially when a model is highly complex. We consider the Cook–Torrance model with multiple specular lobes, a parametric model with the Gaussian-like Beckmann distributions for specular reflectances. Instead of optimizing the multiple parameters simultaneously, we search over all possible surface roughness values based on a branch-and-bound algorithm, and reduce the estimation problem to convex minimization with known fixed surface roughness. Our algorithm guarantees globally optimal solutions. Experiments have been carried out for isotropic surfaces to validate the presented method using the extensive highprecision measurements from the MERL BRDF database. Ó 2011 Elsevier Inc. All rights reserved.
Contents lists available at ScienceDirect
Computer Vision and Image Understanding
journal homepage: /locate/cviu
Global optimization for estimating a multiple-lobe analytical BRDF
Chanki Yu, Yongduek Seo, Sang Wook Lee ⇑
Department of Media Technology, Sogang University, Seoul, Republic of Korea
1. Introduction Accurate description of surface reflection has been a topic of research in computer vision and graphics for decades. The Bidirectional Reflectance Distribution Function (BRDF) is a four-dimensional function that defines how light is reflected at a surface, and a number of BRDFs have been developed to model real-world material surfaces [18]. Analytical models have been popular for their compact representations, and some of those are developed for physical plausibility [1,4,5,10,16,19,23,24]. Since these models generally do not account for all the reflectance properties of all kinds of materials, approaches to representing reflectances on basis functions have recently been developed to describe reflectances from a wider range of material surfaces [2,20,21]. Despite all their comprehensiveness, however, the number of bases needs to be kept large to account for viewing and lighting variability and to maintain high frequency details. Once the analytical model is appropriately selected for a specific type of surface material and its parameters are accurately estimated, the model describes the reflection in a highly compact way. For estimating a BRDF, reflections are measured under various viewing and illumination angles, and the data is usually fitted to an analytical model using conventional (constrained) least-squares nonlinear minimization [13,17,24]. Many real-world surfaces
exhibit complex reflections that can only be adequately represented with multiple lobes if an analytical BRDF is used. Since highly nonlinear BRDFs that include multiple Gaussian-like functions can have a huge number of local minima, the nonlinear local minimization is prone to numerical instabilities and does not guarantee a global minimum. It has been noted that the fitting process becomes highly unstable for multiple lobes [17,27]. In traditional nonlinear optimization, the quality of the fit is dependent on good initial guesses. To make sure the optimization converges to a local minimum yielding a satisfactory result or hopefully the global minimum, the fitting quality of the result is visually/manually inspected and if necessary the optimization is restarted from a different set of initial guesses. However, even this supervised optimization does not necessarily guarantee the globally minimum solution. While global optimization has recently been an area of active research in geometric vision [9,11,12,26], little explicit effort has been made in photometric vision. To our knowledge, no previous work has given globally optimal solutions to the BRDF estimation problems under photometrically meaningful L1 or L2 cost functions. Our work shown in this paper focuses on the globally optimal parameter estimation of the Cook–Torrance model [5] with multiple specular lobes using a set of photometric measurements with known surface orientations and viewing/lighting directions. It is based on our recent work shown in [31]. The Cook–Torrance model has been developed based on the geometrical optics and considered one of the most physically plausible models. Nonlinearity arises from the surface roughness parameter in the Beckmann
⇑ Corresponding author.
E-mail addresses: ckyu@sogang.ac.kr (C. Yu), yndk@sogang.ac.kr (Y. Seo), slee@sogang.ac.kr (S.W. Lee). 1077-3142/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.cviu.2011.05.012
Article history: Received 16 April 2010 Accepted 6 May 2011 Available online 23 July 2011 Keywords: BRDF Analytical BRDF Parameter estimation Branch and bound Global optimization
a r t i c l e
i n f o
ห้องสมุดไป่ตู้a b s t r a c t
We present a global minimization framework for estimating the parameters of multiple-lobe analytical BRDF model using the techniques of convex programming and branch and bound. When an analytical BRDF is used to model a natural or artificial surface, reflections can often be represented accurately with multiple non-Lambertian lobes. A BRDF model with multiple lobes can be highly nonlinear and poses a challenge in data fitting for parameter estimation. Traditional local minimization suffers from local minima and requires a large number of initial conditions and supervision for successful results especially when a model is highly complex. We consider the Cook–Torrance model with multiple specular lobes, a parametric model with the Gaussian-like Beckmann distributions for specular reflectances. Instead of optimizing the multiple parameters simultaneously, we search over all possible surface roughness values based on a branch-and-bound algorithm, and reduce the estimation problem to convex minimization with known fixed surface roughness. Our algorithm guarantees globally optimal solutions. Experiments have been carried out for isotropic surfaces to validate the presented method using the extensive highprecision measurements from the MERL BRDF database. Ó 2011 Elsevier Inc. All rights reserved.