Abstract-Photometric stereo is a method of reconstructing a surface from the amount of light reflected by it. This is done by using prior knowledge of the surface reflectance to estimate the surface normal at all visible points. The theory of photometric stereo has been extensively developed for surfaces and illumination geometries that give rise to a Lambertian reflectance map. For non-Lambertian reflectance maps, the theory has been developed for specific cases, but a general theory has not been presented in the literature. In this paper, we propose a theory of photometric stereo for a large class of non-Lambertian reflectance maps. First, we review the different reflectance maps proposed in the literature for modeling reflection from real-world surfaces. From this, we obtain a mathematical class of reflectance maps to which the maps belong. Next, we show that three lights can be sufficient for a unique inversion of the photometric stereo equation for the entire class of reflectance maps. We also obtain a constraint on the positions of light sources for obtaining this solution. Next, we investigate the sufficiency of three light sources to estimate the surface normal and the illuminant strength. Finally, we address the issue of completeness of reconstruction. We show that if k lights are sufficient for a unique inversion, 2k lights are necessary for a complete inversion. Index Terms-Machine vision, non-Lambertian reflection, photometric stereo, reflectance maps, shape from shading. I. INTRODUC~ION HOTOMETRIC stereo is a fast and reliable 2; -dimensional P surface reconstruction algorithm. Multiple images of a surface are obtained from a camera by illuminating the surface from different directions. The intensities in these images are used to mathematically invert the image formation process, and the surface normal and other characteristics of the surface are obtained at different pixels. Photometric stereo uses prior knowledge about the illumination geometry of the scene and the nature of surface reflection. This knowledge is specified as the reflectance map of the surface. Photometric stereo was developed by Woodham [34], who also developed the theory of photometric stereo for the Lambertian reflectance map in detail. Coleman and Jain [ l l ] proposed the use of four light sources to separate out the specular and Lambertian components in photometric stereo. Ray et al. [23] conducted an error analysis of Lambertian photometric stereo. Ikeuchi [15] used distributed light sources for photometric stereo of specular surfaces and Nayar et al. [18] extended this and used distributed light sources for photometric stereo of surfaces whose reflection is a sum of specular and Lambertian components. Silver [26] showed how photometric stereo could be conducted using experimentally measured reflectance maps. [3] . The resulting nonLambertian reflectance maps can give rise to image formation equations that are significantly nonlinear. A lack of theoretical understanding of these equations can hamper their use in practical applications. Thus, for example, for a given non-Lambertian reflectance map, we may not know how many light sources are sufficient to invert the image formation process. Using an insufficient number of light sources could lead to multiple solutions, causing a loss of confidence in the experimental results. Clearly, a theoretical analysis of non-Lambertian photometric stereo is important, and in this paper we present such an analysis for a large class of non-Lambertian reflectance maps. In choosing the class of reflectance maps, we have tried to satisfy several conflicting demands-that the class of maps be realistic, that it represent a large number of real-world reflectance maps, and yet that it have enough structure to permit theoretical analysis. Fortunately, as we will see in Section 111, it is possible to meet these demands. A survey of reflectance maps proposed in the literature to model real-world reflection reveals that the maps do have common characteristics. Most of the proposed maps have "lobes" and these lobes have a common mathematical structure. We have formalized these properties and propose a class of reflectance maps called "m-lobed reflectance maps." Our theory of photometric stereo is developed for these maps. This class of reflectance maps contains all the proposed reflectance maps and most of their simple extensions. We believe that the most important issue facing a theory of non-Lambertian photometric stereo is that of the number of light sources needed to invert the image formation process for obtaining a unique reconstruction. This issue is also central to Woodham's theory of photometric stereo [34]. For the Lambertian reflectance map, Woodham established that three light sources are sufficient for uniquely inverting the image formation process. In Section V, we show that three light sources are also sufficient to yield a unique surface normal for the entire class of m-lobed reflectance maps. We would like to emphasize that only the knowledge about the class of reflectance maps is used to establish this result. It is possible that, by adding other constraints to the problem, the number of light sources can be reduced. For the Lambertian reflectance map, such an approach has been investigated by Onn and Bruckstein [7]. They showed that, for a Lambertian reflectance map, if the height function z(z,y) of the surface has continuous second derivatives and cannot be written as By non-Lambertian we mean "not necessarily Lambertian" in the same way that nonlinear means "not necessarily linear." 0 162-8828/9 1/0200-0133$0 1 .OO 0 199 1 IEEE
Tagare and deGigueiredo REF developed the theory of photometric stereo for the class of m − lobe reflectance maps.
A theory of photometric stereo for a class of diffuse non-Lambertian surfaces
IEEE Trans. Pattern Anal. Mach. Intell.
IEEE Trans. Pattern Anal. Mach. Intell.
Computer Science