Surface acquisition from single gray-scale images

In this paper we show how a system for performing automatic surface model acquisition from single object views can be designed. The surface acquisition process is a two step one. Firstly, the surface normals are computed using a shape-from-shading algorithm. Secondly, the field of surface normals is integrated into a 3D surface. For the surface integration step, we have performed experiments with two alternatives. The first of these is a geometric surface integration algorithm. The second alternative comprises a graph-spectral surface integration algorithm. We present results on images of classical statues and provide a preliminary quantitative study. 1. . INTRODUCTION Surface acquisition provides a route to automatic object and scene modelling that is of potential practical use to the computer graphics community. Broadly speaking, existing methods may be divided into those that are geometrical in nature and those that are based upon photometric models. Turning our attention first to the geometric methods, the main contribution here has been to exploit projective geometry to develop algorithms for 3D object reconstruction [1]. Methodologically, the idea underpinning these algorithms is that of recovering both planar and curved surfaces from multiple views making use of calibration data and correspondance information. For curved surfaces there has been considerable success in using turntable sequences for surface reconstruction from both detected feature points and occluding contours [2]. Photometric methods, on the other hand, aim to recover surface information from shading or texture variations. This is a classical problem in the computer vision community, which has lead to a considerable literature on the development of algorithms for shape-fromshading [3], photometric stereo [4] and shape-from-texture [5]. However, the drawbacks of these algorithms are twofold. Firstly, they tend to oversmooth surface detail. Secondly, they are prone to error due to their numerical inestability. In an effort to overcome the problems, Worthington and Hancock have developed a new framework for shape-from-shading [3] in which the image irradiance equation is treated as a hard constraint. The method delivers fields of surface normals (Gauss maps) at output. From the field of surface normals, the surface height may be recovered by applying a surface integration algorithm as a post-processing step. Our aim in this paper is to show how a surface acquisition system may be designed. We depart from the fact that a complete 3D representation of the object requires the depth information and the Supported by CONACYT, under grant No. 146475/151752. surface normals to be at hand. This becomes evident ever since the normals are of capital importance for accelerating complex rendering tasks, while the depth data is essential when a mesh or spline representation is required. Following this rationale, our acquisition system makes use of shape-from-shading algorithms and surface height integration methods for recovering the field of surface normals and the surface height of a Lambertian object from a single gray-scale image. 2. . SYSTEM STRUCTURE As mentioned earlier, in order to obtain a meaningful 3D representation of the object it is necessary to recover not only the surface height, but also the surface normals. Following this rationale, the surface acquisition process is performed in two steps. We commence by recovering the surface normals employing the shapefrom-shading algorithm developed by Worthington and Hancock [3]. Once the normals are at hand, we recover the surface height using a surface integration algorithm. For purposes of surface integrations, we have performed experiments with two different alternatives. The first of these is the graph-spectral surface integration method of Robles-Kelly and Hancock [6]. The second alternative consists of a geometric surface integration algorithm of Bors et al. [7]. The former was developed as a general surface integration algorithm that takes as input a field of surface normals delivered by shape-from-shading algorithms, while the later is a method evolved from a study of terrain reconstruction using radar shapefrom-shading. Both techniques share the capability of taking the needle-maps (i.e. Gauss maps) delivered by the Worthington and Hancock [3] algorithm as input. Details of the modules comprising our surface acquisition system are described later in this section. Taking practical considerations into account, the design of our surface acquisition system is such that delivers an OpenGL list at output. A diagram of the surface acquisition system is shown in Figure 1. 2.1. . Surface Normal Extraction In the case of Lambertian reflectance from a matte surface of constant albedo illuminated with a single point light-source, the observed intensity is independent of the viewing direction. The observed intensity depends only on the quantity of absorbed light, and this in turn is proportional to the cosine of the incidence angle. Suppose that is the unit-vector in the direction of the light source and that is the unit-vector in the surface normal direction at the pixel . According to Lambert’s law, the observed image intensity at the pixel is . Lambert’s equation provides insufficient information to uniquely determine the surface normal direction. However, the Fig. 1. Structure of our surface acquisition system. equation does have a simple geometric interpretation which can be used to constrain the direction of the surface normal. The equation specifies that the surface normal must fall on the surface of a right-cone whose axis is aligned in the light-source direction and whose apex angle is . Worthington and Hancock [8] exploit this property to develop an iterative process for shape-from-shading. They show how curvature consistency constraints can be used to recover meaningful topographic surface structure [3]. According to this geometric framework, the surface normals are constrained to fall on an irradiance cone whose axis is in the light source direction and whose apex angle is proportional to the inverse cosine of the measured image brightness. The azimuthal angle of the surface normal on the irradiance cone is determined by local smoothness contraints. 2.2. . Geometric Surface Integration The first alternative for the surface integration module of our surface acquisition system employs a geometric surface height recovery method. Here, we recover the surface height making use of surface height reference and the field of surface normals. The algorithm commences locating a height reference on the pixel lattice. Once the reference is at hand, we proceed to recover the surface height. Let us consider the set comprised by the eight-neighbours of the pixel . Let the surface normal corresponding to the pixel be denoted by . Suppose the line , perpendicular to the image plane at pixel , intersects the surface under study at the point whose height at iteration is "!$#&% . We locate the height reference on the pixel lattice for the pixel making use of the minimum distance ' ( from the pixel *)+ to the plane defined by the line and the surface normal . To do this, we find the pixel-site corresponding to the pixel in whose distance , ( is smallest. Hence, for the pixel , the reference pixelsite is