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Fig. 2.3 Examples of synthetic shape from texture images (a, c) and corresponding surface normal estimates (b, d). Figure courtesy of [17]
the image blur where the amount of defocus can be estimated by averaging the squared gradient in a region.
Single view metrology [13] allows shape recovery from a single perspective view of a scene given some geometric information determined from the image. By exploiting scene constraints such as orthogonality and parallelism, a vanishing line and a vanishing point in a single image can be determined. Relative measurements of shape can then be computed, which can be upgraded to absolute metric measurements if the dimensions of a reference object in the scene are known.
While 3D recovery from a single view is possible, such methods are often not practical in terms of either robustness or speed or both. Therefore, the most commonly used approaches are based on multiple views, which is the focus of this chapter. The first step to understanding such approaches is to understand how to model the image formation process in the cameras of a stereo rig. Then we need to know how to estimate the parameters of this model. Thus camera modeling and camera calibration are discussed in the following two main sections.
2.3 Camera Modeling
A camera is a device in which the 3D scene is projected down onto a 2D image. The most commonly used projection in computer vision is 3D perspective projection. Figure 2.4 illustrates perspective projection based on the pinhole camera model, where C is the position of the pinhole, termed the camera center or the center of projection. Recall that, although the real image plane is behind the camera center, it is common practice to employ a virtual image plane in front of the camera, so that the image is conveniently at the same orientation as the scene.
Clearly, from this figure, the path of imaged light is modeled by a ray that passes from a 3D world point X through the camera center. The intersection of this ray with the image plane defines where the image, xc , of the 3D scene point, X, lies. We can reverse this process and say that, for some point on the image plane, its corresponding scene point must lie somewhere along the ray connecting the center of projection, C, and that imaged point, xc . We refer to this as back-projecting an image point to an infinite ray that extends out into the scene. Since we do not know
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Fig. 2.4 Projection based on a pinhole camera model where a 3D object is projected onto the image plane. Note that, although the real image plane is behind the camera center, it is common practice to employ a virtual image plane in front of the camera, so that the image is conveniently at the same orientation as the scene
how far along the ray the 3D scene point lies, explicit depth information is lost in the imaging process. This is the main source of geometric ambiguity in a single image and is the reason why we refer to the recovery of the depth information from stereo and other cues as 3D reconstruction.
Before we embark on our development of a mathematical camera model, we need to digress briefly and introduce the concept of homogeneous coordinates (also called projective coordinates), which is the natural coordinate system of analytic projective geometry and hence has wide utility in geometric computer vision.
2.3.1 Homogeneous Coordinates
We are all familiar with expressing the position of some point in a plane using a pair of coordinates as [x, y]T . In general for such systems, n coordinates are used to describe points in an n-dimensional space, Rn. In analytic projective geometry, which deals with algebraic theories of points and lines, such points and lines are typically described by homogeneous coordinates, where n + 1 coordinates are used to describe points in an n-dimensional space. For example, a general point in a plane is described as x = [x1, x2, x3]T , and the general equation of a line is given by lT x = 0 where l = [l1, l2, l3]T are the homogeneous coordinates of the line.2 Since the right hand side of this equation for a line is zero, it is an homogeneous equation, and any non-zero multiple of the point λ[x1, x2, x3]T is the same point, similarly any non-zero multiple of the line’s coordinates is the same line. The symmetry in this
2You may wish to compare lT x = 0 to two well-known parameterizations of a line in the (x, y) plane, namely: ax + by + c = 0 and y = mx + c and, in each case, write down homogeneous coordinates for the point x and the line l.