Images are generally produced using a digital camera, which captures a scene by projecting light going through its lens onto an image sensor. The fact that an image is formed by the projection of a 3D scene onto a 2D plane implies the existence of important relationships between a scene and its image and between different images of the same scene. Projective geometry is the tool that is used to describe and characterize, in mathematical terms, the process of image formation. In this chapter, we will introduce you to some of the fundamental projective relations that exist in multiview imagery and explain how these can be used in computer vision programming. You will learn how matching can be made more accurate through the use of projective constraints and how a mosaic from multiple images can be composited using two-view relations. Before we start the recipes, let's explore the basic concepts related to scene projection and image formation.
From the introduction of this chapter, we learned that the essential parameters of a camera under the pin-hole model are its focal length and the size of the image plane (which defines the field of view of the camera). Also, since we are dealing with digital images, the number of pixels on the image plane (its resolution) is another important characteristic of a camera. Finally, in order to be able to compute the position of an image's scene point in pixel coordinates, we need one additional piece of information. Considering the line coming from the focal point that is orthogonal to the image plane, we need to know at which pixel position this line pierces the image plane. This point is called the principal point. It might be logical to assume that this principal point is at the center of the image plane, but in practice, this point might be off by a few pixels depending on the precision at which the camera has been manufactured.
Camera calibration is the process by...
The previous recipe showed you how to recover the projective equation of a single camera. In this recipe, we will explore the projective relationship that exists between two images that display the same scene. These two images could have been obtained by moving a camera at two different locations to take pictures from two viewpoints or by using two cameras, each of them taking a different picture of the scene. When these two cameras are separated by a rigid baseline, we use the term stereovision.
Let's now consider two cameras observing a given scene point, as shown in the following figure:
We learned that we can find the image x of a 3D point X by tracing a line joining this 3D point with the camera's center. Conversely, the scene point that has its image at the position x on the image plane can be located anywhere on this line in the 3D space. This implies that if we want to find the corresponding point of a given image point...
When two cameras observe the same scene, they see the same elements but under different viewpoints. We have already studied the feature point matching problem in the previous chapter. In this recipe, we come back to this problem, and we will learn how to exploit the epipolar constraint between two views to match image features more reliably.
The principle that we will follow is simple: when we match feature points between two images, we only accept those matches that fall on the corresponding epipolar lines. However, to be able to check this condition, the fundamental matrix must be known, but we need good matches to estimate this matrix. This seems to be a chicken-and-egg problem. However, in this recipe, we propose a solution in which the fundamental matrix and a set of good matches will be jointly computed.
The second recipe of this chapter showed you how to compute the fundamental matrix of an image pair from a set of matches. In projective geometry, another very useful mathematical entity also exists. This one can be computed from multiview imagery and, as we will see, is a matrix with special properties.
Again, let's consider the projective relation between a 3D point and its image on a camera, which we introduced in the first recipe of this chapter. Basically, we learned that this equation relates a 3D point with its image using the intrinsic properties of the camera and the position of this camera (specified with a rotation and a translation component). If we now carefully examine this equation, we realize that there are two special situations of particular interest. The first situation is when two views of a scene are separated by a pure rotation. It can then be observed that the fourth column of the extrinsic matrix will be made up...
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