1 Introduction – a Tour of Multiple View Geometry
This chapter introduces the main ideas of the book.
1.1 Introduction – the ubiquitous projective geometry
To understand why we need projective geometry, we start with the familiar Euclidean geometry. Euclidean geometry holds that parallel lines do not intersect in two dimensions, and a common way to solve this problem is to say that parallel lines intersect "at infinity." However, this is not entirely convincing and conflicts with another maxim that infinity does not exist but is simply a convenient fiction. We can solve this problem by augmenting the Euclidean plane by adding these points at infinity where parallel lines intersect, and solving the difficulty at infinity by calling them "ideal points".
By adding these infinite points, the familiar Euclidean space is transformed into a new type of geometric object, projective space. This is a very useful way of thinking because we are familiar with the properties of Euclidean space, involving concepts such as distances, angles, points, lines, and incidence. There's nothing terribly mysterious about projective space - it's just an extension of Euclidean space, in which two lines always meet at a point, although sometimes at a point at infinity.
A simple 2D Euclidean point (x, y) can add an extra coordinate to become (x, y, 1), or it can be expressed as (kx, ky, k). Well, we can observe that although (x, y, 1) represents the same point as the coordinate pair (x, y), there is no point corresponding to (x, y, 0). If we try to divide by the last coordinate, we get an infinite point (x/0, y/0). This is how the point at infinity is created. They are points represented by homogeneous coordinates, where the last coordinate is zero. Therefore, (x, y, 0) represents the point at infinity.
Translation and rotation are called Euclidean transformation, and projective transformation is equivalent to multiplying a homogeneous matrix under homogeneous coordinates. This is a common representation method in machine vision, graphics, and robotics. A more general type of transformation is a linear transformation followed by a Euclidean transformation that moves the origin of space. We can think of it as space moving, rotating, and ultimately linearly stretching in different directions at different scales, and the resulting transformation is called an affine transformation.
1.2 Camera projections
The principle of camera projection can be regarded as the projection from the projective space to the projective plane, and it is enough to convert the homogeneous coordinates through a 3x4 matrix.
A camera can be viewed as a point.
Regarding cameras, if IAC (image of absolute conic) can be obtained in a camera, then we say that the camera has been calibrated.
1.3 Reconstruction from more than one view
Consider the reconstruction of two pictures. Reconstruction often produces many solutions. There must be at least 7 points that are not at critical positions to determine the result of reconstruction.
Our goal is to know the corresponding points in the two pictures and want to obtain their camera coordinates and corresponding 3D coordinates. This solution must be uncertain. The uncertainty can be described by projection transformation. This This kind of reconstruction is called projection reconstruction.
The basic method of reconstruction is to find the fundamental matrix, which means that the two images correspond to the same 3D points.
The main process of reconstruction: find the basic matrix, find the camera matrix, and then use trigonometric method to find the corresponding 3D points.
1.4 Three-view geometry
The basic principle of reconstruction corresponding to three images is similar to that of two-view reconstruction, but the calculation will be more complicated.
1.5 Four view geometry and n-view reconstruction
For multi-view reconstruction, there are different reconstruction methods for different sequences. A basic step is bundle adjustment, which requires repeated iterative adjustments.
1.6 Transfer
We have discussed 3D reconstruction from a set of images. Another useful application of projective geometry is transfer: given the position of a point in one (or more) images, determine where it appears in all other images of the set. To do this, we must first establish the relationship between cameras using (for example) a set of auxiliary point correspondences.
1.7 Euclidean reconstruction
The fundamental task of reconstructing Euclidean geometry is to find the plane where the absolute quadratic curve lies and the plane at infinity. As long as these two planes are found, all Euclidean geometric structures will be known.