Multi-View Geometry--Introduction (I.)

introduction–a Tour of multiple View Geometry

This chapter is an introduction to the main ideas and the informal treatment of these topics. Precise, unambiguous definitions, rigorous algebra, and descriptions of fine-grained estimation algorithms are described in chapters 2nd and beyond. In the entire introduction, we generally do not give the specific first few chapters. Pan-projective geometry

We are all familiar with projective transformations, and when we look at a picture, we see not a square, nor a circular circle. The transformation that maps these planar objects to a picture is an example of a projective transformation.

So what are the geometrical properties that are maintained by projective transformations? Of course, the shape is not, because a circle may be an ellipse. Nor is the length, because the projective transformation, two vertical radii of different circles are elongated by different lengths. Angles, distances, distances are not preserved, and seemingly few geometric properties can be preserved by projective transformations. However, this attribute of straightness can be preserved. The results show that this is the most general requirement for mapping, and we can define a projective transformation of a plane as any mapping of points that maintain a straight line on a plane.

To understand why we need projective geometry, we start with the familiar Euclidean geometry. This is the geometry that describes the angle and shape of the object. Euclidean geometry is a major problem-we need to constantly infer some basic concepts of geometry-such as the intersection of lines. The two lines (which we consider here are two-dimensional geometries) almost always meet at one point, but there are also lines that do not--what we call parallel lines. A general description of the problem is that parallel lines meet "infinity". Yet this is not entirely convincing, and conflicts with another opinion, Infinity does not exist, but just a fictional one. We can strengthen the Euclidean plane by adding these points to the infinity of the parallel line encounters, and solve the infinite difficulties by calling them "ideal points".

By adding these points in infinity, the familiar Euclidean space is transformed into a new geometric object-projective space. This is a very useful way of thinking because we are familiar with the nature of Euclidean space, involving concepts such as distance, angle, point, line, and incidence. Projective space is nothing mysterious, it is only an extension of the Euclidean space, in these two points, the two lines always meet at a point, and sometimes even in the infinity of the mysterious point met.

Coordinate.

A point in the Euclidean space is represented by an ordered real number pair (x, y). We can add an extra coordinate to this pair, give a ternary coordinate (x,y,1), and we declare the same point. This seems to be no different, because we can simply add or remove the last coordinate from one point to another. We now focus on the concept to explain why the last coordinate needs to be 1, after all, the other two coordinates are not constrained. Ternary coordinates (x,y,2). Here, we make a definition, say (x,y,1) and (2x,2y,2) represent the same point, of course, for any non-0 value K, (KX,KY,K) also expressed the same point. Formally, a point is represented by an equivalence class of ternary coordinates, where two triples are equivalent in their different common multiple. This is called the homogeneous coordinate of the point. Given a ternary coordinate (kx,ky,k), we get the original coordinates divided by K (x, y).

The reader will notice that although (x,y,1) represents the same point as the coordinate pair (x, y), there is no point corresponding to the ternary coordinate (x,y,0). If we try to divide the last coordinate, we get an infinite point (x/0,y/0). This is how the infinity point is generated. They are the point at which the coordinates are zero at the end of the homogeneous coordinates.

Now that we know how to do this in two-dimensional Euclidean space, by representing a point as a homogeneous vector and then extending it to the projective space, it is clear that we can do the same thing in any dimension. The Irⁿ of the Euclidean space can be extended to the ipⁿ of the projective space by representing the points as homogeneous vectors. The results show that the Infinity Point in the two-dimensional projective space forms a line, which is often called an infinity line. In three dimensions, they form a plane at infinity.

Homogeneous nature.

In classic Euclidean geometry, all points are the same. No special points. The whole space is homogeneous. When the coordinates increase, a point is specified as the Origin point. However, realize that this is only an artificial choice. We can also find a coordinate plane where the origin is different from the point. In fact, we can consider the coordinate transformations of Euclidean space, in which the axes are moved and rotated to different positions. We can think of this in a different way, because the space itself is transforming and rotating to another location. The resulting operation is called a Euclidean transformation.

The more general type conversion is the linear transform Irⁿ, followed by the Euclidean transform moving space origin. We can think of this as the movement of space, the rotation, and finally the linear stretching in different directions at different scales. The resulting transformation is called an affine transformation.

The result of a Euclidean or affine transformation is that the points in infinity are kept at infinity. Such a point is to some extent, at least as a collection, to be preserved by such conversions. They are in some way prominent, or in the context of Euclidean or affine geometry is special.

From the perspective of projective geometry, Infinity points have no difference from other points. Just as Euclidean space is homogeneous, so is the projective space. At infinity, the last coordinate in the homogeneous coordinate representation is zero, only the artificial selection of the coordinate system is caused. Similar to Euclidean or affine transformations, we can define projective transformations of projective spaces. The linear transformation Irⁿ in Euclidean space is represented by the matrix multiplication represented by coordinates. Similarly, the projective transformation Ipⁿ of projective space is a mapping of the homogeneous coordinates of a point ((n+1) vector), where the coordinate vector is obtained by multiplying a non-singular matrix. Under this mapping, the infinity point (the end coordinate is 0) is mapped to any other point. Infinity points are not retained. Therefore, the projective transformation Ipⁿ of projective space is represented by the linear transformation of homogeneous coordinates.

In the computer vision problem, the projective space can express the real three-dimensional world conveniently by extending to three-dimensional (3D) projective space. In the same way, the image of the real-world space projective to the two-dimensional representation is also easily extended to the two-dimensional projection space. In fact, the real world and its images do not contain infinity, and we need to keep our fingers as imaginary points, that is, the infinite distance of the line in the image and the world's infinitely far plane. For this reason, although we usually use projective space, we know that the infinitely far lines and planes are somewhat special in some way. This violates the spirit of purely projective geometry, but it applies to solving our practical problems. Usually, at the right time, we use both methods to process all points in the projective space at the same time, and to find the line that is infinitely far in space or the plane that is infinitely far in the image when necessary.

1.1.1 Affine and Euclidean geometry
We can obtain projective space by adding a straight line (or plane) to infinity. We are now considering the reverse process. This paper mainly discusses two three-dimensional projective space.

Affine geometry.

We will consider that the projection space is initially uniform and has no preferred coordinate system. In such a space, there is no concept of parallel lines, because parallel lines (or planes in three-dimensional space) are those that never intersect. However, in projective space, no point is the concept of infinity, and all points are equal. We say that parallelism is not the concept of projective geometry. It is meaningless to talk about it.

In order to make this concept meaningful, we need to find out some specific lines and decide which is the Infinity line. This creates a situation in which all points are equal, but others are more equal than others. So, from a blank sheet of paper, imagine that it stretches infinitely, forming a projective space ip². All we see is a small fraction of the space that looks like part of a common Euclidean plane. Now, let's draw a line on the paper and declare it to be a straight line of infinity. Next, we draw another two lines that intersect this line. Because they meet in the "Infinity Line", we define them as parallel. The situation is similar to what people see by observing the infinite plane. Imagine a picture taken in a very flat area on Earth. The infinity point on the plane is displayed as the horizon in the image. Lines like rails appear on the screen like lines that meet on the horizon. The point on the horizon (the image of the sky) in the image is obviously inconsistent with the points on the world plane. However, if we consider extending the corresponding light behind the camera, it will meet the plane at some point behind the camera. Therefore, there is a one by one correspondence between the points in the image and the points on the world plane. The Infinity Point on the world plane corresponds to a true horizon in the image, and the parallels of the world correspond to the lines on the horizon. From our point of view, the world plane and its image are just another way of observing projective plane geometry, plus a special line. The geometry of the projective plane and a special line are called affine geometries, and any projective transformation maps a particular line in a space to a line in another space, called an affine transformation.

By identifying a special line, the "Infinity Line", we can define the parallelism of the straight line on the plane. However, as long as we can define parallelism, some other concepts also make sense. For example, we can define an equal interval between two points of a parallel line. For example, if there are a,b,c and D four points, AB and CD are parallel, if the line AC and BD are also parallel, then we define two interval ab and the length of the CD equal. Similarly, if there is another equal interval on the parallel line, the two intervals on the same row are equal.

Euclidean geometry.

By distinguishing a special line in the projective plane, we get the concept of parallel geometry and radial geometry. Affine geometry is considered to be the specificity of projective geometry, in which we find a particular line (or plane-according to dimensions) based on a dimension called the Infinity Line.

Next, let's talk about Euclidean geometry. The Euclidean geometry is obtained by finding some special features of the line or plane in the affine geometry of infinity. In this article, we present one of the most important concepts in this book, with an absolute two-time curve.

We begin by considering the two-dimensional geometry of the circle. Note that the circle is not the concept of affine geometry, because the arbitrary stretch of the plane maintains a straight line at infinity, making the circle an ellipse. Therefore, affine geometry cannot differentiate circles and ellipses.

However, in Euclidean geometry, they clearly have important differences. Algebra, ellipses are described by a two-time equation. Therefore, we expect and assume that two ellipses usually intersect at four points. However, it is geometrically obvious that two different circles cannot intersect more than two points. Algebra, we make two two curves intersect here, or equivalent to two-time equation solution. We should get four solutions. The question is what is special about circles that intersect only at two points.

The answer to this question, of course, is that there are two other solutions where the two circles intersect at the other two complex points. We don't have to look at these two points.

The equation of the circle in homogeneous coordinates (X,Y,W) is the following form

This means that a circle centered on the homogeneous coordinates is represented, and soon proves that the point is on each circle. To repeat this interesting fact, each circle crosses a point, so they are at the intersection of any two circles. Because they are last zero, the two points are on a straight line at infinity. It is obvious that they are called planar circular dots. It is worth noting that although the two dots are complex points, they satisfy a pair of real equations:.

This discovery gives clues to how we define Euclidean geometry. Euclidean geometry specifies an infinity line in the projective geometry, followed by a two point on the line called the Loop point. Of course, the dots are complex points, but in most cases we're not too worried about that. Now, we can define a circle for any one through two circular point conic sections (curves defined by the equation). Note that in the standard Euclidean coordinate system, the dot coordinates are. When the Euclidean structure is assigned to the projective plane, we can represent any straight line and any two (plural) points on the line as Infinity points and dots.

As an example of applying this view, we note that the General conic section can be seen through five arbitrary points on the plane, such as by calculating the number of coefficients of the two-th equation. On the other hand, a circle is defined only by three points. Another way of looking at it is that the cone curve passes through two special points, a dot, and another three points, so for any conic section, it takes five points to uniquely determine.

It is now known that two dots can be set to get the whole of our familiar Euclidean geometry. In particular, concepts such as angle and length ratios can be defined according to the origin. However, these concepts are most easily defined in terms of some coordinate systems in the Euclidean plane, as described later in this chapter.

Three-dimensional Euclidean geometry.

We understand how the Euclidean plane is defined by the projective plane by specifying a straight line and a pair of dots at infinity. Such ideas can also be applied to three-dimensional geometry. As with the two-dimensional case, people can look closely at the spheres and how they intersect. The two balls intersect into a circle instead of a normal fourth-degree curve, as shown in algebra, and are treated as two general ellipsoid (or two other surfaces). Along this line of thought you will find that all spheres in the homogeneous coordinates intersect with the equation: the curve represented by the equations on an infinitely far plane. This is a two-time curve (conic), which is located on a plane of infinity, consisting only of complex points. It is called the absolute two curve and is one of the most important geometric entities in this book, especially since it is related to camera calibration, which is described later.

The absolute two-time curve is defined by the above equation only in the Euclidean coordinate system. In general, we consider the three-dimensional Euclidean space to originate from a projective space, finding a specific conic section in the projective space on the Infinity plane and designating it as an absolute two-time curve. These entities may have a general description in the coordinate system of the projective space.

Here, we will not detail how the absolute two curve determines the complete Euclidean 3D geometry. Let me give you an example. The verticality of a space line is not an effective concept in affine geometry, but it belongs to Euclidean geometry. The verticality of the line can be defined according to the absolute two curve, as follows. By extending the line to the Infinity plane, we get two points, called the direction of the two lines. The verticality of a line is defined by the relationship between the two directions and the absolute two-time curve. If the two directions are in accordance with the conjugate point of the absolute two curve, then the straight line is perpendicular. The geometrical and algebraic representations of conjugate point are defined in section 2nd. 8.1. In short, if the absolute curve is represented by a 3x3 symmetric matrix ω∞, and the direction is point D1 and D2, then they are conjugate relative to ω∞. In any coordinate system, the angle can be defined according to an absolute two curve, as shown in section 3.23.