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An introduction to

Figure 2.1.1 shows a common perspective phenomenon in life, whose physical essence is “straight line propagation of light”. Perspective can also be described as “near big and far small”, as shown in Figure 2.1.2(a), which corresponds to perspective space; If the size of the object does not change with the distance of the observation point, as shown in Figure 2.1.2(b), it corresponds to the Euclidean space. Since both the human eye and the camera have perspective phenomena when capturing images, computer geometry is studied based on perspective space.

1 Perspective space

For the sake of intuition, two-dimensional perspective space is described first, and its principle can be directly extended to three-dimensional space. Figure 2.2.1 is a two-dimensional Euclidean plane R2\mathbb{R}^2R2, now introduce two-dimensional perspective coordinates P2\mathbb{P}^2P2, perspective coordinates extend the dimension of Euclidean space to three dimensions, the third dimension W ~\tilde{w}w~ represents the distance between the object and the observation point, The convention takes W = 1W =1w=1 as the reference plane.

According to the principle that light travels in straight lines, a line of sight l~\boldsymbol{\tilde{L}} L ~ passes through the point AAA of the Euclidean plane from the origin of the perspective coordinate system. May wish to translate the Euclidean plane up and down, adjust the distance between point AAA and the observation point. The different points AAA and A’A ‘in Euclidean space are the same in perspective space, because L ~\boldsymbol{\tilde{L}} L ~ is the representation set of the same point at different observation distances. Points in Euclidean space are represented by straight lines (lines of sight) L ~\boldsymbol{\tilde{L}} L ~ in perspective space.

Now keep the points in Euclidean space the same and adjust the observation distance again, as shown in Figure 2.2.2. The farther the observation distance, the more scattered the line of sight and the more local the information; The closer the observation distance, the more convergence of the line of sight, the more global information — this phenomenon can be illustrated by the working scene of the projector.

Define the line of sight through space


l ~ = ( x . y . w ) \boldsymbol{\tilde{l}}=\left( x,y,w \right)

Indicates a 0 when w w \  = 0, 0 ne w l ~ \ boldsymbol {\ tilde {l}} l ~ corresponding points in European space l (xw, yw) l \ left (\ frac {x} {w}, \ frac {y} {w} \ right) l (wx, wy), L ~\boldsymbol{\tilde{L}} L ~ also known as the Homogeneous Coordinates of the point LLL, for which w=0w=0w=0, L ~\boldsymbol{\tilde{L}}l~ corresponds to the vector l ⃗=(x,y)\vec{L}=\left(x,y \right) l =(x,y), w=0w=0w=0 in the homogeneous transformation.

The operation of point and vector in perspective space


{ V Plus or minus V = V P Plus or minus V = P P – P = V P + P = M i d P ( P : P o i n t     V : V e c t o r     M i d : midpoint ) \ \ the begin {cases} V PM V = V \ \ \ PM V P = P \ \ \ \ P – P = V P + P = MidP \ {cases} \ \ \ end left (P: Point \ \, V: Vector \, \, Mid: \ text} {middle/right)

Homogeneous coordinates have the following properties:

  • Homogeneity. The geometric essence of homogeneous coordinate, the homogeneous coordinate in perspective space is the expression form of the same point under different observation distance;
  • Linear. In homogeneous coordinates, the transformation of Euclidean space of corresponding dimensions can be linearized. For example, the translation of the midpoint in the two-dimensional Euclidean plane is a nonlinear transformation, but the nonlinear translation in the two-dimensional perspective space can be completed by linear rotation.

In the same way that the points of Euclidean space are represented by the straight lines of perspective space, the straight lines of Euclidean space are represented by the apparent plane in perspective space — since the plane is exclusively determined by the normal vector L ~\boldsymbol{\tilde{L}} L ~ are also represented by the normal vector of the plane L ~\boldsymbol{\tilde{L}} L ~.

From analytic geometry, straight lines and points in perspective space satisfy the following relations:


{ l ~ 1 x l ~ 2 = x ~ x ~ 1 x x ~ 2 = l ~ \begin{cases} \boldsymbol{\tilde{l}}_1\times \boldsymbol{\tilde{l}}_2=\boldsymbol{\tilde{x}}\\ \boldsymbol{\tilde{x}}_1\times \boldsymbol{\tilde{x}}_2=\boldsymbol{\tilde{l}}\\\end{cases}

If the line of sight in perspective space is on the apparent plane (corresponding to the Euclidean space midpoint is on a straight line), it is easy to know


l ~ T x ~ = x ~ T l ~ = 0 \boldsymbol{\tilde{l}}^T\boldsymbol{\tilde{x}}=\boldsymbol{\tilde{x}}^T\boldsymbol{\tilde{l}}=0

2 Perspective transformation

The general form of perspective space transformation is shown in Figure 2.3.1, and the details are listed in Table 2.3.1. Perspective transformation represents the mapping between planes (points on a plane).

It can be seen that all transformations in perspective space are special cases of projection transformation, so it is of great significance to study projection transformation, which is widely usedImage correction, perspective transformation, image Mosaic, augmented reality, as shown in Figure 2.3.2.

Computer vision basics tutorial outline

The chapter content

Color space and digital imaging

Fundamentals of computer geometry

2. Image enhancement, filtering, pyramid

3. Image feature extraction

4. Image feature description

5. Image feature matching

6 Stereovision

7 Actual Project

Welcome to my AI channel “AI Technology Club”.