Spatial transformation is widely used in computer graphics, image and video processing, cryptography, and communication coding. This article summarizes several common transformations and can further refer to textbooks on advanced algebra and analytic ry. In practical applications, it is often necessary to map a point (coordinate) from one space to another. For example, if there are multiple Coordinate Systems in the game design, the point ing relationship between them must be changed. To put it simply, a transformation is a ing from the space to itself (that is, the fixed domain and the value domain are the same), which is completed by multiplying the matrix. Let's repeat the content of Higher Algebra. Here we recommend Meng Yan's article-Understanding matrix. The following are several concepts:
Space
(Space): A collection of objects. The object here can be a point in the ry (referred to in this article), a polynomial, or other abstract objects. Objects in a space are represented as vectors by selecting base vectors and coordinates. To put it bluntly, vector can represent the abstraction of objective things, and now I understand why MATLAB (a matrix computing software) can be applied to almost all engineering fields.
Vector
(Vector): N ordered arrays (A1, A2, A3 ......) It is called an n-dimensional vector.
Base
(Basis): A set of linear irrelevant vectors. Each vector in a linear space can be uniquely represented by a base vector.
Domain
(Field): When discussing space, you must specify the range of numbers in the vector element. In elementary algebra, the value field is an example. In this example, the pixel position field F is a natural number set.
Ing
(Mapping): Set both V and W to the space of domain F. The ing is based on a certain rule F, so that the elements in V find the unique corresponding element in W and record it as F: V ---> W, element a --> B, B is called the image of a, and a is called the original image of B.
Transform
(Transform): The ing between space V and itself is called transformation. transformation is completed through a matrix. For example, a point in a plane space still falls on the plane space after transformation (coordinate change ).
Function
(Function): The ing from space V to domain F is a function. For example, the gray function g (x, y) of the image is from the plane space (two-dimensional integer space) to the gray value (0 ~ A function in the 255 integer. Polynomial evaluate is also a function that maps the polynomial space to a value in the f field. The following is an example. In digital communication, a classic channel code is a linear code. By adding several redundant code words, the system can detect errors and correct errors. In a four-dimensional binary domain vector (A1 ~ Add three components (C1 ~ C3), where c1 = A1 + A2 + A3 C2 = A1 + A2 + A4 C3 = A1 + A3 + A4, then the binary domain (0 and 1) is given) A ing between Vector Space Z4 and z7. This ing is called encoding. Each image is called a code. The elements in the Code are called a code word. The first four components are information bits, and the last three are called verification bits. Map [C1, C2, C3] '= [1 1 1 0] [A1, A2, A3, a4] '[1 1 0 1] [1 0 1 1] We call this code A () linear code. Let's talk about it a little further. For more information, see the digital communication teaching materials. In the next article, we will focus on some common mappings in image processing: affine, bilinear, and perspective.
