数字图像处理-第五章3 (2)

Chapter 5 Discrete Image Transform5.1 Fundamental Concept 5.2 Cosine Transform 5.3 Rectangular Wave Transform 5.4 Principle-Component Analysis and K-L Transform 5.5 Wavelet Transform

5.1 Fundamental Concept5.1.1 One-Dimensional Discrete Linear Transform

Definition. if x is an N 1 vector and T is an N N matrix, then y Tx defines a linear transform of the vector x. The matrix T is also called the kernal matrix of the transform. Example: the rotation of a vector in a two-dimensional coordinate system. y1 cos y sin 2 sin x1 x cos 2

Inversion: the original vector can be recovered by the inverse transform x T 1 y provided that T is nonsingular.

5.1.2 1D discrete orthogonal transform Unitary matrix (酉矩阵): n阶复方阵U的n个列向量是U空间的一个标准正交基，则U是酉矩 阵(Unitary Matrix)。 一个简单的充分必要判别准则是： 方阵U的共扼转置乘以U等于单位阵，则U是酉矩阵。酉矩阵 的逆矩阵与其伴随矩阵相等。

5.1.2 1D discrete orthogonal transform

Unitary transform: y Tx If T is a unitary matrix, then T 1 T * , and TT * T * T I。 Orthogonal transform: If T is a real transform, then the unitary transform is an orthogonal one. T 1 T ,TT I。

Orthogonal basis: each line of the orthogonal matrix T is called its orthonormal basis. This means that any N-by-1 sequence can be viewed as representing a vector from the origin to a point in N-dimensional space. The orthonormal basis are orthogonal to each other.

In summary, a unitary linear transform generates y, a vector of N transform coefficients, each of which is computed as the inner product of the input vector x with one of the rows of the transform matrix T.The forward transform: The inverse transform:

y Txx T 1 y

5.1.3 Two-Dimensional Discrete Linear Transform

The general linear transform that takes the N N matrix F into the transformed N N matrix G is G u , v x, y, u , v F x, y x 0 y 0 N 1 N 1

0 u, v N 1

is the kernal function of the transform, which is a N 2 N 2 block matrix having N rows of N blocks, each of which is an N N matrix. The blocks are indexed by u , v and the elements of each block by x, y.

Separatable: If the kernal function can be separated into the product of rowwise and columnwise component functions. For some (u,v), x, y, u , v Vc y, v Vr x, u then the transform is called separable. It means that it can be carried out in two steps__ N 1 G u , v Vc y, v f x, y Vr x, u x 0 y 0 G Tc ' FTr 'N 1

Vr(x u)

Vc(y v) Tr

Example : 2D function e , x and y takes 0,1. 1 轾0 x2 + y 2 x2 y2 2 犏 e e 2 2 犏 the matrix is 犏 1 . But e = e e 2,

- 1 犏 2 e e 犏 臌 0 轾 1 e 犏 轾 0 2 犏 which is equal to 犏 e . 1 ×e 犏 犏 e 2 臌 犏 臌

x2 + y 2 2

Symmetric: If the two component functions are identical, the transfrom is also called symmetric. N 1 G V y, v f x, y V x, u TFT x 0 y 0 It is a unitary transform if T is a unitary matrix, called the kernal matrixN 1

of the transform. The inverse transform is F T 1GT 1 T * GT *

Orthogonal Transformations: A unitary matrix with real elements is orthogonal. F = T 'GT ' If T is a symmetric matrix, as is often the case, then the forward and inverse transforms are identical, so that G = TFT and F = TGT

5.1.4 Basis Functions And Basis Images

The rows of the kernal matrix of a unitary transform are a set of basis in N -dimensional vector space. TT * I Normally the entire set is derived from the same basic function form. The inverse two-dimensional transform can be viewed as reconstructing the image by summing a set of properly weighted basis images. F x, y ' u , v, x, y G u , v u 0 v 0 N 1 N 1

Each element in the transform matrix, G, is the coefficient by which the corresponding basis image is multiplied in the summation.

Each basis matrix is characterized by a horizontal and a vertical spatial frequency. The matrices shown here are arranged left to right and top to bottom in order of increasing frequencies.

5.2 Cosine Transform 5.2.1 One dimensional Discrete Cosine TransformAs we know, when f(x) is an even function , Fourier transform is only real. How about the Fourier transform if f(x) is not.

设一维离散序列f x , x 0,1, 2,

, N 1,以 1 2为中心反折,形成

N 至 1的序列, 与原序列合并形成2 N的偶序列。此时傅立叶变 换的核函数为e j 2 ux N 改变为e cos 2 x 1 u 2N 这时的变换就叫余弦变换 1 j 2 x u 2 N 2

按傅立叶变换性质, 虚部为0不进行运算, 核函数等价于

因此余弦正变换：F u f x cos 2 x 1 u 2N x 0 为保证每行正交向量模=1,对上式进行归一化处理，N 1

F Cf 1 1 1 f 0 F 0 1 2 4 6 F 1 8 8 8 8 8 8 8 f 1 real ( e ) real ( e ) real ( e ) real ( e ) f 2 F 2 F 3 f 3

F u a u f x cos 2 x 1 u 2N x 0N 1

1 当u 0时 N a u 2 当u 0时 N 余弦变换采用矩阵表示为FC Cf 其中核矩阵C中元素为Cu , x a u cos 2

x 1 u 2N

直流系数DC(u=0时),交流系数AC(其他)

c1 c C= 2 ... cn

余弦变换是正交变换，即 0, l k <cl ,ck >= 1, l k

因为余弦变换是傅立叶变换的特例，傅立叶 反变换的核矩阵即是W阵的共轭矩阵，对于 余弦变换共轭矩阵即等于本身，因此f C T FC

5.2.2、二维余弦变换 思想：如何形成二维偶函数？先水平做对折 镜象，然后再垂直做对折镜象。 偶对称偶函数： f x, y f 1 x, y f x, y f x, 1 y f 1 x, 1 y N 1 M 1

当x, y 0时 当x 0 y 0 当x 0 y 0 当x 0 y 0

2 x 1 u 2 y 1 v FC u , v a u a v f x, y cos cos 2 N 2 M x 0 y 0 它是可分离的，用矩阵表示为FC CfC T

5.2.3、余弦变换的性质* 1 T 1 余弦变换为实正交变换 C C , C C

2 离散序列的余弦变换是DFT的对称扩展形式； 3 和傅立叶变换相同，余弦变换也存在快速变换； 4 和傅立叶变换类似，余弦变换具有将高度相关数据能量集中的优势；

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