数字图像处理-第五章3 (2)
更新时间:2023-08-11 21:36:01 阅读量: 资格考试认证 文档下载
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;
正在阅读:
数字图像处理-第五章3 (2)08-11
学自行车的启示作文600字06-18
妈妈的早餐作文400字06-19
尊师重教作文1000字02-04
可爱的弟弟作文450字07-09
国旗下的讲话-少年强,则国强05-04
我们班主任作文优秀7篇03-25
实验212-23
成为一个优秀军人需要跨越的十个障碍01-19
林场场长扶贫先进事迹材料09-10
- 梳理《史记》素材,为作文添彩
- 2012呼和浩特驾照模拟考试B2车型试题
- 关于全面推进施工现场标准化管理实施的通知(红头文件)
- 江西省房屋建筑和市政基础设施工程施工招标文件范本
- 律师与公证制度第2阶段练习题
- 2019-2020年最新人教版PEP初三英语九年级上册精编单元练习unit6训练测试卷内含听力文件及听力原文
- 小升初数学模拟试卷(十四) 北京版 Word版,含答案
- 认识创新思维特点 探讨创新教育方法-精选教育文档
- 00266 自考 社会心理学一(复习题大全)
- 多媒体在语文教学中的运用效果
- 派出所派出所教导员述职报告
- 低压电工作业考试B
- 18秋福建师范大学《管理心理学》在线作业一4
- 中国铝业公司职工违规违纪处分暂行规定
- 13建筑力学复习题(答案)
- 2008年新密市师德征文获奖名单 - 图文
- 保安员培训考试题库(附答案)
- 银川市贺兰一中一模试卷
- 2011—2017年新课标全国卷2文科数学试题分类汇编 - 1.集合
- 湖北省襄阳市第五中学届高三生物五月模拟考试试题一
- 图像处理
- 数字
- 三年级上英语教案-Unit 3 My friends新译林版
- 2013年无锡招聘会、无锡2013年招聘会
- 四年级科学下复习资料
- 办理建设项目选址意见书(规划意见书)
- 选修双曲线及其标准方程第二讲同步练习题
- Unit 8 Is_life_fair
- 2021年【下载】优秀教师个人事迹50字-优秀word例文 (10页)
- 微机联锁操作手册(修改)
- 罗夏墨迹心理测试
- USB母座连接器
- 乳腺癌诊治新进展
- 半月谈2012年全年评论
- 护理管理学选择题精选资料教学总结
- 国家助学贷款还款流程
- 互换性与技术测量--第5章 光滑工件尺寸的检验与光滑极限量规
- 毛豆先生讲会展:会议管理艺术
- 白英抗肿瘤相关基因及信号通路的研究_王文静
- 第三章 库存管理
- 中财经济学考研真题
- 高炉炼铁工试卷参考答案及评分标准