大家好，又见面了，我是你们的朋友全栈君。
很多人对小波多级分解的wavedec2总是迷惑，今天就详释她！
wavedec2函数:
1.功能:实现图像(即二维信号)的多层分解，多层,即多尺度.
2.格式:[c,s]=wavedec2(X,N,’wname’)
[c,s]=wavedec2(X,N,Lo_D,Hi_D)(我不讨论它)
3.参数说明:对图像X用wname小波基函数实现N层分解,
这里的小波基函数应该根据实际情况选择,具体选择办法可以搜之或者 help WFILTERS
.输出为c,s.
c为各层分解系数,s为各层分解系数长度,也就是大小.
4.c的结构:c=[A(N)|H(N)|V(N)|D(N)|H(N-1)|V(N-1)|D(N-1)|H(N-2)|V(N-2)|D(N-2)|…|H(1)|V(1)|D(1)]
可见,c是一个行向量,即:1*(size(X)),(e.g,X=256*256,then
c大小为:1*(256*256)=1*65536)
A(N)代表第N层低频系数,H(N)|V(N)|D(N)代表第N层高频系数,分别是水平,垂直,对角高频,以此类推,到H(1)|V(1)|D(1).
每个向量是一个矩阵的每列转置的组合存储。原文:Each vector is the vector
column-wise storage of a matrix. 这是你理解A(N) H(N) | V(N) | D(N)
的关键。
很多人对wavedec2和dwt2的输出差别不可理解，后者因为是单层分解，所以低频系数，水平、垂直、对角高频系数就直接以矩阵输出了，没有像wavedec2那样转换成行向量再输出，我想你应该不再迷惑了。
那么S有什么用呢？
s的结构:是储存各层分解系数长度的,即第一行是A(N)的长度(其实是A(N)的原矩阵的行数和列数),
第二行是H(N)|V(N)|D(N)|的长度,
第三行是
H(N-1)|V(N-1)|D(N-1)的长度,
倒数第二行是H(1)|V(1)|D(1)长度,
最后一行是X的长度(大小)
从上图可知道：cAn的长度就是32*32，cH1、cV1、cD1的长度都是256*256。
到此为止，你可能要问C的输出为什么是行向量?
1、没有那一种语言能够动态输出参数的个数，更何况C语言写的Matlab
2、各级详细系数矩阵的大小(size)不一样，所以不能组合成一个大的矩阵输出。
因此，把结果作为行向量输出是最好，也是唯一的选择。
另:MATLAB HELP
wavedec2 里面说得非常明白了,呵呵.
wavedec2
Multilevel 2-D
wavelet decomposition Syntax [C,S] =
wavedec2(X,N,’wname’)
[C,S] = wavedec2(X,N,Lo_D,Hi_D)
Description wavedec2 is a two-dimensional wavelet analysis
function.
[C,S] =
wavedec2(X,N,’wname’) returns the wavelet decomposition of the
matrix X at level N, using the wavelet named in string ‘wname’ (see
wfilters for more information).
Outputs are the
decomposition vector C and the corresponding bookkeeping matrix S.
N must be a strictly positive integer (see wmaxlev for more
information).
Instead of giving the
wavelet name, you can give the filters. For [C,S] =
wavedec2(X,N,Lo_D,Hi_D), Lo_D is the decomposition low-pass filter
and Hi_D is the decomposition high-pass filter.
Vector C is organized
as C = [ A(N) | H(N) | V(N) | D(N) | … H(N-1) | V(N-1) | D(N-1) |
… | H(1) | V(1) | D(1) ]. where A, H, V, D, are row vectors such
that A = approximation coefficients H = horizontal detail
coefficients V = vertical detail coefficients D = diagonal detail
coefficients Each vector is the vector column-wise storage of a
matrix.
Matrix S is such that
S(1,:) = size of approximation coefficients(N) S(i,:) = size of
detail coefficients(N-i 2) for i = 2, …N 1 and S(N 2,:) =
size(X)
Examples% The current
extension mode is zero-padding (see dwtmode).
% Load original
image. load woman; % X contains the loaded image.
% Perform
decomposition at level 2 % of X using db1. [c,s] = wavedec2(X,2,’db1′);
% Decomposition
structure organization. sizex = size(X)
sizex =
256
256
sizec = size(c)
sizec =
1
65536
val_s =
s
val_s =
64
64 64
64 128
128 256
256
Algorithm For images, an algorithm similar to the
one-dimensional case is possible for two-dimensional wavelets and
scaling functions obtained from one-dimensional ones by tensor
product. This kind of two-dimensional DWT leads to a decomposition
of approximation coefficients at level j in four components: the
approximation at level j 1, and the details in three orientations
(horizontal, vertical, and diagonal). The following chart describes
the basic decomposition step for images: So, for J=2, the
two-dimensional wavelet tree has the form See Alsodwt, waveinfo,
waverec2, wfilters, wmaxlev ReferencesDaubechies, I. (1992), Ten
lectures on wavelets, CBMS-NSF conference series in applied
mathematics. SIAM Ed. Mallat, S. (1989), “A theory for
multiresolution signal decomposition: the wavelet representation,”
IEEE Pattern Anal. and Machine Intell., vol. 11, no. 7, pp.
674-693. Meyer, Y. (1990), Ondelettes et opérateurs, Tome 1,
Hermann Ed. (English translation: Wavelets and operators, Cambridge
Univ. Press. 1993.
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