行政院國家科學委員會專題研究計畫成果報告
形態小波轉換理論與應用之研究
A Study on Morphological Wavelet Transform:
Theory and Applications
計畫編號：NSC 90-2213-E-009-133
執行期限：2001 年 8 月 1 日至 2002 年 7 月 31 日
主持人：薛元澤 國立交通大學資訊科學系
計畫參與人員：陳明仁、吳昭賢 國立交通大學資訊科學系
中文摘要 本計畫探討形態小波轉換之理論與應 用。在理論方面，形態小波轉換結合了多 重解析理論，離散小波轉換與形態骨架 法。在應用上，可引進提升法已獲得更佳 之資料壓縮效果。 關鍵詞：多重解析理論、離散小波轉換、 形態骨架法、提升法 AbstractThis article provides guidance for report writing under the Grant of National Science Council beginning from fiscal year 1998.
Keywords: Multiresolution Analysis,
Discrete Wavelet Transform, Morphological Skeletonization, Lifting Scheme
1. INTRODUCTION
Multiresolution decomposition [1, 2, 3] based on discrete wavelet transform [4, 5] has significant applications in signal representation and compression. Since discrete wavelet transform is linear and the construction of a scaling function required Fourier transform, nonlinear approaches to wavelet-like decompositions without using Fourier transform have received many attentions [6-12]. The well-known lifting scheme [13, 14, 15] is one of the most successful nonlinear approaches.
Recently, Goutsias and Heijmans [16, 17] have proposed an axiomatic framework
to unify the linear and nonlinear multiresolution analyses. Under this framework, wavelet decomposition based on mathematical morphology [18] can be constructed.
Mathematical morphology is known as an efficient tool for image analysis and processing. Thus, it is the purpose of this study to investigate the theory and applications of morphological wavelet decomposition.
2. WAVELET DECOMPOSITION
In a multi-resolution analysis, there are nested vector spaces
V1 V0 V1
such that
(1) the closure of the union of all Vj is the space 2()
L and the intersection of all Vj is {0};
(2) f(t)Vj if and only if f(2t)Vj1; (3) there exists a scaling function (t)
such that {(tk):k is an integer} form a basis for V . 0
Suppose Wj is the orthogonal complement of Vj in Vj1. Then the vector space Vj can be written as
Vj1 Wj Wj1 Wjk Vjk for any nonnegative k. It is known that there
exists a function (t), the wavelet, such that {(tk):k is an integer} is a basis for
1
W . Then any function f L2() can be represented as
(2 ) ) (t c , t l f kl k where
f t t l dt c k k l k ( )2 (2 ) 2 2 , is the discrete wavelet transform of f(t). Note that the discrete wavelet transform is linear and the sets Vj and Wj are required to be vector spaces.
Recently, Goutsias and Heijmans proposed an axiomatic framework to unify linear and nonlinear approaches. Let Vj and
j
W be, respectively, the signal space and the detail space at level j. Consider the following analysis and synthesis operators:
(1) signal analysis operator Gj :Vj Vj1
(2) detail analysis operator : 1 j j j V W H
(3) signal synthesis operator Sj:Vj1Wj1 Vj
If the following conditions are satisfied: (1) perfect reconstruction condition Sj(Gj(x),Hj(x))x
for all xVj;
(2) nonredundant decomposition conditions
GjSj(x,y)x and HjSj(x,y)x for all xVj1 and yWj1,
then the recursive analysis scheme x0 {x1,y1}{x2,y2,y1} where xj1 Gj(xj)Vj1 and 1 ( ) 1 j j j j H x W y
is called a wavelet decomposition scheme. Note that x0 can be recursively
reconstructed from x and k y1,y2, ,yk
by
xj Sj(xj1,yj1), jk1,k2, ,0
Moreover, if there are operations *j on Vj and operators Gj Vj Vj 1 : and j j j W V H : 1 such that Sj(x,y) Gj(x)*j 1 Hj(y)
for all xVj1 and yWj1 , then the wavelet decomposition is called uncoupled.
For instances, the lazy wavelet is a uncoupled wavelet decomposition in which
Z j j W V V0 , j = 1, 2, … ) 2 ( ) )( (x n x n G ) 1 2 ( ) )( ( n x n x H ) ( ) 2 )( (x n x n G and G(x)(2n1)0 0 ) 2 )( ( n y H and H(y)(2n1)y(n)
* = the standard addition in Z
.
The one-dimensional morphological Haar wavelet is also a uncoupled wavelet decomposition in which Z j j W V V0 , j = 1, 2, … ) 1 2 ( ) 2 ( ) )( ( n x n x n x G ) 1 2 ( ) 2 ( ) )( ( n x n x n x H ) ( ) 1 2 )( ( ) 2 )( (x n G x n x n G 0 ) ( ) 2 )( ( n y n y H ) 0 ) ( ( ) 1 2 )( ( n y n y H
* = the standard addition in Z.
3. MORPHOLOGICAL WAVELETS
Under the framework alluded above, wavelet decomposition based on morphological operators can be constructed. Indeed, the well-known morphological skeletonization [19] is a uncoupled wavelet decomposition in which Z j j W V V0 , j = 1, 2, … ) ( ) (x E x G w ) ( ) (x x D E x H w w ) ( ) (x D x G w y y H( )
* = the standard addition in Z
where E and w D are respectively the w erosion and dilation with respect to the structuring element w.
In general, suppose there are operators
1 : j j j V V
and j :Vj1 Vj such that )
,
(j j forms an adjunction between Vj
and Vj1, i.e., j (x) yxj(y) for any xVj and yVj1, then the recursive
analysis scheme
x0 {x1,y1}{x2,y2,y1}
is a uncoupled wavelet decomposition in which ) ( ) ( j j j j x x G ) ( ) ( j j j j j j x x x H ) ( ) ( 1 1 j j j j x x G 1 1) ( j j j y y H ) ( ) ( ) , ( j1 j1 j j1 j j1 j x y G x H x S 4. APPLICATIONS
Wavelet decomposition has significant applications in signal representation and
compression. Wavelets are building primitives for general signals on one hand and have the power to decorrelate data on the other hand. In order to quickly find a wavelet decomposition, the lifting scheme has been proposed by Sweldens to construct biorthogonal wavelets without employing the Fourier transform. The lifting scheme consists of three steps: split, predict, and update. It has been shown that any discrete wavelet transform can be factored into lifting steps [20].
In order to possess better compression properties, the lifting scheme can also be applied to morphological wavelets. A general lifting scheme would then consist the following steps: wavelet decomposition, predict, and update. That is, given the wavelet decomposition
x0 {x1,y1}{x2,y2,y1}
if Pj :Vj Wj and Uj :Wj Vj are respectively the predict and update operators, then the lifting scheme is given by
yj yj Pj(xj)
xj xj Uj(xj)
5. CONCLUSIONS
In this study we investigate the morphological approach to wavelet transform. This approach can be viewed as a combination of multiresolution theory, morphological skeletonization, and the lifting scheme. It should have the potential applications in signal representation and compression.
However, in theoretical point of view, it is not clear that what structures should the signal spaces and the detail spaces have in order to satisfy the conditions required in morphological wavelets. We will work on the suitable structures in the future.
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