∑ x(n , n )ψ
m , k2
(6)
ϕm,k
−
2
to be orthogonal, (7)
ϕm,k
and the wavelet function
{ψ m, k2 , m, k2 ∈ Z } is also orthogonal,
2 ( n2 )
= 2 2 ϕ (2− m n2 − k2 )
− 1
Now, we can establish the definition of 2-D DFT-DWT.
Definition 5: For a 2-D signal x( n1 , n2 ) , its 2-D DFT-DWT for
x(n1 , n2 ) is defined as
(14)
X L ( k1 , k 2 , m ) =
2
Definitions and properties of 2-D DFT-DWT Headings
In this paper, we introduce partial transformation for different variable of a 2-D signal x( n1 , n2 ) .
1

−j 2π N1
where
WN1 = e
We express 1-D partial DFT of
x(n1 , n2 )
X (k1 , n2 ) = Fn1 [ x(n1 , n2 )]
Definition 2: For a 2-D hybrid signal
(11)
i.e.
1 [ X L (n1, k2 , m) + X H (n1, k2 , m)] x(n1, n2 ) = Wk− 2
2

x ( n1 , n 2 ) = +
m∈Z k2∈Z
∑ ∑
m∈Z k2∈Z H
∑ ∑
X L ( n1 , k 2 , m )ϕ m , k 2 ( n 2 )
(2)
X (k1 , n2 ) , its inverse 1-D partial DFT is defined as
N1 −1 k =0
1 x(n1 , n2 ) = N
where
∑ X (k , n )W
1 2
− n1k1 N1
(3)
WN1 = e
−j
2π N1
We express the inverse 1-D DFT
m
ψm,k (n2 ) = 2 2 [ϕm−1,2k (n2 ) −ϕm−1,2k +1(n2 )]
2 2 2
(8)
We express 1-D partial DWT of
x(n1 , n2 )
(9)
X L (n1 , k2 , m) = WLn2 [ x(n1 , n2 )]
and
X H (n1 , k2 , m) = WHn2 [ x(n1 , n2 )]
X L (n1 , k2 , m)
X H (n1 , k2 , m) , its inverse 2-D DFT-DWT is defined as
(17)
1 1 x ( n1 , n 2 ) = F W [ X L ( k1 , k 2 , m )] + Fk− W k− [ X H ( k1 , k 2 , m )] 1 2
We notice that
(10)
X L (n1 , k2 , m)
and
Definition 4: For a 3-D hybrid signal
X H (n1 , k2 , m) are 3-D hybrid signals under the transform. X L (n1 , k2 , m) and X H (n1 , k2 , m) , its inverse 1-D partial DWT is defined as
1
Introduction
2-D DFT and 2-D DWT have been introduced and discussed by many classical books and papers [1-3]. Presented 2-D signal processing [4-10] utilizes 2-D DFT or 2-D DWT to remove noise or frequency interference, not carefully considers that the remove noise or frequency interference may exist in some direction (horizontal and vertical, or space and time) only, or to be completely different in two directions. Though the wavelet transform (WT) has very good localized specialty in time-frequency domain, it is difficult for to remove narrow band frequency interference, due to the lack of frequency resolution although it does have some advantages for very small signals due to its high temporal resolution. Compared to DWT, DFT has an advantage to remove single frequency interference. Since DFT is not very valid for cancel the wide band noise like WT, it is necessary to apply them for different direction of 2-D signals with complicated noise and interference. To our knowledge, there is still no paper involved this problem, though there are many books are papers about WT and DFT such as [1-10]. To solve the problem, the paper develops the 2-D hybrid transform (DFT-DWT), with definitions, properties and algorithms.
N 1 −1 n1 = 0
Definition 1: For a 2-D signal, its 1-D DFT for the first variable of is defined as
X ( k1 , n2 ) =
∑ x(n , n
1
2
)W Nn11k1
(1)
* This work is supported by the National Natural Science Foundation of China under grant: 60572093. This paper has been partly published on Proc. ICSP2006.

2-D Hybrid DFT-DWT Application to Two-dimensional Signal Processing*
XIAO Tan, XIAO Yang and HU Shao-hai Institute of Information Science, Beijing Jiaotong University Beijing 100044, P. R. China E-mail:xiaotan_cn@; yxiao@
Abstract
Some two-dimensional signals may have different probability distribution of noise and frequency interference in different directions (horizontal and vertical, or space and time). It poses the question of how to extract desired signals from such complicated environment of noise and interference. It is difficult to extract by using classical DFT and DWT algorithms alone. To solve the problem, a hybrid 2-D transform, discrete Fourier transform-discrete wavelet transform (2-D DFT-DWT), with definitions, properties and algorithms, is developed for the hybrid 2-D signal processing. The hybrid transform can be used to remove the noise or interference in 2-D signals from different directions. Under the transform, desired results can be obtained to take advantage of merits of both DFT and DWT. Also, simulation illustrates the application. Keywords: 2-D hybrid transform, DFT, DWT.