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【Abstract】In this paper , application of two -dimensional continuous wavelet transform to image processes is studied. We first show that the convolution and correlation of two continuous wavelets satisfy the required admissibility and regularity conditions ,and then we derive the convolution and correlation theorem for two -dimensional continuous wavelet transform. Finally, we present numerical example showing the usefulness of applying the convolution theorem for two -dimensional continuous wavelet transform to perform image restoration in the presence of additive noise.
【Key words】wavelet transform, Fourier transform, convolution,correlation
1.Introduction
The transmission, storage and process of image may lead to degraded. The aim of image restoration is to recover the degrade image as more as better, and use an objective judgment system to judge the . Wavelet at present is extensively used in different areas such as signal analysis, image processing, qiantum mechnics, bioinformatics, etc. Image restoration is one of the most important problems in image processing. If the degrade system has the character of linear and time-invariant, and the blurrde noise is additive noise, then we can use linear filter to restoer the degrade image, because there is a filter which satisfiles convolution theorem which the .
Traditional methods such as Winner and Kalman filter perform in spectrum , all of them use Fourier Transform(in shot:FT). But the ranges of FT uses is restricted, it is better to stationary signal. However the degrade image signal are usually non-stationary. If we use perform this kind of image, then the result is not as good as what it is expected. Wavelet Transform(in shot:WT) solves these problems . A.F.Perez-Rendon and R.Robles referred to the convolution theorem for continuous wavelet transform and used it to signal . In this paper, we construct a convolution theorem for two –dimensional continuous wavelet transform and use it to perform the degrade image.
2.Preliminaries
Definition 1. The Fourier transform of a function is
(1)
The inverse formula is given by
(2)
If , then is a bounded in .
Definition 2. The convolution of and is
(3)
Let , it is easy to show that (4)
It is possible to convolve just one variable as follows:
(5)
(6)
Definition 3. The correlation of and is
(7)
Let , then a simple calculate shows the following result (8)
It is possible to convolve just one variable as follows:
(9)
(10)
Definition 4. We say a fuction is a two –dimensional continuous wavelet, if
(11)
Where
Two –dimensional continuous wavelet transform is defined as
(12)
The inverse formula is
(13)
(14)
is called admissibility condition of the wavelet.
Definition 5. vanishing moments of a wavelat is defined as
(15)
Where
3.Convolution theorem
Theorem 1. Let and be two admissible continuous wavelets with and vanishing moments respectively, and consider two wavelets and then both and are admissible wavelets too, and have vanishing moments.
Proof: Since , , so and are in too. From (14) the admissible condition can be defines as
(16)
From (4) and (8), we have
(17)
(18)
then
(19)
Therefore
(20)
By definition 1,so
(21)
Therefore, the new defined wavelets and satisfy admissible condition.
Now calculate the vanishing moments of wavelet that defined as
(22)
By inserting (3) for into above equation, we arrive at
(23)
Let then expand every components of according to Newton binomial expansion, it is easy to prove
Where
Let , if , then , because and vanishing moments; otherwise, so , because and vanishing moments. Therefore, for all . Calculate the vanishing moments about is similar as .
Theorem 2.(Convolution theorem) Let and be two admissible continuous wavelets, and denote the continuous wavelet transform of two functions and with wavelets and respectively. If and , then
(25)
Where .
Proof: From definition 4, the wavelet transform of can be written as
(26)
Let then
Theorem 3.(Correlation theorem) Let and be two admissible continuous wavelets, and denote the continuous wavelet transform of two functions and with wavelets and respectively. If and , then
(28)
Where .
The proof of this theorem is similar as theorem2, only change is the subtraction sign in theorem2.
4.Numerical example
In this section, one numerical example is given to analyze the test image. Two wavelets which satisfy the admissibility and regularity condition are given by
(29)
Let and .
Example. In this example, we will compare the image reconstruct in Fourier and wavelet domain respectively. The test image is “sinsin” in Matlab whose plot shown in Fig.1 (a). This image has een blurred with a Gaussian filter with standard deviation 0.15 and random white noise. Athe edge expands method is zero model and the directional angel is zero in this example, Fig.1 (b)shows the dgraded image, We first study the degrade image in Fourier domain with Winner filter defined as
(30)
Where is the power spectrum of the input sigal, and is the power spectrum of the .Fig.1 (c) shows the restored image in Fourier domain.
Now we study the degrade image in wavelet domain. From theorem2 the Gaussian blur is analyzed by and the degrade image by , so the restored image is analyzed with . The computation formula is as follows
(31)
Fig.1: (a) Test image used in the numerical example.
(b) Degrade image obtained by a Gaussian filter
and adding white noise.
(c) Restored image with winner filter in Fourier domain.
(d) Restored image with filter in wavelet domain.
Fig.1 (d) shows the restored image wavelet domain. The most commonly used measure, PSNR of the reconstructed image, is given as an indication of the image quality which is defined as
(32)
Where and denote the original and reconstructed image, respectively, M and N are the vertical and horizontal dimension of the image. A reconstructed image with better quality usually has a higher PSNR value. Restored image in Fourier domain has PSNR=14.46, while PSNR=24.18 in wavelet domain.
5.Conclusion
In this paper, we have first shown that the convolution snd correlation of two wavelet function satisfy the required admissibility and regularity condition, and then we have derived convolution and correlation theorem for continuous wavelet transform. We have also shown the result from one numerical example that illustrated the validity of the theorems proved in this paper. Comparisons between restoration of the test image in Fourier and Wavelet domain have demonstrated the benefit of using the wavelet convolution theorem for performing image restore. Future work may rely on the search of analogous convolution theorem for discrete wavelet transform, and search optimal example methods and directional angle in image process.
References：
[1]R.C.Gonzalez and R.E.Woods, Digital Image Processing, 175-219(2002).
[2]Dongwook.Cho snd TienD.Bui, Signal procssing:Image Communication Multiveriate statistical modeling for image denoising using wavelet transforms, 77-89(2005).
[3]Boris Escalante-Ramirez and Jose L.Slivan-Cardenas, Signal procssing:
[4]Image Communication. A multi-resolution directional-oriented image transform based on Gaussian derivative, 801-812(2005).
[5]Antonio F.Perez-Rendon and Rafael Robles, Signal procssing: The convolution theorem for the continuous wavelet transform, 195-208(1999).
【Key words】wavelet transform, Fourier transform, convolution,correlation
1.Introduction
The transmission, storage and process of image may lead to degraded. The aim of image restoration is to recover the degrade image as more as better, and use an objective judgment system to judge the . Wavelet at present is extensively used in different areas such as signal analysis, image processing, qiantum mechnics, bioinformatics, etc. Image restoration is one of the most important problems in image processing. If the degrade system has the character of linear and time-invariant, and the blurrde noise is additive noise, then we can use linear filter to restoer the degrade image, because there is a filter which satisfiles convolution theorem which the .
Traditional methods such as Winner and Kalman filter perform in spectrum , all of them use Fourier Transform(in shot:FT). But the ranges of FT uses is restricted, it is better to stationary signal. However the degrade image signal are usually non-stationary. If we use perform this kind of image, then the result is not as good as what it is expected. Wavelet Transform(in shot:WT) solves these problems . A.F.Perez-Rendon and R.Robles referred to the convolution theorem for continuous wavelet transform and used it to signal . In this paper, we construct a convolution theorem for two –dimensional continuous wavelet transform and use it to perform the degrade image.
2.Preliminaries
Definition 1. The Fourier transform of a function is
(1)
The inverse formula is given by
(2)
If , then is a bounded in .
Definition 2. The convolution of and is
(3)
Let , it is easy to show that (4)
It is possible to convolve just one variable as follows:
(5)
(6)
Definition 3. The correlation of and is
(7)
Let , then a simple calculate shows the following result (8)
It is possible to convolve just one variable as follows:
(9)
(10)
Definition 4. We say a fuction is a two –dimensional continuous wavelet, if
(11)
Where
Two –dimensional continuous wavelet transform is defined as
(12)
The inverse formula is
(13)
(14)
is called admissibility condition of the wavelet.
Definition 5. vanishing moments of a wavelat is defined as
(15)
Where
3.Convolution theorem
Theorem 1. Let and be two admissible continuous wavelets with and vanishing moments respectively, and consider two wavelets and then both and are admissible wavelets too, and have vanishing moments.
Proof: Since , , so and are in too. From (14) the admissible condition can be defines as
(16)
From (4) and (8), we have
(17)
(18)
then
(19)
Therefore
(20)
By definition 1,so
(21)
Therefore, the new defined wavelets and satisfy admissible condition.
Now calculate the vanishing moments of wavelet that defined as
(22)
By inserting (3) for into above equation, we arrive at
(23)
Let then expand every components of according to Newton binomial expansion, it is easy to prove
Where
Let , if , then , because and vanishing moments; otherwise, so , because and vanishing moments. Therefore, for all . Calculate the vanishing moments about is similar as .
Theorem 2.(Convolution theorem) Let and be two admissible continuous wavelets, and denote the continuous wavelet transform of two functions and with wavelets and respectively. If and , then
(25)
Where .
Proof: From definition 4, the wavelet transform of can be written as
(26)
Let then
Theorem 3.(Correlation theorem) Let and be two admissible continuous wavelets, and denote the continuous wavelet transform of two functions and with wavelets and respectively. If and , then
(28)
Where .
The proof of this theorem is similar as theorem2, only change is the subtraction sign in theorem2.
4.Numerical example
In this section, one numerical example is given to analyze the test image. Two wavelets which satisfy the admissibility and regularity condition are given by
(29)
Let and .
Example. In this example, we will compare the image reconstruct in Fourier and wavelet domain respectively. The test image is “sinsin” in Matlab whose plot shown in Fig.1 (a). This image has een blurred with a Gaussian filter with standard deviation 0.15 and random white noise. Athe edge expands method is zero model and the directional angel is zero in this example, Fig.1 (b)shows the dgraded image, We first study the degrade image in Fourier domain with Winner filter defined as
(30)
Where is the power spectrum of the input sigal, and is the power spectrum of the .Fig.1 (c) shows the restored image in Fourier domain.
Now we study the degrade image in wavelet domain. From theorem2 the Gaussian blur is analyzed by and the degrade image by , so the restored image is analyzed with . The computation formula is as follows
(31)
Fig.1: (a) Test image used in the numerical example.
(b) Degrade image obtained by a Gaussian filter
and adding white noise.
(c) Restored image with winner filter in Fourier domain.
(d) Restored image with filter in wavelet domain.
Fig.1 (d) shows the restored image wavelet domain. The most commonly used measure, PSNR of the reconstructed image, is given as an indication of the image quality which is defined as
(32)
Where and denote the original and reconstructed image, respectively, M and N are the vertical and horizontal dimension of the image. A reconstructed image with better quality usually has a higher PSNR value. Restored image in Fourier domain has PSNR=14.46, while PSNR=24.18 in wavelet domain.
5.Conclusion
In this paper, we have first shown that the convolution snd correlation of two wavelet function satisfy the required admissibility and regularity condition, and then we have derived convolution and correlation theorem for continuous wavelet transform. We have also shown the result from one numerical example that illustrated the validity of the theorems proved in this paper. Comparisons between restoration of the test image in Fourier and Wavelet domain have demonstrated the benefit of using the wavelet convolution theorem for performing image restore. Future work may rely on the search of analogous convolution theorem for discrete wavelet transform, and search optimal example methods and directional angle in image process.
References：
[1]R.C.Gonzalez and R.E.Woods, Digital Image Processing, 175-219(2002).
[2]Dongwook.Cho snd TienD.Bui, Signal procssing:Image Communication Multiveriate statistical modeling for image denoising using wavelet transforms, 77-89(2005).
[3]Boris Escalante-Ramirez and Jose L.Slivan-Cardenas, Signal procssing:
[4]Image Communication. A multi-resolution directional-oriented image transform based on Gaussian derivative, 801-812(2005).
[5]Antonio F.Perez-Rendon and Rafael Robles, Signal procssing: The convolution theorem for the continuous wavelet transform, 195-208(1999).