The quaternion wavelet transform is a new multiscale analysis tool. Firstly, this paper studies the standard orthogonal basis of scale space and wavelet space of quaternion wavelet transform in spatial
Wavelet analysis is a rapidly developing branch of mathematics since 1980s; its research has just been unfolding. As a mathematical tool, wavelet transform is a major breakthrough of the Fourier transform and Fourier transform window known to many people; it has good timefrequency features and multiple resolution and wavelet analysis theory has become one of the most useful tools in signal analysis, image processing, pattern recognition and other fields. In image processing, the basic idea of the wavelet transform is to decompose image multiresolution; the original image is decomposed into different space and different frequency subimage, and then coefficients of subimage are processed. Commonly used wavelet transforms are real discrete wavelet transform and complex wavelet transform and so on. The real discrete wavelet and complex wavelet transform have two generals shortcoming; first, the real discrete wavelet transform signal small shift will produce the energy of wavelet coefficient distribution change; second, dualtree complex wavelet although overcame the first problem, it can generate signal phase ambiguity when represented twodimensional image’s features. While the quaternion wavelet transform is a new multiscale analysis image processing tool, it is based on the Hilbert twodimensional transform theory, which has approximate shift invariance and can well overcome the above drawbacks [
At present, quaternion wavelet research is divided into two branches, one is based on quaternion numerical function multiresolution analysis theory of quaternion wavelet, using a single tree structure, the earliest in 1994, Mitrea gave quaternion wavelet form concept [
In most cases, the image corruption is commonly modeled by a zeromean additive white Gaussian random noise leading to the following additive degradation model:
This paper gives and proves the properties of the Hilbert transform and the standard orthogonal basis of scale space and wavelet space of quaternion wavelet transform in spatial
Set
The quaternion is proposed by W. R. Hamilton in 1843; quaternion is a complex promotion, and it can be regarded as a special Clifford algebra.
Set
The conjugate of quaternion
So
Quaternion wavelet transform is based on quaternion analytic signal, and we give the concept.
Set
where
This section is based on the study in [
Let
According to the theorem of the paper in [
According to the lemma, it is not difficult to prove the following.
Let
We know that, in real orthogonal wavelet form 2D tensor product of wavelet, having a scale function and form of standard orthogonal basis in scale space, there are three wavelet functions form of standard orthogonal basis in wavelet space. For a function, it can be made along the
In fact, by Theorem
Let
Note
So
Further one has the following.
Let
And
One can get concept of the quaternion wavelet transform.
For for all
The above discussion shows that quaternion wavelet transform by using four real discrete wavelet transforms, the first real discrete wavelet corresponding quaternion wavelet real part, the other real discrete wavelets are formed by the first real discrete wavelet transform by Hilbert transform, corresponding to the three imaginary parts of quaternion wavelet, respectively. In the last section, we know that
Figure
The decomposition and reconstruction of quaternion wavelet.
Decompose structure
Reconstruct structure
One level QWT decomposition on the Barbara image
An image via quaternion wavelet transform, the coefficients can constitute a matrix
The quaternion wavelet transform of image Barbara. (a)–(e) Original image, magnitude, and the 3 terms of phase
Original image
Magnitude
Phase
Phase
Phase
In order to calculate the coefficients of QWT, looking from the structure, the quaternion wavelet filters’ system is similar to dualtree complex wavelet, quaternion wavelet filters’ coefficients is quaternion, and it is realized by using the dualtree algorithm, using an analytic quaternion wavelet bases in order to satisfy the Hilbert transform. Quaternion wavelet filters are dualtree filters, each filters’ subtree part comprises 2 analysis filters and 2 synthesis filters, respectively. In this paper, we use the quaternion wavelet with nearsymmetric orthogonal “Farrs” filters [
We find that the QWT coefficients of natural image is mainly distributed in the near to zero, and the two sides have a long tail, the traditional Gauss distribution is not accurate modeling of the QWT coefficients’ distribution. Through a large number of examples research, generalized Gaussian distribution (GGD) model can be used as the prior model of QWT coefficients in the high frequency subbands. The GGD [
And
Probability histogram and corresponding GGD distribution for Lena image.
Real part
Imaginary part
Imaginary part
Imaginary part
In common cases, the clear image is corrupted by zero mean additive white Gaussian noise, and the degradation model is
The histogram of magnitudes subtracting its mean of boat image.
In order to achieve better denoising effect, we amended formula (
The average PSNR varies with
First we estimate the variance of noise
Perform the 5 levels QWT decomposition on the noisy image.
Calculate the magnitude and phase of each subbands coefficients, and using (
Compute noisefree coefficients’ magnitude using the multiplied threshold and soft thresholding function.
Perform inverse transform on the estimated magnitude and the original phase to get QWT coefficients, and conduct the inverse QWT with estimated coefficients, and get the denoised image.
In the experiment, we have tested various denoising method for a representative set of standard 8bit grayscale images such as Lena, Barbara, Boat, Flinstones (size
The PSNR values are listed in Table
Performance comparison of different denoising methods (PSNR).

Donoho’s HT  LAWML  SURELET  PBivariate  Our method  

10  30.05  34.59  34.56  34.77  35.18  
Lena  20  27.29  31.32  31.37  31.72  32.10 
30  25.85  29.28  29.56  29.89  30.18  
 
10  26.02  32.28  32.16  33.16  33.34  
Barbara  20  23.38  28.81  27.96  29.75  29.36 
30  22.38  26.83  25.82  27.77  27.07  
 
10  27.42  32.11  32.91  32.72  33.15  
Boat  20  25.08  29.04  29.47  29.66  29.88 
30  23.83  27.18  27.63  27.83  27.92  
 
10  24.57  30.54  31.17  30.51  31.45  
Flinstones  20  21.72  27.17  27.40  27.28  28.12 
30  19.95  25.02  25.29  25.35  26.04  
 
10  30.07  33.18  34.27  34.28  34.75  
House  20  27.09  29.98  30.90  30.89  31.76 
30  25.49  27.96  28.96  28.94  29.88  
 
10  26.09  30.92  32.07  31.63  32.87  
Mit  20  22.78  27.08  27.88  27.77  28.62 
30  20.95  24.91  25.60  25.60  26.22  
 
10  26.87  31.47  32.35  31.81  33.08  
Cameraman  20  24.05  27.94  28.51  28.27  29.13 
30  22.54  26.00  26.48  26.38  27.05 
Denoised image using different denoising methods (
Noisy image
Hard
LAWML
SURELET
PBivariate
Our method
Denoised image using different denoising methods (
Noisy image
Hard
LAWML
SURELET
PBivariate
Our method
Denoised image using different denoising methods (
Noisy image
Hard
LAWML
SURELET
PBivariate
Our method.
Denoised image using different denoising methods (
Noisy image
Hard
LAWML
SURELET
PBivariate
Our method
From the experimental data and the denoised images, it can be seen that the proposed method almost provides the highest objective data. The proposed method has obvious advantage over hard thresholding shrinkage, and the PSNR has been greatly improved, compared with LAWML and SURELET which also has a certain degree of increase; the denoised Barbara image of PBivariate method obtained higher PSNR values than our method; it is mainly due to that PDTDFB has multidirection that it can well represent more texture images. From that Figures
Quaternion wavelet transform is established based on the quaternion algebra, quaternion Fourier transform, and Hilbert transform; using four real discrete wavelet transform (DWT), the first real discrete wavelet corresponding to quaternion wavelet real part, the other real discrete wavelet is obtained by the first real discrete wavelet transform’s Hilbert transform, corresponding to quaternion wavelet three imaginary part, respectively, the four real wavelet composed of quaternion analytic signal. It can be understood as the improved real wavelet and complex wavelet’s promotion, which have approximate shift invariance, abundant phase information, and limited redundancy and so forth, while still retaining the traditional wavelet timefrequency localization ability, filters design using Hilbert transform pair of dualtree structure, and it is easy to be realized.
This paper mainly studies some of the concepts and properties of quaternion wavelet transform, gives quaternion wavelet scale and wavelet functions, and applies the quaternion wavelet in image denoising, puts forward Bayesian denoising method based on quaternion wavelet transform, considering wavelet coefficient’s correlation, and generalized Gaussian distribution is used to model the probability distribution function of wavelet coefficients’ magnitude and the best range of the Bayesian thresholding parameter is found out. The experimental results show that our method both in visual effect and PSNR are better than many current denoising methods.
The authors are grateful for the support received from the National Natural Science Foundation of China (NSFC no. 11172086), and this work is supported by the Natural Science Foundation of Anhui (no. 11040606M06).