Dual-tree complex wavelet coefficient magnitude modelling using the bivariate Cauchy–Rayleigh distribution for image denoising

doi:10.1016/j.sigpro.2014.03.028

Signal Processing

Volume 105, December 2014, Pages 464-472

https://doi.org/10.1016/j.sigpro.2014.03.028 Get rights and content

Highlights

•
Accurately model the univariate complex magnitude statistics of the DT-CWT transform.
•
Accurately model the bivariate complex magnitude statistics of the DT-CWT transform.
•
Produce a denoising application based on the above.

Abstract

Wavelet shrinkage is a standard technique for denoising natural images. Originally proposed for univariate shrinkage within the Discrete Wavelet Transform (DWT) domain it has been made more effective through the use of (approximately) translationally invariant wavelet decompositions such as the Dual-Tree Complex Wavelet Transform (DT-CWT) and bivariate shrinkage. Techniques using the DT-CWT benefit from both (approximate) translational invariance and improved directionality. However, these techniques have used the real and imaginary components of the complex valued subband coefficients separately. The proposed work instead uses coefficient magnitudes to produce a novel bivariate shrinkage technique based on a heavy tailed bivariate distribution (of magnitudes) to provide a quantitative improvement in image denoising.

The results were compared to state of the art non-local means denoising technique BM3D. The PSNR results for small amounts of noise were comparable and within a small range for larger amounts of noise. However, when using the perceptually based structural similarity metric (SSIM) our developed technique offers improved results across the range of noise inputs when compared to BM3D in many cases. Perceptually, the developed technique is able to retain a greater quantity of the high frequency elements of the input image compared to BM3D.

Introduction

Denoising is a key image processing application within medical, military and general image analysis domains. It is impossible to completely recover an exact representation of original data when observed in combination with noise. However, a good representation of the original signal must retain as much perceptually important information as possible while efficiently attenuating noise. Many different techniques have been developed over the last 20 years for denoising within the domain of digital images [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. The most successful of these techniques fall into two categories: multiscale coefficient shrinkage and collaborative filtering/non-local processing. Our work uses a multiscale shrinkage method but we have compared it quantitatively and qualitatively with previous multiscale methods and non-local denoising techniques.

Transform domain denoising has achieved good results over the last few decades. Hard thresholding can be used within multiscale decompositions such as the Discrete Wavelet Transform (DWT). Optimised denoising can be achieved using “soft thresholding”. Many algorithms have been developed that are based on this idea as originally presented by Donoho [14]. Additionally, many algorithms have been recently developed that exploit the interdependence of related subband coefficients within wavelet transforms. For example Sendur and Selesnick [11] have developed a Maximum a-Posteriori (MAP) estimator based wavelet shrinkage technique that estimates the bivariate prior of a coefficient and its parent as a circularly symmetric Laplacian model. Recently developed better statistical models of wavelet coefficients have led to improved denoising performance. For example, it has been shown that the heavy tailed nature of wavelet coefficients can be well modelled using alpha-stable distributions [5]. Alpha stable distributions are a family of heavy-tailed densities that offer the ability to flexibly model the wavelet subband statistics for various applications [15]. Kwitt and Uhl [16] have modelled the distribution of complex wavelet coefficient magnitudes using various forms of PDFs such as the Rayleigh distribution although they have only used this for characterisation and retrieval not denoising.

Portilla et al. [10] have used Gaussian Scale Mixtures (GSMs) to produce an alternative wavelet based denoising method using larger neighbourhoods of wavelet coefficients across and within scales. This method has been further extended using the DT-CWT together with using coefficients with derotated phase to improve correlation between coefficients near edges and high contrast image features [8], [9]. Phase distributions of complex wavelet coefficients have been further examined by Vo and Oraintara [17] and Vo et al. [18]. A similar method to Miller and Kingsbury has also been proposed by Rakvongthai et al. [19]. This technique extends the Gaussian Scale Mixtures denoising algorithm to use complex coefficients, the so-called Complex Gaussian Scale Mixtures (CGSM) method, within the DT-CWT and other complex wavelet transforms.

Non-local methods of denoising attempt to find a group of similar patches to the considered image region. Through collaborative filtering, common transform elements can be retained and replaced in the original image.

The comparison metric for denoising has conventionally been a simple computational comparison such as PSNR or MSE. It has been shown in recent publications (e.g. [20]) that these metrics do not have a direct correlation with the human visual system in terms of perceived quality. Specifically, two different images can easily have the same PSNR and yet be dramatically different perceptually. The structural similarity metric SSIM (Structural Similarity Image Metric [20]) has recently been developed to provide a computational measure that better aligns with a perceptual quality comparison of two images.

There have been few papers trying to optimise denoising algorithms in terms of perceptual measures such as SSIM [7], [21]. We have therefore given our results in terms of both PSNR and SSIM measures together with a qualitative visual assessment.

Thresholding multiscale transforms to denoise images has given good results. Originally, the real valued Discrete Wavelet Transform (DWT) was used giving a baseline performance (e.g. [14]). However, as the DWT is translationally variant, thresholded coefficients do not give optimal results. Translationally invariant transforms such as the Undecimated Discrete Wavelet Transform (UDWT) [22] and the Dual Tree Complex Wavelet Transform (DT-CWT) [23], [24] have therefore been used to give better results. Although the UDWT is translationally invariant, it suffers from being considerably over-complete and lacks directional selectivity. The directional selectivity of the DT-CWT and its approximate shift invariance have provided good results for a range of denoising algorithms (e.g. [11]). However, within the bivariate shrinkage methods introduced below, a correspondence relationship needs to be established between co-located coefficients at neighbouring scales within the transform. This has usually entailed the upsampling of the lower scale subbands. Recently the Un-Decimated Dual Tree Wavelet Transform (UDT-CWT) [6] has been introduced that retains the directionality of the dual tree complex structure but does not down sample at each stage of the transform. There is therefore a direct one-to-one relationship between all of the co-located coefficients at all scales enabling improved (and more localised) bivariate shrinkage. We have used this transform (UDT-CWT) for all of our subsequent denoising methods.

Section snippets

Statistical modelling of dual tree complex wavelet transform coefficient components

The real and imaginary components of DT-CWT coefficients have often been modelled using normal and Laplacian distributions [11]. It has been observed that such wavelet coefficients are not well modelled using these distributions due to their heavy tails [4]. Observing the heavy tailed nature of the real and imaginary components of the DT-CWT complex subbands, considering the generalised central limit theorem and following related arguments given in [15], [25] we can assume that these complex

Statistical modelling of dual tree complex wavelet transform coefficient magnitudes

The method described in Section 2.1 denoises the real and imaginary components of the DT-CWT coefficients separately. However, this method has two disadvantages. Firstly, the relationship between the complex coefficients’ real and imaginary components is altered without any regard to the original phase relationship. This will introduce more phase noise into the denoised coefficients. Phase has been recognised as being vital in carrying important visual information [2]. Secondly, the bivariate

MAP denoising using bivariate magnitude model

The DT-CWT (and UDT-CWT) coefficient–parent relationship in the presence of noise can be expressed as follows: $y = x + n$ where $y$ = $(y_{1}, y_{2})$ is a vector of the actual noisy (observed) coefficient values where the elements y₁ and y₂ represent the coefficients and the co-located parent coefficients respectively. $x$ = $(x_{1}, x_{2})$ and $n$ = $(n_{1}, n_{2})$ are similarly defined for the clean coefficients and the noise values respectively. $x$ and $n$ are assumed to be statistically independent.

Denoising is achieved by defining an

Simulation results

Table 1 shows the results of the proposed method in terms of its performance in denoising standard test images in the presence of additive Gaussian noise. Zero mean Gaussian noise (with a range of σ values shown in the table) was added to two test images in two different resolutions. The same images were chosen to be tested over two resolutions since the content was broadly the same, but the level of high frequency content was higher with the higher resolution in textured regions.

The proposed

Conclusion

Wavelet shrinkage is a standard denoising algorithm for denoising image data. Wavelet shrinkage techniques based on a Maximum-a-Posteriori (MAP) framework have been able to provide a principled soft-thresholding system for optimised denoising results. Bivariate shrinkage techniques have further improved denoising where the joint statistics of a coefficient and its parent are modelled to generate an accurate prior. Approximate shift invariance and greater orientation selectivity have made the

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Dual-tree complex wavelet coefficient magnitude modelling using the bivariate Cauchy–Rayleigh distribution for image denoising

Highlights

Abstract

Introduction

Section snippets

Statistical modelling of dual tree complex wavelet transform coefficient components

Statistical modelling of dual tree complex wavelet transform coefficient magnitudes

MAP denoising using bivariate magnitude model

Simulation results

Conclusion

Signal Process. Image Commun.

Signal Process.

Appl. Comput. Harmon. Anal.

Signal Process.

Image denoising by sparse 3-D transform-domain collaborative filtering

IEEE Trans. Image Process.

Nonlinear filtering for phase image denoising

IEE Proc. Vis. Image Signal Process.

Ideal spatial adaptation by wavelet shrinkage

Biometrika

Image denoising using bivariate α-stable distributions in the complex wavelet domain

IEEE Signal Process. Lett.

SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling

IEEE Trans. Geosci. Remote Sens.

Image modeling using interscale phase properties of complex wavelet coefficients

IEEE Trans. Image Process.

Image denoising using derotated complex wavelet coefficients

IEEE Trans. Image Process.

Image denoising using scale mixtures of Gaussians in the wavelet domain

IEEE Trans. Image Process.

Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency

IEEE Trans. Signal Process.