Dual Tree Complex Wavelet based Regularized ... - Semantic Scholar

GVIP Journal, Volume 7, Issue 1, April, 2007

Dual Tree Complex Wavelet based Regularized Deconvolution for Medical Images R. Murugesan 1 , V. Thavavel2 and B. Meenakshi Sundaram3 Department of Physical Chemistry, Madurai Kamaraj University, Madurai - 625021, India 2 Department of Applied Sciences, Sethu Institute of Technology, Kariapatti - 626 106, India Email: [email protected]

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regularized Wiener inverse filter commonly implemented in the Fourier domain is not well suited for signals that contain localized phenomena such as edges [5]. The use of wavelet transforms overcomes this problem due to the ability of wavelets to capture localized phenomena. The Wavelet Vaguellete Decomposition (WVD) method developed by Donoho [6], performs a simple inversion in the Fourier domain using H-1(f) to obtain a noisy, unbiased estimate of the input, followed by wavelet shrinkage in which some of the wavelet coefficients are reduced depending on the variance of the noise. The main drawback with WVD method is that the noise variance becomes large when the system function contains zeros; making the method ill-posed. The present work proposes an improved hybrid wavelet based regularized deconvolution algorithm suitable for use with any ill-conditioned system. This tandem processing exploits Fourier-domain regularization adapted to the convolution system to control the noise but uses it sparingly to keep the accompanying smearing distortions to the minimum required. The bulk of the noise removal and signal estimation is achieved using complex wavelet shrinkage. The present work has also exploited the properties of DT-CWT viz. excellent directionality and explicit phase information to remove blur and noise without the need for assuming a specific noise model. By means of images corrupted with different amounts of blur and noise, the proposed method was compared with adaptive Wiener filtering [7] and Fourier-Wavelet Regularized Deconvolution (ForWaRD) [8]. The result has proved that the present method yields far better result than those existing methods.

Abstract Deblurring in the presence of noise is a hard problem, especially in Ultrasound and CT images. In this paper, we propose a hybrid approach of wavelet-based image deconvolution that incorporates Fourier-domain system inversion followed by wavelet-domain noise suppression. In contrast to conventional wavelet-based deconvolution methods, the algorithm employs a regularized inverse filter to operate when the system is non-invertible as well as exploits the properties of dual tree complex wavelet transform (DT-CWT) to remove blur and noise without the need for assuming a specific noise model. Furthermore, the proposed approach uses an adaptive shrinkage function based on median, mean and standard deviation of wavelet coefficients to suppress noise while preserving the sharpness of the image. Its application on ultrasound and CT images has shown a clear improvement over alternative methods. Keywords: Image Deblurring, Regularized Inverse Filter, Dual Tree Complex Wavelet, and Wavelet denoising.

1. Introduction Generally medical images are blurred due to the finite resolution of the imaging system. For instance, an ultrasound image can be viewed as a distorted version of original image, where the distortion operator is a convolution with a point spread function (PSF) of the imaging system [1, 2]. Recently it has been shown that CT scanners cannot resolve many important bone details, particularly miniature sized features. For resolving these details, Deblurring can be an effective strategy [3]. In addition to blur the inclusiveness of noise heavily corrupts the images. Illustratively ultrasound images are assumed to be degraded by speckle noise and CT images are supposed to be corrupted by Poisson distributed random noise [4]. Corruption of blurred images with noise makes a poor visual analysis.

This paper is organized as follows: section-2 discusses the regularization of inverse filters in Fourier domain. The section-3 describes the theory, properties of the DT-CWT and complex wavelet based regularized deconvolution with adaptive thresholding scheme. The workflow and experimental results for the proposed algorithm is depicted along with existing competitive algorithms in section-4. The proposed research ends up with conclusion in section-5.

Deblurring of noisy images commonly require regularization techniques due to the presence of noise as well as zeros in the system transfer function. The

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GVIP Journal, Volume 7, Issue 1, April, 2007

structures with wavelet filters ho, h1 for the real part and go, g1 for the imaginary part. The DT-CWT is implemented using separable transforms and by combining subband signals appropriately [11]. Although it is non-separable, it inherits the computational efficiency of separable transforms. Specifically, the 1-D DT-CWT is implemented using two filter banks in parallel operating on the same data. Thus the dual tree does not appear to be a complex transform at all. However, the outputs from the two trees are interpreted as the real and imaginary parts of complex coefficients. In 2-D, the CWT decomposes an image f(x,y) using dilation and translations of a complex scaling function and six complex wavelet functions

2. Regularized Inverse Filters The noisy medical images with blurs due to the finite resolution of the imaging system are modeled as, (1) z ( x ) = ( y ⊗ v )( x ) + η ( x ) where v(x) is the linear operator that causes the blur in the true image y(x) and η(x) is zero mean white Gaussian noise with the variance σ2. In the 2-D Discrete Fourier Transform (DFT) domain, we have for (1) (2) Z ( f ) = Y ( f )V ( f ) + η( f ) where Z(f),Y(f),V(f) and η(f) are the DFT of the corresponding signals. In principle, an unbiased estimate of Y(f) can be obtained as a straightforward pure (“naïve”) inverse solution of the (2), given as,

 Y ( f ) = Z( f ) V ( f )

(3) However the inverse transform does not exist for V as a bounded linear operator; such inverse problems are called ill posed or ill conditioned [9]. It is now a standard to approach the inverse problem by the method of applying Regularized Inverse (RI) operator [9] which is defined as,

 2 Y ( f ) = Z ( f )V ( − f ) V ( f ) + ε 2

f (x, y) = ∑sjo,kφjo,k (x, y) +∑∑∑cθj,k ψθj,k (x, y) (6)

Impulse response of these six wavelet subbands of the 2-D DT-CWT are strongly oriented in θ = {+15o, +45o, +75o, -15o, -45o, -75o} direction and captures image information in that direction. Frequency-domain partition of complex wavelet transforms resulting from two levels decomposition can discriminate between features at positive and negative frequencies. Hence, there are six subbands capturing features along lines at angles of {+15o, +45o, +75o, -15o, -45o, -75o}. These advantages of DT-CWT provoked us to modify the two common basic steps of standard deconvolution algorithms [12, 13] which are summarized below:

(4)

where ε is a regularization parameter that controls the tradeoff between the amount of noise suppression and the amount of signal distortion. Setting ε = 0 gives an unbiased but noisy estimate. Setting ε = 1 completely suppresses the noise, but also totally distorts the signal. Hence, the present algorithm uses the RI estimates (4) with ε as small as possible leaving the main filtering to be done in the complex wavelet domain.

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3. Complex Wavelets and Deconvolution Real DWT has poor directional selectivity and lacks shift invariance. These problems are solved effectively by the complex wavelet transform (CWT) by introducing limited redundancy into the transform. In CWT, filters have complex coefficients and generate complex output samples. However, a further problem arises in achieving perfect reconstruction for complex wavelet decomposition beyond level 1. To overcome this, Kingsbury [10] have recently developed the DT-CWT, which allows perfect reconstruction while still providing the other advantages of complex wavelets.

l∈Z

∑ ∑ c j,l ψ j,l ( x)

Regularization by complex wavelet denoising: The actual denoising is achieved by thresholding the coefficients with thresholds that are scale-wise adaptive, depending on standard deviation (σ), absolute mean (μ) and absolute median (M) of wavelet coefficients of the scale. To compute a complex threshold, the method has been extended to both real and imaginary domain, as in [14]. The value of threshold for real part is calculated as,

Treal =

1 σ real M real 2 μ real

(7)

The threshold value for imaginary part is calculated as,

Dual Tree Complex Wavelets: In DT-CWT, to achieve perfect reconstruction and good frequency characteristics, two parallel fully decimated trees with real filter coefficients are used [10]. The 1-D DT-CWT that decomposes a signal f(x) in terms of a complex shifted and dilated mother wavelet ψ(x) and scaling function φ(x),is defined as,

f ( x) = ∑ s jo,l φ jo,l ( x) +

b∈θ j≥jok∈Z2

k∈Z2

Timag = 2.

(5)

j ≥ jo l∈Z

where sjo,l is scaling coefficient and cj,l is complex wavelet coefficient with φjo and ψj complex: φjo =φrjo,l + −1φijo,l r i and ψ j = ψrj,l + − 1ψij,l . The ψ j,l and ψ j,l are themselves real wavelets: the complex wavelet transform is a combination of two real wavelet transforms, in 1-D the r i i { φ j,l φ j,l ψ rj ,l , ψ j,l } form a tight wavelet frame with two times redundant. The real and imaginary parts of the DT-CWT are computed using separate filter bank

1 σ imag M imag 2 μ imag

Wavelet Filtering: After computing the complex threshold values, the wavelet coefficients are filtered in wavelet domain to give real and imaginary part of wavelet coefficients for deblurred image as follows

Wd _ real = Wo _ real Wd _ imag = Wo _ imag

2

(8)

2 Treal 2 Treal + σ 2n 2 Timag 2 Timag + σ 2n

(9)

(10)

GVIP Journal, Volume 7, Issue 1, April, 2007

Metric RMSE

SNR

ISNR

designed with the menu-driven approach as shown in figure1 (A) to perform the task effectively. The proposed algorithm with workflow depicted in figure 1(B) takes around 2.5 seconds in Intel Pentium III 800 MHz PC with 128 MB RAM for 256 x 256 pixel images. Thereby, the total computational complexity is O(N log N). It is the combination of two computational complexities namely Regularization in Fourier domain with O(N log N) and complex wavelet domain with O(N).

Definition

1 MN

M

N

∑∑ [ I (i, j ) − Iˆ(i, j)]2 i =1 j =1

10 log10 20 log 10

1 MN

σˆ

M

N

∑∑ | I (i, j ) |

2

i =1 j =1

RMSE

RMSE

Table 1. Image Quality Assessment Metrics

4. Simulation Results Quality Measures: Image quality measures play an important role in various image-processing applications. A great deal of effort has been made in recent years to develop several image quality metrics in addition to visual analysis, to predict the visible differences between a pair of images, the input image I ( x , y ) and the resultant image Iˆ ( x , y ) . Many measures are used to compare the performance of the various image processing systems and to select the appropriate system for any given applications. The widely used metrics such as Signal to Noise ratio (SNR) and improvement in SNR (ISNR) in decibels [15] tabulated in table 1, were used to evaluate the potential of the proposed method. Higher values of SNR and ISNR represents better quality image.

A

B

Figure 2. Noise free Ultrasound (Column A) and CT (Column B) image used for simulation study.

Results and Discussion: In this section, we illustrate the performance of the proposed method on noisy blurred Ultrasound and CT image, and compare it to the existing methods such as spatially adaptive Wiener filter and ForWaRD method. The reference noise free Ultrasound (Column A) and CT (Column B) image used for investigation is shown in figure 2. In simulations, these images are contaminated with a mixture of blurs and noises as shown in (Column A) of figure 3 and figure 4. Nevertheless, the potency of the proposed method was assessed particularly for the effect of different types blur on speckled ultrasound and poison corrupted CT image. Further the proposed method employs Daubechies least asymmetric compactlysupported wavelet with 5 vanishing moments with 3 scales of orthogonal decomposition for complex wavelet transform.

A

Figure 3. Visual performance of the proposed method, compared to other methods. Input image (Column A) is an Ultrasound image with mixture of speckle noise and various blur types box-car, circular blur and convolution blur respectively. Resultant image of deconvolution methods Wiener, ForWaRD and the proposed is in consecutive columns (B, C, D).

B Figure 1. GUI environment (A) and workflow (B) of DT-CWT based Deconvolution system

System Implementation: A WYSIWYG editor using MATLAB libraries to read input images and parameters to view the output is implemented. To increase usability the selection of blur type, noise type and Deblurring filter type were

In order to evaluate the performance of the proposed algorithm subjectively, we have depicted the restored 3

GVIP Journal, Volume 7, Issue 1, April, 2007

ultrasound image of coronal view of brain as well as CT image of inner ear in figure 3 and figure 4 respectively. In these figures, Column B, C, D refers the resultant images obtained using Wiener, ForWaRD and the proposed method respectively. The input image contaminated with mixture of noise and various blur types such as box-car, circular blur and convolution blur are shown in Column A of Row 1, 2, 3 respectively.

As well to illustrate the superiority of the proposed method, its performance was analyzed quantitatively by computing the quality metrics tabulated in table 1 for all resultant image shown in figure 3 and figure 4. The computed metrics are collected in table 2(A) for ultrasound image and in table 2(B) for CT image. From table 2, it is clear that for both the reference images, the proposed method provides significant improvement over all the other approaches, as reflected by good SNR and ISNR. Overall, the simulation results show that the proposed algorithm demonstrates a good performance and outperforms some of the best competitive algorithms with reduced computational complexity.

5. Conclusion In this paper, we have proposed a new framework for deblurring of medical images that have been blurred and corrupted by noise. The proposed method uses a combination of Fourier-domain regularized inversion and complex wavelet-domain based noise suppression to provide a good estimate of the original image even with ill-conditioned systems. The results of our proposed method were also compared with previous reported methods for medical images and our proposed method was found to perform better than those existing methods both in visual quality and quality metrics. .

Figure 4. Visual performance of the proposed method, compared to other methods. Input image (Column A) is an CT image with mixture of poisson noise and various blur types box-car, circular blur and convolution blur respectively. Resultant image of Deconvolution methods Wiener, ForWaRD and the proposed is in consecutive columns (B, C, D).

6. Acknowledgements R.Murugesan thanks Department of Science and Technology, New Delhi, India for providing a major research grant. A part of this work was carried out under the UGC sponsored Center for Potential in Genomic Sciences Programme. Support in part by the Scientist Exchange Program of the Office of International Affairs, National Cancer Institute, NIH, DHHS, USA is also gratefully acknowledged.

From these figures, it can be observed that the restored image in figure 3(D) and figure 4(D) has recovered most visual clarity from the blurred image. When compared with the result obtained using Wiener and ForWaRD method, the proposed method is sharper and possesses less residual noise for all types of noise and blurs. The reason that the proposed method is so desirable in preserving the clinically interesting features is that the thresholding scheme applied for complex wavelet filtering adapts to spatial variations in both signal and noise. Method

SNR (in Decibels) Radial Blur

LPF Blur

Wiener

Boxcar Blur 23.77

22.7

ForWaRD

23.97

24.46

Proposed

25.72

26.75

7. References [1] T. Taxt, “Comparison of Cepstrum-based methods for Radial Blind Deconvolution of Ultrasound Images,” IEEE Trans. on Ultrasonics, Feraoelectrics, and FrequencyControl, vol. 44, pp. 666 - 674, 1997. [2] O. Michailovich and D. Adam, “Phase Unwrapping for 2-D Blind Deconvolution of Ultrasound Images,” IEEE Trans. on Medical Imaging, vol. 23, no. 1, pp.7-25, 2005. [3] M. Jiang, G. Wang, M. Skinner, J. Rubinstien, and M. Vannier, “Blind deblurring of spiral ct images,” IEEE Trans. on Medical Imaging, vol. 22, no. 7, pp.837-45, 2003. [4] A. Khare and U. S. Tiwary, “Soft-thresholding for Denoising of Medical Images - A Multiresolution Approach,” International Journal of Wavelet, Multiresolution and Information Processing, vol. 3, pp. 49 - 81, 2005. [5] S. Wan, B. I. Raju, and M. A. Srinivasan, “Robust Deconvolution of High-Frequency Ultrasound Images using Higher-Order Spectral Analysis and Wavelets,” IEEE Trans. on Ultrasonics,

ISNR (in Decibels) Radial Blur

LPF Blur

31.85

Boxcar Blur 6.6

8.36

14.77

33.02

7.3

10.24

16.11

34.61

8.56

12.4

17.53

A Method

Wiener

SNR (in Decibels) Boxcar Blur 24.52

Radial Blur

LPF Blur

23.58

32.59

ISNR (in Decibels) Boxcar Blur 4.35

Radial Blur

LPF Blur

8.19

12.39

ForWaRD

27.64

27.94

36.21

9.62

13.27

17.63

Proposed

32.29

33.56

41.17

12.12

18.18

20.96

B Table 2. Quantitative performance of the proposed method, compared to other methods on Ultrasound (A) and CT image (B)

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GVIP Journal, Volume 7, Issue 1, April, 2007

Feraoelectrics, and Frequency Control,vol. 50, no. 10, pp. 1286 - 1295, 2003. [6] D. L. Donoho, “Nonlinear Solution of Linear Inverse Problems by Wavelet-Vaguellete Decomposition,” App. Comp. Harmonic Anal., vol. 2, pp. 101 - 126, 1995. [7] M. Sapia, “Deconvolution of Ultrasonic Waveforms using an Adaptive Wiener Filter,” Review of Progress in Quantitative Nondestructive Evaluation, vol. 13, pp. 855 - 862, 1994. [8] R. Neelamani, H. Choi, and R. Baraniuk, “ForWaRD: Fourier-Wavelet Regularized Deconvolution for IllConditioned Systems,” IEEE Trans. on Signal Processing, vol. 52, no. 2, pp. 418 - 433, 2004. [9] A. N. Tikhonov and V. Y. Arsenin, “Solution of IllPosed Problems,” Wiley and sons, New York, 1977. [10] N. G. Kingsbury, “Image Processing with Complex Wavelet,” Phil. Trans. Roy Soc. London A, vol. 357, pp. 2543 - 2560, 1999. [11] I. W. Selesnick, “The Design of Approximate Hilbert Transform pairs of Wavelet bases,” IEEE Trans. Signal Processing, vol. 50, no. 5, pp. 1144 - 1152, 2002. [12] R. D. Nowak, “A Fast Wavelet-Vaguelette Algorithm for Discrete LSI problems,” Tech. Rep., Michigan State University, 1997. [13] J. Kalifa, S. Mallat, and B. RougB, “Image Deconvolution in Mirror Wavelet bases,” in IEEE Int. Conf.Image Processing - ICIP 98, pp. 565 - 569, October 1998. [14] A. Khare and U. S. Tiwary, “A New Method for Deblurring and Denoising of Medical Images using Complex Wavelet Transform,” in Proceedings of the IEEE, pp. 1897 – 1900, September 2005. [15] Vladimir Katkovnik, Karen Egiazarian,, and Jaakko Astola, “Spatially Adaptive Nonparametric Regression Image Deblurring,” IEEE Trans. Image Processing,, vol. 14, no. 10, pp. 1469 - 1478, October 2005.

V. Thavavel received M.C.A and M.Phil Degrees from Madurai Kamaraj University, India. She is an Assistant Professor in Computer Science at Sethu Institute of Technology, Kariapatti, India. She is currently doing her Ph.D., in Computer Science at Madurai Kamaraj University, India. Her research area of interest involves medical image reconstruction in an Object Oriented approach and application of Genetic algorithms to medical image analysis. B. Meenakshi Sundaram received his M.Sc., M.Phil in computer science from Madurai Kamaraj University. He has served as a Faculty and Head of Academics with an Institution in UAE, affiliated to University of Portsmouth and Scottish Qualifications Authority, UK. Currently he is working as a Senior Lecturer in Sethu Institute of Technology. His area of research interest involves semantic context based image retrieval systems, Ontological knowledge base for medical imaging.

Ramachandran Murugesan received his Ph.D Degree from Regional Instrumentation Center, Indian Institute of Technology, Chennai, India. He has been involved in developing novel Electron Magnetic Resonance (EMR) instrumentation such as Flash Photolysis CW EMR, Zero Field EMR spectroscopy etc., during his post doctoral research in US and Australia. With his experience in EMR instrumentation, he is pursuing collaborative research with the Radiation Biology Branch, National Cancer Institute, in the development of pulsed radio frequency FT EMR imaging, Fluorine Electron Double Resonance Imaging (FEDRI) and Overhauser enhanced Magnetic Resonance Imaging (OMRI) techniques for biomedical applications. The imaging technology developments are covered by a number of patents and research publications. His research interests include EMR spectroscopy and imaging, and medical image processing.

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