International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 1
ISSN 2229-5518
Denoising of Computed Tomography Images using Curvelet Transformation
with Log Gabor Filter
Gaurav Gupta, Tanjeet Kaur
Abstract— Digital Imaging plays important role in major areas of life such as clinical diagnosis etc. But it faces problem of Guassian noise. Noise corrupts both images and videos. The purpose of the denoising algorithm is to remove such noise. Image denoising is needed because a noisy image is not pleasant to view. In addition, some fine details in the image may be confused with the noise or vice-versa. The work presented herein focuses on a zero mean additive white Gaussian noise (AWGN). Zero mean does not lose generality, as a non- zero mean can be subtracted to get to a zero mean model. The purpose of this paper is to carry out the performance assessment of the noise reduction methods on the brain Computed Tomography (CT) images. This Work proposes a Curvelet Transformation based image denoising, which is combined with Gabor filter in place of the low pass filtering in the transform domain. This study is foc used not only on the noise suppression but also on fine details and edge preservation. The quality assessment parameters used in this paper are Peak - signal-to-noise-ratio (PSNR), Coefficient of Correlation (CoC), Mean Square Error (MSE).
Index Terms— Coefficient of Collelation (CoC), Computed Tomography, Curvelet Transform, Denoising, Gabor Filter, Mean Square Error
(MSE), Peak signal-to-noise ratio (PSNR).
—————————— ——————————
Computed tomography (CT scan) or computed axial to- mography (CAT scan), can be used for medical imaging and industrial imaging methods employing tomography created by computer processing. Digital geometry processing is used to generate a three dimensional image of the inside of an ob- ject from a large series of two-dimensional X-ray images taken around a single axis of rotation. CT produces a volume of data that can be manipulated, through a process known as "win- dowing", in order to demonstrate various bodily structures based on their ability to block the X-ray beam. Although his- torically the images generated were in the axial or transverse plane, perpendicular to the long axis of the body, modern scanners allow this volume of data to be reformatted in vari- ous planes or even as volumetric (3D) representations of struc- tures. Although most common in medicine, CT is also used in other fields, such as nondestructive materials testing. Another example is archaeological uses such as imaging the contents of sarcophagi [17,18]. Computed Tomography (CT) and magnetic resonance imaging (MRI) are the two modalities that are regu- larly used for brain imaging. CT imaging is preferred over MRI due to lower cost, short imaging times, widespread avail- ability, ease of access , optimal detection of calcification and hemorrhage and excellent resolution of bony detail. CT scans of internal organs, bone and soft tissue provide greater clear- ness than conventional x-ray. However, the limitations for CT
————————————————
Tanjeet Kaur is currently pursuing masters degree program in Computer Engineering at University College of Engineering, Punjabi University, Patiala (Punjab)-147002, India, PH-+91-9780725377.
E-mail: : tanjeet_arora19@yahoo.co.in
scanning of head images are due to partial volume effects which affected the edges and produce low brain tissue con- trast [1]. CT is a radiographic inspection method that gener- ates a 3-D image of the inside of an object from a large series of
2-D images taken on a cross sectional plane of the same object. CT generates thin slices of the body with a narrow X-ray beam, which rotates around the body of the stationary patient [2].
CT images are generally of low contrast and they often have a Gaussian noise due to various acquisitions, transmis- sion storage and display devices [3], [4]. In most of the image processing applications, a suitable noise removal phase is of- ten required before any relevant information could be extract- ed from analyzed images. In this area initial effort was started with ideas based on statistical filtering in spatial domain [5].
Over the last decade there has been wide attention for noise removal in signals and images based on wavelet methods. The wavelet coefficients at different scales could be obtained by taking Discrete Wavelet Transform (DWT) of the image. The small coefficients in the subbands are dominated by noise, while coefficients with large absolute value carry more signal information than noise. Replacing noisy coefficients by zero and an inverse wavelet transform may lead to reconstruction that has lesser noise. Normally Hard thresholding (Hard_WT) and Soft thresholding techniques are used for such denoising process. Although the hard and soft thresholding methods are widely used for noise removal purpose but they have some disadvantages. In case of hard thresholding, the wavelet coef- ficients are not continuous at the preset threshold so that it may lead to the oscillation of the reconstructed signal. The wavelet coefficients in case of soft thresholding method have good continuity, but it may cause constant deviations between the estimated wavelet coefficients and original wavelet coeffi- cients. Thus the accuracy of the reconstructed image might
IJSER © 2012
International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 2
ISSN 2229-5518
suffer. There has been a lot of research on wavelet threshold- ing and threshold selection for signal and image denoising [6],[7],[8],[9]. To overcome above mentioned disadvantages, considerable improvements in perceptual quality were ob- tained by translation invariant methods based on thresholding of an undecimated wavelet transform [10],[11],[12].However as an optimal tool for 1-D signals, wavelet in 2-D images is only better at isolating the discontinuities at edge points but cannot detect the smoothness along the edges. Wavelet can only capture limited directional information. Due to these dis- advantages of wavelet, theory of multiscale geometric analysis has been developed. In 1999 Donoho and others [13] proposed the concept of curvelet transform. Unlike wavelets, curvelets are localized not only in position and scale but also in orienta- tion. This localization provides the curvelet frame with sur- prising properties; it is an optimally sparse representation for singularities supported on curves in two dimensions and has become a promising tool for various image processing applica- tions[13],[14],[15],[16].
In this paper curvelet transformation with log gabor filter is used to denoise computed tomography images. This Work proposes a Curvelet Transformation based image denoising, which is combined with Gabor filter in place of the low pass filtering in the transform domain. This analysis is not only focused on the suppression of noise but also on preservation of fine details and edges. The quality assessment metrics used are Peak-signal-to-noise-ratio (PSNR), Coefficient of Correla- tion (CoC), Mean Square Error (MSE).
The Curvelet transform is a higher dimensional generali- zation of the Wavelet transform designed to represent images at different scales and different angles. Curvelets enjoy two unique mathematical properties,namely:
•Curved singularities can be well approximated with very few coefficients and in a non-adaptive manner - hence the name "curvelets". Curvelets remain coherent waveforms un- der the action of the wave equation in a smooth medium. Curvelets are a non-adaptive technique for multi-scale object representation. Being an extension of the wavelet concept, they are becoming popular in similar fields, namely in image processing and scientific computing.Curvelets are an appro- priate basis for representing images (or other functions) which are smooth apart from singularities along smooth curves, where the curves have bounded curvature, i.e. where objects in the image have a minimum length scale. This property holds for cartoons, geometrical diagrams, and text. As one zooms in on such images, the edges they contain appear in- creasingly straight. Curvelets take advantage of this property, by defining the higher resolution curvelets to be skinnier the lower resolution curvelets. However, natural images (photo- graphs) do not have this property; they have detail at every scale. Therefore, for natural images, it is preferable to use some sort of directional wavelet transform whose wavelets have the same aspect ratio at every scale. Curvelet Transform give a superior performance in image denoising due to prop- erties such as sparsity and multiresolution structure.but for
the CT and medical images , information need to refined for better way, So that radiologist could give the better opinion for the diagonosis
The basic process of the digital realization for curvelet trans- form consist the following four stages:
P0 – Low-pass filter.
1, 2, … Using Log Gabor filter to approximate the
frequencies .
The original image can be reconstructed from the sub-bands,
So need to perform Sub-band decomposition
For each Q, the operator TQ is defined. Before the Ridgelet Transform .The s f layer contains objects with frequencies near domain.
the ridgelet system. The ridge fragment has an aspect ratio of
2-2s2-s. After the renormalization, it has localized frequency
in band [2s, 2s+1].A ridge fragment needs only a very few
ridgelet coefficients to represent it.
Gabor filters are commonly recognized as one of the best choices for obtaining localized frequency information. They offer the best simultaneous localization of spatial and frequen- cy information. There are two important characteristics of log gabor filter. Firstly, Log-Gabor function has no DC compo- nent, which contributes to improve the contrast ridges and edges of images. Secondly, the transfer function of the Log- Gabor function has an extended tail at the high frequency end, which enables to obtain wide spectral information with local- ized spatial extent and consequently helps to preserve true ridge structures of images[19]. The Gabor filter bank is a well known technique to determine a feature domain for the repre- sentation of an image. However, a Gabor filter can be de- signed for a bandwidth of 1 octave maximum with a small DC component in the filter. A Log-Gabor filter has no DC compo- nent and can be constructed with any arbitrary bandwidth.
IJSER © 2012
International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 3
ISSN 2229-5518
There are two important characteristics in the Log-Gabor filter. Firstly the Log-Gabor filter function always has zero DC com- ponents which contribute to improve the contrast ridges and edges of images. Secondly, the Log-Gabor function has an ex- tended tail at the high frequency end which allows it to en- code images more efficiently than the ordinary Gabor func- tion. To obtain the phase information log Gabor wavelet is used for feature extraction. It has been observed that the log filters can code natural images better than Gabor filters. Statis- tics of natural images indicate the presence of high-frequency components. Since the ordinary Gabor filters under-represent high frequency components, the log filters is a better choice [20]
The methodology of work will start with the overview of image denoising algorithms. The results of various algorithms will be interpreted on the basis of different quality metrics. Thus, the methodology for implementing the objectives can be summarized as follows: -
(a) Study of Image denoising Techniques.
(b) Design and Implementation of Image denoising based up-
on curvelet transformation using log Gabor filter
(c) Check the developed Image Denoising Technique on dif-
ferent CT images like brain, head, kidney, lungs.
(d) Comparison with other state-of-art techniques.
(e) Deducing conclusion
The four CT images (brain, head, kidney, lungs) at differ- ent noise levels have been discussed in the paper. The various parameters collected include the CoC, PSNR and MSE which will give the quality related outcomes of the experiment. Peak Signal-to-Noise Ratio (PSNR) is considered to be the least complex metric, as it defines the image quality degradation as a plain pixel by pixel error power estimate. PSNR is an engi- neering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that af- fects the fidelity of its representation. In order to quantify the achieved performance improvement, this measure was com- puted based on the original and the denoised data. Mean Squared Error for both noisy and de-noised images was iden- tified. The correlation is defined only if both of the standard deviations are finite and both of them are nonzero. It is a cor- ollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in absolute value. The correlation is 1 in the case of an increasing linear relationship and - 1 in the case of a decreasing linear relationship. Its value lies in between in all other cases,indicating the degree of linear dependence be- tween the images. The closer the coefficient is to either -1 or 1, the stronger the correlation between the images.
The Curvelet using log Gabor approach is a powerful method for image denoising. In this research, a very simple and expert image denoising system has been implemented for the grey scale images based upon curvelet using Gabor filter- ing approach.
The first objective was to propose an algorithm which is simple and effective for image denoising based upon curvelet Transformation approach. It is based on log Gabor filter meth- od.
The second objective was to compare the proposed meth- od with existing state-of-art techniques. In this thesis work results of curvelet transformation is compared with log gabor filter. PSNR, CoC, MSE quality metrics have been used for calculating results to compare quantitatively these techniques. Experimental results show that proposed method performs well than the Curvelet method in terms of quality of images. The proposed method increases the quality significantly, while preserving the important details or features. This also gives the better results in terms of visual quality.
For future work, the other stages of curvelet transfor- mation may be improved to find out the better results. This algorithm can be used in other type of images like Remote sensing images, Ultrasound images, SAR images etc. Other quality metrics can be used to judge the performance of this algorithm. And further improvements can also be done in the algorithm to improve the quality. Instead of log gabor ap- proach, algorithm can be modified to improve the quality of the images.
[1] P. Gravel, G. Beaudoin, and J. A. De Guise, “A method for modeling
noise in medical images,” IEEE Trans. Med. Imag.,vol. 23, no. 10, pp.
1221–1232, Oct. 2004.
[2] C. J. Garvey and R. Hanlon, “Computed tomography in clinical prac-
tice,” BMJ, vol. 324, no. 7345, pp. 1077–1080, May 2002.
[3] H. Lu, X. Li, I. T. Hsiao, and Z. Liang, “Analytical noise treatment for low-dose CT projection data by penalized weighted least-squares smoothing in the K-L domain,” Proc. SPIE, Medical Imaging 2002, vol. 4682, pp. 146–152.
IJSER © 2012
International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 4
ISSN 2229-5518
[4] H. Lu, I. T. Hsiao, X. Li, and Z. Liang, “Noise properties of low-dose CT projections and noise treatment by scale transformations,” in Conf. Rec. IEEE Nuclear Science Symposiump., Nov. 2001, vol. 3, pp.
1662–1666.
[5] J.S. Lee, “Refined filtering of image noise using local statistics” Com- put. Graphics Image Process, vol. 15, pp. 380–389, 1981.
[6] D.L. Donoho, “Denoising by soft thresholding,” IEEE Trans. Inform.
Theory, vol. 41, pp. 613–627, 1995.
[7] S. Grace Chang, Bin Yu and M. Vatterel, “Wavelet Thresholding for Multiple Noisy Image Copies” IEEE Transaction. Image Processing, vol. 9, pp.1631- 1635, 2000.
[8] D.L. Donoho and I.M. Johnstone. “Adapting to unknown smoothness via wavelet shrinkage” Journal of American Statistical Association, Vol. 90, no. 432, pp1200-1224,1995.
[9] A. M. Alsaidi, M. T. Ismail, “A comparision of some thresholding selection methods for wavelet regression” World Academy of Sci- ence, Engineering and Technology, Vol. 62,pp 119-125, 2010.
[10] R. R. Coifman and D. L. Donoho, "Translation-Invariant De- Nois- ing," In Wavelets and Statistics, Springer Lecture Notes in Statistics
103. New York: Springer-Verlag, 1994, pp. 125-150.
[11] I.A.Ismail, and T.Nabil, "Applying Wavelet Recursive Translation- Invariant to Window Low-Pass Filtered Images," International Jour- nal of Wavelets, Multiresolution And Information Processing, Vol.2, No.1, 2004 , p.p.99-110.
[12] S. Qin, C. Yang, B. Tang and S. Tan, "The denoise based on transla- tion invariance wavelet transform and its applications," A Confer- ence on Structural Dynamics, Los Angeles, vol. 1, pp.783 - 787, Feb.
2002.
[13] D. L. Donoho, and M. R. Duncan “Digital Curvelet Transform: Strat- egy, Implementation and experiments”Standford University. No- vember 1999.
[14] J. L. Starck, F. Murtagh, E. J. Candes, and D. L. Donoho, “Gray and color image contrast enhancement by the curvelet transform” IEEE Trans. on Image Processing, 12(6):706–717, June 2003.
[15] J. L. Starck, E. J. Candes, and D. L. Donoho, “ The curvelet transform
for image denoising” IEEE Trans. on Image Processing, 11(6):670–
684, June 2002.
[16] E. J. Candes, L. Demanet, D. L. Donoho and L Ying, “Fast Discrete Curvelet Transforms” 2005 http://www.curvelet.org.papers.FDCT.pdf
[17] Herman, G. T., Fundamentals of Computerized tomography: Image
reconstruction from projection, 2nd edition, Springer, 2009
[18] "computed tomography—Definition from the Merriam-Webster
Online Dictionary"
[19] www.csse.uwa.edu.au/~pk/research/matlabfns/.../convexpl.html
[20] Mehrotra, H . Majhi , B and Gupta , P.(2009), “Multi-algorithmic Iris Authentication System”, International Journal of Electrical and Com- puter Engineering,pp 78—82.W.-K. Chen, Linear Networks and Sys-
tems. Belmont, Calif.: Wadsworth, pp. 123-135, 1993. (Book style).
IJSER © 2012