Noise Reduction in MR brain image via various transform domain schemes


Bhawna Goyal1*, Sunil Agrawal1, B.S. Sohi2, Ayush Dogra1

1Department of Electronics and Communications, UIET, Panjab University, Chandigarh

2Vice Chancellor, Chandigarh University, Chandigarh

*Corresponding Author E-mail:



Despite the phenomenal progress in the field of image denoising it continues to be an active area of research and still holds margin in improving the standard of the denoising techniques. Image denoising has emerged as a significant tool in medical imaging specifically. In this article we have compared and evaluated three transform domain techniques on an MRI test image subjectively and objectively. The performance of Curvelet, Shearlet, and Tetrolet transform with a selective thresholding is evaluated. Shearlet is able to yield the best quality of image denoising. The study aims at analysing the performance of transform domain methods on MRI image at low and high levels of noise. 


KEYWORDS: Curvelet, Shearlet, Tetrolet, Magnetic Resonance image, Denoising, thresholding.




As the number of pixels per unit area keeps on increasing, the modern image acquisition devices are becoming increasing sensitive to noise [1]. Therefore there grows a huge dependability on image denoising algorithms to reduce the effect of noise and artefacts in the resultant image. Noise can be perceived as a random fluctuation in the colour information or the intensity value of the image pixels. Image denoising is a well studied field by researchers. A large sum of image denoising techniques has been presented so far [2]. Removal of noise is a fundamental operation in image processing and its applications range from the direct i.e. photographic enhancement to the technical i.e. as a sub problem in image reconstruction algorithms. It serves as a pre processing as well as a post processing technique in image registration [3, 4] and image enhancement. 


The degradation model of an image contaminated with noise can be described as:


where h and f are column vectors and represent the true and noisy signal respectively and n is the added noise[5].


During image acquisition two predominant sources for noise are the detectors which posses the stochastic nature of photon count and electronic fluctuations of the electronic devices. When there is enough illumination which is usually the case the second source of noise gives rise to Additive White Gaussian Noise (AWGN). These leads to the assumption of Gaussian noise models as a common practise. Under the conditions with lower power light source, short exposure time and phototoxity the source of noise is signal dependant and it forms the reasonable basis to model the output of the detectors as a Poisson distributed random vector. Other commonly known noise is impulse noise [6].


Image denoising finds application in remote sensing, surveillance, medical imaging etc. Image denoising can be considered as an area of vital and critical importance in medical imaging as noise can be intercepted as an image deformity and it can lead to false diagnosis. The basic fundamental of image denoising is to reconstruct a plausible estimate of the original image from the distorted image and maintain a trade off between removal of noise and feature preservation. The image denoising techniques have been forwarding parallel in three domains i.e. spatial filters, transform domain methods and dictionary learning methods.


The underlying property of sparse representation in image transforms i.e. an image can be represented as a sum of few high value coefficients makes transform domain methods like Directionlet, Shearlet, Curvelet, Tetrolet, Ridgelet, Ripplet ,Wedgelet very popular for image denoising [2, 7, 8, 9]. In this article three popular transform domain methods i.e. Curvelet, Shearlet and Tetrolet with a selected thresholding have been implemented and contrast is withdrawn amongst them. The transforms are modified Wavelet systems with more directional sensitivity. Shearlet was able to overcome the limitations possessed by Curvelet transform i.e. not to resolve 2D singularities. Some of the other image transforms like Bandlets, Wedgelets, Grouplets and Tetrolet were coined to work on image structures based on averaging in the adaptive data neighbourhood point. In this category Tetrolet is the most recent transform is the most recent transform and has been discussed in context of denoising in this article.


This paper is organised as follows: Section 2 discusses the basic and related work in the field of denoising. Section 3 gives a brief insight into the Curvelet, Shearlet and Tetrolet transform. Section 4 discusses the experimental setup and objective evaluation metrics. Results and discussions are presented in section 5. Section 6 finally concludes the article and gives the future scope.


Related work:

There exists a plethora of denoising methods originating from various great disciplines such as partial differential equations, spectral and multiresolution analysis probability theory and statistics etc [10]. All the image denoising methods rely on some implicit and explicit assumptions in order to separate the true image from the noise. In the field of denoising one can exploit the spatial domain filters, can work with filters in frequency domain and can also rely on dictionary learning methods. Spatial domain methods include local and non local filters which exploit the similarities between either pixels or patches in an image [11].


Some of the local filters include Gaussian filter, Weiner filter trained filter, bilateral filter, steering kernel regression method (SKR), MSKR, KSPR [12, 13, 14]. A filter is said to be local if it exploits the similarity between the pixels restricted by spatial distance. On the other the non local filters exploit the similarity between the pixels in the entire image and unrestricted by the spatial distance. However in the presence of high noise, the similarity between pixels adjacent to each other gets corrupted due to which these techniques are not able to perform well. Gaussian filter was designed to provide image denoising. Since then its several improvisations have been proposed to provide better edges preservation. Anisotropic filter removes the blurring effect caused by the Gaussian filter. Total variation minimization technique was designed to denoise the homogenous regions in the image but the edges. Then the SUSAN filter was proposed for further better edge preservation. This filter can average the pixels in the neighbourhood those are same spatial distance as the centre of the pixel. Apart from these parametric methods SKR exploits the kernel regression fact to denoise an image and remove artifacts. The motivation behind SKR is pixels on the same side of the edge have a higher impact on the filtering of the pixel then those on the other side of the edge. Trained filter: the basis of this filter is a non parametric process in which weight shave been obtained by the offline training of the large set of images [2].


In non local filters the weights are only marked by the similarity. Since the proposal of the basic non local filter many further improvements have been presented based on acceleration in the implementation of the technique and improvement in qualitative and quantitative results. An effort was made by M.Mahamoudi [15] by computing mean and gradient values to fasten the non local manner. Further SVD and local variance were introduced to eradicate dissimilar pixels. Also when FFT was used to calculate weights it made the algorithm 50 times faster than the innate one [16]. Now taking into consideration the improvement of qualitative and quantitative results Seven et al. proposed to increase the number of weight values for the target patches by designing a rotationally invariant block matching for non local filters. NLM was the first filter designed taking into consideration the similarity of the entire image. The basic idea behind non local filters can be mapped as the modification of the bilateral filter by calculating the Euclidean distance between two patches instead of two pixels. In 2008 Goosen et al. proposed improved non local filters in various aspects like the variance in noise is recorded at every location and the extra noise was removed as a post processing technique. The INLM can be used to denoise colour images as well. The major drawback associated with this method it tends to blur the details due to post processing involved, though it removes the extra noise[2,10].


Image transforms consider transforming the images into other domains in which the similarities of the transformed coefficients are employed. These methods are particularly based on the assumption that the true signal can be well approximated by a linear combination of few basis elements. Hence the basic fundamental is to preserve few high magnitude transform coefficients that contain the true signal intensity and thresholding rest of the coefficients that contain noise so as to calculate the total estimated signal. Transform domain has been researched in context of image denoising for decades. A large number of variants have been proposed in this category like DCT, Wavelets, BM3D ,Directionlet Curvelet, Coutourlet, Ridgelet, Tetrolet, Shearlet and corresponding thresholding techniques like sure shrink Visu shrink and Bayes shrink  [7,17]. Wavelet is a multi resolution analysis tool. As far as the coining of wavelet transform is concerned there are primitively fifteen variants of wavelet namely Haar wavelet, called the mother wavelet, Symlet, Mexican Hat Wavelet, Morlet, Mayerlet, Daubechies, Symlet, Coiefflet, Spline Wavelet, Battle Lemarie Wavelet, DWT, CWT,  Gaussian Wavelet [18]. It uses multiscale pyramid representation method. During wavelet thresholding images are decomposed into low frequency and high frequency subbands. The low frequency subbands are called the approximation level and high frequency subbands are called detailed levels. The subbands are processed by hard and soft thresholding. Visu-shrink, Sure-shrink And Bayes Shrink are the popularly known thresholding formulas. FT and WT are better to represent 1-d signals but when it comes to 2-d signals edges and contours are the main concerns to be addressed in an image processing task. Since wavelet was a tensor product of 1D wavelet transform which is able resolve 1D singularity but not 2D. In the beginning to overcome the problems possessed by wavelet transform Ridgelet transform was proposed. Then in progression came Curvelet and Ripplet transform. Ripplet transform another MGA tool is a higher order generalization of the parabolic scale law. Ridgelet resolves the discontinuities along lines where as Counterlet and Ripplet is confined to resolve discontinuities along smooth curves. But these transforms are incapable to resolve discontinuities along edges and contours. There for the rescue Vladan proposed a multidirectional anisotropic transform based on integer lattices named Direction-let. This is the most promising technique of image processing manuover[7]. PURE-let[1,5] (poison unbiased risk estimate linear expansion of thresholds) is a denoising function for shot noise. It was devised to do two functions one is signal preservation and the nosie suppression in the presence of shot noise. PURE denoising function uses unnormalized haar wavelet transform. The advantages of PURE-let are lesser computations, less memory and less complexity. The main idea behind dictionary learning methods is sparse representation. They perform denoising by learning a large number of group of pixels blocks from an image database, such that each block can be expressed as a linear combination of the few patches from the dictionary so obtained. K-SVD is the well known dictionary denoising method. In contrast in another method namely K-LLD classification is done on the basis of SKR in order to the disguise of different patches as similar patches. The other state of art dictionary method is WESNR; it targets the mixed noise that is the Gaussian and the impulse noise [2,7,10]


Image Transforms

(a). Curvelet Transform:

To overcome the limitations of Wavelet transform, the Ridgelet and Curvelet transforms were developed with an attempt to put a  halt on the inherent limit of wavelet denoising of images. Ridgelet transform was based on radon transform and was able to provide information regarding orientation of linear edges. However Ridgelet was not able to resolve the discontinuities in the two dimensions. So in Candes and Donoho proposed the second generation Curvelet Transform [19]. Curvelet transform with a purpose to achieve anisotropic directionality uses a parabolic scaling law. Curvelet is also a special case of Ripplet Type 1 Transform with support c=1 and degree d=2. Combining multiscale Ridgelets with a spatial bandpass filtering leads to the formation of Curvelet transform with a purpose to isolate different scales. Curvelets are able to work at all scales, locations and orientations [20]. The mathematical background has been developed by Candes in [19] clearly. Curvelet decomposition consists of four steps: subband filtering that is the object under consideration is decomposed into subbands. Then each subband is smoothly windowed into squares of an approximate scale. Each resulting square is normalized to unit scale. Then finally hard thresholding rule is followed for estimating the unknown Curvelet coefficients. According to [21] Curvelet reconstructions are able to display a higher sensitivity then the wavelet based reconstruction also all the structures which are visually detectable in the noisy image were clearly seen in the Curvelet reconstructed image.


Shearlet Transform:

Multivariate functions like images that are governed typically by anisotropic features such as edges are not optimally presented by classical wavelets. Since then various directional representations systems have been proposed like Curvelets, Contourlets and Shearlets [22]. However only Shearlet systems were able to provide a unified treatment for the continuum and digital world along with giving optimally sparse approximations. Shearlet lead to improvements in many image processing applications such as denoising, image enhancement [23, 24] and image fusion. The discrete wavelet transform is a special type of composite wavelet transforms [25]. There are basically two types of Shearlet systems i.e. band limited Shearlet and compactly supported Shearlet transform [22,25]. The representation of the discrete Shearlet transform can be summarized as a cascade of three steps: the first step includes transformation to classical Fourier transform and changing of variables to pseudo polar coordinates which is followed by weighting by a radical density compensation factor. Then it is decomposed into rectangular tiles and inverse transform is applied to each of tiles after using the Bayes thresholding method. The main reason Shearlet is able to perform well then Curvelet and Counterlet is the fact that the Shearlets are affine systems that leads to parameterization of directions by slopes instead of angles which helps in digital setting along extensive theoretical framework. The mathematical equations are fully explained by Gitta. K  in [22,25].


Tetrolet Transform:

the construction of Tetrolet transform is similar to the idea of digital Wedgelets [26] which works on image structures. Tetrolet is based on adaptive Haar wavelet transform. Tetrolet was designed visualising the non redundancy of the basis functions for sparse image approximation. Tetrolets have four equal sized squares which are called tetrimonies. The algorithm of Tetrolet transform includes the division of the image into 4*4 blocks. Then out of them the sparsest Tetrolet representation is chosen. Then the rearranging of the high and the low pass coefficients of each block into 2*2 blocks the various coefficients are finally stored. The output image obtained is a decomposed image and this is an adaptive decomposition algorithm. There are a number variants of Tetrolet transform available namely Tetrolet 16, Tetrolet Rel, Tetrolet Edge and Tetrolet 16 Rel Edge in context of denoising along with Tensor Biorthogonal 9/7 wavelet filter bank. In this article we have worked with basic form of Tetrolet Transform.


Objective Evaluation Metrics

The objective evaluation metrics calculated for the quality of results so obtained in the context of this article are PSNR (peak signal to noise ratio) and MSE (mean square error). PSNR is measured in decibels. It is used as an objective image quality metrics between the source image and the denoised image. Higher is the PSNR value better is the quality of the denoised image so obtained. MSE is the difference or cumulative error between the original and the denoised image. Lower is the calculated MSE value better is the result of the denoised image [27].


(b) Experimental setup

A MRI was taken as the input image from the Google database. Then the Gaussian noise was added to this image varying the standard deviation σ progressively for the amount of noise added ranging as 5, 10, 20, 25 and 40. Different denoising schemes i.e. Curvelet, Shearlet and Tetrolet are used to denoise the image while ranging the noise distribution from minimum to maximum values. The entire analysis has been implemented using MATLAB software.


The all three algorithms have been tested from low noise values i.e. σ = 5 and 10   to high range noise values i.e. σ = 20, 25 and 40. The flow chart for the implemented methodology is given in Fig.1


(b) Experimental setup

A MRI was taken as the input image from the Google database. Then the Gaussian noise was added to this image varying the standard deviation σ progressively for the amount of noise added ranging as 5, 10, 20, 25 and 40. Different denoising schemes i.e. Curvelet, Shearlet and Tetrolet are used to denoise the image while ranging the noise distribution from minimum to maximum values. The entire analysis has been implemented using MATLAB software.

The all three algorithms have been tested from low noise values i.e. σ = 5 and 10   to high range noise values i.e. σ = 20, 25 and 40. The flow chart for the implemented methodology is given in Fig.1


The basics steps in image denoising (as shown in Fig. 1)


·        Pre-process the MRI test image

·        Add Gaussian noise standard deviation at σ= 5, 10, 20, 25 and 40.

·        The image is transformed using Curvelet, Shearlet and Tetrolet representation

·        Apply various thresholding schemes to modify the images coefficients (i.e., Soft/hard/ Bayes Shrink thresholding).

·        Apply the inverse image transform is applied to reconstruct the synthesised image

·        Denoised image is obtained at the output.




To evaluate the performance of the state of art denoising techniques they are applied to the synthetic data i.e. a MRI image shown in Fig2. A comparison is made amongst Curvelet, Shearlet and Tetrolet Transform as how well does the decomposition of a noisy image into the respective transform coefficients can be utilized with a type of thresholding to estimate a sum of linear threshold to obtain a denoised image. The MRI image added with Gaussian noise with standard deviation σ = 5, 10, 20, 25 and 40 are shown in Fig 3(a-e) respectively. The denoised image using Curvelet transform and soft thresholding are shown in Fig 4 (a-e) with standard deviation σ = 5, 10, 20, 25 and 40 respectively. The noise free estimated inverse transformed image coefficients by using Shearlet transform with Byes shrink are shown in Fig 5 (a-e) with intensity of uniformly distributed Gaussian noise ranging as σ = 5, 10, 20, 25 and 40. The varying the similar parameters the results for Tetrolet transform are shown in Fig6 (a-e). The calculated PSNR and MSE values between the input image and denoised image are listed in Table 1.


Table1: PSNR and MSE values with three schemes


deviation σ=

Curvelet  transform

Shearlet transform

Tetrolet transform













































As the value of noise level increases from σ= 5, 10, 20, 25 and 40 the PSNR values keep on decreasing while the MSE value keeps on increasing after employing all three denoising techniques. Higher the noise level in the image lower will be the restoration ability of the denoising technique. At high noise levels the techniques tends to smoothen away the fine details along with the elimination of noise, there is deterioration in the image details. In all cases Shearlet transform outperforms Curvelet and Tetrolet transform in terms of PSNR and MSE- at least for moderate and large values of the noise levels.


In addition the Shearlet transform output images are visually more pleasant. From the visual inspection and as well as from table1 it can be seen that Tetrolet is able to work at moderate noise levels, whereas its performance degrades at high noise levels. As far as Curvelet denosing scheme is concerned it yields better results than Tetrolet based denoising scheme. This fact can be clearly validated from the subjective and objective (Table1) evaluation.  So it can be stated that Shearlet transform is able to work well on denoising a MRI image in terms of PSNR and MSE on moderate and high levels of noise.



The two directionally sensitive image transforms that is Curvelet and Shearlet transform are able to work well on the denoising of the MRI image as well preservation of the fine details whereas Tetrolet transform which works on image structures is not able to protect the edges and fine details in the image at high noise levels. Shearlet so far is the best denoising technique within the scope of this article. However further improvisations can be tried by using other types of thresholding schemes in combination with other state of art image transforms.



1.       Chatterjee, Priyam, and Peyman Milanfar. "Is denoising dead?” IEEE Transactions on Image Processing 19.4 (2010): 895-911

2.       Shao, Ling, et al. "From heuristic optimization to dictionary learning: a review and comprehensive comparison of image denoising algorithms." IEEE Transactions on Cybernetics 44.7 (2014): 1001-1013.

3.       Dogra, A., and M. S. Patterh. "CT and MRI Brain Images Registration for Clinical Applications." J Cancer Sci Ther 6 (2014): 018-026.

4.       Dogra, Ayush, and Parvinder Bhalla. "CT and MRI Brain Images Matching Using Ridgeness Correlation." Biomedical and Pharmacology Journal, Vol 7., Issue 2, 20

5.       Talebi, Hossein, and Peyman Milanfar. "Global image denoising." IEEE Transactions on Image Processing 23.2 (2014): 755-768.

6.       Luisier. F., Thierry Blu, and Michael Unser. "Image denoising in mixed Poisson–Gaussian noise." IEEE Transactions on Image Processing 20.3 (2011): 696-708.

7.       Xu, Jun, Lei Yang, and Dapeng Wu. "Ripplet: A new transform for image processing." Journal of Visual Communication and Image Representation21.7 (2010): 627-639.

8.       Sethunadh, R., and Tessamma Thomas. "Spatially adaptive image denoising using inter-scale dependence in directionlet domain." IET Image Processing9.2 (2015): 142-152.

9.       Dogra, A., and Sunil Agrawal.”Efficient Image Representation Based on Ripplet Transform and Pure-Let.”Int. J.Pharm. Sci. Rev. Res. , 34(2),September – October 2015: Article No. 16, Pages :93-97

10.     Buades, Antoni, Bartomeu Coll, and Jean-Michel Morel. "A review of image denoising algorithms, with a new one." Multiscale Modeling & Simulation 4.2 (2005): 490-530.

11.      X. Li, Y. Hu, X. Gao, D. Tao, and B. Ning, “A multi-frame image superresolution method,” Signal Process., vol. 90, no. 2, pp. 405–414, Feb.2010.

12.     N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationar Time Series. New York, NY, USA: Wiley, 1949.

13.     B. Widrow and S. Haykin, Least-Mean-Square Adaptive Filters. New York, NY, USA: Wiley-IEEE, 2003

14.     C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” in Proc.6th Int. Conf. Comput. Vision, 1998, pp. 839–846.

15.     M. Mahmoudi and G. Sapiro, “Fast image and video denoising via nonlocal means of similar neighborhoods,” IEEE Signal Process. Lett., vol. 12, no. 12, pp. 839–842, Dec. 2005.

16.     J. Wang, Y. Guo, Y. Ying, Y. Liu, and Q. Peng, “Fast nonlocal algorithmfor image denoising,” in Proc. IEEE Int. Conf. Image Process., 2006,pp. 1429–1432.

17.     Kumar. Manoj, Manoj. Diwakar, “ CT image denoising using locally adaptive shrinkage  rule in tetrolet domain., “ Journal of King Saud University Computer and Information Sciences, 2016.

18.     Lin, Pao-Yen. "An introduction to wavelet transform." Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC(2007).

19.     Starck, Jean-Luc, Emmanuel J. Candès, and David L. Donoho. "The curvelet transform for image denoising." IEEE Transactions on image processing 11.6 (2002): 670-684.

20.     Candes, Emmanuel, and Laurent Demanet. "Curvelets and Fourier integral operators." Comptes Rendus Mathematique 336.5 (2003): 395-398

21.     Candes, Emmanuel J., and David L. Donoho. "Continuous curvelet transform: II. Discretization and frames." Applied and Computational Harmonic Analysis19.2 (2005): 198-222.

22.     Karami, Azam, Rob Heylen, and Paul Scheunders. "Band-Specific Shearlet-Based Hyperspectral Image Noise Reduction." IEEE Transactions on Geoscience and Remote Sensing 53.9 (2015): 5054-5066.

23.     Dogra, A., and Sunil Agrawal.” 3-Stage Enhancement Of Medical Images Using Ripplet Transform,high Pass Filters and Histogram Equalization Techniques.”International Journal Of Pharmacy And Technology,Dec  2015 | Vol. 7 | Issue No.3 |9748-9763

24.     Dogra, A., and Sunil Agrawal.” Efficient representation of texture details in medical images by fusion of  Ripplet and DDCT transformed images”.” tropical journal of pharmaceutical research”(accepted for publication).”

25.     W. Q. Lim, “The discrete shearlets transform: A new directional transform compactly supported shearlets frames,” IEEE Trans. Image Process., vol. 19, no. 5, pp. 1166–1180, May 2010

26.     Krommweh, Jens. "Tetrolet transform: A new adaptive Haar wavelet algorithm for sparse image representation." Journal of Visual Communication and Image Representation 21.4 (2010): 364-374

27.     Dogra, Ayush, and Parvinder Bhalla. "Image Sharpening By Gaussian And Butterworth High Pass Filter." Biomedical and Pharmacology Journal.





Received on 26.06.2016          Modified on 06.07.2016

Accepted on 16.07.2016        © RJPT All right reserved

Research J. Pharm. and Tech. 2016; 9(7):919-924.

DOI: 10.5958/0974-360X.2016.00176.1