Performance Comparison of Tetrolet Transform and Wavelet-Based Transforms for Medical Image Denoising

Abstract: Noise reduces the quality of medical images and raise the difficulties of diagnosis. Although the wavelet transform has already been used in medical noise removal applications extensively, there are many other multi-resolution analysis methods proposed in recent years for denoising. The main goal of this study is comparing the image denoising abilities of some of these methods with wavelet transform. In this paper, image denoising is implemented by a three-stage methodology. Effectiveness of the multiresolution analysis methodologies has been investigated for standard test images beside magnetic resonans, mammography and fundus images. Performances of the transforms are compared by using peak signal to noise ratio, mean square error, mean structural similarity index and feature similarity index. The best results are obtained by tetrolet transform for random and rician noises with the benchmark images. Medical image denoising performance of Tetrolet transform is compared to other multi resolution analysis methods for the first time in the literature with this study. It surpassed ridgelet and haar wavelet transforms while the noise ratio was low. On the other hand, it is seen that curvelet transforms are effectively produce the best results for all rates of noise on medical images.


Introduction
The main argument of image processing, Fourier analysis, is inadequate to solve denoising problems because it is not able to determine the time of frequency changes. Wavelet transform (WT) was proposed to overcome this problem [1,2]. Although it is a popular method for processing medical images [3][4][5][6], it suffers from limited direction. Ridgelet transform (RT) uses angular windows instead of directional implementation of WT [7]. Although RT was used for denoising [8,9], it is not good enough for denoising medical images [10,11]. Curvelet transform (CT) that uses curves instead of lines for windowing was improved [12][13][14]. Since CT is efficiently detects edge information of images, it is commonly used for denoising [15][16][17]. Ma and Plonka [18], presented a review on CT including its evolution history, theory and its correlation to other multiresolution methods. CT is implemented on a variety of medical images for segmenting and denoising and had efficient results [19,20]. Raju and Kumar [21] also compared denoising performances of dual tree complex WT and CT on medical images and CT had better results. Tetrolet transform (TT) was came out as a new generation multiresolution method [22]. In this local analysis method, image regions are rearranged according to five different shapes called tetrominoes and haar WT is implemented on these regions. WT and TT are compared for denoising of benchmark images and realized that TT is superior than WT [23,24]. Ceylan and Öztürk [25], studied on optimizing the tetrominoe combinations in denoising algortihms and shortened the process time. There is no study which compares the denoising performance of TT with the other transforms (RT and CT) in the literature.
Although the most frequent noise type of medical images is gaussian [26], MR images follow a rician distribution [27][28]. So, recent studies about medical image denoising focus on removing gaussian and rician noise [29][30]. In this paper, we perform denoising on medical and standard test images. We removed the artificially implemented noise by using different multi-resolution analysis (MRA) methods and evaluated and compared the performance of each method.Thisis the first paper introducing Tetrolet transform into comparative medical image denoising systems.

Materials and Methods
MRA techniques in this study, are given in Section 2.1. Noise addition and thresholding methods are included in Section 2.2. Performance evaluation methods are given in Section 2.3.

Wavelet Transform (WT)
WT is the basis form of the MRA and has been implemented for many years. Generally, daubechies wavelets (especially db4) are preferred for denoising algorithms in literature [23,31] We use db4 and also haar wavelet to the images to decide whether TT, as a haar WT based method, surpasses haar WT or not. WT coefficient matrix of an image is computed as implementing discrete form of WT to rows and columns as seen in Fig. 1(a). At the end of the WT decomposition level, an approximation and three detail components are obtained and located in the coefficient matrix as seen in Fig. 1(b).

Tetrolet Transform (TT)
Tetrolet transform (TT) is an effective, local and scalable algorithm. In order to implement TT, image matrix is first separated into blocks each of which include 4x4 pixels. If the image size cannot be divided by four, zero-padding should be applied. There are five shapes called tetrominoes which of each contain four pixels, are used for TT ( Fig. 2. (a)). Four of them are chosen and the selected tetrominoes should be located into a 4x4 sized pixel blocks. 117 different combinations are possible and one of them is given in Fig. 2. (b). After locating the tetrominoes, pixels of the 4x4 sized blocks are rearranged according to TT rules with the selected combination of tetrominoes and haar WT is applied on the reformed part of the image locally. This procedure is applied on whole 4x4 blocks in the image. At the end of the decomposition level, a coefficient matrix whose size is the same as the original image is obtained (Fig. 3).

Ridgelet Transform (RT)
Poor directionality of WT is overcame with RT by using an angle parameter. This method is successful at catching one-dimensional (1D) line singularities in multi-dimensional signals like images. RT is based on radon transform (RAT). So, it is possible to comprehend RT by understanding RAT. Mathematical background of RAT is given in [9,13]. The finite RAT of an image ( , ) x i j is given as follows where , k l L represents the lines and 0 y is the local mean: (1) Finite RT is defined by performing 1D WT to matrix [ ] k r l , column by column [8]. As another way, RT can be achieved in Fourier domain by using 2D fast Fourier transform (FFT) [11]. Initially 2D FFT is applied, after that Fourier coefficients are interpolated along lines which pass through the frequency space and 1D reverse FFT is calculated. Finally, 1D WT is applied and RT coefficients matrix of the image is obtained (Fig. 4).

Curvelet Transform (CT)
Curvelet transform (CT) that combines MRA with parabolic scaling is enhanced by improving RT. It has the ability of catching quadratic curves on images. In order to apply CT, firstly the image is decomposed into different scaled subbands. Then, each subband is windowed into squares and each square is renormalized [15]. Lastly, discrete RT is applied on each square (Fig. 5). There are two generations of CT. First one is efficiently used on images for denoising but its process takes a long time compared to second generation [14]. First generation CT (FGCT) based on RT is limited in digital application because of the geometry of ridgelets [18]. On the other hand, second generation CT (SGCT-fast ∑ discrete CT) uses unequally-spaced FFT and wrapping (Fig. 6). Thus, fast discrete CT can be implemented faster, less redundantly and simpler than FGCT [14].

Noise Addition and Thresholding Methods
In this study, random, gaussian and rician noise are applied on all the medical and test images. We applied random noise ( ) r N to the images with Eq. (2). Here, sigma is the noise ratio which is an important factor to specify the amount of noise. * n r X X sigma N = + The gaussian distribution of a random variable x , could be defined as follows [30]: Additionally, if m is the pixel intensity of the noisy image and s is the pixel intensity of noiseless image, while 0 Γ represents the modified order of the first kind of Bessel function, the rician probability distribution for m is given by [28,30] (3) In our study, we used three sigma values for random noise and three signal to noise ratios (snr) for gaussian and rician noise. The main difference between random, rician and gaussian noise, is that gaussian noise is inversely proportional with snr value. So, if snr increases, noise ratio decreases for gaussian. A thresholding process is implemented on detail coefficients obtained by transforms. We used universal and hard thresholding methods. The threshold value T in universal thresholding is given by: (4) where M is the size of the image and 2 σ is the noise variance. In hard thresholding, if absolute value of a coefficient is greater than the specified threshold value, it remains unchanged. However, if the value is less than the threshold, it is supposed to be zero: where X represents the calculated coefficients, Y states the noisy coefficients and T is the threshold.

Performance Measurement
Each implemented process causes corruptions on images. The purpose of denoising algorithms is reconstructing images with minimum decay. In order to compare performances of the MRA methods, we have calculated PSNR, MSE, MSSIM and FSIM values for each image.

Mean Square Error (MSE)
The square of the numeric difference between two images is calculated. Lower MSE means cleaner image. While R and C state number of rows and columns of images x and y , MSE is denoted as:

Peak Signal to Noise Ratio (PSNR)
It is an objective performance test which ensures to score lost information in dB. Higher PSNR means less decay. Considering x is the original image and y is the obtained image, PSNR is denoted as: 2 10 ( , ) 10.log (255 / ( , )) PSNR x y MSE x y = (7)

Mean Structural Similarity Index (MSSIM)
Recently, MSSIM has begun to be used extensively in image/video processing applications through its simple process as an alternative of MSE [32]. SSIM combines three components called luminance distortion term ( , ) l x y , contrast distortion term ( , ) c x y and correlation term ( , ) s x y [33]. SSIM of the two images are calculated as given in Eq. (9), where and are the similarity maps. It can take values between [-1, 1] and if ( , ) 1 SSIM x y = , it means the given images are the same. It is applied on local regions using a sliding window [34]. This procedure is implemented from top-left to bottom-right corner of the images till all image is operated. MSSIM is denoted as Eq. (10) while is the number of windows:

Feature Similarity Index (FSIM)
Another novel performance evaluation method is FSIM. It is calculated by using phase congruence ( ( )) m PC x and a specified similarity measure ( ( )) L S x while Ω states the whole image (Eq. (11)). Mathematical background could be find in [35]. (10)

Experimental Results
In this study, we use five test images and ninety biomedical images which of size is 256x256. Firstly, they are corrupted by noise artificially. Then coefficient matrices are obtained for each image, separately. A thresholding procedure is applied on high frequency components and then images are reconstructed. Finally, the reconstructed and the original images are compared by the performance evaluation criteria (Fig. 7). Singh [23] came up with the idea of decreasing numbers of tetrominoe orders used for denoising could be shorten the operation time of TT and proposed to use 65 combinations of tetrominoes instead of 117. We also research whether 65 is an optimal number or not for our images. TT is implemented to the noisy images by using an increasing number of tetrominoe combinations from 1 to 117 for each noise ratio and PSNR values are calculated. It is seen that using all combinations is unnecessary, because the PSNR takes a constant value when number of combinations increases as similar to [25]. So, optimal numbers of combinations are specified for different noise rates and given in Table 1.  Optimum decomposition level is determined as two for WT and one for TT. Optimal scale number is specified as four for both RT and CT. Sample denoised images for different kinds of noise, are given in Fig. 8.

(a)-(d).
Performance test results of benchmark images are given in Table  2. The best results are in bold. Three different image sets are used to have an objective decision about medical image denoising performances of the MRA methods. Liver MR images were taken from Selcuk University Faculty of Medicine (Department of Radiology), mammography images were taken from MIAS database [36] and fundus images were taken from DRIVE database [37].The evaluation criteria are calculated for all noisy medical images to compare the performances of the methods. The averaged results of the image sets are given in Table 3-5 for each noise ratio, separately. The best results are in bold.

Conclusion
In this study, denoising ability of four MRA techniques are compared. Although the PSNR values are so close to each other for snr=3 for Gaussian noise in Table 2, it is quite obvious that FGCT and SGCT visually outperformed the other transforms in Fig. 5.(a). It means traditional procedures are not adequate enough for evaluation. Results are corroborated with new-generation criteria (MSSIM, FSIM). On the other hand, TT has overwhelmingly better results than the other transforms for random and rician noisy test images. So, it is clear that TT and CT are suitable methods for denoising studies of benchmark images. The cleanest results are obtained by FGCT for medical images, visually ( Fig.5.(b-d)). Moreover, the almost all best quantita-tive results are also achieved with FGCT. So, the best method for denoising the medical images is FGCT. On the other hand, TT surpasses RT, Haar WT and SGCT for almost all rates of rician noise for both liver MR and fundus images ( Table 3, 5). In addition to that, TT gets better results than db4 WT and FGCT as long as the noise rates of liver MR images reduce. Considering that MR images follow a rician distribution noise, TT is more efficient than the others. Lastly, although SGCT has a short process time, FGCT achieves better results than it for almost all types of noise and images.
TT is applied on biomedical images for the first time in literature and it can be an alternative to haar considering the results. Although the performance of TT decreases in high noise rates and takes much more time to perform, generally it has the drop on haar WT. As a future study, different thresholding methods can be applied to improve the performances of the transforms. TT and FGCT have the longest-running processes, so a time-comparative study between these transforms could be realized.