Non Parametric Denoising Methods Based on Wavelets: Application to Electron Microscopy Images in Low Exposure Time Sid Ahmed Soumia1, a) Zoubeida Messali1, 2, b) Abdeldjalil Ouahabi 3, c) Sylvain Trépout 4, 5, d) Cedric Messaoudi4, 5, d) Sergio Marco4, 5, d) 1 Science and Technology Faculty, El Bachir El Ibrahimi University, Bordj Bou Arreridj 2 Laboratory of Electrical Engineering (LGE), University of M'sila 3 Polytechnic School University of Tours (EPU - Polytech Tours), EPU - Energy and Electronics Department 4 Curie Institute, University Campus Orsay 91405 Orsay Cedex, France 5 INSERM U759, University Campus Orsay, 91405 Orsay Cedex, France a) samasoumia@hotmail.fr b) messalizoubeida@yahoo.fr c) abdeldjalil.ouahabi@univ-tours.fr d) {cedric.messaoudi, sylvain.trepout, sergio.marco}@curie.fr Abstract. The 3D reconstruction of the Cryo-Transmission Electron Microscopy (Cryo-TEM) and Energy Filtering TEM images (EFTEM) hampered by the noisy nature of these images, so that their alignment becomes so difficult. This noise refers to the collision between the frozen hydrated biological samples and the electrons beam, where the specimen is exposed to the radiation with a high exposure time. This sensitivity to the electrons beam led specialists to obtain the specimen projection images at very low exposure time, which resulting the emergence of a new problem, an extremely low signal-to-noise ratio (SNR). This paper investigates the problem of TEM images denoising when they are acquired at very low exposure time. So, our main objective is to enhance the quality of TEM images to improve the alignment process which will in turn improve the three dimensional tomography reconstructions. We have done multiple tests on special TEM images acquired at different exposure time 0.5s, 0.2s, 0.1s and 1s (i.e. with different values of SNR)) and equipped by Golding beads for helping us in the assessment step. We herein, propose a structure to combine multiple noisy copies of the TEM images. The structure is based on four different denoising methods, to combine the multiple noisy TEM images copies. Namely, the four different methods are Soft, the Hard as Wavelet-Thresholding methods, Bilateral Filter as a non-linear technique able to maintain the edges neatly, and the Bayesian approach in the wavelet domain, in which context modeling is used to estimate the parameter for each coefficient. To ensure getting a high signal- to -noise ratio, we have guaranteed that we are using the appropriate wavelet family at the appropriate level. So we have chosen “sym8” wavelet at level 3 as the most appropriate parameter with what is required. Whereas, for the bilateral filtering many tests are done in order to determine the proper filter parameters represented by the size of the filter, the range parameter and the spatial parameter respectively. The experiments reported in this paper demonstrate the best performance of the Bilateral Filtering and the Bayesian approaches in terms of improving the SNRout and the image quality, so that there was no big change in the golden beads diameter compared to the thresholding methods where the soft method was the best choice. Taken together, these results suggest that the Bayesian process has a potential to outperform all previous methods, where in the multiple noisy copies structure it gave us the best SNRout without change the golden beads diameter. The Bayesian approach gives us an enhanced average image without needing a huge amount of copies. The upshot of these tests revealed the importance of the Bayesian denoiser in attending the subjects we were asked about. Introduction Transmission electron microscopy (TEM) is a microscopy technique able of imaging biological samples at high resolutions in biological sciences. TEM’s technique is based on shedding an electrons beam at ultra-thin specimen, this latter interacts with the electrons beam. This interaction between the atoms of the specimen and the electron beam causes a deviation of the beams from their initial trajectory. This technique permits TEM's to be used to see molecular structure of proteins and large molecules. Cryo electron microscopy involves viewing unaltered macromolecular assemblies by vitrifying them, placing them on a grid and obtaining images by the electrons transmitted through the specimen [1]. However, there is a drawback in TEM technique if the specimen is biological, where the electrons beam, may be causes damages in the specimen because of the sensitivity of the biological samples to this radiation. This leads the biologist to image the specimen at very low electrons doses, which creates a new problem, the noise in the obtained images. This hinders the Alignment process during the 3D reconstruction of the TEM images, which requires finding a way to reduce as much as possible this noise in order to get a good 3D image quality. For this, there are many scientists interesting in creating new methods to minimize the noise in its different form, so several methods now exist to eliminate the noise in images. Some of the denoised methods succeed in eliminate the noise but damaged the image by blurring the edges as the Gaussian filtering techniques does. In contrast some methods are capable to preserve the image edges like the Bilateral Filtering which is based on both spatial and intensity distances. The effectiveness of the method refers to the capacity of distinguishing between the information and the noise, which is an important point in giving a good quality images after the denoising. In the wavelet domain, Donoho and Johnston proposed the famous wavelet thresholding methods, which widely applied in the signal denoising [2], [3], specifically, the Soft and the Hard thresholding methods. These methods based on choosing a thresholding value, usually calculated from the details coefficients to maintain the approximations values, and then applied it to separate the significant coefficients from the noise coefficients. There are also a nonparametric methods used in image denoising, such as nonparametric Bayesian estimator in the wavelet domains [4], [5], where a prior statistical model based on the -stable densities used to detect the sparseness of the wavelet detail coefficients. In this paper, we have proposed to solve the problem of the noise in the TEM images four methods combined each one with averaging operator in multiple noisy copies. The main objective is to enhance the 3D quality after the TET (Transmission electron tomography) by using the denoised projections. We have tested the proposed structures on a set of experimental TEM test images acquired with different exposure time and organized in zones, each zone differ from the other in the acquisition order of the images. We showed the effectiveness of the Bayesian estimator beside the wavelet thresholding and the bilateral filtering algorithms in removing the noise and enhancing the SNR of these images, so improving the quality of the TEM images, before the alignment step during the tomographic reconstruction. The remainder of the paper is outlined as follows: in Section II, we point out the noise in the Transmission Electron Microscopy. Section III is devoted to expose the development of the considered methods. Section IV compares the performance of the designed structure using multiple noisy copies with previously published denoisers on an experimental dataset of TEM images. Finally, concluding remarks illustrate the capabilities of our proposal. Noise Sources in TEM Technique Getting inside the structural molecular biology was a challenge for the scientists because of its importance in avoiding surgical intervention. The emergence of the electron microscope allowed them to achieve their ambitions. They would be able to reveal the structure of smaller objects because of the electron microscope which has a higher resolving power than a light microscope. However, the TEM technique suffer from a number of drawbacks, it necessitates ultra-thin specimen which require an extensive preparation plus the sensitivity of samples during the interaction with the electrons beam. For the sake of maintain the specimen from the electrons radiation and avoid the damaged happened in case if it highlighted with high dose electrons, the biologist uses low electron doses (measured in electrons per square Angstrom). This makes the TEM images noisy and poor in their contrast. There are two types of noise (background) in electron microscopy. The first one which is very low comes from the sensor such as the CCD camera while the second comes from the inelastic interactions of the electrons beam with the specimen. The noise from the camera is very low, so we can neglect it. In these experiments, the test TEM images are first acquired at different exposure time to get multiple noisy copies of each test images. We denoised the TEM images before their alignment to enhance the 3D reconstruction. We have considered the noise in our test images as a white Gaussian noise and applied the proposed methods combined with averaging operator for the case of multiple noisy copies. Note that the acquired test images are regrouped in different zones. The Proposed Methods for the TEM Images Denoising Denoising Using Thresholding Methods Thresholding technique is one of the simplest methods in the denoising domain, either for 1D or 2D signals, it is based on separating the low coefficients which are contaminated by the noise from the high coefficients, depending on the calculated thresholding value and the used methods. We have used in our studies two methods the Hard and Soft thresholding. These two methods differ in the way how the coefficients selected, where in the first methods (keep or kill) the coefficients that their values less than the thresholding value considered as zero, whereas the higher ones kept. This technique is usually used in medical image processing. While the second method the shrinkage process is calculated according to Eq. (6) [3]. In our study, we considered the observations model as follows: xy (1) Where y is the measured image, x is the desired image, and is random independent noise. Then we applied the wavelet transform on our observations: )x(D)y(D (2) The operator D significant that the model is in the wavelet domain, where the good choices of the mother wavelet, the threshold type, the decomposition level and the way how the thresholding value calculated are a critical parameters in attending a high signal-to-noise ratio (SNR) after the denoising. Knowing that we have tested all these parameters before doing this comparison, where in our experiment after analyzing previous results we have chosen ‘sym8’ as the mother wavelet beside ‘level 3’ as the decomposition level which gave us good results without affecting the images and for getting the optimal thresholding value we applied the universal rule proposed by Donoho and Johnstone [2] given by: )log(2 nT (3) is the standard deviation of the data values, whereas n is the length of the analyzed image. This popular estimate of the noise level is based on the last level of the detail coeﬃcients, and used the median absolute deviation during the calculation, according to the following equation: 6745.0 ))(( yDmedian (4) The factor in the denominator is equal to 0.6745 for a normally distributed data. The thresholded wavelet coeﬃcients are obtained using either hard or soft thresholding rule given respectively by: otherwise TyDyD yD jj j )( 0 )( )( (5) otherwise TyDTyDyDsign yD jjj j )( 0 ))()).((( )( (6) Where )(yDj are the noisy wavelet coefficients at level j, )(yDj the denoised wavelet coefficients and T is the threshold value. Denoising Using Nonparametric Bayesian Estimator Over the last decade, various estimation approaches are proposed under the wavelet domain in the context of Bayesian paradigms, where a prior distribution is imposed on the wavelet coefficients [4, 5].These works proved the efficiency of the Bayesian wavelet estimators which outperform the classical wavelet term- by-term thresholding estimators. In our work, the applied Bayesian Denoiser in the wavelet domain is based on adapting a prior statistical model for )x(D in Eq. (2), and then imposes it on the wavelet coefficients to describe their distribution [4, 5]. It follows from Eq. (1) that: 12,...,0,1,..., j cmn oj mn oj mn mnmnmn nmJJjsd ac (7) Where mna is the approximation coefficient of the true image x (resp. y ) at location (m,n), oj mns the details coefficients of the original image x in the wavelet domain, j and o are the scale and the orientation respectively. The complete process of the Bayesian Denoiser in wavelet domain shows in the following diagram in Fig.1. FIGURE 1. Flowchart of the used Bayesian Denoiser We can resume the denoising process by using the Bayesian Denoiser, in the following steps: Step1 Calculation of the wavelet coefficients of the noisy data as in Eq. (2); Step2 Apply the Bayesian denoising algorithm to estimate the denoised wavelet coefficients. For the sake of clarity, we should report here more details of this step. Different choices of loss function lead to different Bayesian rules and hence to different nonlinear wavelet shrinkage and wavelet thresholding rules. For example, it is well known that the L1-loss function corresponds to the maximum a posteriori (MAP) estimator. However, except some special cases of SαS distributions (e.g. α = 2), it is not easy to derive a general analytical form of the corresponding Bayesian shrinkage rule even with the scale mixture approximation. Alternatively, the L2-based Bayes rules are used which correspond to Posterior Conditional Means (PCM) estimates. The general expression, using the approximate prior PDF, of the PCM estimates of the wavelet coefficients sˆ is j Bj j Bj Bj );d()j(P );d( d )j(p dsˆ 22 22 22 2 (8) Where is the hypermarameters set, j),j(P 1 and );d( 2 is the normal noise PDF with variance 2 = 2 B ; Step3 Reconstruction of the denoised image xˆ by computing the inverse wavelet transform IWT from the estimated wavelet coefficients. Denoising Using Bilateral Filtering The bilateral ﬁlter is a nonlinear ﬁlter that does spatial averaging which can blur the image without smoothing its edges [6]. The key idea of the bilateral ﬁlter is that for takes the weighted sum of the nearby pixels; they also should have similar values. It means that the weights depend on both the spatial distance and the intensity distance, which preserves the edges and averages the noise. The bilateral ﬁlter at a pixel location x, is deﬁned by: )( 1 )( ~ 2 2 2 2 2 )()( )( 2 yIee c xI rd xIyI xNy xy (9) Where d and r are parameters controlling the fall-off of the weights in spatial and intensity domains, respectively, )(xN is a spatial neighborhood of x and C is the normalization constant [6]. 2 2 2 2 22 rd )x(I)y(I )x(Ny xy eeC (10) In our study, we applied the bilateral filter to denoise our TEM noisy data. In order to compare the performance of this filter to the other proposed methods, we have done previous analyses we choose the appropriate parameters for the bilateral filter we want to use (window size, d and r ). The proposed structure of multiple noisy copies, based on wavelet transform, consists of three major modules: (i) a subband representation function that utilizes the wavelet transform, (ii) applying one of the considered denoising algorithms and (iii) the traditional averaging operation. As shown in Fig. 4, the denoised copies (the out puts of the denoiser blocs) are combined by averaging operation in order to obtain the final recovered image. Experimental Results In this section, the assessments of the denoising results are reported. Due to space limitation, we expose some obtained results. Test Images Dataset In order to verify the proposed methods, we handled with Cryo Microscopy images acquired by using a JEOL 2200 FS transmission electron microscope operating at 40000X (Name microscope Mag 10000x) at different exposure times, and different zones. The order of acquisitions of each zone is as follows: zone1: 2s(1x), 1s(2x), 0.5s(4x), 0.2s(10x), 0.1s(20x), zone2: 1s(2x), 0.5s(4x), 0.2s(10x), 0.1s(20x), 2s(1x), zone3: 0.5s(4x), 0.2s(10x), 0.1s(20x), 2s(1x), 1s(2x), zone4: 0.2s(10x), 0.1s(20x), 2s(1x), 1s(2x), 0.5s(4x), zone5: 0.1s(20x), 2s(1x), 1s(2x), 0.5s(4x), 0.2s(10x). We should note that the order of acquisitions is taken in account. This order put intentionally by the biologists because it is affecting on the images which area golden beads whose size is a quality control of the methods, and frozen water in the glassy state. Denoising Quality, in the Context of Computational Performance -Evaluation using circularity coefficient: As it is mentioned above we calculated the gold beads circularity for the sake of getting a good assessment of the used methods. We calculate this parameter by using ImageJ 1.47 according to the following equation: 2 )(perimetre Air 4.=ycircularit (11) If the circularity equal to 1 this means that the beads are circular and we should keep it as one or improve it if is not equal to one. We can also check the circularity of the Gold beads through the aspect ratio parameter AR. It is defined as the ratio of the Axis Major to the Axis minor of the Gold beads by using Eq. (12). This parameter is also measured by using ImageJ 1.47. Minor)(Axis Major)(Axis =AR (12) If the resulted aspect ratio equals to one, the Gold beads are round. These two parameters, the Circularity or the AR indicate the accuracy of the proposed denoising structure. Experimental Denoising Results -For one copy: We first, show the results where we denoised each image separately using the Soft, Hard Thresholding, Bilateral filtering and the Bayesian Approach. The results are shown in Tab.1. The first row contains the name of the original images from zone1, where we chose to take one image as an example from each series due to limitation space, 0.1s-001 from 0.1s series, 0.2s-001 from 0.2s series, 0.5s-001 from 0.5s and 2s. We can see that the proposed method successfully enhance the SNRout compared to the SNRin of these images, also, it is seen that the Bayesian Approach gave higher SNRout for 0.5s, 1s and 2s images than the obtained by applying Bilateral Filtering and the Thresholding methods. The original images are shown in Fig.2. Fig.3 shows the denoised ones. TABLE 1. The SNRin and SNRout Results of the considered denoising methods for one nosy copy Zone1/ Images