Iterative Bayesian denoising based on variance stabilization using Contourlet Transform with Sharp Frequency Localization: application to EFTEM images
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Abstract
Background
Due to the presence of high noise level in tomographic series of energy filtered transmission electron microscopy (EFTEM) images, alignment and 3D reconstruction steps become so difficult. To improve the alignment process which will in turn allow a more accurate and better three dimensional tomography reconstructions, a preprocessing step should be applied to the EFTEM data series.
Results
Experiments with real EFTEM data series at low SNR, show the feasibility and the accuracy of the proposed denoising approach being competitive with the best existing methods for Poisson image denoising. The effectiveness of the proposed denoising approach is thanks to the use of a nonparametric Bayesian estimation in the Contourlet Transform with Sharp Frequency Localization Domain (CTSD) and variance stabilizing transformation (VST). Furthermore, the optimal inverse Anscome transformation to obtain the final estimate of the denoised images, has allowed an accurate tomography reconstruction.
Conclusion
The proposed approach provides qualitative information on the 3D distribution of individual chemical elements on the considered sample.
Keywords
Image denoising Variance stabilizing transformation Bayesian estimation Contourlet transform EFTEMAbbreviations
 AT
Anscombe transformation
 AWGN
Additive white Gaussian noise
 CTSD
Transform with sharp frequency localization
 DFB
Directional filter bank
 EFTEM
Energy filter transmission electron microscopy
 SNR
Signaltonoise ratio
 STD
Standard deviation
 TET
Transmission electron tomography
 VST
Variance stabilizing transformation
Backround
Transmission Electron Tomography (TET) is one of the most widely used methods for structural analysis in biology and is capable to reveal subcellular structures at the nanometric scale. The combination of TET with chemical mapping (such energy filter transmission electron microscopy: EFTEM) gives qualitative information on the distribution of the chemical elements by the generation of 3D chemical maps in the analyzed samples [1] thus overcoming the limitation of 2D maps. In an EFTEM mode, the transmitted electrons lose different energies according to their interaction with the atoms present in the sample. These energies are characteristic of each type of interaction where electron magnetic fields can be used to separate these electrons. Thus, it is possible to construct a filtered image using only those electrons having lost a precise energy. This approach allows for the computation of elemental maps as images calculated after removing the unspecific signals. The inherent presence of low signaltonoise ratio (SNR) in biological specimens when an EFTEM is performed, remains a major issue to generate high resolution and good quality EFTEM3D maps. thus limiting the use of 3D chemical mapping in biology. This paper aims to improve the quality of the acquired images by applying denoising approaches respecting the physical significance of the pixel values of EFTEM maps (which represent the number of electrons having lost a characteristic energy) to produce 3D chemical maps of very high quality of the sample to be analyzed. There is much interest in developing novel methods to remove the noise in its different forms from images in such a way that the original image is discernible and the signal quality is not modified. However, existing imageenhancement methods amplify noise when they amplify weak edges since they cannot distinguish noise from weak edges [2, 3]. Here, we extend our preliminary work, by considering more general optimal inverses for the Anscombe transformation in an iterative process. on the other hand, it has been shown that there are two types of noise in electron microscopy [4, 5]. The first one 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 dominant and is modeled as a Poisson process. Therefore, we have assumed that the EFTEM images are corrupted by additive Poisson noise. Therefore, EFTEM images are denoised using a Bayesian denoiser in the Contourlet Transform with Sharp Frequency Localization (CTSD) [6] domain iteratively in order to improve progressively the effectiveness of the Anscombe transformation (i.e. variance stabilizing transformation VST) [7, 8]. Furthermore, we demonstrate that the assumption of a Poisson noise with a combination of a Bayesian denoiser in CTSD domain and the Anscombe transformation allow for a significant enhancement of the chemical map computation which in turn will enhance the 3D reconstructed volume of EFTEM images with a computational cost at worst twice that of our previous noniterative Bayesian denoiser [9]. We demonstrate through experiments with real EFTEM images contaminated by additive Poisson noise that the performance of the proposed method substantially surpasses that of previously published methods. The proposed method is qualitatively evaluated in an observer study to assess the improvement of 3D visualizations of EFTEM series and quantitatively in terms of SNR.
This paper is organized as follows: “Results” section defines the evaluation criteria considered and the computed maps including a comparative analysis of the performance of the proposed denoising method in this study with previously published denoising methods [3, 9, 10, 11, 12] on different real data sets. Furthermore, numerical experiments in this section are presented to demonstrate the effectiveness of the proposed method over recent denoising approaches. “Discussion” section discusses the performance and effectiveness of the proposed method. Concluding remarks are given in “Conclusion” section. Finally, “Methods” section describes first the EFTEM images used in this work and which are a specific data collected at different energies 650, 680 and 710 eV from a biological sample, namely Fonsecaea pedrosoi. It also describes the propose iterative denoising method for the purpose to perform chemical maps computation and therefore to enhance the quality of the 3D reconstructed volume of the EFTEM images.
Results
In order to assess the performance of the proposed method described in “Methods” section, a quantitative evaluation has been carried out against our previously published denoising approach[9] including recent denoising methods. For the sake of comparison, we have only chosen denoising methods using the same Bayesian denoiser with the scalemixture approximation to the alphastable prior, called " αstable mixture" in different domains. The three domains that we have considered are the Wavelet transform [13], the Contourlet transform [14] and the CTSD domains, respectively, as shown in the workflow at the end of this paper (Fig. 5). Knowing that the bloc of Hot spot in the workflow represents a preprocessing of removing the aberrant pixels from the EFTEM images using the ImageJ plugin EFTEMTomoJ [1, 15]. The EFTEMTomoJ and TomoJ blocs are the plugins under ImageJ used to compute the elemental map and the 3D tomography reconstruction of our tilt series respectively.
where \(\overline {W}\) and \(\overline {R}\) are the average values of the amplitude of the net wall signal and the resin, respectively, α_{resin} is the standard deviation of the resin.
Figure 5 summarizes all the methods that we have used in this study where (A) is the reference. We reconstructed the 3D volume of the original images (i.e. without denoising) to compare its quality with the quality of those with denoising. The outputs of (B), (C), (D) and (E) are the tilt series denoised using the Bayesian denoiser in the wavelet, the contourlet SD, the contourlet SD in iterative way and in the contourlet domains, respectively. The bloc of Hot spot in the workflow represents a preprocessing of removing the aberrant pixels from the EFTEM images using the ImageJ plugin EFTEMTomoJ [1, 14]. This step is applied before and after the denoising step to make the alignement process during the reconstruction easier. The EFTEMTomoJ and TomoJ blocs are the plugins under ImageJ used to compute the elemental map using the 3window technique which requires three energyfiltered images and the 3D tomography reconstruction of our tilt series, respectively.
SNR and contrast C_{W} of the wall area
Projections  SNR _{ wall}  C _{ W}  \(\overline {W}\)  \(\overline {R}\)  resin _{ std} 

Original  2.66  0.06  21.30  20.00  0.96 
Baysian denoiser in WT domain  9.99  0.10  22.01  19.95  0.65 
Baysian denoiser in CTSD domain  8.05  0.07  21.33  19.91  0.56 
Baysian denoiser in CT domain  9.01  0.08  21.63  19.91  0.61 
Proposed iterative Bayesian denoiser with VST in CTSD domain  19.21  0.10  22.08  19.95  0.23 
We should note, that the accurate and judicious assumption of the Poisson distribution instead of the Gaussian one to model the additive noise in the observation data EFTEM, helped to improve the considered Bayesian estimators.
Discussion
After analysing the results, one can see that the SNR and the CW are enhanced in all the applied methods and the Bayesian estimator in the wavelet and the contourlet transform domains is comparable to the Bayesian estimator in the CTSD domain. One can also notice that the proposed iterative denoiser outperforms the previous methods, especially our previous work [9] and gives much better results in terms of both SNR and CW, where the SNR is enhanced by about 11 dB compared to the Bayesian estimator in the CTSD domain [9]. The main reason is that the iterative combination with a previous estimate refines the stabilization and helps to tackle the problem of the low SNR for this type of images. These findings suggest that the proposed iterative Bayesian denoising in the CTSD domain with VST is an accurate method adapted to capture the fine details that are hidden because of the Poisson noise. We should note, that the accurate and judicious assumption of the Poisson distribution instead of the Gaussian one to model the additive noise in the observation data EFTEM, helped to improve the considered Bayesian estimators.
Conclusion
This paper has proposed a novel iterative method based on a nonparametric Bayesian estimator in CTSD domain with VST which is capable to denoise EFTEM images. The iterative combination with a previous estimate (denoised image) refines the stabilization which leads to a better quality of the images in terms of a higher SNR and contrast which in turn enhances the 3D tomographic reconstruction. In order to illustrate the potential of the proposed denoising method and analyze the importance of embedding the VST framework within the iterations, we have compared our results using simplified version of the developed algorithm (without iteration and without VST) in different domains with the proposed denoising algorithm. After applying the non iterative Bayesian estimator in the different domains, we have obtained good results where the SNR is considerably enhanced. To further address the problems associated with missing details in the denoised images, we have refined our previous method by taking into account the geometrical information of the images (i.e. contours). Therefore, we have applied iteratively the Bayesian denoiser in the CTSD domain where we have used the Anscombe transform to normalize the image noise. Then denoising the EFTEM images with a nonlinear nonparametric Bayesian estimator is performed to reconstruct the images to their original range via an optimal inverse transformation. This algorithm gave us better results as shown in Fig. 2, where details hidden after previous denoising approach, are now preserved, as shown in Fig. 3. Our future will focus on studying other nonparametric Bayesian estimators, in particular, the estimator based on BesselKform (BKF) density [18, 19, 20].
Methods
Nature of data
Proposed denoiser
where y and x are respectively the noisy EFTEM image and the original clean image to recover, ε is an additive Gaussian noise.
Basic assumption
Proposed iterative algorithm
where CTSD_{k}(a),s_{k} and ε_{k} are the contourlet coefficients in the k^{th} directional subband of the observed noisy image, noisefree image and noise respectively.
Because the contourlet has the similar characteristics as the wavelet, so we can straightforwardly extended the Bayesian denoiser proposed in the wavelet domain [11, 12], into the contourlet domain.
In our study, similarly to the wavelet domain, the applied Bayesian denoiser in the contourlet domain is based on adapting a prior statistical model for s_{k} and then imposes it on the contourlet coefficients to describe their distribution.
In the other hand, it has been shown that the statistical behavior of contourlet coefficients is successfully modeled by families of heavytailed distributions such as the αstable. More precisely, Sadreazami et al. [23] demonstrated through the plots of histograms and the computation of kurtosis of the contourlet coefficients that symmetric αstable family, is more appropriate distribution for modeling the contourlet coefficients of natural images than families with exponential tails such as the generalized Gaussian. In view of this, we propose to use the αstable prior with the scale mixture approximation, called " αstable mixture" to model the contourlet subband coefficients [9].
The denoised contourlet coefficients of the image are then estimated by the L_{2}based Bayes rules, which correspond to posterior conditional mean (PCM) estimate as shown in our previous work [9]. The inverse contourlet transform is computed through the processed contourlet coefficients to get the denoised image).
where BD_{CTSD} denotes the Bayesian denoiser in the CTSD proposed in [9].

Step 1: Normalize the variance noise of the observed EFTEM data by applying the VST to each image of the three tilt series. This step produces an EFTEM data set such that each image y_{AT} like it is contaminated with AWGN.

Step 2: Apply the Bayesian denoiser in the CTSD domain (BD_{CTSD}) [9] to the transformed noisy data. The (BD_{CTSD}) consists on: (a) calculate the CTSD coefficients of the y_{AT}, (b) denoise the detail coefficients of the CTSD at each scale and each orientation, (c) reconstruct the denoised image by applying the inverse CTSD to the estimated coefficients. This is done for each image separately. We should recall that for the Bayesian denoiser in the contourlet transform and the contourletSD, we selected the number of levels for the Directional Filter Bank (DFB) at each pyramidal level equal to (2, 3, 4, 5) pkva filters and we did not downsample the lowpass subband at the first level of decomposition, based on [6].

Step 3: Apply the optimal inverse AT to generate the denoised image to the original range of y.
Notes
Acknowledgements
The authors wish to thank Sylvain Trépout for valuable discussions and suggestions concerning the biological data.
Funding
The project was supported by the High Ministry of Education of the Algerian Republic and Campus France, project 33257ZB, Huber Curien PHC Tassili, with grant number 15MDU950 and by Agence Nationale de la Recherche ANR11BSV8016. The authors want to acknowledge the PICTIBiSA for providing access to chemical imaging equipment. The funding bodies had no role in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.
Availability of data and materials
All data generated or analysed during this study are included in this published article and its supplementary information files.
Authors’ contributions
SSA, ZM and LB initiated the contribution. SSA implemented the algorithms in Matlab code and got the quantitative results. ZM and SM performed concept experiments and workflows. S.M and C.M performed EFTEM image acquisitions. ZM, SM, LB, and AB coordinated the team. All authors contributed in drafting and reviewing the manuscript; also in analyzing, discussing and interpreting of the results. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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