Dept. Computer Architecture and Electronics, University of Almería, 04120 Almería, Spain

Centro Nacional de Biotecnologia – CSIC, Campus de Cantoblanco, 28049 Madrid, Spain

Abstract

Background

Noise filtering techniques are needed in electron tomography to allow proper interpretation of datasets. The standard linear filtering techniques are characterized by a tradeoff between the amount of reduced noise and the blurring of the features of interest. On the other hand, sophisticated anisotropic nonlinear filtering techniques allow noise reduction with good preservation of structures. However, these techniques are computationally intensive and are difficult to be tuned to the problem at hand.

Results

TOMOBFLOW is a program for noise filtering with capabilities of preservation of biologically relevant information. It is an efficient implementation of the Beltrami flow, a nonlinear filtering method that locally tunes the strength of the smoothing according to an edge indicator based on geometry properties. The fact that this method does not have free parameters hard to be tuned makes TOMOBFLOW a user-friendly filtering program equipped with the power of diffusion-based filtering methods. Furthermore, TOMOBFLOW is provided with abilities to deal with different types and formats of images in order to make it useful for electron tomography in particular and bioimaging in general.

Conclusion

TOMOBFLOW allows efficient noise filtering of bioimaging datasets with preservation of the features of interest, thereby yielding data better suited for post-processing, visualization and interpretation. It is available at the web site

Background

The advent of bioimaging technology has made it possible to observe the molecular and cellular architecture and interactions that underlie essential functions within cells and tissues. The availability of bioimaging techniques (e.g. light, confocal, X-ray, electron microscopies) in laboratories is growing rapidly. So is the need for advanced image processing methods that facilitate analysis and interpretation at different scales of resolution and complexity.

Electron tomography (ET), which combines electron microscopy with the power of three-dimensional (3D) imaging, is the leading technique to elucidate the molecular architecture of biological specimens in a close-to-native state

Noise reduction is paramount for proper interpretation or post-processing of multidimensional images in bioimaging in general, and electron tomography in particular. Standard linear filtering techniques based on local averages or Gaussian kernels succeed in reducing the noise, but at the expense of blurring edges and features

TOMOBFLOW is a program for noise reduction with feature-preserving capabilities based upon geometric flow, particularly the so-called Beltrami flow. The fact that this approach is parameter-free is one of its main advantages and makes it user-friendly. Therefore, TOMOBFLOW combines the power of diffusion-based noise filtering approaches with the easiness from the user's point of view. Furthermore, the program has been implemented efficiently in order to minimize the memory requirements and reduce the computation time.

Implementation

Several approaches for noise reduction in multidimensional image processing are based on considering images as maps that are embedded into a higher dimension and that flow towards minimal surfaces

Principles of Beltrami flow

**Principles of Beltrami flow**. An image **Z **direction. At each point of the surface, the projection of the normal **n **(arrows in blue) to the **Z **direction (arrows in black) is the edge indicator **n **runs parallel to **Z **and the projection thus yields maximum value.

The Beltrami flow is an efficient geometric diffusion flow approach that aims to minimize the area of the image manifold, driving the flow towards a minimal surface solution while preserving edges. The Beltrami flow is formulated as follows

where _{t }= ∂**I **is the gradient vector, that is ▽**I **≡ (_{x}, _{y}) for 2D images whereas ▽**I **= (_{x}, _{y}, _{z}) for 3D volumes, being _{x }= ∂**f **= (_{x}, _{y}, _{z}) as div(**f**) = ∂_{x}/∂_{y}/∂_{z}/∂**I**|^{2}.

The term

Moreover, the term

In TOMOBFLOW, the implementation of the partial differential equation derived from Equation (1) is based on finite differences, using an Euler forward difference approximation for _{t }and central differences to approximate the spatial derivatives (for brevity, only the numerical approximation for the 2D case is shown):

where ^{k }is the image in the _{t }is the time step (for stability, the maximum value is the inverse of the squared number of dimensions, i.e. 0.25 for 2D images), _{x }is the derivative with respect to _{xx }is the second order derivative with respect to _{xy }is the mixed second order partial derivative with respect to ^{k-1}, and are numerically approximated by central differences, as shown for

where

An efficient implementation has been carried out using single processor optimization

Sliding window for processing of 3D volumes

**Sliding window for processing of 3D volumes**. The sliding window keeps the data needed for the processing of the current slice. This allows TOMOBFLOW to allocate memory only for one copy of the dataset, which is progressively updated during the processing. The solid lines show the information transfer during the processing of the slice n: (1) the slice n+1 from the current volume is got from the volume, (2) the processing of the slice n is carried out using only the data in the sliding window, (3) the processed version of the slice n is updated and stored in the volume and (4) the slices in the sliding window are pushed backward to make space for a new slice coming from the volume. The dotted lines show how the sliding window is pushed forward for the processing of a new slice. The working principle of the sliding window for processing 2D images is similar.

To make it suitable for bioimaging in general, TOMOBFLOW is capable of dealing with most image formats in electron microscopy (e.g. EM, MRC, Spider), in other microscopies (e.g. Biorad) and general formats (e.g. TIF, JPG, PNG) by using the Bsoft library

Results

The performance of TOMOBFLOW is illustrated with its application to a number of experimental datasets obtained from electron tomography. Tomograms (3D volumes) of (a) spiny dendrite, (b) algae chloroplast, (c) mitochondrion, (d) small unilamellar liposomes with integrin, (e) vaccinia virion and (f) human immunodeficiency virions (strain HIV-1) were tested. Different contrast and signal-to-noise ratio were present in those datasets as they were obtained by using different preparation techniques. The specimens in (a-c) were stained before imaged, hence their much better contrast in the original dataset compared to the other specimens in (d-f), which were imaged while frozen in close-to-native conditions without stain. The datasets in (a, b) were taken from the Cell Centered Database

Figure

Filtering results on the tomograms of stained specimens

**Filtering results on the tomograms of stained specimens**. (a) spiny dendrite; (b) algae chloroplast; (c) mitochondrion. The original tomograms (left), the results with TOMOBFLOW (centre), and the results with three iterations of the median filtering (right) are shown. Only a representative slice of the tomograms is presented. The number of iterations of TOMOBFLOW were (a) 150, (b) 150 and (c) 70. The datasets (a) and (b) were taken from the Cell Centered Database (accession codes 13 and 3408, respectively). The dataset (c) was kindly provided by Dr. G. Perkins (National Center for Microscopy and Imaging Research-NCMIR, UCSD, USA). In the three datasets, the results with TOMOBFLOW have the background particularly flat with respect to the original tomogram, and also with respect to the results with median filtering. Moreover, TOMOBFLOW outperforms the median filtering in the preservation of the structural features. The arrows point to areas where the sharpness of the features is especially apparent after the processing with TOMOBFLOW.

Figure

Filtering results on the tomograms of unstained specimens

**Filtering results on the tomograms of unstained specimens**. (a) small unilamellar liposomes with integrin; (b) Vaccinia virion, (c) HIV-1 virions. The original tomograms (left), the results with TOMOBFLOW (centre), and the results with three iterations of the median filtering (right) are shown. Only a representative slice of the tomograms is presented. The number of iterations of TOMOBFLOW were (a) 100, (b) 50 and (c) 70. The datasets (a) and (c) were taken from the Electron Microscopy Data Bank (EMDB) at the European Bioinformatics Institute (accession codes 1487 and 1155, respectively). The dataset (b) comes from a previous work

Isosurface of the tomograms of unstained HIV-1 virions

**Isosurface of the tomograms of unstained HIV-1 virions**. From left to right, the 3D visualization of the original tomogram of unstained HIV-1 virions and the denoised versions with TOMOBFLOW (70 iterations) and the iterative median filtering (3 iterations) are shown. A slice of each tomogram was previously shown in Figure 4c. Both filtering methods allow the 3D inspection of the dataset, though TOMOBFLOW preserves more details at the membranes of the virions.

The evolution of the denoising with the iterations was then studied on the HIV-1 dataset. Figure _{s }- _{b})/_{b}, where _{s }and _{b }denote the average intensity in the structure of interest and in the background, respectively, and _{b }is the standard deviation of the background. This SNR metric is similar to the contrast-to-noise ratio (CNR) used in other disciplines

Evolution as a function of the iterations

**Evolution as a function of the iterations**. From left to right, the original HIV-1 tomogram and the results with TOMOBFLOW at 10, 25, 50, 100 and 150 iterations are shown. The background is progressively flattened with the iterations whereas the structural features remain sharp. At high number of iterations, some edges begin to look blurred.

Effect of the denoising on the SNR of the HIV-1 dataset

**Tomogram**

**SNR**

Original

1.23

Median filtering

2.63

TOMOBFLOW 10 it.

2.31

TOMOBFLOW 25 it.

2.48

TOMOBFLOW 50 it.

2.79

TOMOBFLOW 100 it.

3.35

TOMOBFLOW 150 it.

3.55

TOMOBFLOW 200 it.

3.38

TOMOBFLOW 250 it.

3.04

TOMOBFLOW 300 it.

2.67

The SNR was computed for the original dataset and the denoised versions. For the median filtering, the standard of three iterations were used. For TOMOBFLOW, several tests with different iterations (from 10 to 300) were performed in order to study the evolution with time and to identify the decrease in resolution at high number of cycles.

The SNR metric was also used to assess the results shown in Figure

TOMOBFLOW and the iterative median filtering were also compared in terms of computation time. The average time per iteration was computed in both methods (in a standard computer based on Intel Core 2 processor 2.4 GHz running under linux) and the ratio between both was then calculated. For the six datasets, which had very different sizes (from 14 MB to 390 MB), it turned out that a single iteration in the median filtering took around 20 times more than a single iteration of TOMOBFLOW, regardless of the data size. As the number of iterations of TOMOBFLOW is usually between 50 and 150, this involves that the computation times for both methods are of the same order of magnitude (1–3 minutes for the datasets and the computer tested here). As far as memory consumption is concerned, TOMOBFLOW only used space for one copy of the dataset, as described above. It thus required half the amount of memory allocated by the median filtering (two copies of the volume) as implemented in Bsoft.

Discussion

TOMOBFLOW allows efficient noise reduction with levels of background smoothing and feature preservation better than other comparable standard nonlinear filtering methods. TOMOBFLOW applies an isotropic nonlinear filtering method based on the Beltrami flow, which tunes the strength of the smoothing according to a local edge indicator. In contrast to anisotropic nonlinear filtering (e.g. AND), there is no enhancement of features since the direction of the smoothing is not tuned. Therefore, it must not be expected that TOMOBFLOW will outperform AND. In this regard, the comparison with AND carried out in this work suggests that the method based on the Beltrami flow lies between the median filtering and the AND methods.

The main advantage of the method implemented in TOMOBFLOW stems from the fact that there is no need for complicated parameter tuning. Nevertheless, it is indeed an iterative method and one thus needs to specify a number of iterations. But this does not pose a serious inconvenience as the program easily allows an experiment to be continued with further iterations, if necessary. On the other hand, there has been intense investigation on objective stopping criteria for iterative noise reduction methods (e.g.

On the other hand, the computational burden involved by sophisticated diffusion-based filtering methods precludes their integration on interactive environments

Conclusion

TOMOBFLOW allows efficient noise filtering of datasets with preservation of the features of interest, thereby yielding data better suited for post-processing, visualization and interpretation. The program is versatile to deal with different types and formats of multidimensional images produced by bioimaging techniques.

Availability and requirements

**Project name**: TOMOBFLOW

**Project home page**:

**Operating system(s)**: Unix-based (linux, OS X, cygwin under Windows).

**Programming language**: C.

**Other requirements**: none.

**License**: public domain binaries.

**Any restrictions to use by non-academics**: none.

Authors' contributions

JJF conceived and designed the work, developed the program, carried out the experiments, interpreted the resulting data and wrote the manuscript.

Acknowledgements

Dr. G. Perkins kindly provided the mitochondrion dataset. The anonymous reviewers provided helpful suggestions to improve the manuscript. Work partially supported by grants MCI-TIN2008-01117, MCI-PR2008-0273, JA-P06-TIC-01426 and EU-LSHG-CT-2004-502828.