FEATURE

NOISE REDUCTION IN 3D MAPS

ANISOTROPIC NONLINEAR FILTERING OF CELLULAR STRUCTURES IN CRYOELECTRON TOMOGRAPHY Cryoelectron tomography attempts to allow the visualization of complex biological specimens’ molecular architecture. This technique yields 3D maps with an extremely low signal-to-noise ratio. A new approach based on anisotropic nonlinear diffusion offers a strategy to reduce the noise and enhance local structures.

C

ryoelectron tomography (CryoET) combines the power of 3D imaging with the best possible preservation method for structural analysis of large complex biological specimens at molecular resolution, which is critical to understanding cellular function.1,2 Researchers have achieved several breakthroughs using CryoET, including the elucidation of eukaryotic cell architectures3,4 and large complex viruses.5,6 Although researchers have understood the principles of electron microscope tomography for several years,7 recent technological advances have made it feasible for them to produce extremely low-contrast 3D density maps (or tomograms) of biological materials embedded in ice (cryomicroscopy), thereby ensuring their preservation in near-physiological conditions. In CryoET, tomograms’ poor signal-to-noise ratio (SNR), which is approximately 0.1, severely hinders their visualization and interpretation, precluding researchers from applying automatic image-analysis tech-

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niques, such as segmentation and pattern recognition. Sophisticated filtering techniques are thus indispensable for properly interpreting tomograms.8 Anisotropic nonlinear diffusion (AND) is currently one of the most powerful noise-reduction techniques in the computer vision field.9 It accounts for the local structures found in the image to filter noise, preserve edges, and enhance certain features, thus increasing the SNR considerably with no significant signal distortions. Pioneered in 1990 by Pietro Perona and Jitendra Malik,10 AND has become a well-established tool in the past decade.9–13 Achilleas Frangakis and Reiner Hegerl introduced it in CryoET in 2001,14 and we further developed it in 2003;15 it’s been crucial to some of the latest breakthroughs in the field.3–6 We propose an AND filtering approach to CryoET that combines structure-preserving noise reduction with a strategy for enhancing both planar and curvilinear local structures. Moreover, we’ve provided it with a background-filtering mechanism that highlights the interesting biological structural features along with a new stopping criterion for the iterative diffusion process.

´ FERNA´ NDEZ JOSÉ-JESUS

Anisotropic Nonlinear Diffusion

University of Almeria, Spain

AND accomplishes sophisticated, edge-preserving denoising that takes into account the structures at local scales. Conceptually speaking, AND tunes the smoothing’s strength along different directions

SAM LI MRC Laboratory of Molecular Biology

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v3

v3

based on the local structure estimated at every point of the multidimensional image.

v1

Estimating Local Structure

The structure tensor is the mathematical tool that lets us estimate the local structure in a multidimensional image. The structure tensor of a 3D image I is a symmetric positive semidefinite matrix given by  I2  x J( I ) =  I x I y  I I  x z

IxI y I 2y I y Iz

Ix Iz   I y Iz  ,  I z2  

(1)

where Ix = 聿I/聿x, Iy = 聿I/聿y, and Iz = 聿I/聿z are the derivatives of the image with respect to x, y, and z. The components of J are usually averaged with a Gaussian convolution kernel to represent the local structure at a higher scale. The eigen-analysis of the structure tensor helps us determine the image’s local structural features:9  µ1  J( I ) = [ v1v 2 v 3 ] ⋅  0  0

0 0  µ2 0  ⋅ [ v1v 2 v 3 ]T . (2) 0 µ3 

The orthogonal eigenvectors v1, v2, v3 provide the preferred local orientations, and the corresponding eigenvalues ␮1, ␮2, ␮3 (assume ␮1
␮2
␮3) provide the average contrast along these directions. The first eigenvector v1 represents the direction of the maximum variance, whereas v3 points to the direction with the minimum variance. Based on the eigenvalues’ relative values, we can characterize basic local structures (see Figure 1): • Line-like structures have a preferred direction exhibiting a minimum variation that has an eigenvalue much lower than the other two—that is, ␮1  ␮2 » ␮3. The third eigenvector v3 represents the linear structure’s major direction, whereas v1 and v2 are perpendicular to the line. • Plane-like structures have two preferred directions exhibiting similar small contrast variation, with eigenvalues much lower than the first one— that is, ␮1 » ␮2  ␮3. The first eigenvector represents the direction perpendicular to the plane-like structure; v2 and v3 define the plane that best fits the local structure. • Isotropic structures occur when the two previous conditions don’t hold. In general, for these structures, the eigenvalues have values of similar magnitude or order: ␮1  ␮2  ␮3.

v1

(a)

v1

v2

v2

␮1 ≈ ␮2 >> ␮3

v3

v2

␮1 >> ␮2 ≈ ␮3

␮1 ≈ ␮2 ≈ ␮3

(b)

(c)

Figure 1. Eigen-analysis. The structure tensor reveals basic local structures for (a) line, (b) plane, and (c) isotropic structures. ␮1, ␮2, and ␮3 are the eigenvalues, and v1, v2, and v3 are the corresponding eigenvectors.

These local structures help us tune the smoothing’s strength and directions at every point of the image, which lets us preserve and enhance the structural features of interest. Diffusion in Image Processing

Diffusion is a physical process that equilibrates concentration differences as a function of time, without creating or destroying mass. In image processing, density values play the role of concentration. This observation is expressed by the diffusion equation:9 It = div(D  I),

(3)

where It = 聿I/聿t denotes the derivative of the image I with respect to time t, I is the gradient vector, D is a square matrix called diffusion tensor, and div is the divergence operator. The diffusion tensor D lets us tune the smoothing (the strength and direction) across the image. We define D as a function of the structure tensor J: λ1 0 0    D = [ v1v 2 v 3 ] ⋅  0 λ2 0  ⋅ [ v1v 2 v 3 ]T ,  0 0 λ3 

(4)

where vi denotes the structure tensor’s eigenvectors. The values of the eigenvalues ␭i define the smoothing’s strength along the direction of the corresponding eigenvector vi. The values of ␭i rank from 0 (no smoothing) to 1 (strong smoothing). Depending on the definition of the eigenvalues ␭i of the diffusion tensor D, we can apply different diffusion processes. Figure 2 illustrates noise reduction on a test 2D image using these processes: • A linear diffusion process is accomplished if ␭1 = ␭2 = ␭3 = 1. In this case, the diffusion equation performs a Gaussian filtering. Smoothing is applied equally in all directions regardless of the

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(a)

(b)

(c)

(d)

Figure 2. Denoising. A test image shows different diffusion processes: (a) the original noisy image denoised using (b) a linear diffusion process, (c) nonlinear diffusion, and (d) anisotropic nonlinear diffusion.

structure, without preserving the structure at all. As a result, this approach severely blurs the edges. • A nonlinear diffusion occurs if ␭1 = ␭2 = ␭3 = g(|I|), where g is a monotonically decreasing function. In this case, the smoothing’s strength depends on the value of the gradient |I|, so it preserves the edges because the higher the gradient, the lower the smoothing. This approach’s limitation is its isotropy, which makes smoothing equal in all directions. Consequently, edges are preserved, but they still remain noisy because the smoothing stops on them. • With AND, smoothing depends on the gradient’s strength and its direction. Here, the values of ␭i are set up independently so that the smoothing is anisotropically adapted to the image’s local structure. Consequently, this approach allows smoothing on the edges. Smoothing runs along the edges so that they are preserved and smoothed. AND has turned out, by far, to be the most effective denoising method because of its capabilities for structure preservation and feature enhancement.9,14,15 Common Diffusion Approaches

AND can function differently, by either filtering noise or enhancing some structural features, depending on the definition of ␭i of the diffusion tensor D. Currently, the most common ways of setting up the diffusion tensor give rise to two diffusion approaches. The primary effects of edge-enhancing diffusion (EED) are edge preservation and enhancement.9 Here, strong smoothing is applied along the direction corresponding to the minimum change (the eigenvector v3), while the strength of the smoothing along the other eigenvectors depends on the gradient: the higher the value, the lower the

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smoothing strength. As such, we set up ␭i as λ1 = g ( ∇I )  λ2 = g ( ∇I ) λ = 1  3

(5)

where g is a monotonically decreasing function.9 Figure 2d shows the denoised version of the original image using 50 iterations of EED. Coherence-enhancing diffusion (CED), on the other hand, exploits the curvilinear continuity and can connect interrupted lines and improve flow-like structures.11 In this approach, there’s no smoothing except along the preferred direction v3. The strength of this smoothing depends on the local structure’s anisotropy (␮1 – ␮3): again, the higher the anisotropy, the higher the smoothing, so we set up ␭i as λ1 ≈ 0   λ2 ≈ 0 λ = h( µ − µ ) 1 3  3

(6)

where h is a monotonically increasing function.11 This approach enhances line-like structures because the smoothing is applied along their major direction, v3. Figure 3 illustrates the results of applying CED to a 2D fingerprint image. CED improves the lines in the fingerprint and can even fill in gaps in interrupted lines (see Figure 3b). Figure 3c gives the results of applying EED to the same image for comparison purposes.

Anisotropic Nonlinear Diffusion in CryoET Researchers typically use a hybrid diffusion approach in CryoET to combine the advantages of EED and CED.14,15 This strategy is based on the

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fact that the anisotropy (␮1 – ␮3) reflects the local relation of structure and noise, so we can use this value as a switch: a voxel is processed with CED if the anisotropy is larger than a suitably chosen threshold; otherwise, it’s processed with EED. The threshold tec is derived ad hoc as the maximum anisotropy found in a subvolume of the image that only contains noise. This approach performs an efficient denoising that highlights edges, connects lines, and enhances flow-like structures. Enhancing Plane-Like Structures

The standard CED approach enhances line-like structures (where ␮1  ␮2 » ␮3) by diffusing along v3. Nevertheless, a significant amount of structural features are plane-like at local scales in biological specimens—membranes, surfaces of organelles, vesicles, and so forth. In such structures, a similar small contrast variation exists along v2 and v3—that is, ␮1 » ␮2  ␮3—so for a better enhancement, CED diffusion should run two-dimensionally across the plane defined by both v2 and v3.15 The smoothing’s strength along v2 must be tightly coupled to the plane-ness, given by (␮1 – ␮2), whereas the smoothing along v3 depends on the anisotropy ( ␮1 – ␮3). So, we can set up ␭i as λ1 ≈ 0  λ2 = h( µ1 − µ2 ) λ = h( µ − µ ) 1 3  3

(a)

1. 2.

where h is a monotonically increasing function.11

Our diffusion approach includes a novel strategy to further smooth out the background. Because the interesting structural features usually have higher density levels than the background, we just consider those voxels with density values below a threshold as background and, hence, apply linear Gaussian filtering. The threshold tg is computed from the average gray level in a subvolume of the tomogram that only contains noise—that is, only background. Consequently, these voxels we consider background are significantly smoothed thanks to the Gaussian filtering. We can lay out our AND approach with a simple outline: 0. Compute statistics of the subvolume containing noise. 0.1 Compute the threshold tec used to switch between EED and CED.

(c)

Figure 3. Coherence-enhancing diffusion (CED). With (a) a fingerprint test image, 9 the lines appear noisy and even interrupted. Contrast this to (b) the result after 20 iterations of CED and (c) 20 iterations of edgeenhancing diffusion (EED).

(7)

Background Smoothing with Gaussian Filtering

(b)

3. 4.

0.2 Compute the threshold tg used to apply Gaussian filtering. Compute the structure tensor J. Compute the diffusion tensor D. For every voxel, 2.1. Analyze the local structure. The method decides if the voxel is to be processed as EED, CED, or background. It considers the voxel background if its gray level is lower than the threshold tg, and applies CED if the local anisotropy (␮1 – ␮3) is larger than the threshold tec. Otherwise, it applies EED. 2.2. Computation. If background, the method applies Gaussian filtering: if EED, it computes the diffusion tensor D according to Equations 4 and 5, but if CED, it computes the diffusion tensor D according to Equations 4 and 7. Solve the partial differential equation of diffusion (Equation 3). Iterate: go to step (0).

Numerical Discretization of the Diffusion Equation

We can numerically solve Equation 3 by using finite differences. Specifically, we can replace the term It = 聿I/聿t with an Euler forward difference approximation. The resulting explicit scheme allows calculation of subsequent versions of the image iteratively: I(k+1) = I(k) + ␶  (聿/聿x(D11Ix) + 聿/聿x(D12Iy) 聿/聿x(D13Iz) + 聿/聿y(D21Ix) 聿/聿y(D22Iy) + 聿/聿y(D23Iz) 聿/聿z(D31Ix) + 聿/聿z(D32Iy) 聿/聿z(D33Iz)),

+ + + +

where ␶ denotes the time-step size, I(k) denotes the

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(a)

(b)

(c)

Figure 4. Denoising Vaccinia virus tomograms with anisotropic nonlinear diffusion (AND). (a) A slice of the original tomogram, (b) the result of denoising with our AND approach, and (c) the result of our denoising AND approach without using the background smoothing. The diffusion process stopped at the fifth and 10th iteration (␶ = 0.4), respectively, according to the stopping criterion.

image at time tk = k␶, and the Dmn terms represent the components of the diffusion tensor D. The standard scheme to approximate the spatial derivatives (聿/聿x, 聿/聿y, and 聿/聿z) is based on central differences, but here, we approximate them by using filters with optimally directional invariance13 because of their abilities to structurally preserve local features. In our previous work,15 we derived the 3D version of these derivative filters, resulting in three 3  3  3 kernels, similar to the Sobel operators in image processing. These derivative filters let us better preserve fine details because the gradients account for curvatures in the structures.13 Furthermore, the discretization scheme is more stable and allows up to a four times larger time-step size (␶ = 0.4) than the traditional explicit scheme (␶ = 0.1).13 Consequently, our scheme requires up to four times fewer iterations to obtain similar improvements in SNR.

Noise Estimate Variance AND is an iterative method that yields successively smoother versions of an image, gradually removing noise and details. Determining the optimal time to stop the iterative denoising process is an important issue: essentially, the process should stop before the denoising significantly affects the signal in the image. So far, researchers have proposed several objective stopping criteria,16 but most of them aren’t successful in CryoET, as we’ve shown in previous work.15 The criterion based on the relative variance12—the ratio between the variances of the filtered image and the original noisy image—underestimates the optimal stopping time.12,16 In

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turn, the decorrelation criterion16 generally doesn’t exhibit a unimodal tendency, so it can’t yield a reliable stopping time. Our previous work introduced a criterion based on the relative noise variance,15 in which the diffusion stops when the variance in the noise subvolume is reduced by a given factor, typically 90 percent. However, this criterion doesn’t guarantee that the denoising isn’t affecting the signal because the stopping criterion doesn’t consider the whole volume. Our new stopping criterion, the noise-estimate variance (NEV) criterion, estimates the noise at time t as the difference between the original noisy image I 0 and its current filtered version It. This noise estimate’s variance var(I 0 – I t) increases monotonically from 0 to var(I 0) during diffusion. We can then define a stopping criterion with a suitable threshold over that variance. The threshold should represent an estimation of the amount of noise contained in the original tomogram. So, this criterion makes diffusion stop as soon as var(I 0 – I t) reaches the noise in the tomogram. We can define the NEV’s threshold as the variance of the noise subvolume used to set up the tec and tg thresholds in our AND approach. This noise subvolume provides a reliable estimation of the noise variance in the tomogram. We can analytically express the optimal stopping time for the NEV criterion as

{

}

0 t stop = arg min var( I N ) − var( I 0 − I t ) , t

where IN0 denotes the subvolume containing only noise in the original image, and || represents the

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absolute value. This formula states that the optimal stopping time is the time slot in which var(I 0 – I t ) reaches var(IN0 ). We applied the AND approach we present here to tomograms of three complex biological specimens to illustrate its performance: the Vaccinia virus,6 microtubules, and eukaryotic cells (Dictyostelium discoideum cells).3 The results we show with the Vaccinia virus especially illustrate the novel features we describe here—the background smoothing strategy and the NEV stopping criterion—whereas the tomograms of all the specimens illustrate the AND approach’s global performance.

600 Noise estimate variance

Experimental Applications

700

500 400 300 200 Background smoothing No background smoothing Variance of noise subvolume

100

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (seconds)

Vaccinia Virus

The results with the Vaccinia tomograms show the performance gained when we apply Gaussian filtering to the background. For comparison’s sake, Figure 4 shows the results for a slice of the tomogram with and without the background smoothing. In this tomogram, the 2D CED approach plays an essential role in enhancing the virus’s membranes and other planar features. Furthermore, the strategy to smooth the background has successfully highlighted the virus’s features over the background. The result we obtained without using this strategy (Figure 4c) has a significant remaining background. Figure 5 compares the time it takes the different AND versions (with and without the background smoothing technique) to reach the final denoised solution. In the noise subvolume of the original Vaccinia tomogram, the variance was 648.6 density units, and that value was used as NEV’s threshold. Figure 5 shows that the stopping time was 2.0 seconds (5 iterations at ␶ = 0.4) and 4.0 seconds (10 iterations at ␶ = 0.4), respectively. Our AND approach makes it possible to visualize the virus in the 3D space (Figure 6), helping us interpret and discern the virus’s architecture, which includes the outer membrane, the core (made up of a membrane and a palisade), and other interesting features.6 Microtubules

Figure 7 shows the results of denoising a microtubule tomogram. The noise reduction that AND provides also gives us continuity along the microtubule and a better delimitation of their boundaries. Figure 7c shows the microtubules and other aggregates in the tomogram, which can now be traced and delineated, thus helping researchers interpret the tomogram and make quantitative mea-

Figure 5. Solution time. The comparison of the time it takes for each strategy to reach the denoised version of the Vaccinia virus’s tomogram shows that AND’s strategy for background smoothing requires significantly less iterations and shorter diffusion time.

(a)

(b)

Figure 6. 3D isosurface representation of the Vaccinia virus. This visualization is made possible by our AND approach, which lets us discern the virus’s main features. The original tomogram is extremely noisy, and its 3D visualization (not shown here) is impossible to interpret.

sures. The visualization of the original raw tomogram (not shown here) is extremely noisy, and no interpretation is possible from it. Dictyostelium Discoideum Cells

Lastly, we applied our AND approach to tomograms of D. discoideum cells. Visualizing the in vivo structure of these cells’ eukaryotic cytoskeleton has been one of the most recent breakthroughs achieved via CryoET:3 researchers can visualize the actin filament network, which plays an essential role in maintaining cell shape and motility. Figure 8 shows the results of applying our AND approach to a tomogram of D. discoideum cells. Our approach significantly attenuates the noise and sub-

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(a)

(b)

(c)

Figure 7. Denoising microtubule tomograms with AND. (a) A slice of the original tomogram, (b) the result using AND, (c) and an isosurface visualization of the denoised tomogram clearly shows the microtubules and other aggregates contained in the tomogram. The diffusion process stopped at the fifth iteration, according to our stopping criterion.

(a)

(b)

(c)

Figure 8. Denoising tomograms of D. discoideum cells. (a) A slice from the original tomogram, (b) the same slice from the denoised tomogram, and (c) the volume texture representation of the denoised tomogram shows the actin filament network in the cell’s cytoplasm.

stantially highlights the cellular components’ structural organization; the cell’s membranes and the fibrous structures that compose the cell’s cytoplasm now stand out from the background. Figure 8c shows the volume texture representation of a piece of the tomogram. The actin filament network is now visible in the 3D space, showing the interaction between individual filaments and the assembly at the cell membrane.

standard tool in the CryoET field. Researchers normally use denoising to prepare tomograms for subsequent automatic image-analysis techniques. Some examples include the segmentation of structural components of interest3,5,6 or pattern-recognition techniques to automatically identify molecular features.4 In the future, we plan to apply this denoising technique to electron tomography of nanoparticles in the materials science field.17

Acknowledgments

W

e’ve addressed one of the most challenging computational problems in CryoET: signal-preserving noise removal from tomograms to allow visualization and interpretation of the information about the molecular organization of complex biological specimens. The denoising technique based on AND is now becoming a

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We thank Richard A. Crowther from the Medical Research Council Laboratory of Molecular Biology for fruitful discussions during the entire work; Ohad Medalia and Wolfgang Baumeister for providing us with the data sets of the Dictyostelium discoideum cells; and Jose L. Carrascosa, Cristina Risco, Marek Cyrklaff, and Wolfgang Baumeister for supplying the Vaccinia virus data sets. This work is supported by grants CICYT-

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TIC2002-00228, HFSP2003-ST00107, MEC-PR20040367, EU-FP6-502828, and the Medical Research Council. Sam Li was supported by a Human Frontier Science Program (HFSP) long-term fellowship.

References 1. W. Baumeister, “Electron Tomography: Towards Visualizing the Molecular Organization of the Cytoplasm,” Current Opinion in Structural Biology, vol. 12, no. 5, 2002, pp. 679–684. 2. A. Sali et al., “From Words to Literature in Structural Proteomics,” Nature, vol. 422, no. 6928, 2003, pp. 216–225. 3. O. Medalia et al., “Macromolecular Architecture in Eukaryotic Cells Visualized by Cryoelectron Tomography,” Science, vol. 298, no. 5596, 2002, pp. 1209–1213. 4. M. Beck et al., “Nuclear Pore Complex Structure and Dynamics Revealed by Cryoelectron Tomography,” Science, vol. 306, no. 5700, 2004, pp. 1387–1390. 5. K. Grunewald et al., “Three-Dimensional Structure of Herpes Simplex Virus from Cryo-Electron Tomography,” Science, vol. 302, no. 5649, 2003, pp. 1396–1398. 6. M. Cyrklaff et al., “Cryo-Electron Tomography of Vaccinia Virus,” Proc. Nat’l Academy of Sciences of USA, vol. 102, no. 8, 2005, pp. 2772–2777. 7. J. Frank, ed., Electron Tomography: Three-Dimensional Imaging with the Transmission Electron Microscope, Plenum Press, 1992. 8. A.S. Frangakis and F. Forster, “Computational Exploration of Structural Information from Cryo-Electron Tomograms,” Current Opinion in Structural Biology, vol. 14, no. 3, 2004, pp. 325–331. 9. J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, 1998. 10. P. Perona and J. Malik, “Scale Space and Edge Detection using Anisotropic Diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, 1990, pp. 629–639. 11. J. Weickert, “Coherence-Enhancing Diffusion Filtering,” Int’l J. Computer Vision, vol. 31, nos. 2–3, 1999, pp. 111–127. 12. J. Weickert, “Coherence-Enhancing Diffusion of Colour Images,” Image and Vision Computing, vol. 17, nos. 3–4, 1999, pp.

201–212. 13. J. Weickert and H. Scharr, “A Scheme for Coherence-Enhancing Diffusion Filtering with Optimized Rotation Invariance,” J. Visual Communication and Image Representation, vol. 13, nos. 1–2, 2002, pp. 103–118. 14. A.S. Frangakis and R. Hegerl, “Noise Reduction in Electron Tomographic Reconstructions Using Nonlinear Anisotropic Diffusion,” J. Structural Biology, vol. 135, no. 3, 2001, pp. 239–250. 15. J.J. Fernandez and S. Li, “An Improved Algorithm for Anisotropic Nonlinear Diffusion for Denoising Cryo-Tomograms,” J. Structural Biology, vol. 144, nos. 1–2, 2003, pp. 152–161. 16. P. Mrazek and M. Navara, “Selection of Optimal Stopping Time for Nonlinear Diffusion Filtering,” Int’l J. Computer Vision, vol. 52, nos. 2–3, 2003, pp. 189–203. 17. M. Weyland and P.A. Midgley, “Electron Tomography,” Materials Today, vol. 12, no. 2, 2004, pp. 32–40.

José-Jesús Fernández is an associate professor of computer architecture and is a member of the supercomputing-algorithms research group at the University of Almeria, Spain. His research interests include image processing, tomographic reconstruction, and highperformance computing applied to structural biology. Fernández has a PhD in computer science from the University of Granada, Spain. Contact him at jjfdez@ ual.es. Sam Li is a postdoctoral research fellow for the Medical Research Council Laboratory of Molecular Biology in Cambridge, UK. His research is focused on the structure and function of the yeast spindle pole body and on electron tomography. Li has a PhD in biochemistry from the University of Utah. Contact him at samli@ mrc-lmb.cam.ac.uk.

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