Intrinsic Classification of Single Particle Images by

Spectral Clustering

Yutaka Ueno

, Masaki Kawata

and Shinji Umeyama

Neuroscience Research Institute,

Grid Technology Research Center,

National Institute of Advanced Industrial Science and Technology, 1 Umezono Central-2,

Tsukuba, 305-8568, Japan

Abstract. An application of spectral clustering to single particle analysis of a

biological molecule is described. Using similarity scores for the given data set,

clustering was performed in a factor space made by the eigenvector of the

normalized similarity matrix. Image data was thus classified by means of

information intrinsic to the ensemble of given data. The method was tested on a

simulated transmission electron microscopy image and a real image data set of

70S ribosome. The average images of clusters were obtained by iterative

alignment, which successfully represented characteristic views of the target

molecules. Comparisons with traditional methods and techniques in practical

implementation are discussed.

1 Introduction

With the increasing attention given to single particle analysis for structural analysis of

biological molecules, novel classification algorithms and mathematical studies are in

demand for automated image processing. A currently popular image processing

method first tries to align all images in rotation and translations, then performs

multivariate statistic analysis –– in particular, correspondence analysis –– to classify

images into sub classes [1]. Then, images for a class are searched from the whole set

of images and the members of the class refined. This process is called multi-reference

alignment. While many studies in structural biology have benefited from this method,

very careful assessment of classification results are required [2].

Unfortunately, the obtained class average images are often not sufficient for

subsequent structural analysis of the molecule. Under the minimum dose condition in

transmission electron microscopy, the obtained images are immersed in substantial

background noise, and image alignment is a critical step [1]. Although a reference

free alignment [3] was proposed to align images without prior knowledge of the

particle structure, the classification task is hardly trivial. Invariant classification [4]

also addresses this problem of unsupervised clustering; however, its performance is

insufficient for wider application in this field [5].

In contrast, data mining studies –– for example, information discovery in large text

database and trade information database systems –– have researched various

Ueno Y., Kawata M. and Umeyama S. (2005).

Intrinsic Classiﬁcation of Single Particle Images by Spectral Clustering.

In Proceedings of the 1st International Workshop on Biosignal Processing and Classiﬁcation, pages 60-67

DOI: 10.5220/0001193800600067

 SciTePress

algorithms to meet the emerging demand. Spectral clustering [6], which groups data

by an eigenvalue method, has demonstrated advanced performance in text data

clustering [7] and image segmentation problems [8]. The kernel principal component

analysis [9] which has been developed differently, shares the same concept of

eigenvalue solutions, emphasizing the efficiency of the nonlinear kernel function

introduced to the distance measure for the observed data set.

In this paper, we consider the unsupervised clustering

of molecular images for

alignment within a consistent set group. Instead of a reference-based method, which

involves potential bias to preferences, we have applied spectral clustering, and

evaluated its performance with simulated transmission electron microscope images

and a data set from a real observation. Our results illustrate that appropriate groups

are resolved through intrinsic classification of images with a common motif. We also

discuss a mechanism for clustering images and indispensable techniques in practical

implementations for single particle analysis.

Fig. 1 illustrates an overview of single particle analysis with spectral clustering.

Digitized micrographs are windowed into individual particle images. Noise must then

be reduced by averaging images, grouping and alignment of images. Spectral

clustering involves mapping images onto a factor space, which is different from one

used in correspondence analysis [10], then the images are grouped by means of their

coordinates. The obtained images in a group are aligned and averaged by the iterative

alignment method [3].

The word "clustering" is used for unsupervised grouping, while "classification" implies

supervised grouping of a data set.

Fig. 1. Overview of single particle analysis with spectral clustering. Similar pairs of images are

mapped in near locations on a factor space

2 Method

2.1 Spectral Clustering

Below are the steps for the Ng-Jordan-Weiss algorithm [6] for spectral clustering of k

subsets for a given set of points S={s

,...,s

} in R

1. From the similarity matrix

A in R

defined by A

= exp(-||s

-s

/σ

) for i≠j and

=0.

2. Define D to be the diagonal matrix with D

= Σ

j=1,n

3. Normalize N = D

-1/2

4. Find x

,...,x

the k largest eigenvectors of N and form the matrix X=[x

,...,x

]

5. Normalize the rows of X to be of unit length

6. Treating each row of X as a point in R

, cluster into k clusters using k-means or

any other sensible clustering algorithm.

7. Assign the original point s

to cluster j if and only if row i of X was assigned to

cluster j.

The scaling parameter σcontrols how rapidly the similarity values fall off. There

are other studies for normalization step 3, which can adapt to individual problems [7].

2.2 A Fast Calculation of Similarity Between Images

The cross correlation of images is used for the similarity measure between images.

The similarity of two images, x and y, is evaluated by maximizing the cross

correlation in rotation R and translation t of image y.

Crr(x, y) = (x

− x )(y

− y )

pixels

∑

(1)

= max

R,t

Crr(x,T(y;R,t)) (2)

is the i-th pixel of image x; x and σ

is the mean and the standard deviation of

image x; T(y;R,t) designates the transformed image. Then, the similarity matrix is

defined as follows.

= exp((1- c

)/σ

) (3)

Spectral clustering requires these distance values between all pairs of images with

the best matching rotation and translations, that distinguish our method from the

traditional method. Since it requires extensive computation, we devised a fast

calculation method. Both rotation and translation searches are performed in two steps,

coarse and fine. In addition, a table of pixels addressing all pixels and rotation angles

is pre-calculated to reduce computation.

The term “affinity matrix” in the original paper was superseded by “similarity matrix”

because the former term was confusing in the field of molecular biology.

2.3 Dimension Reduction of Similarity Matrix

For very large numbers of images the calculation of a similarity matrix is

overwhelmed by the vast number of image-pair similarity scores. There are ways to

reduce the dimensions of the similarity matrix, however. For example, random

selection of images is sufficient in most cases for the first step of clustering images.

In addition, a significant portion of the similarity matrix can be estimated. For

example, eliminating odd images is a conventional method, which usually gives a low

correlation value to any of good images. The average correlation to randomly selected

images is useful to rank images and find odd images.

3 Results

3.1 Test with Simulated Images

In order to evaluate the performance of spectral clustering, we first applied it to

simulated projection images for transmission electron microscopy of a protein. A

trimer of blue tongue virus coat protein was chosen as well as a sample data set in

Fig. 2. Clustering result for simulated images of the blue tongue virus coat protein trimer. Seven

groups except three mirror images for 2,3,4 are depicted. (a) average images for each clusters

(b) two sample of images assigned to each clusters

Fig. 3. The distribution of the predefined projection angles (α, β) of individual images in

obtained clusters. With three fold symmetry of this particle, projection angles were limited

between 30 and 150 for

EMAN software [19]. Volume density was calculated from the atomic coordinates

data entry 1BTV in the protein databank. Using random projection angles, a total of

300 of synthetic images with size 100x100 was processed by the contrast transfer

function filter [5], and random noises were added.

Fig. 2 illustrates the cluster images obtained by averaging images of the clusters

with iterative alignment, and their member images. The common image features

among the cluster images were successfully extracted. Fig. 3 shows a distribution of

cluster members in the projection angle of individual images. Due to the three-fold

symmetry of the molecule, the imposed cluster number 10 seemed to be adequate to

distinguish classes of images. The scaling parameter was set to σ=1/√2 .

3.2 Test with Ribosome Images

Among successful results in application to real data set from transmission electron

microscopy, we demonstrates a result of 70S ribosome by cryo-electron microscopy

[11], which is provided by SPIDER software package [17] including the

three-dimensional volume data reconstructed from the data set; therefore, it allows us

to evaluate the results of clustering performance by comparisons with the projection

of the volume at various angles (Fig. 5c).

At first, odd images without particle images were eliminated from the 6996 total

images. The result of spectral clustering using 700 randomly selected images with

size 75x75 to form 20 clusters, with the scaling parameter σ=0.5, is illustrated in Fig.

4. The iterative alignment of the obtained cluster images nicely exhibited most

characteristic views of the ribosome published by the original study [11]. Different

values of the scaling parameter and various calculation methods for similarity matrix

were also tested, but results were similar and less distinct.

From raw images of the cluster members, it was difficult for the images to be

Fig. 4. The cluster average images of 70S ecoli ribosome [11] obtained by our spectral

clustering. Orientations of images are roughly adjusted to demonstrates similar views of particle

orientations, The top line lists five images small and globular; the second line showed the

famous portrait of the ribosome; in the third line particles look larger; and the fourth line are the

others

classified by hand (Fig. 5a). In addition, we failed reproduce the original result by

means of traditional method on software package IMAGIC [18]. Using the obtained

cluster images as seed images, similar images were searched in the observed images

by means of multi-reference alignment. The result improved the resolution of the

average image (Fig. 5b). They were consistent to the projection images from the three

dimensional reconstruction (Fig. 5c). Thus, it seems to be satisfactory for the

subsequent three-dimensional reconstruction, indicating that clusters had been

resolved appropriately.

While clustering results are usually checked by the resolution assessment such as

Fourier shell correlation [1], we tried to estimate projection angles of the observed

images from a reference volume of the structure. However, they were too noisy to

confirm their accuracy. The distribution of estimated projection angles showed less

clear separation of clusters than was the case with the previous test (Fig.3). Many

clusters occupied two different regions in the projection angle coordinates, which also

correspond to a pseudo mirror symmetry of the image.

4 Discussion and Conclusion

4.1 Comparison with the Traditional Method

The traditional method of correspondence analysis [12] takes a vector of the image

pixel; then clustering results also provide information about important pixels, which is

used as a mask to improve classification. With this method, small differences between

Fig. 5. (a) Raw images for selected distinct clusters in Fig. 4. (b) The average images refined by

multi-reference alignment against the whole image data set. (c) Corresponding projection

images of the volume data after a final refinement of three dimensional reconstruction

projection images were superbly clarified [13]. However, particle images with various

orientations are difficult to align in general, and the subsequent supervised

classification by preferred seed images tends to fail. In this procedure, we need to

align images to produce trends of classification. Moreover, images with different

features, for example, a top view and side views of the molecule, are all compared in

a single alignment, so that it is easy to encounter an artifact image.

We consider that images with different features should be separated in the

traditional framework with multivariate statistic analysis using aligned images. The

features of images should be resolved intrinsically from the context of the image

ensemble. Spectral clustering is an algorithm for addressing this kind of problem.

4.2 Mechanism of Clustering

Spectral clustering only takes into account distances within data sets, once the pixel

data have been used to construct the similarity matrix. One worries that a single

similarity value between two images provides little information as to the optimum

image alignment, and that it can be disturbed by noise. However, on the other hand,

there are other similarity values against other images which support the trend of

clustering from different points of view.

Therefore, once several images are found with very high similarity, and they are

confirmed as different from other images, they obtain a certain magnitude in factor

coordinates to be distinguished as a cluster. The factor space is constructed by the

eigenvector of normalized similarity matrix. The vector is not directly calculated from

pixels of a single image, but defined as the mutual relationship of images within the

ensemble of the data set. Images can be located at a position in space constructed by

cooperation of all image data, ensuring an unsupervised method of clustering. Using

this factor space, the distance of images becomes Euclidian, which is required for

stable convergence in the k-means algorithm [14].

4.3 Techniques in Practical Implementation

In our framework of spectral clustering, the calculation of a similarity matrix requires

extensive computation, which had been a disadvantage in comparison to other

methods [5]. However, our fast implementation allowed performance improvements

in feasible degrees on current microprocessors. For example, it takes about 45

minutes for 700 images of size 75x75 on a 3GHz Pentium4 (Intel Corp.). Besides our

code, there are ways to accelerate computations [15].

The accurate evaluation of similarity is key to the successful clustering of noisy

images, even though the dimension of the similarity matrix is reduced. Besides

random sampling of images, we can rank images so that significant images will

represent the most important part of the whole similarity matrix.

We demonstrated our successful results with spectral clustering for intrinsic

classification of single particle images. To establish a robust software system, the

scaling parameter of spectral clustering and other options need to be optimized.

Moreover, a learning algorithm to automate the parameter tuning [7],[8] is also an

intriguing topic for the future development of our software system [16].

Acknowledgement

The authors would like to thank Dr. Patrick Schultz, Dr. Bruno P. Klaholz and Dr.

Dino Moras at IGBMC/University Louis Pasteur for their helpful discussions. We are

greatly encouraged by Dr. Andreas Engel and Dr. Chikara Sato. In this study we used

electron microscopy images of the 70S ribosome, distributed with the software

package SPIDER, with kind permission by Dr. Joachim Frank.

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