Object Detection by Correlation Coefficients Using ... - Semantic Scholar
2006
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004
Object Detection by Correlation Coefficients Using Azimuthally Averaged Reference Projections William V. Nicholson
Abstract—A method of computing correlation coefficients for object detection that takes advantage of using azimuthally averaged reference projections is described and compared with two alternative methods—computing a cross-correlation function or a local correlation coefficient versus the azimuthally averaged reference projections. Two examples of an application from structural biology involving the detection of projection views of biological macromolecules in electron micrographs are discussed. It is found that a novel approach to computing a local correlation coefficient versus azimuthally averaged reference projections, using a rotational correlation coefficient, outperforms using a cross-correlation function and a local correlation coefficient in object detection from simulated images with a range of levels of simulated additive noise. The three approaches perform similarly in detecting macromolecular views in electron microscope images of a globular macrolecular complex (the ribosome). The rotational correlation coefficient outperforms the other methods in detection of keyhole limpet hemocyanin macromolecular views in electron micrographs. Index Terms—Electron microscopy, object detection, single particle reconstruction.
I. INTRODUCTION
C
RYO-ELECTRON microscopy of biological macromolecules is a field which could benefit from effective object detection. Cryo-electron microscopy and single-particle reconstruction are emerging as powerful techniques in structural biology. In particular, these techniques can be applied to large macromolecular assemblies lacking symmetry, which are difficult to study by other methods. Ever improving resolution is currently being achieved for a diverse range of macromolecular complexes. Among recent examples, resolutions of 11.5 Å have been reported for the E. coli ribosome [1], a very large, multisubunit particle; 22 Å for bovine complex I [2], a 890 kD membrane protein; and 21 Å for the catalytic subunit of the DNA-dependent protein kinase [3], a relatively small, 460 kD soluble protein assembly. As radiation damage imposes strong limitations on the electron doses used to collect micrographs of biological macromolecules, images of individual particles have a low signal-to-noise ratio. The low signal-to-noise ratio can only be Manuscript received August 9, 2002; revised February 29, 2004. This work was supported in part by the National Science Foundation (NSF) (Cooperative Agreement ACI-9619019) through the University of Illinois under PACI subaward 776. The author is with the School of Biomedical Sciences, University of Leeds, Worsley Building, Leeds LS2 9JT, U.K. (e-mail: [email protected]) Digital Object Identifier 10.1109/TBME.2004.834271
overcome by aligning and averaging a large number of images using the techniques of single-particle reconstruction. The number of particle images required for a three-dimensional (3-D) reconstruction increases dramatically with the desired resolution (e.g., 73 000 particles were used in the ribosome reconstruction at 11.5-Å resolution). It is estimated that the number must be increased to about 1 million before it is even physically possible to reach “atomic” resolutions, i.e., better than 4 Å [4], [5], when using images of the currently available quality. As a first step in image processing, selection of particles from each micrograph is performed either manually, using software for interactive particle selection [6], [7], or by computer assisted, semi-automated methods. Either way, particle selection becomes a very labor-intensive step in image processing, as one works toward ever-increasing resolutions. Thus, automation of particle selection will be necessary to avoid this stage from becoming a serious bottleneck. Therefore, several approaches have been proposed for automatic particle selection, which have met with varying degrees of success. The approaches include methods that make use of various forms of template matching, local comparison of intensity values, edge detection, quantitative measures of the local image texture/statistics and neural networks and were recently reviewed [8]. Various template-matching methods based on correlation have been tried including cross-correlation with an azimuthally averaged [9], [10] and the synthetic discriminant filter [11]. In this paper, some alternative correlation-based methods of object detection are compared, including the cross-correlation function and the computation of a local correlation coefficient, and their performance with a simulated image and an experimental image [from electron microscopy (EM)] are discussed. It has been suggested that computing a local correlation coefficient can overcome problems of sensitivity of the cross-correlation function to intensity variations in the image [12]. Computing the cross-correlation function versus azimuthally averaged reference projections [9], [10], [13], [14] can be an effective method for object detection that trades off some efficiency in object detection in order to speed up the computation in comparison to computing the cross-correlation for each orientation. Also described is an alternative approach to computing a local correlation coefficient that attempts to address some problems, discussed below, with the “naïve” approach to computing a local correlation coefficient when using azimuthally averaged references.
0018-9294/04$20.00 © 2004 IEEE
NICHOLSON: OBJECT DETECTION BY CORRELATION COEFFICIENTS USING AZIMUTHALLY AVERAGED REFERENCE PROJECTIONS
II. METHODS The cross-correlation function is given by (1) is the image and is the reference. In where this paper, azimuthally averaged reference images were used to avoid the computational cost of examining each possible orientation of the (unaveraged) reference images. Computing a map of correlation coefficients involves computing a cross-correlation function (using an FFT-based algorithm) and normalizing for local variation due to high intensity in the test image [15], [16]. The correlation coefficient is given by (2), shown at the bottom of the page, where is the image, in which one wishes to carry out object detection, is the reference image. is a local average, and and the summation is carried out within a small area, defined by mask , for each point in the micrograph image. The term is the local variance in the micrograph image multiplied by the number of points under the mask , and it can be computed efficiently using one of two algorithms described by van Heel [17]. In this work the local variance was computed using the FFT-based algorithm described in van Heel’s paper as this is easier to implement for a circular mask although the sliding sums algoalgorithm than FFT-based order rithm is a faster order algorithm. The use of multiple azimuthally averaged references with the correlation coefficient is problematic. Round and globular projection views will match their azimuthally average better than other projection views that are not globular. How well an image matches its azimuthal average depends on the amount of power in the zeroth order circular harmonic. Stoschek and Hegerl proposed detecting nonglobular particles by correlating with other circular harmonics [11]. An alternative is to compute the correlation coefficient versus nonazimuthally averaged reference in each orientation; however, this is computationally intensive. The rotational correlation coefficient is an alternative approach that attempts to deal with these problems. Assume that and are two images, represented by functions and are the same functions in polar in register. coordinates. Azimuthal averages may be obtained for and (3) (4)
2007
Feature vectors may be obtained by taking the azimuthal av, with erage at discrete values of . For example, and where is the radius of the mask used. A correlation coefficient can be evaluated to compare and
(5)
(6) is the reference. The correlation coefficient Suppose as defined above can be evaluated for different translations of . Then, the numerator is a cross-correlation function and the azimuthally averaged reference. The between terms in the denominator are rotational variances multiplied by the area of the region of integration and can be obtained by azimuthally averaging an image and then evaluating its variance. Computing the rotational variance terms is equivalent to subtracting the mean and then computing the power in the zeroth order circular harmonic. The rotational correlation coefficient can be used to detect an object in a larger image by evaluating it locally, under a mask of the same size as the reference image. Normalized azimuthally averaged references may be used so . The rotational variance term is that obtained by integrating with respect to the radius and can be approximated by using a composite open Newton–Cotes formula
(7) with
(8) (9)
(2)
2008
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004
So, an effective method for evaluating the rotational variance term in an image is to use
(10) It is carried out by cross-correlating smoothed circles of the , in the summation with the image different radii to obtain for the local azimuthal average at a given position, subtracting the local mean and squaring. The cross-correlations may be carried out using an fast Fourier transform (FFT)-based algorithm. Computing the local rotaalgorithm for tional variance term is an order a square image of side pixels when rings are used. Computing rotational correlation coefficients is of interest when the method performs better at detection than the cross-correlation function or the “naive” local correlation coefficients and also when a larger number of reference projections are used (as in that case, obtaining the rotational variance does not add a substantial extra computational cost). The computational cost of the cross-correlation function for multiple azimuthally averfor a square image of aged references is order projection views are used. side when Typically, multiple references, for different projection views of the target object (either obtained by computation from a 3-D model or from experimental particle images), are used for the various template-matching functions (cross-correlation function, local correlation coefficient and rotational correlation coefficient). The template-matching function is computed for each reference for each position in the test image and the maximum value of the template-matching function is used at the given position, in the final map. Peaks in the maps for the various template-matching functions are detected by determination of a suitable threshold and subsequent pruning based on proximity of peaks to one another. Peaks within a user-chosen distance of one another are pruned in favor of the highest peak. Image processing and other computations were carried out using SPIDER an image processing package developed for biological EM [6] and other Fortran programs that implemented some of the new required functionality. In order to generate reference projections and projections of test objects in the simulated image, the projections are integrated in the projection direction. A projection in the direction is given by (11) and the computation is carried out by summation along the axis. The formation of experimental (bright-field) images of thin biological specimens from transmission EM is described by the weak phase approximation [18]. In the weak phase approximation, the image is a projection as defined above, convoluted with a point spread function. In EM, the Fourier transform of the point-spread function is the product of the contrast transfer
function (dependent on electron optical parameters such as the defocus, spherical aberration, and astigmatism) and the envelope function (dependent on the aperture and partial coherence effects). Reference images were chosen from projections from an artificial 3-D model in the case of the simulated images. The references were chosen from particle images in the case of the micrograph image. The chosen references were rotationally averaged. Then, for each micrograph the following is done. A map of correlation coefficients is computed for each reference. All of the peaks above a user set threshold are detected. Benchmarks were run of the different methods to obtain computation times on a single processor of a dual-processor PC with 1 GB of memory and AMD Athlon MP 2200-MHz processors. Simulated images and micrograph images of ice-embedded ribosome particles and keyhole limpet hemocyanin (KLH) were used as test data. The micrographs of ribosomes were part of a dataset made available to participants at the “Single Particle Reconstruction from Electron Microscope Images” course held at Pittsburgh Supercomputing Center on July 21–24, 1999. Comparable data is available as part of the SPIDER software distribution [6]. The micrographs of KLH were part of a dataset, made available by Zhu et al.,1 for testing automatic particle detection methods [19]. References in the cross-correlation function-based and the local correlation coefficient particle detection methods were obtained by selecting example particles (from another image than was used in the tests) and azimuthally averaging the particles. III. RESULTS Fig. 1(a) shows a test object (a man) and Fig. 1(b) shows another test object (a car). The man test object has views which are not very globular and the car test object is quite globular in all directions. Higher intensity values were used for the car. Projections in random orientations from each test object were placed at random locations to generate the simulated image in Fig. 1(c). The curves in the graph in Fig. 2 were generated for the cross-correlation function, the local correlation coefficient (versus azimuthally averaged references) and the rotational correlation coefficient to evaluate these methods for object detection. A parametric plot of hits (as a percentage of true particles) versus false particles (as a percentage of true particles), with each variable depending on the threshhold to obtain the free-response receiver operator characteristic (FROC) curves in the graph [20]–[24]. The number of hits and the number of false alarms are determined for a range of different choices of threshhold (from the height of the lowest detection peak to the height of the highest detection peak) and this is used to plot a point of the curve. At low threshholds, a high level of hits with more false positives is obtained and at high threshholds there is a lower level of hits with less false positives. Coordinates from automatic particle detection are accepted as hits when they are within some distance of the coordinates of the true particles which were used to generate the simulated image. The distance used is of the same order as the size of the particles. 1Available:
http://ami.scripps.edu/prtl_data/klh/klh_1k/index.htm
NICHOLSON: OBJECT DETECTION BY CORRELATION COEFFICIENTS USING AZIMUTHALLY AVERAGED REFERENCE PROJECTIONS
2
Fig. 1. (a) Projection from “man” test object in a 64 64 pixel image. (b) Projection from “car” test object in a 64 64 pixel image. (c) A 512 512 pixel field from the larger simulated test image (which was 2048 2048 pixels) with projections from the “man” and “car” test objects in random orientations and positions.
2
2
2
Fig. 2. Plot of %Hits versus False alarms for detection of projections of the “man” object in the simulated image (Fig. 1) using the cross-correlation function, the local correlation coefficient (versus azimuthally averaged references), and the rotational correlation coefficient. %Hits is the percentage of true particles that are successfully detected. False alarms is the number of false positives in an image.
The curves in Fig. 2 for the original simulated image show that the performance of the cross-correlation function and the “naive” local correlation coefficient is about the same. Also, the rotational correlation coefficient outperforms the other two methods for a range of threshholds—there are more hits for a given level of false positives. A fully automated method of object detection requires an automatic method of choosing a
2009
threshhold for peak detection. One approach to automatically determining a threshhold that is applicable to object detection using correlation coefficients was described by Goshtaby, et al. [16]. Fig. 3 shows similar curves for the simulated image with varying levels of additive Gaussian noise introduced. The rotational correlation coefficient outperforms the other methods for most choices of threshhold with the introduction of simulated noise. At higher levels of noise, when the projections of the man test object become difficult to distinguish by eye, the rotational correlation coefficient is less successful in detecting man test objects in preference to car test object projections. The same trend is observed at lower noise levels for the other two methods tested with the simulated data. Detection by the different methods was tested in 2048 2048 pixel simulated images. The reference projection images were 64 64 pixels with a 30-pixel-radius mask imposed. Running times were 52, 85, and 144 s for detection by the cross-correlation function, local correlation coefficient, and rotational correlation coefficient, respectively. The simulated image in Fig. 1 was generated using a choice of test objects that illustrates the advantage of using the rotational correlation coefficient for detecting a test object that is nonglobular in most projection views and that has a low variance. Fig. 4 shows an experimental image from EM. Fig. 5 shows curves for particle detection in the EM image. Coordinates from automatic particle detection are accepted as hits when they are within some distance of the coordinates from interactive particle selection. Particle images were obtained from another EM image which had been taken under similar experimental conditions for use as references. The EM image shows projections of the ribosome macromolecular complex that are relatively globular in appearance and the different methods perform similarly in detection to one another. Detection was carried out in a 2048 2048 pixel image with 100 reference projection images of 75 75 pixels, using a circular mask radius of 35 pixels. Running times for detection by the cross-correlation function, local correlation coefficient, and the rotational correlation coefficient were 204, 234, and 292 s, respectively. Results more similar to those with the simulated images were obtained in detecting particles in an EM image of KLH (Fig. 6). A dataset of KLH images has been made available for testing automatic particle detection methods [19]. The particles are more varied in shape and include a rectangular side view, a roughly circular top view, and intermediate views. Side views and intermediate views were detected, excluding the top views in order to avoid a reconstruction artefact due to an overabundant type of views. There are also some contaminants in the image including broken molecules and aggregates. Fig. 7 shows FROC curves for detection of KLH particles in the EM image. It is found that the rotational correlation coefficient outperforms the other two methods in detecting particles in this image. Detec2048 pixel image with 100 tion was carried out in a 2048 reference projection images of 272 272 pixels, using a circular mask radius of 112 pixels. Running times for detection by the cross-correlation function, local correlation coefficient, and the rotational correlation coefficient were 263, 300, and 566 s, respectively. The running time for the rotational correlation coefficient is relatively high in this case; because the large mask
2010
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004
Fig. 3. Plots of %Hits versus False alarms for detection of projections of the “man” object in the simulated image with varying levels of additive Gaussian noise introduced using the cross-correlation function, the local correlation coefficient (versus 20 azimuthally averaged reference projections) and the rotational correlation coefficient. %Hits is the percentage of true particles that are successfully detected. False alarms is the number of false positives in an image. To generate each of the noisy simulated images in (a)–(f), an image with Gaussian noise was created with a variance some multiple of the variance of the simulated image and added to the original simulated image (from Fig. 1). In (a)–(f), the factor was 1, 2, 4, 6, 8, and 10, respectively.
radius requires using a large number of rings in the computation of the rotational variance. IV. DISCUSSION Some simple algorithms were described for selecting particles, against a background of similar undesired particles, from cryo-electron micrographs. As discussed in Section I, various methods [9]–[11], [17], [25]–[27] have been proposed in the literature for automatic particle selection, the majority of these have been based on cross-correlation [9], [10], [17]. A number of the methods described have entered routine use in a semi-automated mode [6], [7], [17], [27]. Particles are selected automatically and then a human operator is required to reject false positives (or adjust parameters such as the correlation peak threshold controlling which particles will be selected). Moving to higher resolutions and increasing the number of particles will require the elimination of even this level of human input. ROC curves have been used in the object-detection literature to evaluate alternative methods [20], [21], [23], in a similar manner to that described in this paper and also in medicine to
evaluate the diagnostic value of tests [22]. A localized receiver operator characteristic curve was described by Abu-Naser, et al. [20]. FROC curves are considered more suitable for evaluating methods for detection in images where there are multiple instances of target objects and also multiple false alarms may occur [24]; so FROC curves were used to evaluate the methods described in this paper. It is not clear what the number of potential negatives in an image is; so the false alarm rate cannot be normalized as is usual in obtaining a ROC curve; so FROC curves, with the hit rate plotted versus the number of false alarms, are used to evaluate detection in images. Also, the ROC approach cannot readily be applied to detection tasks with more than one target per image—a problem to which the FROC approach can be applied [24]. The rotational correlation coefficient approach with azimuthally averaged references is more effective in detecting particles than the cross-correlation function and local correlation coefficient in the simulated images and the EM image of KLH particles but obtains similar results in the case of the EM image of ribosome particles. As described earlier, the rotational correlation coefficient is intended to address the problem that the local correlation coefficient with azimuthally averaged
NICHOLSON: OBJECT DETECTION BY CORRELATION COEFFICIENTS USING AZIMUTHALLY AVERAGED REFERENCE PROJECTIONS
Fig. 4. A micrograph showing ice-embedded ribosome particles. The image shows a 1024 1024 pixel field from the larger 2048 2048 test image. The micrograph was part of a dataset made available to participants at the “Single Particle Reconstruction from Electron Microscope Images” Course held at Pittsburgh Supercomputing Center on July 21–24, 1999. Comparable data is available as part of the SPIDER software distribution (Frank et al., 1996). Thanks are due to Joachim Frank for permission to use the ribosome test image.
2
2
Fig. 5. Plot of %Hits versus False alarms for detection of ribosome particles with 100 experimental particles as reference images using the cross-correlation function, the local correlation coefficient (versus azimuthally averaged references) and the rotational correlation coefficient. %Hits is the percentage of true particles that are successfully detected. False alarms is the number of false positives in an image.
reference projections tends to match round and globular views better. This is less useful for improving the performance in detecting ribosome particles as they are round and globular so they are already effectively detected by the local correlation coefficient. However, the simulated images include targets which are not round and globular and also the EM image of KLH particles shows projection views which are of more
2011
Fig. 6. Micrograph showing ice-embedded KLH particles. The micrograph was part of a dataset made available by Y. Zhu et al. [19] to the research community for testing automatic particle detection methods.
Fig. 7. Plot of %Hits versus false alarms for detection of KLH particles with 100 experimental particles as reference images using the cross-correlation function, the local correlation coefficient (versus azimuthally averaged references) and the rotational correlation coefficient. %Hits is the percentage of true particles that are successfully detected. False alarms is the number of false positives in an image.
varied shape. The rotational correlation coefficient is useful in reducing false alarms due to the circular top view (which it is desirable to eliminate as the over-represented view can cause an artefact in the subsequent single particle reconstruction). A simple method has been described for automatic particle selection using a template-matching approach, with suggested improvements on methods which use cross-correlation. Work on template-matching for pattern recognition and previous work on automatic particle selection for EM was briefly reviewed.
2012
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004
ACKNOWLEDGMENT The author would like to thank R. M. Glaeser for many invaluable discussions and J. Frank for permission to use the ribosome test image shown in Fig. 4. REFERENCES [1] I. S. Gabashvilli, R. K. Agrawal, C. M. T. Spahn, R. A. Grassucci, D. I. Svergun, J. Frank, and P. Penczek, “Solution structure of the E. coli 70S ribosome at 11.5 angstrom resolution,” Cell, vol. 100, pp. 537–549, 2000. [2] N. Grigorieff, “Three-dimensional structure of bovine NADH: Ubiquinone oxidoreductase (Complex I) at 22 angstrom in ice,” J. Mol. Biol., vol. 277, pp. 1033–1046, 1998. [3] C. Y. Chiu, R. B. Cary, D. J. Chen, S. R. Peterson, and P. L. Stewart, “Cryo-EM imaging of the catalytic subunit of the DNA-dependent protein kinase,” J. Mol. Biol., vol. 284, pp. 1075–1081, 1998. [4] R. M. Glaeser, “Electron crystallography: Present excitement, a nod to the past, anticipating the future,” J. Struct. Biol., vol. 128, pp. 3–14, 1999. [5] R. Henderson, “The potential and limitations of neutrons, electrons and X-rays for atomic resolution microscopy of unstained biological molecules,” Quart. Rev. Biophys., vol. 28, pp. 171–193, 1995. [6] J. Frank, M. Radermacher, P. Penczek, J. Zhu, Y. Li, M. Ladjadj, and A. Leith, “SPIDER and WEB: Processing and visualization of images in 3D electron microscopy and related fields,” J. Struct. Biol., vol. 116, pp. 190–199, 1996. [7] S. J. Ludtke, P. R. Baldwin, and W. Chiu, “EMAN: Semi-automated software for high resolution single particle reconstruction,” J. Struct. Biol., vol. 128, pp. 82–97, 1999. [8] W. V. Nicholson and R. M. Glaeser, “Automatic particle detection in electron microscopy,” J. Struct. Biol., vol. 133, pp. 90–101, 2001. [9] J. Frank and T. Wagenknecht, “Automatic selection of molecular images from electron micrographs,” Ultramicroscopy, vol. 12, pp. 169–176, 1984. [10] P. Thuman-Commike and W. Chiu, “Automatic detection of spherical particles in spot-scan electron cryomicroscopy images,” J. Microsc. Soc. Amer., vol. 1, pp. 181–201, 1995. [11] A. Stoschek and R. Hegerl, “Automated detection of macromolecules from electron micrographs using advanced filter techniques,” J. Microsc. (Oxford), vol. 185, pp. 76–94, 1997. [12] D. H. Ballard and C. M. Brown, Computer Vision. Englewood Cliffs, NJ: Prentice-Hall, 1982. [13] H. H. Arsenault and Y. Hsu, “Rotation-invariant discrimination between almost similar objects,” Appl. Opt., vol. 22, pp. 130–132, 1983. [14] Y. N. Hsu, H. H. Arsenault, and G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt., vol. 21, pp. 4012–4015, 1982.
[15] R. C. Gonzalez and R. E. Woods, Digital Image Processing. Reading, MA: Addison-Wesley, 1992. [16] A. Goshtaby, S. H. Gage, and J. F. Bartholic, “A two-stage cross-correlation approach to template matching,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 374–378, 1984. [17] M. van Heel, “Detection of objects in quantum-noise-limited images,” Ultramicroscopy, vol. 7, pp. 331–342, 1982. [18] J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies. San Diego, CA: Academic, 1996. [19] Y. Zhu, B. Carragher, F. Mouche, and C. S. Potter, “Automatic particle detection through efficient Hough transforms,” IEEE Trans. Med. Imag., vol. 22, pp. 1053–1062, Sept. 2003. [20] A. Abu-Naser, N. P. Galatsanos, and M. N. Wernick, “Object recognition based on impulse restoration with use of the expectation-maximization algorithm,” J. Opt. Soc. Amer. A, vol. 15, pp. 2327–2340, 1998. [21] R. S. Caprari, “Method of target detection in images by moment analysis of correlation peaks,” Appl. Opt., vol. 38, pp. 1317–1324, 1999. [22] H. A. Kestler, “ROC with confidence—A perl program for receiver operator characteristic curves,” Comput. Meth. Programs Biomed., vol. 64, pp. 133–136, 2001. [23] P. C. Miller and R. S. Caprari, “Demonstration of improved automatic target-recognition performance by moment analysis of correlation peaks,” Appl. Opt., vol. 38, pp. 1325–1331, 1999. [24] D. P. Chakraborty and L. H. L. Winter, “Free-response methodology: Alternative analysis and a new observer-performance experiment,” Radiology, vol. 174, pp. 873–881. [25] I. M. B. Martin, D. C. Marinescu, R. E. Lynch, and T. S. Baker, “Identification of spherical virus particles in digitized images of entire electron micrographs,” J. Struct. Biol., vol. 120, pp. 146–157, 1997. [26] G. Harauz and A. Fong-Lochovsky, “Automatic selection of macromolecules from electron micrographs,” Ultramicroscopy, vol. 31, pp. 333–344, 1989. [27] K. R. Lata, P. Penczek, and J. Frank, “Automatic particle picking from electron micrographs,” Ultramicroscopy, vol. 58, pp. 381–391, 1995.
William V. Nicholson received the B.Sc. degree from the University of Edinburgh, Edinburgh, U.K. in 1990 and the Ph.D. degree from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1995. He is currently working on projects in single particle reconstruction and image processing in electron microscopy in the School of Biomedical Sciences at the University of Leeds, Leeds, U.K.
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004
Object Detection by Correlation Coefficients Using Azimuthally Averaged Reference Projections William V. Nicholson
Abstract—A method of computing correlation coefficients for object detection that takes advantage of using azimuthally averaged reference projections is described and compared with two alternative methods—computing a cross-correlation function or a local correlation coefficient versus the azimuthally averaged reference projections. Two examples of an application from structural biology involving the detection of projection views of biological macromolecules in electron micrographs are discussed. It is found that a novel approach to computing a local correlation coefficient versus azimuthally averaged reference projections, using a rotational correlation coefficient, outperforms using a cross-correlation function and a local correlation coefficient in object detection from simulated images with a range of levels of simulated additive noise. The three approaches perform similarly in detecting macromolecular views in electron microscope images of a globular macrolecular complex (the ribosome). The rotational correlation coefficient outperforms the other methods in detection of keyhole limpet hemocyanin macromolecular views in electron micrographs. Index Terms—Electron microscopy, object detection, single particle reconstruction.
I. INTRODUCTION
C
RYO-ELECTRON microscopy of biological macromolecules is a field which could benefit from effective object detection. Cryo-electron microscopy and single-particle reconstruction are emerging as powerful techniques in structural biology. In particular, these techniques can be applied to large macromolecular assemblies lacking symmetry, which are difficult to study by other methods. Ever improving resolution is currently being achieved for a diverse range of macromolecular complexes. Among recent examples, resolutions of 11.5 Å have been reported for the E. coli ribosome [1], a very large, multisubunit particle; 22 Å for bovine complex I [2], a 890 kD membrane protein; and 21 Å for the catalytic subunit of the DNA-dependent protein kinase [3], a relatively small, 460 kD soluble protein assembly. As radiation damage imposes strong limitations on the electron doses used to collect micrographs of biological macromolecules, images of individual particles have a low signal-to-noise ratio. The low signal-to-noise ratio can only be Manuscript received August 9, 2002; revised February 29, 2004. This work was supported in part by the National Science Foundation (NSF) (Cooperative Agreement ACI-9619019) through the University of Illinois under PACI subaward 776. The author is with the School of Biomedical Sciences, University of Leeds, Worsley Building, Leeds LS2 9JT, U.K. (e-mail: [email protected]) Digital Object Identifier 10.1109/TBME.2004.834271
overcome by aligning and averaging a large number of images using the techniques of single-particle reconstruction. The number of particle images required for a three-dimensional (3-D) reconstruction increases dramatically with the desired resolution (e.g., 73 000 particles were used in the ribosome reconstruction at 11.5-Å resolution). It is estimated that the number must be increased to about 1 million before it is even physically possible to reach “atomic” resolutions, i.e., better than 4 Å [4], [5], when using images of the currently available quality. As a first step in image processing, selection of particles from each micrograph is performed either manually, using software for interactive particle selection [6], [7], or by computer assisted, semi-automated methods. Either way, particle selection becomes a very labor-intensive step in image processing, as one works toward ever-increasing resolutions. Thus, automation of particle selection will be necessary to avoid this stage from becoming a serious bottleneck. Therefore, several approaches have been proposed for automatic particle selection, which have met with varying degrees of success. The approaches include methods that make use of various forms of template matching, local comparison of intensity values, edge detection, quantitative measures of the local image texture/statistics and neural networks and were recently reviewed [8]. Various template-matching methods based on correlation have been tried including cross-correlation with an azimuthally averaged [9], [10] and the synthetic discriminant filter [11]. In this paper, some alternative correlation-based methods of object detection are compared, including the cross-correlation function and the computation of a local correlation coefficient, and their performance with a simulated image and an experimental image [from electron microscopy (EM)] are discussed. It has been suggested that computing a local correlation coefficient can overcome problems of sensitivity of the cross-correlation function to intensity variations in the image [12]. Computing the cross-correlation function versus azimuthally averaged reference projections [9], [10], [13], [14] can be an effective method for object detection that trades off some efficiency in object detection in order to speed up the computation in comparison to computing the cross-correlation for each orientation. Also described is an alternative approach to computing a local correlation coefficient that attempts to address some problems, discussed below, with the “naïve” approach to computing a local correlation coefficient when using azimuthally averaged references.
0018-9294/04$20.00 © 2004 IEEE
NICHOLSON: OBJECT DETECTION BY CORRELATION COEFFICIENTS USING AZIMUTHALLY AVERAGED REFERENCE PROJECTIONS
II. METHODS The cross-correlation function is given by (1) is the image and is the reference. In where this paper, azimuthally averaged reference images were used to avoid the computational cost of examining each possible orientation of the (unaveraged) reference images. Computing a map of correlation coefficients involves computing a cross-correlation function (using an FFT-based algorithm) and normalizing for local variation due to high intensity in the test image [15], [16]. The correlation coefficient is given by (2), shown at the bottom of the page, where is the image, in which one wishes to carry out object detection, is the reference image. is a local average, and and the summation is carried out within a small area, defined by mask , for each point in the micrograph image. The term is the local variance in the micrograph image multiplied by the number of points under the mask , and it can be computed efficiently using one of two algorithms described by van Heel [17]. In this work the local variance was computed using the FFT-based algorithm described in van Heel’s paper as this is easier to implement for a circular mask although the sliding sums algoalgorithm than FFT-based order rithm is a faster order algorithm. The use of multiple azimuthally averaged references with the correlation coefficient is problematic. Round and globular projection views will match their azimuthally average better than other projection views that are not globular. How well an image matches its azimuthal average depends on the amount of power in the zeroth order circular harmonic. Stoschek and Hegerl proposed detecting nonglobular particles by correlating with other circular harmonics [11]. An alternative is to compute the correlation coefficient versus nonazimuthally averaged reference in each orientation; however, this is computationally intensive. The rotational correlation coefficient is an alternative approach that attempts to deal with these problems. Assume that and are two images, represented by functions and are the same functions in polar in register. coordinates. Azimuthal averages may be obtained for and (3) (4)
2007
Feature vectors may be obtained by taking the azimuthal av, with erage at discrete values of . For example, and where is the radius of the mask used. A correlation coefficient can be evaluated to compare and
(5)
(6) is the reference. The correlation coefficient Suppose as defined above can be evaluated for different translations of . Then, the numerator is a cross-correlation function and the azimuthally averaged reference. The between terms in the denominator are rotational variances multiplied by the area of the region of integration and can be obtained by azimuthally averaging an image and then evaluating its variance. Computing the rotational variance terms is equivalent to subtracting the mean and then computing the power in the zeroth order circular harmonic. The rotational correlation coefficient can be used to detect an object in a larger image by evaluating it locally, under a mask of the same size as the reference image. Normalized azimuthally averaged references may be used so . The rotational variance term is that obtained by integrating with respect to the radius and can be approximated by using a composite open Newton–Cotes formula
(7) with
(8) (9)
(2)
2008
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004
So, an effective method for evaluating the rotational variance term in an image is to use
(10) It is carried out by cross-correlating smoothed circles of the , in the summation with the image different radii to obtain for the local azimuthal average at a given position, subtracting the local mean and squaring. The cross-correlations may be carried out using an fast Fourier transform (FFT)-based algorithm. Computing the local rotaalgorithm for tional variance term is an order a square image of side pixels when rings are used. Computing rotational correlation coefficients is of interest when the method performs better at detection than the cross-correlation function or the “naive” local correlation coefficients and also when a larger number of reference projections are used (as in that case, obtaining the rotational variance does not add a substantial extra computational cost). The computational cost of the cross-correlation function for multiple azimuthally averfor a square image of aged references is order projection views are used. side when Typically, multiple references, for different projection views of the target object (either obtained by computation from a 3-D model or from experimental particle images), are used for the various template-matching functions (cross-correlation function, local correlation coefficient and rotational correlation coefficient). The template-matching function is computed for each reference for each position in the test image and the maximum value of the template-matching function is used at the given position, in the final map. Peaks in the maps for the various template-matching functions are detected by determination of a suitable threshold and subsequent pruning based on proximity of peaks to one another. Peaks within a user-chosen distance of one another are pruned in favor of the highest peak. Image processing and other computations were carried out using SPIDER an image processing package developed for biological EM [6] and other Fortran programs that implemented some of the new required functionality. In order to generate reference projections and projections of test objects in the simulated image, the projections are integrated in the projection direction. A projection in the direction is given by (11) and the computation is carried out by summation along the axis. The formation of experimental (bright-field) images of thin biological specimens from transmission EM is described by the weak phase approximation [18]. In the weak phase approximation, the image is a projection as defined above, convoluted with a point spread function. In EM, the Fourier transform of the point-spread function is the product of the contrast transfer
function (dependent on electron optical parameters such as the defocus, spherical aberration, and astigmatism) and the envelope function (dependent on the aperture and partial coherence effects). Reference images were chosen from projections from an artificial 3-D model in the case of the simulated images. The references were chosen from particle images in the case of the micrograph image. The chosen references were rotationally averaged. Then, for each micrograph the following is done. A map of correlation coefficients is computed for each reference. All of the peaks above a user set threshold are detected. Benchmarks were run of the different methods to obtain computation times on a single processor of a dual-processor PC with 1 GB of memory and AMD Athlon MP 2200-MHz processors. Simulated images and micrograph images of ice-embedded ribosome particles and keyhole limpet hemocyanin (KLH) were used as test data. The micrographs of ribosomes were part of a dataset made available to participants at the “Single Particle Reconstruction from Electron Microscope Images” course held at Pittsburgh Supercomputing Center on July 21–24, 1999. Comparable data is available as part of the SPIDER software distribution [6]. The micrographs of KLH were part of a dataset, made available by Zhu et al.,1 for testing automatic particle detection methods [19]. References in the cross-correlation function-based and the local correlation coefficient particle detection methods were obtained by selecting example particles (from another image than was used in the tests) and azimuthally averaging the particles. III. RESULTS Fig. 1(a) shows a test object (a man) and Fig. 1(b) shows another test object (a car). The man test object has views which are not very globular and the car test object is quite globular in all directions. Higher intensity values were used for the car. Projections in random orientations from each test object were placed at random locations to generate the simulated image in Fig. 1(c). The curves in the graph in Fig. 2 were generated for the cross-correlation function, the local correlation coefficient (versus azimuthally averaged references) and the rotational correlation coefficient to evaluate these methods for object detection. A parametric plot of hits (as a percentage of true particles) versus false particles (as a percentage of true particles), with each variable depending on the threshhold to obtain the free-response receiver operator characteristic (FROC) curves in the graph [20]–[24]. The number of hits and the number of false alarms are determined for a range of different choices of threshhold (from the height of the lowest detection peak to the height of the highest detection peak) and this is used to plot a point of the curve. At low threshholds, a high level of hits with more false positives is obtained and at high threshholds there is a lower level of hits with less false positives. Coordinates from automatic particle detection are accepted as hits when they are within some distance of the coordinates of the true particles which were used to generate the simulated image. The distance used is of the same order as the size of the particles. 1Available:
http://ami.scripps.edu/prtl_data/klh/klh_1k/index.htm
NICHOLSON: OBJECT DETECTION BY CORRELATION COEFFICIENTS USING AZIMUTHALLY AVERAGED REFERENCE PROJECTIONS
2
Fig. 1. (a) Projection from “man” test object in a 64 64 pixel image. (b) Projection from “car” test object in a 64 64 pixel image. (c) A 512 512 pixel field from the larger simulated test image (which was 2048 2048 pixels) with projections from the “man” and “car” test objects in random orientations and positions.
2
2
2
Fig. 2. Plot of %Hits versus False alarms for detection of projections of the “man” object in the simulated image (Fig. 1) using the cross-correlation function, the local correlation coefficient (versus azimuthally averaged references), and the rotational correlation coefficient. %Hits is the percentage of true particles that are successfully detected. False alarms is the number of false positives in an image.
The curves in Fig. 2 for the original simulated image show that the performance of the cross-correlation function and the “naive” local correlation coefficient is about the same. Also, the rotational correlation coefficient outperforms the other two methods for a range of threshholds—there are more hits for a given level of false positives. A fully automated method of object detection requires an automatic method of choosing a
2009
threshhold for peak detection. One approach to automatically determining a threshhold that is applicable to object detection using correlation coefficients was described by Goshtaby, et al. [16]. Fig. 3 shows similar curves for the simulated image with varying levels of additive Gaussian noise introduced. The rotational correlation coefficient outperforms the other methods for most choices of threshhold with the introduction of simulated noise. At higher levels of noise, when the projections of the man test object become difficult to distinguish by eye, the rotational correlation coefficient is less successful in detecting man test objects in preference to car test object projections. The same trend is observed at lower noise levels for the other two methods tested with the simulated data. Detection by the different methods was tested in 2048 2048 pixel simulated images. The reference projection images were 64 64 pixels with a 30-pixel-radius mask imposed. Running times were 52, 85, and 144 s for detection by the cross-correlation function, local correlation coefficient, and rotational correlation coefficient, respectively. The simulated image in Fig. 1 was generated using a choice of test objects that illustrates the advantage of using the rotational correlation coefficient for detecting a test object that is nonglobular in most projection views and that has a low variance. Fig. 4 shows an experimental image from EM. Fig. 5 shows curves for particle detection in the EM image. Coordinates from automatic particle detection are accepted as hits when they are within some distance of the coordinates from interactive particle selection. Particle images were obtained from another EM image which had been taken under similar experimental conditions for use as references. The EM image shows projections of the ribosome macromolecular complex that are relatively globular in appearance and the different methods perform similarly in detection to one another. Detection was carried out in a 2048 2048 pixel image with 100 reference projection images of 75 75 pixels, using a circular mask radius of 35 pixels. Running times for detection by the cross-correlation function, local correlation coefficient, and the rotational correlation coefficient were 204, 234, and 292 s, respectively. Results more similar to those with the simulated images were obtained in detecting particles in an EM image of KLH (Fig. 6). A dataset of KLH images has been made available for testing automatic particle detection methods [19]. The particles are more varied in shape and include a rectangular side view, a roughly circular top view, and intermediate views. Side views and intermediate views were detected, excluding the top views in order to avoid a reconstruction artefact due to an overabundant type of views. There are also some contaminants in the image including broken molecules and aggregates. Fig. 7 shows FROC curves for detection of KLH particles in the EM image. It is found that the rotational correlation coefficient outperforms the other two methods in detecting particles in this image. Detec2048 pixel image with 100 tion was carried out in a 2048 reference projection images of 272 272 pixels, using a circular mask radius of 112 pixels. Running times for detection by the cross-correlation function, local correlation coefficient, and the rotational correlation coefficient were 263, 300, and 566 s, respectively. The running time for the rotational correlation coefficient is relatively high in this case; because the large mask
2010
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004
Fig. 3. Plots of %Hits versus False alarms for detection of projections of the “man” object in the simulated image with varying levels of additive Gaussian noise introduced using the cross-correlation function, the local correlation coefficient (versus 20 azimuthally averaged reference projections) and the rotational correlation coefficient. %Hits is the percentage of true particles that are successfully detected. False alarms is the number of false positives in an image. To generate each of the noisy simulated images in (a)–(f), an image with Gaussian noise was created with a variance some multiple of the variance of the simulated image and added to the original simulated image (from Fig. 1). In (a)–(f), the factor was 1, 2, 4, 6, 8, and 10, respectively.
radius requires using a large number of rings in the computation of the rotational variance. IV. DISCUSSION Some simple algorithms were described for selecting particles, against a background of similar undesired particles, from cryo-electron micrographs. As discussed in Section I, various methods [9]–[11], [17], [25]–[27] have been proposed in the literature for automatic particle selection, the majority of these have been based on cross-correlation [9], [10], [17]. A number of the methods described have entered routine use in a semi-automated mode [6], [7], [17], [27]. Particles are selected automatically and then a human operator is required to reject false positives (or adjust parameters such as the correlation peak threshold controlling which particles will be selected). Moving to higher resolutions and increasing the number of particles will require the elimination of even this level of human input. ROC curves have been used in the object-detection literature to evaluate alternative methods [20], [21], [23], in a similar manner to that described in this paper and also in medicine to
evaluate the diagnostic value of tests [22]. A localized receiver operator characteristic curve was described by Abu-Naser, et al. [20]. FROC curves are considered more suitable for evaluating methods for detection in images where there are multiple instances of target objects and also multiple false alarms may occur [24]; so FROC curves were used to evaluate the methods described in this paper. It is not clear what the number of potential negatives in an image is; so the false alarm rate cannot be normalized as is usual in obtaining a ROC curve; so FROC curves, with the hit rate plotted versus the number of false alarms, are used to evaluate detection in images. Also, the ROC approach cannot readily be applied to detection tasks with more than one target per image—a problem to which the FROC approach can be applied [24]. The rotational correlation coefficient approach with azimuthally averaged references is more effective in detecting particles than the cross-correlation function and local correlation coefficient in the simulated images and the EM image of KLH particles but obtains similar results in the case of the EM image of ribosome particles. As described earlier, the rotational correlation coefficient is intended to address the problem that the local correlation coefficient with azimuthally averaged
NICHOLSON: OBJECT DETECTION BY CORRELATION COEFFICIENTS USING AZIMUTHALLY AVERAGED REFERENCE PROJECTIONS
Fig. 4. A micrograph showing ice-embedded ribosome particles. The image shows a 1024 1024 pixel field from the larger 2048 2048 test image. The micrograph was part of a dataset made available to participants at the “Single Particle Reconstruction from Electron Microscope Images” Course held at Pittsburgh Supercomputing Center on July 21–24, 1999. Comparable data is available as part of the SPIDER software distribution (Frank et al., 1996). Thanks are due to Joachim Frank for permission to use the ribosome test image.
2
2
Fig. 5. Plot of %Hits versus False alarms for detection of ribosome particles with 100 experimental particles as reference images using the cross-correlation function, the local correlation coefficient (versus azimuthally averaged references) and the rotational correlation coefficient. %Hits is the percentage of true particles that are successfully detected. False alarms is the number of false positives in an image.
reference projections tends to match round and globular views better. This is less useful for improving the performance in detecting ribosome particles as they are round and globular so they are already effectively detected by the local correlation coefficient. However, the simulated images include targets which are not round and globular and also the EM image of KLH particles shows projection views which are of more
2011
Fig. 6. Micrograph showing ice-embedded KLH particles. The micrograph was part of a dataset made available by Y. Zhu et al. [19] to the research community for testing automatic particle detection methods.
Fig. 7. Plot of %Hits versus false alarms for detection of KLH particles with 100 experimental particles as reference images using the cross-correlation function, the local correlation coefficient (versus azimuthally averaged references) and the rotational correlation coefficient. %Hits is the percentage of true particles that are successfully detected. False alarms is the number of false positives in an image.
varied shape. The rotational correlation coefficient is useful in reducing false alarms due to the circular top view (which it is desirable to eliminate as the over-represented view can cause an artefact in the subsequent single particle reconstruction). A simple method has been described for automatic particle selection using a template-matching approach, with suggested improvements on methods which use cross-correlation. Work on template-matching for pattern recognition and previous work on automatic particle selection for EM was briefly reviewed.
2012
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 11, NOVEMBER 2004
ACKNOWLEDGMENT The author would like to thank R. M. Glaeser for many invaluable discussions and J. Frank for permission to use the ribosome test image shown in Fig. 4. REFERENCES [1] I. S. Gabashvilli, R. K. Agrawal, C. M. T. Spahn, R. A. Grassucci, D. I. Svergun, J. Frank, and P. Penczek, “Solution structure of the E. coli 70S ribosome at 11.5 angstrom resolution,” Cell, vol. 100, pp. 537–549, 2000. [2] N. Grigorieff, “Three-dimensional structure of bovine NADH: Ubiquinone oxidoreductase (Complex I) at 22 angstrom in ice,” J. Mol. Biol., vol. 277, pp. 1033–1046, 1998. [3] C. Y. Chiu, R. B. Cary, D. J. Chen, S. R. Peterson, and P. L. Stewart, “Cryo-EM imaging of the catalytic subunit of the DNA-dependent protein kinase,” J. Mol. Biol., vol. 284, pp. 1075–1081, 1998. [4] R. M. Glaeser, “Electron crystallography: Present excitement, a nod to the past, anticipating the future,” J. Struct. Biol., vol. 128, pp. 3–14, 1999. [5] R. Henderson, “The potential and limitations of neutrons, electrons and X-rays for atomic resolution microscopy of unstained biological molecules,” Quart. Rev. Biophys., vol. 28, pp. 171–193, 1995. [6] J. Frank, M. Radermacher, P. Penczek, J. Zhu, Y. Li, M. Ladjadj, and A. Leith, “SPIDER and WEB: Processing and visualization of images in 3D electron microscopy and related fields,” J. Struct. Biol., vol. 116, pp. 190–199, 1996. [7] S. J. Ludtke, P. R. Baldwin, and W. Chiu, “EMAN: Semi-automated software for high resolution single particle reconstruction,” J. Struct. Biol., vol. 128, pp. 82–97, 1999. [8] W. V. Nicholson and R. M. Glaeser, “Automatic particle detection in electron microscopy,” J. Struct. Biol., vol. 133, pp. 90–101, 2001. [9] J. Frank and T. Wagenknecht, “Automatic selection of molecular images from electron micrographs,” Ultramicroscopy, vol. 12, pp. 169–176, 1984. [10] P. Thuman-Commike and W. Chiu, “Automatic detection of spherical particles in spot-scan electron cryomicroscopy images,” J. Microsc. Soc. Amer., vol. 1, pp. 181–201, 1995. [11] A. Stoschek and R. Hegerl, “Automated detection of macromolecules from electron micrographs using advanced filter techniques,” J. Microsc. (Oxford), vol. 185, pp. 76–94, 1997. [12] D. H. Ballard and C. M. Brown, Computer Vision. Englewood Cliffs, NJ: Prentice-Hall, 1982. [13] H. H. Arsenault and Y. Hsu, “Rotation-invariant discrimination between almost similar objects,” Appl. Opt., vol. 22, pp. 130–132, 1983. [14] Y. N. Hsu, H. H. Arsenault, and G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt., vol. 21, pp. 4012–4015, 1982.
[15] R. C. Gonzalez and R. E. Woods, Digital Image Processing. Reading, MA: Addison-Wesley, 1992. [16] A. Goshtaby, S. H. Gage, and J. F. Bartholic, “A two-stage cross-correlation approach to template matching,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 374–378, 1984. [17] M. van Heel, “Detection of objects in quantum-noise-limited images,” Ultramicroscopy, vol. 7, pp. 331–342, 1982. [18] J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies. San Diego, CA: Academic, 1996. [19] Y. Zhu, B. Carragher, F. Mouche, and C. S. Potter, “Automatic particle detection through efficient Hough transforms,” IEEE Trans. Med. Imag., vol. 22, pp. 1053–1062, Sept. 2003. [20] A. Abu-Naser, N. P. Galatsanos, and M. N. Wernick, “Object recognition based on impulse restoration with use of the expectation-maximization algorithm,” J. Opt. Soc. Amer. A, vol. 15, pp. 2327–2340, 1998. [21] R. S. Caprari, “Method of target detection in images by moment analysis of correlation peaks,” Appl. Opt., vol. 38, pp. 1317–1324, 1999. [22] H. A. Kestler, “ROC with confidence—A perl program for receiver operator characteristic curves,” Comput. Meth. Programs Biomed., vol. 64, pp. 133–136, 2001. [23] P. C. Miller and R. S. Caprari, “Demonstration of improved automatic target-recognition performance by moment analysis of correlation peaks,” Appl. Opt., vol. 38, pp. 1325–1331, 1999. [24] D. P. Chakraborty and L. H. L. Winter, “Free-response methodology: Alternative analysis and a new observer-performance experiment,” Radiology, vol. 174, pp. 873–881. [25] I. M. B. Martin, D. C. Marinescu, R. E. Lynch, and T. S. Baker, “Identification of spherical virus particles in digitized images of entire electron micrographs,” J. Struct. Biol., vol. 120, pp. 146–157, 1997. [26] G. Harauz and A. Fong-Lochovsky, “Automatic selection of macromolecules from electron micrographs,” Ultramicroscopy, vol. 31, pp. 333–344, 1989. [27] K. R. Lata, P. Penczek, and J. Frank, “Automatic particle picking from electron micrographs,” Ultramicroscopy, vol. 58, pp. 381–391, 1995.
William V. Nicholson received the B.Sc. degree from the University of Edinburgh, Edinburgh, U.K. in 1990 and the Ph.D. degree from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1995. He is currently working on projects in single particle reconstruction and image processing in electron microscopy in the School of Biomedical Sciences at the University of Leeds, Leeds, U.K.
