## Solving structure with sparse, randomly-oriented x-ray data |

Optics Express, Vol. 20, Issue 12, pp. 13129-13137 (2012)

http://dx.doi.org/10.1364/OE.20.013129

Acrobat PDF (968 KB)

### Abstract

Single-particle imaging experiments of biomolecules at x-ray free-electron lasers (XFELs) require processing hundreds of thousands of images that contain very few x-rays. Each low-fluence image of the diffraction pattern is produced by a single, randomly oriented particle, such as a protein. We demonstrate the feasibility of recovering structural information at these extremes using low-fluence images of a randomly oriented 2D x-ray mask. Successful reconstruction is obtained with images averaging only 2.5 photons per frame, where it seems doubtful there could be information about the state of rotation, let alone the image contrast. This is accomplished with an expectation maximization algorithm that processes the low-fluence data in aggregate, and without any prior knowledge of the object or its orientation. The versatility of the method promises, more generally, to redefine what measurement scenarios can provide useful signal.

© 2012 OSA

## 1. Introduction

3. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E **80**, 026705 (2009). [CrossRef]

4. N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. **104**, 225501 (2010). [CrossRef] [PubMed]

^{−4}photons per pixel per frame). For this demonstration, in order to emulate realistic detector noise performance, we use the same pixel array detector chip that makes up the detector installed at the LCLS Coherent X-ray Imaging (CXI) beamline for the protein imaging experiment [5, 6

6. H. T. Philipp, M. W. Tate, and S. M. Gruner, “Low-flux measurements with Cornell’s LCLS integrating pixel array detector.” J. Inst. **6**, C11006 (2011). [CrossRef]

3. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E **80**, 026705 (2009). [CrossRef]

## 2. Expectation maximization algorithm

7. L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist. **41**, 164–171 (1970). [CrossRef]

*w*(

*i*), where each pixel

*i*is assigned a random value uniformly in the range [0, 1]. These values are iteratively updated by a rule that can only increase the likelihood of the model. The initial model is random because no information about the model is known.

*f*, is assigned a probability distribution,

*p*(

_{f}*r*), with respect to its unknown rotation,

*r*, relative to the current intensity model. The rotations are sampled in increments of 2

*π*/

*N*, where

*N*defines the angular resolution of the reconstruction. Each frame comprises photon occupancy,

*k*(

_{f}*i*), at pixel

*i*, which in our low-fluence experiment are almost all zero, the exceptions being equal to 1. Because the photon counts are independent Poisson samples of the intensity at each pixel, the probability is where

*i*+

*r*is rotation

*r*, applied to pixel

*i*,

*I*is the set of pixels recording photons in frame

_{f}*f*, with the final expression applying in the low-fluence limit. After normalizing the distributions,

*p*(

_{f}*r*), the algorithm proceeds to the second step.

*w*′(

*i*) is an average of the photon counts in all frames with the appropriate distribution of rotations applied to each one. Linear interpolation is used in both steps, when mapping one square grid (model) onto one that has been rotated (detector). We consider the reconstruction to have converged when the root-mean-square difference between successive models is below a cutoff, chosen to be 0.0001 times the mean pixel value. The EM algorithm is still valid when the data is winnowed by a structure-neutral criterion, such as the photon occupancy. In our implementation, for example, we discarded all frames with zero occupancy.

8. G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. **144**, 219–227 (2003). [CrossRef] [PubMed]

*c*

_{ff}_{′}are essentially all zero, and in any event do not distinguish frames derived from like or unlike particle orientations. A proposal [9

9. R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, and A. Singer, “Graph Laplacian tomography from unknown random projections,” IEEE Trans. Image Proc. **17**, 1891–1899 (2008). [CrossRef]

*c*

_{ff′}> 0). The EM algorithm, by contrast, compares each frame not with other frames but with a model. A greater sensitivity of rotation determination in the EM algorithm can be traced to the multiplicative nature of the comparison expressed by Eq. (1).

*σ*is the fraction of uncovered pixels.

*σ*of uncovered pixels, describes this signal prior. When modeling a true diffraction signal, the prior would instead be the Wilson distribution. To model the background, we used a single-valued distribution corresponding to uniform background counts across the detector.

*σ*= 0.6 gives

*R*≈ 0.01. A low fluence experiment with 2 signal photons per frame and this poor signal-to-noise would therefore be like a zero-background experiment with only 2

*R*= 0.02 photons per frame. Applying Poisson statistics to this low rate we find that only about 1 in 5000 frames would have 2 or more photons and be actually useful for the reconstruction.

## 3. Data collection

*K*of the tube spectrum to produce an approximately monochromatic x-ray beam of 8-keV Cu

_{β}*K*radiation. The rotation stage and aperture were mounted on the end of a 45 cm flight-path to produce a nearly flat-field illumination of x-rays across the 19 mm sample.

_{α}6. H. T. Philipp, M. W. Tate, and S. M. Gruner, “Low-flux measurements with Cornell’s LCLS integrating pixel array detector.” J. Inst. **6**, C11006 (2011). [CrossRef]

## 4. Results

3. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E **80**, 026705 (2009). [CrossRef]

*R*quoted above, which equals 0.26 for our chosen signal-to-noise. With this level of background our data set with 11.5 signal-photons/frame corresponds to a zero-background data set with only 3 photons/frame. The resulting reconstruction by the EM algorithm was therefore similar to that of our 2.5 photons/frame background-free reconstruction in both image quality and number of iterations (Fig. 3). Repetition of the algorithm starting with another initially random guess always results in a reconstruction that is identical to the eye, but with an arbitrary rotation of the final image. This has been verified by repeated trials (data not shown).

## 5. Conclusion

11. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science **321**, 379–382 (2008). [CrossRef] [PubMed]

12. D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. **12**, 035011 (2010). [CrossRef]

## Appendix: Information reduction by background

*w*be the x-ray flux integrated over one pixel in the time interval of a single frame. In our experiment

*w*takes two values: the background value

*ν*when the pixel is covered by mask, and

*ν*+

*μ*when the pixel is uncovered. Because the mask is given random rotations, the prior distribution on

*w*is, where

*σ*represents the fraction of open area in the mask.

*w*as a discrete number of photons

*k*. This is a Poisson process with conditional probability In the communications analogy

*k*is the message that is received when

*w*is sent. The information obtained in the measurement (transmitted by the channel) equals the mutual information

*I*associated with the joint probability distribution The mutual information is the difference of entropies where is the entropy of the photon counts and is the average entropy of the counts when the flux is given.

*ν*→ 0,

*μ*→ 0) where we can neglect

*k*> 1 in the sums: Here SN =

*μ*/

*ν*is the signal-to-noise. The zero background limit at low signal flux corresponds to the limit of infinite SN in the expression above: The ratio of these quantities depends only on SN and represents the ratio of the information rate is acquired with and without background: The function

*R*is linear at small SN and monotonically approaches 1 at large SN. Figure 4 shows a plot for the case

*σ*= 1/2.

## Acknowledgments

## References and links

1. | R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature |

2. | V. Elser, “Noise limits on reconstructing diffraction signals from random tomographs,” IEEE Trans. Inf. Theory |

3. | N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E |

4. | N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. |

5. | H. T. Philipp, L. J. Koerner, M. S. Hromalik, M. W. Tate, and S. M. Gruner, “Femtosecond radiation experiment detector for x-ray free-electron laser (XFEL) coherent x-ray imaging,” IEEE Trans. Nucl. Sci. |

6. | H. T. Philipp, M. W. Tate, and S. M. Gruner, “Low-flux measurements with Cornell’s LCLS integrating pixel array detector.” J. Inst. |

7. | L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist. |

8. | G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. |

9. | R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, and A. Singer, “Graph Laplacian tomography from unknown random projections,” IEEE Trans. Image Proc. |

10. | D. Giannakis, P. Schwander, and A. Ourmazd, “The symmetries of image formation by scattering. I. Theoretical framework,” arXiv:1009.5035 (2010). |

11. | P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science |

12. | D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. |

13. | C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J., |

**OCIS Codes**

(000.2190) General : Experimental physics

(040.0040) Detectors : Detectors

(040.7480) Detectors : X-rays, soft x-rays, extreme ultraviolet (EUV)

(110.7440) Imaging systems : X-ray imaging

(110.3055) Imaging systems : Information theoretical analysis

(110.4155) Imaging systems : Multiframe image processing

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 23, 2012

Revised Manuscript: April 30, 2012

Manuscript Accepted: May 16, 2012

Published: May 25, 2012

**Virtual Issues**

Vol. 7, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Hugh T. Philipp, Kartik Ayyer, Mark W. Tate, Veit Elser, and Sol M. Gruner, "Solving structure with sparse, randomly-oriented x-ray data," Opt. Express **20**, 13129-13137 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13129

Sort: Year | Journal | Reset

### References

- R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature406, 752–757 (2000). [CrossRef] [PubMed]
- V. Elser, “Noise limits on reconstructing diffraction signals from random tomographs,” IEEE Trans. Inf. Theory55, 4715–4722 (2009). [CrossRef]
- N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E80, 026705 (2009). [CrossRef]
- N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett.104, 225501 (2010). [CrossRef] [PubMed]
- H. T. Philipp, L. J. Koerner, M. S. Hromalik, M. W. Tate, and S. M. Gruner, “Femtosecond radiation experiment detector for x-ray free-electron laser (XFEL) coherent x-ray imaging,” IEEE Trans. Nucl. Sci.57, 3795–3799 (2010).
- H. T. Philipp, M. W. Tate, and S. M. Gruner, “Low-flux measurements with Cornell’s LCLS integrating pixel array detector.” J. Inst.6, C11006 (2011). [CrossRef]
- L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist.41, 164–171 (1970). [CrossRef]
- G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol.144, 219–227 (2003). [CrossRef] [PubMed]
- R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, and A. Singer, “Graph Laplacian tomography from unknown random projections,” IEEE Trans. Image Proc.17, 1891–1899 (2008). [CrossRef]
- D. Giannakis, P. Schwander, and A. Ourmazd, “The symmetries of image formation by scattering. I. Theoretical framework,” arXiv:1009.5035 (2010).
- P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science321, 379–382 (2008). [CrossRef] [PubMed]
- D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys.12, 035011 (2010). [CrossRef]
- C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J.,27, 379–423, 623–656 (1948).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.