## Reliable method for calculating the center of rotation in parallel-beam tomography |

Optics Express, Vol. 22, Issue 16, pp. 19078-19086 (2014)

http://dx.doi.org/10.1364/OE.22.019078

Acrobat PDF (3968 KB)

### Abstract

High-throughput processing of parallel-beam X-ray tomography at synchrotron facilities is lacking a reliable and robust method to determine the center of rotation in an automated fashion, i.e. without the need for a human scorer. Well-known techniques based on center of mass calculation, image registration, or reconstruction evaluation work well under favourable conditions but they fail in cases where samples are larger than field of view, when the projections show low signal-to-noise, or when optical defects dominate the contrast. Here we propose an alternative technique which is based on the Fourier analysis of the sinogram. Our technique shows excellent performance particularly on challenging data.

© 2014 Optical Society of America

## 1. Introduction

2. B. Zitová and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. **21**(11), 977–1000 (2003). [CrossRef]

3. M. Yang, H. Gao, X. Li, F. Meng, and D. Wei, “A new method to determine the center of rotation shift in 2D-CT scanning system using image cross correlation,” NDT Int. **46**, 48–54 (2012). [CrossRef]

4. A. Brunetti and F. de Carlo, “A robust procedure for determination of center of rotation in tomography,” Proc. SPIE **5535**, 652–659 (2004). [CrossRef]

5. T. Donath, F. Beckmann, and A. Schreyer, “Automated determination of the center of rotation in tomography data,” J. Opt. Soc. Am. A **23**(5), 1048–1057 (2006). [CrossRef] [PubMed]

6. D. S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, “Calculation of the rotational centers in computed tomography sinograms,” IEEE Trans. Nucl. Sci. **37**(4), 1525–1540 (1990). [CrossRef]

## 2. Method

### 2.1 Proposed method

*f(x,y)*, is reconstructed from a set of its 1D projections,

*P,*at different angles

*θ*which is defined by [1]:Here,

*δ*is the Dirac’s delta-function, and

*t*is the projection co-ordinate at the detector plane. This is commonly known as a sinogram. Figure 4 illustrates a computationally generated phantom (Fig. 4(a)) and corresponding sinogram (Fig. 4(b)). The origin of

*t (t = 0)*coincides with the location of the CoR projected onto the detector plane.

*u*is the spatial frequency and

*ν*is the angular harmonic number [7

7. P. R. Edholm, R. M. Lewitt, and B. Lindholm, “Novel properties of the Fourier decomposition of the sinogram,” Proc. SPIE **671**, 8–18 (1986). [CrossRef]

7. P. R. Edholm, R. M. Lewitt, and B. Lindholm, “Novel properties of the Fourier decomposition of the sinogram,” Proc. SPIE **671**, 8–18 (1986). [CrossRef]

*r*is the outermost radius of the object. The origin of the double-wedge may be understood by considering the origin of the intensity in the Fourier transform at the given point in Fourier space

*(u,ν)*. The value arises as the convolution of a plane wave whose peaks lie at a slope defined by this point in frequency space. Where a feature in the sinogram lies parallel to this angle, there will be some significant value arising from this convolution. In a perfect sinogram, the sinusoidal trajectory corresponding to a feature in the object will have a minimum slope defined by the distance of that feature from the rotation center. The smallest slopes occurring in the complete sinogram are therefore defined by the outermost radius of the whole object, and so points in the Fourier transform corresponding to lesser slopes do not have significant intensity. A mathematical derivation is fully presented in the reference [7

7. P. R. Edholm, R. M. Lewitt, and B. Lindholm, “Novel properties of the Fourier decomposition of the sinogram,” Proc. SPIE **671**, 8–18 (1986). [CrossRef]

*t = 0*to cover the angles [π; 2π] (see Fig. 4(b)). As the location of the origin of

*t*is yet to be determined, it is not possible from

*a priori*knowledge to calculate a correct full revolution sinogram. Instead, an approximate mirror axis is assumed as shown in Figs. 4(c) and 4(d) at some arbitrary location 5 or 10 pixels away from

*t = 0*. In practice, one may use the horizontal centre of the image (HCoI). It is here that the properties of a projection sinogram in Fourier space become crucially important. Any estimated full revolution sinogram that has been calculated with a mirror axis displaced from the origin will show non-negligible Fourier coefficients outside the double-wedge region described in Eq. (3) (see Figs. 4(f) and 4(g)).

- 1- Create a reflection of the [0; π] sinogram about the HCoI axis. Shift the copy horizontally by
*s*in the search range*[s*. Then append the shifted copy vertically with the original one to form an estimated [0; 2π] sinogram._{min}; s_{max}] - 2- Apply Fourier transform to the estimated sinogram, multiply the result with a binary mask,
*M(u,ν)*, which is 1 outside the double-wedge region and 0 elsewhere, then calculate the average of coefficients, which we call the sinogram Fourier metric*Q*._{SF}

*r,*and so the method may be fully automated. As can be seen in Figs. 4(f) and 4(g) non-negligible Fourier coefficients caused by misalignment mainly distribute around the vertical line

*ν = 0*, thus we can safely choose

*r*much larger than the radius of the object. Particularly, in our work we use

*r*equal to the width of the image, corresponding to an object of diameter twice the width of the detector.

### 2.2 Prior methods

5. T. Donath, F. Beckmann, and A. Schreyer, “Automated determination of the center of rotation in tomography data,” J. Opt. Soc. Am. A **23**(5), 1048–1057 (2006). [CrossRef] [PubMed]

#### 2.2.1 Image registration methods

*s,*in the search range

*[s*to find the position giving the best correlation with the projection at 0°. Then, the CoR is calculated by Eq. (6). Following are the definitions of two different correlation coefficients:

_{min}; s_{max}]8. K. Pearson, “Contributions to the mathematical theory of evolution, III, Regression, Heredity, and Panmixia,” Philos. Trans. R. Soc. Lond. Ser. A **187**, 253–318 (1896). [CrossRef]

*P*and

_{ij}*P*is the numerical description of the projection at 0° and the flipped and shifted projection at 180°.

_{ij}^{s}9. C. Spearman, “The proof and measurement of association between two things,” Am. J. Psychol. **15**(1), 72–101 (1904). [CrossRef] [PubMed]

*P*is the intensity of a 1D array formed by ordering intensities of

_{k}*P*from the smallest to the largest.

_{ij}*K*is the total number of elements. The function

*R*returns the rank of intensity in the array.

*F*.

*s*is determined by locating the maximum value of

*P*.

_{pc}11. V. Argyriou and T. Vlachos, “Estimation of sub-pixel motion using gradient cross-correlation,” Electron. Lett. **39**(13), 980–982 (2003). [CrossRef]

#### 2.2.2 Reconstruction based methods

5. T. Donath, F. Beckmann, and A. Schreyer, “Automated determination of the center of rotation in tomography data,” J. Opt. Soc. Am. A **23**(5), 1048–1057 (2006). [CrossRef] [PubMed]

*w*is given byThe CoR is given by searching the global minimum of

*Q*and

_{SA}*Q*in the range of estimated CoRs.

_{SN}## 3. Results

### 3.1 Performance on a good data set

_{4}scintillator, and a PCO.4000 camera with an optic system giving a pixel size of 5 µm. The sample was positioned at 53 m from the source and the sample-detector distance was 1 m to take advantage of edge enhancement from refraction [12

12. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. **66**(12), 5486–5492 (1995). [CrossRef]

### 3.2 Performance on a challenging data set

*Q*and

_{SA}*Q*give the CoR values of 1851.5 and 1852.5, respectively. However, our sinogram Fourier metric technique returns a result identical with the human scorer.

_{SN}### 3.3 Sub-pixel performance

13. T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: Interpolation methods in medical image processing,” IEEE Trans. Med. Imaging **18**(11), 1049–1075 (1999). [CrossRef] [PubMed]

_{4}scintillator optically coupled to the PCO.4000 camera giving a pixel size of 12 µm. The sample-detector distance was 2.22 m. As indicated in Figs. 9(c) and 9(d) single pixel accuracy is not sufficient for finding the correct CoR.

*Q*values while linear interpolation shows fluctuations. However, both interpolators return the same CoR value of 357.8 over 713 pixel image width in a searching range of [-2; 6] and a step-size of 0.2. This CoR value is confirmed by visual inspection (Fig. 9(e)) at the border between the two half-sinograms. Although the correct value can be found with a linear interpolation scheme, the smoother approach to the minimum provided by cubic interpolation could lead to a more robust automatic calculation.

_{SF}## 4. Discussion and conclusion

*Q*, exhibits a global minimum at the correct alignment, allowing robust automatic location of this value. For practical automatic use when the object size,

_{SF}*r,*is not known to the system, an over-estimate of

*r*may be used.

## Acknowledgments

## References and links

1. | A. C. Kak and M. Slaney, |

2. | B. Zitová and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. |

3. | M. Yang, H. Gao, X. Li, F. Meng, and D. Wei, “A new method to determine the center of rotation shift in 2D-CT scanning system using image cross correlation,” NDT Int. |

4. | A. Brunetti and F. de Carlo, “A robust procedure for determination of center of rotation in tomography,” Proc. SPIE |

5. | T. Donath, F. Beckmann, and A. Schreyer, “Automated determination of the center of rotation in tomography data,” J. Opt. Soc. Am. A |

6. | D. S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, “Calculation of the rotational centers in computed tomography sinograms,” IEEE Trans. Nucl. Sci. |

7. | P. R. Edholm, R. M. Lewitt, and B. Lindholm, “Novel properties of the Fourier decomposition of the sinogram,” Proc. SPIE |

8. | K. Pearson, “Contributions to the mathematical theory of evolution, III, Regression, Heredity, and Panmixia,” Philos. Trans. R. Soc. Lond. Ser. A |

9. | C. Spearman, “The proof and measurement of association between two things,” Am. J. Psychol. |

10. | C. D. Kuglin and D. C. Hines, “The phase correlation image alignment method,” in |

11. | V. Argyriou and T. Vlachos, “Estimation of sub-pixel motion using gradient cross-correlation,” Electron. Lett. |

12. | A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. |

13. | T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: Interpolation methods in medical image processing,” IEEE Trans. Med. Imaging |

14. | Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL, 2013. |

15. | R. C. Gonzalez and R. E. Woods, |

**OCIS Codes**

(100.6950) Image processing : Tomographic image processing

(110.7440) Imaging systems : X-ray imaging

(340.6720) X-ray optics : Synchrotron radiation

**ToC Category:**

Image Processing

**History**

Original Manuscript: June 23, 2014

Revised Manuscript: July 18, 2014

Manuscript Accepted: July 18, 2014

Published: July 30, 2014

**Citation**

Nghia T. Vo, Michael Drakopoulos, Robert C. Atwood, and Christina Reinhard, "Reliable method for calculating the center of rotation in parallel-beam tomography," Opt. Express **22**, 19078-19086 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-19078

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### References

- A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).
- B. Zitová and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput.21(11), 977–1000 (2003). [CrossRef]
- M. Yang, H. Gao, X. Li, F. Meng, and D. Wei, “A new method to determine the center of rotation shift in 2D-CT scanning system using image cross correlation,” NDT Int.46, 48–54 (2012). [CrossRef]
- A. Brunetti and F. de Carlo, “A robust procedure for determination of center of rotation in tomography,” Proc. SPIE5535, 652–659 (2004). [CrossRef]
- T. Donath, F. Beckmann, and A. Schreyer, “Automated determination of the center of rotation in tomography data,” J. Opt. Soc. Am. A23(5), 1048–1057 (2006). [CrossRef] [PubMed]
- D. S. G. Azevedo, D. J. Schneberk, J. P. Fitch, and H. E. Martz, “Calculation of the rotational centers in computed tomography sinograms,” IEEE Trans. Nucl. Sci.37(4), 1525–1540 (1990). [CrossRef]
- P. R. Edholm, R. M. Lewitt, and B. Lindholm, “Novel properties of the Fourier decomposition of the sinogram,” Proc. SPIE671, 8–18 (1986). [CrossRef]
- K. Pearson, “Contributions to the mathematical theory of evolution, III, Regression, Heredity, and Panmixia,” Philos. Trans. R. Soc. Lond. Ser. A187, 253–318 (1896). [CrossRef]
- C. Spearman, “The proof and measurement of association between two things,” Am. J. Psychol.15(1), 72–101 (1904). [CrossRef] [PubMed]
- C. D. Kuglin and D. C. Hines, “The phase correlation image alignment method,” in IEEE 1975 Conference on Cybernetics and Society, New York (1975), pp. 163–165.
- V. Argyriou and T. Vlachos, “Estimation of sub-pixel motion using gradient cross-correlation,” Electron. Lett.39(13), 980–982 (2003). [CrossRef]
- A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.66(12), 5486–5492 (1995). [CrossRef]
- T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: Interpolation methods in medical image processing,” IEEE Trans. Med. Imaging18(11), 1049–1075 (1999). [CrossRef] [PubMed]
- Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL, 2013.
- R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002).

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