## Isotropic scalar image visualization of vector differential image data using the inverse Riesz transform |

Biomedical Optics Express, Vol. 5, Issue 3, pp. 907-920 (2014)

http://dx.doi.org/10.1364/BOE.5.000907

Acrobat PDF (5544 KB)

### Abstract

X-ray Talbot moiré interferometers can now simultaneously generate two differential phase images of a specimen. The conventional approach to integrating differential phase is unstable and often leads to images with loss of visible detail. We propose a new reconstruction method based on the inverse Riesz transform. The Riesz approach is stable and the final image retains visibility of high resolution detail without directional bias. The outline Riesz theory is developed and an experimentally acquired X-ray differential phase data set is presented for qualitative visual appraisal. The inverse Riesz phase image is compared with two alternatives: the integrated (quantitative) phase and the modulus of the gradient of the phase. The inverse Riesz transform has the computational advantages of a unitary linear operator, and is implemented directly as a complex multiplication in the Fourier domain also known as the *spiral phase transform*.

© 2014 Optical Society of America

## 1. Introduction

*deflection*rather that the X-ray

*absorption*of conventional radiography. Clinical differential phase provides fine detail about the soft-tissue structure, detail simply not present in normal X-ray images. However, there is a catch: differential phase gives a highly asymmetric and anisotropic depiction of the optical path (or integrated refractive index) through a specimen. Many systems (using linear gratings) provide just one oriented differential phase image - the x derivative for example. More recent systems have been developed with crossed-gratings (or 2-D arrays) that simultaneously provide two differential phase images; the x and the y derivatives for example.

- • integration may encounter inconsistent gradient data between two components, resulting in damaging image artifacts,
- • integration typically increases dynamic range beyond the range of LCD displays,
- • integration introduces blur due to the intrinsic low-pass nature of 2-D integrator.

- Section 2 outlines the Fourier theory of differential imaging.
- Section 3 introduces the Riesz transform and the related complex embedding (known as the
*spiral phase transform)* - Section 4 outlines the main algorithmic definition of the process.
- Section 5 presents inverse Riesz visualization in comparison with conventional images.
- Section 6 concludes with a discussion of the main results.

## 2. Differential phase imaging prerequisites

3. H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express **19**(4), 3339–3346 (2011). [CrossRef] [PubMed]

5. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. **2**(4), 258–261 (2006). [CrossRef]

*black-box*system (such as two separate, orthogonal, Nomarski DIC images).

### 2.1 Differential phase from 2-D X-ray grating Fourier side-lobes

### 2.2 Phase gradients and Fourier correspondence

## 3. Riesz transform

### 3.1 Origin of the Riesz transform

15. R. S. Strichartz, “Radon inversion - variations on a theme,” Am. Math. Mon. **89**(6), 377–384 (1982). [CrossRef]

16. K. T. Smith and F. Keinert, “Mathematical foundations of computed tomography,” Appl. Opt. **24**(23), 3950–3957 (1985). [CrossRef] [PubMed]

17. A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math. **52**(2), 459–484 (1992). [CrossRef]

**R**is the Riesz transform, and its Fourier multiplier has unit magnitude and odd symmetry, very like the Hilbert multiplier, and can be compactly expressed in terms of the Fourier polar angle

### 3.2 Embedding of the vector Riesz transform in the complex scalar domain

*spiral phase transform*[13

13. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A **18**(8), 1862–1870 (2001). [CrossRef] [PubMed]

### 3.3 Discrepancy term of the inverse Riesz transform

20. K. G. Larkin and P. A. Fletcher, “A coherent framework for fingerprint analysis: are fingerprints Holograms?” Opt. Express **15**(14), 8667–8677 (2007). [CrossRef] [PubMed]

## 4. Algorithmic definition inverse Riesz transform applied to differential images

## 5. Application to differential images

### 5.1 Example from X-ray Talbot moiré interferometry

3. H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express **19**(4), 3339–3346 (2011). [CrossRef] [PubMed]

22. D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. **25**(10), 1653–1660 (1986). [CrossRef] [PubMed]

23. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**(1), 156–160 (1982). [CrossRef]

3. H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express **19**(4), 3339–3346 (2011). [CrossRef] [PubMed]

### 5.2 Visual assessment

### 5.3 Optical (Craik-O'Brien-Cornsweet) illusion

### 5.4 Discrepancy image

## 6. Discussion

## Acknowledgments

**19**(4), 3339–3346 (2011). [CrossRef] [PubMed]

## References and links

1. | M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. |

2. | M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc. |

3. | H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express |

4. | A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2 , |

5. | F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. |

6. | D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping (John Wiley and Sons, New York, 1998). |

7. | R. N. Bracewell, |

8. | K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A |

9. | R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. |

10. | D. C. Ghiglia and L. A. Romero, “Robust Two-Dimensional Weighted and Unweighted Phase Unwrapping That Uses Fast Transforms and Iterative Methods,” J. Opt. Soc. Am. A |

11. | S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. |

12. | E. M. Stein, |

13. | K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A |

14. | M. Felsberg and G. Sommer, “The Monogenic Signal,” IEEE Trans. Signal Process. |

15. | R. S. Strichartz, “Radon inversion - variations on a theme,” Am. Math. Mon. |

16. | K. T. Smith and F. Keinert, “Mathematical foundations of computed tomography,” Appl. Opt. |

17. | A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math. |

18. | M. Reisert and H. Burkhardt, “Complex derivative filters,” IEEE Trans. Image Process. |

19. | M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. Imaging Sciences |

20. | K. G. Larkin and P. A. Fletcher, “A coherent framework for fingerprint analysis: are fingerprints Holograms?” Opt. Express |

21. | K. Nagai, H. Itoh, G. Sato, T. Nakamura, K. Yamaguchi, T. Kondoh, and S. H. T. Den, “New phase retrieval method for single-shot x-ray Talbot imaging using windowed Fourier transform,” Optical Modeling and Performance Predictions V, Proc. SPIE |

22. | D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. |

23. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

24. | M. W. Levine and J. M. Shefner, |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(110.6760) Imaging systems : Talbot and self-imaging effects

(180.3170) Microscopy : Interference microscopy

(340.7450) X-ray optics : X-ray interferometry

(330.4595) Vision, color, and visual optics : Optical effects on vision

(100.4994) Image processing : Pattern recognition, image transforms

**ToC Category:**

X-Ray Microscopy and Imaging

**History**

Original Manuscript: December 13, 2013

Revised Manuscript: February 4, 2014

Manuscript Accepted: February 17, 2014

Published: February 26, 2014

**Citation**

Kieran G. Larkin and Peter A. Fletcher, "Isotropic scalar image visualization of vector differential image data using the inverse Riesz transform," Biomed. Opt. Express **5**, 907-920 (2014)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-5-3-907

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### References

- M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc.214(1), 7–12 (2004). [CrossRef] [PubMed]
- M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, “Using the Hilbert transform for 3D visualization of differential interference contrast microscope images,” J. Microsc.199(1), 79–84 (2000). [CrossRef] [PubMed]
- H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express19(4), 3339–3346 (2011). [CrossRef] [PubMed]
- A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2, 42, 866–868 (2003).
- F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys.2(4), 258–261 (2006). [CrossRef]
- D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping (John Wiley and Sons, New York, 1998).
- R. N. Bracewell, The Fourier Transform and its Applications, McGraw-Hill Electrical and Electronic Engineering Series (McGraw Hill, New York, 1978).
- K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A3(11), 1852–1861 (1986). [CrossRef]
- R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell.10(4), 439–451 (1988). [CrossRef]
- D. C. Ghiglia and L. A. Romero, “Robust Two-Dimensional Weighted and Unweighted Phase Unwrapping That Uses Fast Transforms and Iterative Methods,” J. Opt. Soc. Am. A11(1), 107–117 (1994). [CrossRef]
- S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett.30(3), 245–247 (2005). [CrossRef] [PubMed]
- E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton mathematical series (Princeton University Press, Princeton, N.J., 1970).
- K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A18(8), 1862–1870 (2001). [CrossRef] [PubMed]
- M. Felsberg and G. Sommer, “The Monogenic Signal,” IEEE Trans. Signal Process.49(12), 3136–3144 (2001). [CrossRef]
- R. S. Strichartz, “Radon inversion - variations on a theme,” Am. Math. Mon.89(6), 377–384 (1982). [CrossRef]
- K. T. Smith and F. Keinert, “Mathematical foundations of computed tomography,” Appl. Opt.24(23), 3950–3957 (1985). [CrossRef] [PubMed]
- A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math.52(2), 459–484 (1992). [CrossRef]
- M. Reisert and H. Burkhardt, “Complex derivative filters,” IEEE Trans. Image Process.17(12), 2265–2274 (2008). [CrossRef] [PubMed]
- M. Unser and N. Chenouard, “A Unifying Parametric Framework for 2D Steerable Wavelet Transforms,” SIAM J. Imaging Sciences6(1), 102–135 (2013). [CrossRef]
- K. G. Larkin and P. A. Fletcher, “A coherent framework for fingerprint analysis: are fingerprints Holograms?” Opt. Express15(14), 8667–8677 (2007). [CrossRef] [PubMed]
- K. Nagai, H. Itoh, G. Sato, T. Nakamura, K. Yamaguchi, T. Kondoh, and S. H. T. Den, “New phase retrieval method for single-shot x-ray Talbot imaging using windowed Fourier transform,” Optical Modeling and Performance Predictions V, Proc. SPIE 8127, San Diego, California, August 21, 2011.
- D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt.25(10), 1653–1660 (1986). [CrossRef] [PubMed]
- M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982). [CrossRef]
- M. W. Levine and J. M. Shefner, Fundamentals of Sensation and Perception (Pacific Grove, CA: Brooks/Cole, 1991) p. 675.

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