## Fourier-wavelet regularization of phase retrieval in x-ray in-line phase tomography

JOSA A, Vol. 26, Issue 8, pp. 1876-1881 (2009)

http://dx.doi.org/10.1364/JOSAA.26.001876

Enhanced HTML Acrobat PDF (273 KB)

### Abstract

Phase sensitive x-ray imaging extends standard x-ray microscopy techniques by offering up to a thousand times higher sensitivity than absorption-based techniques. If an object is illuminated with a sufficiently coherent beam, phase contrast is achieved by moving the detector downstream from the object. There is a quantitative relationship between the phase shift induced by the object and the recorded intensity. This relationship can be used to retrieve the phase shift induced by the object through the solution of an inverse problem. Since the phase shift can be considered as a projection through the 3D refractive index, the latter can be reconstructed using standard tomographic inversion techniques. However, the determination of the phase shift from the recorded intensity is an ill-posed inverse problem. We investigate the application of Fourier-wavelet regularized deconvolution (ForWaRD) to this problem. The method is evaluated using simulated and experimental data and is shown to increase the quality of reconstructions, in terms of normalized RMS error and compared with standard Tikhonov regularization, at a three times increase in computational cost.

© 2009 Optical Society of America

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(110.7440) Imaging systems : X-ray imaging

(180.0180) Microscopy : Microscopy

(110.7410) Imaging systems : Wavelets

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 13, 2009

Revised Manuscript: June 8, 2009

Manuscript Accepted: June 24, 2009

Published: July 29, 2009

**Virtual Issues**

Vol. 4, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Max Langer, Peter Cloetens, and Françoise Peyrin, "Fourier-wavelet regularization of phase retrieval in x-ray in-line phase tomography," J. Opt. Soc. Am. A **26**, 1876-1881 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=josaa-26-8-1876

Sort: Year | Journal | Reset

### References

- A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. (N.Y.) 2, 473-475 (1996).
- 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, 5486-5492 (1995). [CrossRef]
- J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttgart) 46, 121-125 (1977).
- K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961-2964 (1996). [CrossRef] [PubMed]
- P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999). [CrossRef]
- X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33-42 (2003).
- J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617-1619 (2007). [CrossRef] [PubMed]
- M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556-4566 (2008). [CrossRef] [PubMed]
- R. Neelamani, H. Choi, and R. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418-433 (2004). [CrossRef]
- A. N. Tikhonov and V. A. Arsenin, Solution of Ill-Posed Problems (Winston, 1977).
- S. Zabler, P. Cloetens, J. P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. 76, 1-7 (2005). [CrossRef]
- D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika 81, 425-455 (1994). [CrossRef]
- D. L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613-627 (1995). [CrossRef]
- D. L. Donoho and I. M. Johnstone, “Asymptotic minimaxity of wavelet estimators with sampled data,” Stat. Sin. 9, 1-32 (1999).
- S. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997). [CrossRef]
- S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).
- I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992). [CrossRef]
- J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007). [CrossRef] [PubMed]

## 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.