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Phase imaging coupled to micro-tomography acquisition has emerged as a powerful tool to investigate specimens in a non-destructive manner. While the intensity data can be acquired and recorded, the phase information of the signal has to be “retrieved” from the data modulus only. Phase retrieval is an ill-posed non-linear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several linear phase recovery methods have been proposed and it is expected that some limitations resulting from the linearization of the direct problem will be overcome by taking into account the non-linearity of the phase problem. To achieve this goal, we propose and evaluate a non-linear algorithm for in-line phase micro-tomography based on an iterative Landweber method with an analytic calculation of the Fréchet derivative of the phase-intensity relationship and of its adjoint. The algorithm was applied in the projection space using as initialization the linear mixed solution. The efficacy of the regularization scheme was evaluated on simulated objects with a slowly and a strongly varying phase. Experimental data were also acquired at ESRF using a propagation-based X-ray imaging technique for the given pixel size 0.68 μm. Two regularization scheme were considered: first the initialization was obtained without any prior on the ratio of the real and imaginary parts of the complex refractive index and secondly a constant a priori value was assumed on . The tomographic central slices of the refractive index decrement were compared and numerical evaluation was performed. The non-linear method globally decreases the reconstruction errors compared to the linear algorithm and is achieving better reconstruction results if no prior is introduced in the initialization solution. For in-line phase micro-tomography, this non-linear approach is a new and interesting method in biomedical studies where the exact value of the a priori ratio is not known.

Hard X-ray imaging with a high spatial resolution is nowadays a powerful tool to investigate specimens in 2D or 3D in a non-destructive manner. For an object illuminated by partially coherent light sources, a simple and effective technique known as propagation-based phase contrast has a particular interest because of its simple imaging set-up. Additional optical elements are not required and the phase contrast images can be recorded by simply letting the X-ray beam propagate in free space after interaction with the sample [

Compared with attenuation-based imaging techniques, the main interest in X-ray phase imaging is the pos- sibility to study objects with either negligible absorption or dense objects with small density variations. More- over, in the hard X-ray region, the phase shift for low-Z elements improves the sensitivity with three orders of magnitude [

The most common algorithms for the phase retrieval problem for short propagation distances rely on the linearization of the Fresnel diffraction relationship [

In this paper, we consider the case of in-line phase tomography using different propagation distances. We extend to the tomographic case a modified non-linear algorithm proposed in [

the Landweber iterative algorithm is modified by replacing this term

paper, we first summarize this new multi-image non-linear

For a parallel, partially coherent, monochromatic X-ray wave with wavelength

with

where

are considered as the projections of the absorption index

The intensity distribution

where

volution of the complex transmission function

As detailed in [

function of the phase

previously proposed Landweber iterative approach [

The aim of this non-linear approach

where

The minimizer of the cost functional is calculated iteratively with a non-linear Landweber type scheme [

The classical Landweber method is modified with a variable step

that can be obtained using the explicit calculation [

where

Thanks to the analytical expressions of the Fréchet derivative and of its adjoint, it was possible to decrease the computation time and to obtain a better convergence in the radiographic case. In order to take into account the intensity maps, better results were obtained when the propagation distances were used in a random way and not in a successive way during the iterations.

It was shown that the non-linear algorithm improves the solution obtained with a linear algorithm in the radio- graphic case for simulated data [

For a projection angle

where

It is well known that the regularization parameter plays a crucial role in the convergence of the iterative regularization methods, therefore it has to be chosen carefully. In all the studies of this work, large and small values of the parameter leading to poor reconstruction results are first chosen. Then, the optimal value of the re- gularization parameter is gradually refined by trial-and-error with a decreasing interval. For a well-chosen para- meter, the errors on the intensity maps for all the three propagation distances are decreased. If the convergence is not achieved for all the propagation distances, the value of the regularization parameter is refined till the convergence is achieved.

The new non-linear inversion method has been tested on simulated images and on experimental data for a multi- material object.

Two phantoms were defined in a deterministic fashion [

4πβ/λ (cm^{−1}) | 2πδ_{r/}λ (×100 cm^{−1}) | |
---|---|---|

Aluminium | 5.130 | 11.4 |

Ethanol | 0.305 | 4.00 |

Oil | 0.262 | 4.36 |

PMMA | 0.425 | 5.63 |

Water | 0.482 | 4.87 |

Polymer | 0.306 | 5.00 |

white noise with various peak-to-peak signal to noise ratios (PPSNR) of 24 dB. The peak-to-peak signal to noise ratio is defined by:

where

In order to asses the performance of the new regularization scheme, two types of objects were considered for short propagation distances and weak absorption, one with a slowly varying phase and another with a strongly varying phase.

The experimental set-up used is equivalent to the one for the standard propagation based technique described in [

The efficiency of the proposed new regularization scheme was analysed by comparing the numerical results obtained with the NL method with the phases retrieved with the CTF, TIE and mixed linear approach in the radiographic case. The four methods were tested for weakly and strongly varying phases, and for noise-free and noisy data.

Since the ideal reconstruction image is available, direct comparisons can be made. The method will be quantitatively evaluated by measuring the normalized mean square error (NMSE) using the

where

The NMSE (Equation (11)) for all the methods are presented in

The evolution of the NMSE as a function of the iterations number is displayed in

TIE | CTF | Mixed | Nonlinear | |
---|---|---|---|---|

Strong phase without noise | 25.54 | 42.52 | 26.81 | 7.57 |

Weak phase without noise | 1.5 | 24.37 | 3.16 | 2.35 |

Strong phase PPSNR = 24 dB | 262.13 | 56.54 | 27.78 | 11.58 |

Weak phase PPSNR = 24 dB | 459 | 54.67 | 12.36 | 8.69 |

obtained for the three distances. The initialization of the NL algorithm for these four situations was given by the linear mixed solution. These curves show that the proposed algorithm has good convergence properties. Very few iterates are necessary to obtain an improved stationary point.

The reconstructed projections for the angle of view

The tomographic central slices of the refractive index decrement, in the case of the mixed algorithm with a standard Tikhonov regularization without any a priori knowledge on the ratio

Al | PETE | PP | |||
---|---|---|---|---|---|

367 | 570 | 2203 | 2930 | ||

1220 | 1793 | 701.7 | 408.5 | ||

553.43 ± 221.92 | 934.92 ± 199.45 | 101.11 ± 74.99 | 154.61 ± 65.58 | ||

1184.77 ± 470.37 | 2000.23 ± 431.02 | 219.28 ± 158.92 | 333.72 ± 139 | ||

1204.22 ± 61.10 | 1313 ± 116.33 | 149.79 ± 56.78 | 190.79 ± 45.75 | ||

1351.2 ± 69.66 | 1473.99 ± 131.84 | 169.83 ± 63.18 | 215.81 ± 50.85 |

Al | Al_{2}O_{3} | PETE | PP | TOTAL | ||||||
---|---|---|---|---|---|---|---|---|---|---|

%NE | %RSD | %NE | %RSD | %NE | %RSD | %NE | %RSD | %NE | %RSD | |

(A) mixed, no prior | −54.63 | 40.1 | −47.85 | 21.33 | −85.59 | 74.16 | −62.14 | 42.41 | 62.55 | 44.5 |

NL, initialization (A) | −2.88 | 39.7 | 11.55 | 21.54 | −68.74 | 72.47 | −18.30 | 41.65 | 25.36 | 43.8 |

(B) mixed | −1.29 | 5.07 | −26.75 | 8.85 | −78.65 | 37.90 | −53.29 | 23.97 | 39.99 | 18.95 |

NL, initialization (B) | 10.75 | 5.15 | −17.79 | 8.94 | −75.79 | 37.2 | −47.17 | 23.56 | 37.87 | 18.7 |

and

where SD represents the standard deviation,

The tomographic central slice obtained using the mixed approach with a prior value of the ratio

Comparing this reconstruction (_{2}O_{3} is reduced with 33.5% (

In this paper, we have considered a non-linear phase retrieval method for phase tomography. The method has been evaluated quantitatively on simulated images and from experimental data acquired at three different propagation distances on a synchrotron X-ray micro-CT set-up. The proposed NL algorithm is achieving better results if no prior is introduced in the initialization solution. On the other hand, if the approach is initialized with the mixed solution including an a prior value, the improvement is not significant in terms of normalized errors. The proposed method decreases globally the reconstruction errors compared to the mixed algorithm applied with various priors [