Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region

Optimum plane selection criteria for single-beam phase retrieval techniques based on the contrast transfer function

In this work, an optimum plane selection methodology is reported that can be applied to a wide range of single-beam phase retrieval techniques, based on the contrast transfer function. It is shown that the optimum measurement distances obtained by this method form a geometric series that maximizes the range of spatial frequencies to be recovered using a minimum number of planes. This allows a noise-robust phase reconstruction that does not rely on regularization techniques, i.e., an extensive search for a regularization parameter is avoided. Measurement systems that employ this optimization criteria give an instant deterministic noise-robust phase reconstruction with higher accuracy, and enable the phase retrieval of the entire object spectrum, including lower frequency components.

Phase retrieval for object and probe using a series of defocus near-field images

Full field x-ray propagation imaging can be severely deteriorated by wave front aberrations. Here we present an extension of ptychographic phase retrieval with simultaneous probe and object reconstruction suitable for the near-field diffractive imaging setting. Update equations used to iteratively solve the phase problem from a set of near-field images in view of reconstruction both object and probe are derived. The algorithm is tested based on numerical simulations including photon shot noise. The results indicate that the approach provides an efficient way to overcome restrictive idealizations of the illumination wave in the near-field (propagation) imaging.

High-speed transport-of-intensity phase microscopy with an electrically tunable lens

We present a high-speed transport-of-intensity equation (TIE) quantitative phase microscopy technique, named TL-TIE, by combining an electrically tunable lens with a conventional transmission microscope. This permits the specimen at different focus position to be imaged in rapid succession, with constant magnification and no physically moving parts. The simplified image stack collection significantly reduces the acquisition time, allows for the diffraction-limited through-focus intensity stack collection at 15 frames per second, making dynamic TIE phase imaging possible. The technique is demonstrated by profiling of microlens array using optimal frequency selection scheme, and time-lapse imaging of live breast cancer cells by inversion the defocused phase optical transfer function to correct the phase blurring in traditional TIE. Experimental results illustrate its outstanding capability of the technique for quantitative phase imaging, through a simple, non-interferometric, high-speed, high-resolution, and unwrapping-free approach with prosperous applications in micro-optics, life sciences and bio-photonics.

How minute sooglossid frogs hear without a middle ear

Acoustic communication is widespread in animals. According to the sensory drive hypothesis [Endler JA (1993) Philos Trans R Soc Lond B Biol Sci 340(1292):215-225], communication signals and perceptual systems have coevolved. A clear illustration of this is the evolution of the tetrapod middle ear, adapted to life on land. Here we report the discovery of a bone conduction-mediated stimulation of the ear by wave propagation in Sechellophryne gardineri, one of the world's smallest terrestrial tetrapods, which lacks a middle ear yet produces acoustic signals. Based on X-ray synchrotron holotomography, we measured the biomechanical properties of the otic tissues and modeled the acoustic propagation. Our models show how bone conduction enhanced by the resonating role of the mouth allows these seemingly deaf frogs to communicate effectively without a middle ear.

Compressive x-ray phase tomography based on the transport of intensity equation

We develop and implement a compressive reconstruction method for tomographic recovery of refractive index distribution for weakly attenuating objects in a microfocus x-ray system. This is achieved through the development of a discretized operator modeling both the transport of intensity equation and the x-ray transform that is suitable for iterative reconstruction techniques.

Nonlinear approaches for the single-distance phase retrieval problem involving regularizations with sparsity constraints

The phase retrieval process is a nonlinear ill-posed problem. The Fresnel diffraction patterns obtained with hard x-ray synchrotron beam can be used to retrieve the phase contrast. In this work, we present a convergence comparison of several nonlinear approaches for the phase retrieval problem involving regularizations with sparsity constraints. The phase solution is assumed to have a sparse representation with respect to an orthonormal wavelets basis. One approach uses alternatively a solution of the nonlinear problem based on the Fréchet derivative and a solution of the linear problem in wavelet coordinates with an iterative thresholding. A second method is the one proposed by Ramlau and Teschke which generalizes to a nonlinear problem the classical thresholding algorithm. The algorithms were tested on a 3D Shepp-Logan phantom corrupted by white Gaussian noise. The best simulation results are obtained by the first method for the various noise levels and initializations investigated. The reconstruction errors are significantly decreased with respect to the ones given by the classical linear phase retrieval approaches.

A robust general phase retrieval method for medical applications

From medical imaging perspective the robustness of a phase retrieval method is of critical importance. In this presentation we compare the robustness of two general phase retrieval methods, namely the transport of intensity equation inversion (TIE-inversion) method and the attenuation partition based (AP-based) method. We showed that the TIE-inversion method, regardless if being assisted with the Tikhonov regularization, failed to retrieve the phase maps in two experimental studies. The failure exposes this method's weakness as being unstable against the noise. In contrast, the sample phase maps are retrieved successfully by using the AP-based method. The stark performance differences of the two methods are rooted in their different techniques dealing with the singularity problem. This comparison shows that the robust AP-based phase retrieval method will be superior to the TIE-inversion method for medical imaging applications where radiation doses are stringently limited.

Total variation minimization approach in in-line x-ray phase-contrast tomography

The reconstruction problem in in-line X-ray Phase-Contrast Tomography is usually approached by solving two independent linearized sub-problems: phase retrieval and tomographic reconstruction. Both problems are often ill-posed and require the use of regularization techniques that lead to artifacts in the reconstructed image. We present a novel reconstruction approach that solves two coupled linear problems algebraically. Our approach is based on the assumption that the frequency space of the tomogram can be divided into bands that are accurately recovered and bands that are undefined by the observations. This results in an underdetermined linear system of equations. We investigate how this system can be solved using three different algebraic reconstruction algorithms based on Total Variation minimization. These algorithms are compared using both simulated and experimental data. Our results demonstrate that in many cases the proposed algebraic algorithms yield a significantly improved accuracy over the conventional L2-regularized closed-form solution. This work demonstrates that algebraic algorithms may become an important tool in applications where the acquisition time and the delivered radiation dose must be minimized.

Comparison of single distance phase retrieval algorithms by considering different object composition and the effect of statistical and structural noise

Phase retrieval is a technique for extracting quantitative phase information from X-ray propagation-based phase-contrast tomography (PPCT). In this paper, the performance of different single distance phase retrieval algorithms will be investigated. The algorithms are herein called phase-attenuation duality Born Algorithm (PAD-BA), phase-attenuation duality Rytov Algorithm (PAD-RA), phase-attenuation duality Modified Bronnikov Algorithm (PAD-MBA), phase-attenuation duality Paganin algorithm (PAD-PA) and phase-attenuation duality Wu Algorithm (PAD-WA), respectively. They are all based on phase-attenuation duality property and on weak absorption of the sample and they employ only a single distance PPCT data. In this paper, they are investigated via simulated noise-free PPCT data considering the fulfillment of PAD property and weakly absorbing conditions, and with experimental PPCT data of a mixture sample containing absorbing and weakly absorbing materials, and of a polymer sample considering different degrees of statistical and structural noise. The simulation shows all algorithms can quantitatively reconstruct the 3D refractive index of a quasi-homogeneous weakly absorbing object from noise-free PPCT data. When the weakly absorbing condition is violated, the PAD-RA and PAD-PA/WA obtain better result than PAD-BA and PAD-MBA that are shown in both simulation and mixture sample results. When considering the statistical noise, the contrast-to-noise ratio values decreases as the photon number is reduced. The structural noise study shows that the result is progressively corrupted by ring-like artifacts with the increase of structural noise (i.e. phantom thickness). The PAD-RA and PAD-PA/WA gain better density resolution than the PAD-BA and PAD-MBA in both statistical and structural noise study.

Phase retrieval in in-line x-ray phase contrast imaging based on total variation minimization

State-of-the-art techniques for phase retrieval in propagation based X-ray phase-contrast imaging are aiming to solve an underdetermined linear system of equations. They commonly employ Tikhonov regularization - an L2-norm regularized deconvolution scheme - despite some of its limitations. We present a novel approach to phase retrieval based on Total Variation (TV) minimization. We incorporated TV minimization for deconvolution in phase retrieval using a variety of the most common linear phase-contrast models. The results of our TV minimization was compared with Tikhonov regularized deconvolution on simulated as well as experimental data. The presented method was shown to deliver improved accuracy in reconstructions based on a single distance as well as multiple distance phase-contrast images corrupted by noise and hampered by errors due to nonlinear imaging effects.

Three-dimensional synchrotron virtual paleohistology: a new insight into the world of fossil bone microstructures

The recent developments of phase-contrast synchrotron imaging techniques have been of great interest for paleontologists, providing three-dimensional (3D) tomographic images of anatomical structures, thereby leading to new paleobiological insights and the discovery of new species. However, until now, it has not been used on features smaller than 5-7 μm voxel size in fossil bones. Because much information is contained within the 3D histological architecture of bone, including an ontogenetic record, crucial for understanding the paleobiology of fossil species, the application of phase-contrast synchrotron tomography to bone at higher resolutions is potentially of great interest. Here we use this technique to provide new 3D insights into the submicron-scale histology of fossil and recent bones, based on the development of new pink-beam configurations, data acquisition strategies, and improved processing tools. Not only do the scans reveal by nondestructive means all of the major features of the histology at a resolution comparable to that of optical microscopy, they provide 3D information that cannot be obtained by any other method.

Three-dimensional quantification of capillary networks in healthy and cancerous tissues of two mice

A key issue in developing strategies against diseases such as cancer is the analysis of the vessel tree in comparison to the healthy one. In the search for parameters that might be characteristic for tumor capillaries we study the vascularization in mice for cancerous and healthy tissues using synchrotron radiation-based micro computed tomography in absorption and phase contrast modes. Our investigations are based on absorption tomograms of casted healthy and cancerous tissues as well as a phase tomogram of a fixated tumor. We demonstrate how the voxel-based tomography data can be vectorized to assess the capillary networks quantitatively. The processing includes segmentation, skeletonization, and vectorization to finally extract the vessel parameters. The mean diameter of capillaries in healthy and cancerous tissues corresponds to (8.0±1.1) μm and (3.9±1.1) μm, respectively. Further evaluated parameters show marginal or no differences between capillaries in healthy and cancerous tissues, namely fractal dimension 2.3±0.3 vs. 2.3±0.2, tortuosity (SOAM) 0.18 rad/μm vs. 0.24 rad/μm and vessel length 20 μm vs. 17 μm. The bifurcation angles exhibit a narrow distribution around 115°. Furthermore, we show that phase tomography is a powerful alternative to absorption tomography of casts for the vessel visualization omitting any invasive specimen preparation procedure.

X-ray in-line phase tomography of multimaterial objects

We present a method for phase retrieval from x-ray Fresnel diffraction patterns for multimaterial objects. Previously, homogeneous object assumptions have been used and have been introduced in the Radon domain. Here, we apply prior knowledge in the object domain, which permits the introduction of multiple materials. This is achieved first by a tomographic reconstruction of an attenuation scan and then introduction of the prior followed by a forward projection step to yield the a priori phase maps. The method is applied to the reconstruction of an object of known composition consisting of both soft and hard materials and is shown to perform better than previously proposed algorithms.

Attenuation, scattering, and absorption of ultrasound in the skull bone

PURPOSE

Measured values of ultrasound attenuation in bone represent a combination of different loss mechanisms. As a wave is transmitted from a fluid into bone, reflections occur at the interface. In the bone, mode conversion occurs between longitudinal and shear modes and the mechanical wave is scattered by its complex internal microstructure. Finally, part of the wave energy is absorbed by the bone and converted into heat. Due to the complexity of the wave propagation and the difficulty in performing measurements that are capable of separating the various loss mechanisms, there are currently no estimates of the absorption in bone. The aim of this paper is, thus, to quantify the attenuation, scattering, and thermal absorption in bone.

METHODS

An attenuating model of wave propagation in bone is established and used to develop a three-dimensional finite difference time domain numerical algorithm. Hydrophone and optical heterodyne interferometer measurements of the acoustic field as well as a x-ray microtomography of the bone sample are used to drive the simulations and to measure the attenuation. The acoustic measurements are performed concurrently with an infrared camera that can measure the temperature elevation during insonication. A link between the temperature and the absorption via a three-dimensional thermal simulation is then used to quantify the absorption coefficients for longitudinal and shear waves in cortical bone.

RESULTS

We demonstrate that only a small part of the attenuation is due to absorption in bone and that the majority of the attenuation is due to reflection, scattering, and mode conversion. In the nine samples of a human used for the study, the absorption time constant for cortical bone was determined to be 1.04 μs ± 28%. This corresponds to a longitudinal absorption of 2.7 dB/cm and a shear absorption of 5.4 dB/cm. The experimentally measured attenuation across the approximately 8 mm thick samples was 13.3 ± 0.97 dB/cm.

CONCLUSIONS

This first measurement of ultrasound absorption in bone can be used to estimate the amount of heat deposition based on knowledge of the acoustic field.

Non-linear iterative phase retrieval based on Frechet derivative

Several methods of phase retrieval for in-line phase tomography have already been investigated based on the linearization of the relation between the phase shift induced by the object and the diffracted intensity. In this work, we present a non-linear iterative approach using the Frechet derivative of the intensity recorded at a few number of propagation distances. A Landweber type iterative method with an analytic calculation of the Frechet derivative adjoint is proposed. The inverse problem is regularized with the smoothing L₂ norm of the phase gradient and evaluated for several different implementations. The evaluation of the method was performed using a simple phase map, both with and without noise. Our approach outperforms the linear methods on simulated noisy data up to high noise levels and thanks to the proposed analytical calculation is suited to the processing of large experimental image data sets.

Interface-specific x-ray phase retrieval tomography of complex biological organs

We demonstrate interface-specific propagation-based x-ray phase retrieval tomography of the thorax and brain of small animals. Our method utilizes a single propagation-based x-ray phase-contrast image per projection, under the assumptions of (i) partially coherent paraxial radiation, (ii) a static object whose refractive indices take on one of a series of distinct values at each point in space and (iii) the projection approximation. For the biological samples used here, there was a 9-200 fold improvement in the signal-to-noise ratio of the phase-retrieved tomograms over the conventional attenuation-contrast signal. The ability to 'digitally dissect' a biological specimen, using only a single phase-contrast image per projection, will be useful for low-dose high-spatial-resolution biomedical imaging of form and biological function in both healthy and diseased tissue.

Apparent Linear Attenuation Coefficients in Phase Contrast X-Ray Tomography

In the inline phase contrast x-ray tomography the reconstructed apparent linear attenuation coefficient values may be greatly larger than sample's linear attenuation coefficients or even be negative. In this work we present a general formula to quantitatively relate the apparent linear attenuation coefficient values in cone-beam phase contrast tomography to sample's linear attenuation coefficients and refractive indices. This formula overcomes the gross inaccuracy of the existing formula in the literature in analyzing high-resolution phase contrast tomography, and it will be useful for correctly interpreting and quantifying the apparent linear attenuation coefficients in cone-beam x-ray phase contrast tomography.

Whispering to the deaf: communication by a frog without external vocal sac or tympanum in noisy environments

Atelopus franciscus is a diurnal bufonid frog that lives in South-American tropical rain forests. As in many other frogs, males produce calls to defend their territories and attract females. However, this species is a so-called "earless" frog lacking an external tympanum and is thus anatomically deaf. Moreover, A. franciscus has no external vocal sac and lives in a sound constraining environment along river banks where it competes with other calling frogs. Despite these constraints, male A. franciscus reply acoustically to the calls of conspecifics in the field. To resolve this apparent paradox, we studied the vocal apparatus and middle-ear, analysed signal content of the calls, examined sound and signal content propagation in its natural habitat, and performed playback experiments. We show that A. franciscus males can produce only low intensity calls that propagate a short distance (<8 m) as a result of the lack of an external vocal sac. The species-specific coding of the signal is based on the pulse duration, providing a simple coding that is efficient as it allows discrimination from calls of sympatric frogs. Moreover, the signal is redundant and consequently adapted to noisy environments. As such a coding system can be efficient only at short-range, territory holders established themselves at short distances from each other. Finally, we show that the middle-ear of A. franciscus does not present any particular adaptations to compensate for the lack of an external tympanum, suggesting the existence of extra-tympanic pathways for sound propagation.

Phase retrieval in quantitative x-ray microtomography with a single sample-to-detector distance

Phase retrieval extracts quantitative phase information from x-ray propagation-based phase-contrast images. Notwithstanding inherent approximations, phase retrieval using a single sample-to-detector distance (SDD) is very attractive, because it imposes no setup complications or additional radiation dose compared to absorption-based imaging. Considering the phase-attenuation duality (ε=δ/β, where ε is constant), a simple absorption correction factor is proposed for the modified Bronnikov algorithm in x-ray propagation-based phase-contrast computed tomography (PPCT). Moreover, a practical method for calculating the optimal ε value is proposed, which requires no prior knowledge of the sample. Tests performed on simulation and experimental data successfully distinguished different materials in a quasihomogeneous and weakly absorbing sample from a single SDD-PPCT data point.

Optimization of image recording distances for quantitative X-ray in-line phase contrast imaging

Compared to phase retrieval from single sample-to-detector distance (SDD) image, phase retrieval with multiple SDD images could improve the precision in quantitative X-ray in-line phase contrast imaging (QXIPCI). Among all the related phase retrieval approaches, the two-SDD-image-based one is the simplest and well compromises between precision and dose. However, how to optimize the recording distances for the two images to achieve highest precision, remains unsolved. In this paper, the problem was investigated systematically based on digital simulation and related experiments. Spectral correlation degree (SCD) is introduced to evaluate the pertinence between the two SDD images. The simulation results show that the highest retrieving precision could be obtained while the SDD of the second image is three times that of the first image. The best retrieval could be achieved when SDD of the first image is selected properly, meanwhile the SCD occurs with a typical damping oscillation. Experiments, carried out at the X-ray imaging beamline of SSRF, demonstrated the simulation results.

Phase and amplitude imaging from noisy images by Kalman filtering

We propose and demonstrate a computational method for complex-field imaging from many noisy intensity images with varying defocus, using an extended complex Kalman filter. The technique offers dynamic smoothing of noisy measurements and is recursive rather than iterative, so is suitable for adaptive measurements. The Kalman filter provides near-optimal results in very low-light situations and may be adapted to propagation through turbulent, scattering, or nonlinear media.

Higher-order phase shift reconstruction approach

PURPOSE

Biological soft tissues encountered in clinical and preclinical imaging mainly consists of atoms of light elements with low atomic numbers and their elemental composition is nearly uniform with little density variation. Hence, x-ray attenuation contrast is relatively poor and cannot achieve satisfactory sensitivity and specificity. In contrast, x-ray phase-contrast provides a new mechanism for soft tissue imaging. The x-ray phase shift of soft tissues is about a thousand times greater than the x-ray absorption over the diagnostic x-ray energy range, yielding a higher signal-to-noise ratio than the attenuation contrast counterpart. Thus, phase-contrast imaging is a promising technique to reveal detailed structural variation in soft tissues, offering a high contrast resolution between healthy and malignant tissues. Here the authors develop a novel phase retrieval method to reconstruct the phase image on the object plane from the intensity measurements. The reconstructed phase image is a projection of the phase shift induced by an object and serves as input to reconstruct the 3D refractive index distribution inside the object using a tomographic reconstruction algorithm. Such x-ray refractive index images can reveal structural features in soft tissues, with excellent resolution differentiating healthy and malignant tissues.

METHODS

A novel phase retrieval approach is proposed to reconstruct an x-ray phase image of an object based on the paraxial Fresnel-Kirchhoff diffraction theory. A primary advantage of the authors' approach is higher-order accuracy over that with the conventional linear approximation models, relaxing the current restriction of slow phase variation. The nonlinear terms in the autocorrelation equation of the Fresnel diffraction pattern are eliminated using intensity images measured at different distances in the Fresnel diffraction region, simplifying the phase reconstruction to a linear inverse problem. Numerical experiments are performed to demonstrate the accuracy and stability of the proposed approach.

RESULTS

The proposed reconstruction formula is a generalization of the transport of intensity equation (TIE). It has the second-order accuracy compared to the linear model used in the conventional phase retrieval approach. The numerical experiments demonstrate that the accuracy and stability of the proposed phase reconstruction method outperforms the TIE-based reconstruction method.

CONCLUSIONS

A novel approach has been proposed to retrieve an x-ray phase shift image induced by an object from intensity images measured at different distances in the Fresnel diffraction region. The authors' approach has the second-order accuracy and is able to retrieve the phase shift of an object stably, overcoming the restriction of slow phase variation assumed by the conventional phase retrieval techniques.

Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography

We consider the phase retrieval problem in 3-D holotomography for strongly absorbing objects. Holotomography combines phase retrieval from Fresnel diffraction patterns with tomographic reconstruction to reconstruct the 3-D refractive index distribution. The main interest is the increase in sensitivity of up to three orders of magnitude compared to standard, absorption based tomography. Most existing algorithms are based upon linearization of the forward problem. This is motivated by the large problem size, since it yields computationally efficient solutions. Here, the mixed approach is used, which allows for both strong absorption and long propagation distances. Previous implementations have shown promising results, but in practice often suffer from strong low frequency artifacts. To address this problem, we introduce a homogeneous object assumption through a regularizing term based upon the absorption image. This allows the homogeneous object assumption to be introduced only in the low frequency range. The proportionality constant between absorption and refractive index is assumed to be known. The regularizing parameter is found using the standard L-curve technique. The benefits of our approach are illustrated using data measured at the European Synchrotron Radiation Facility. Low frequency noise in the reconstruction is alleviated, but the result is only quantitative in the areas of the sample where the homogeneous object assumption is fulfilled.

Performance analysis of the attenuation-partition based iterative phase retrieval algorithm for in-line phase-contrast imaging

The phase retrieval is an important task in x-ray phase contrast imaging. The robustness of phase retrieval is especially important for potential medical imaging applications such as phase contrast mammography. Recently the authors developed an iterative phase retrieval algorithm, the attenuation-partition based algorithm, for the phase retrieval in inline phase-contrast imaging [1]. Applied to experimental images, the algorithm was proven to be fast and robust. However, a quantitative analysis of the performance of this new algorithm is desirable. In this work, we systematically compared the performance of this algorithm with other two widely used phase retrieval algorithms, namely the Gerchberg-Saxton (GS) algorithm and the Transport of Intensity Equation (TIE) algorithm. The systematical comparison is conducted by analyzing phase retrieval performances with a digital breast specimen model. We show that the proposed algorithm converges faster than the GS algorithm in the Fresnel diffraction regime, and is more robust against image noise than the TIE algorithm. These results suggest the significance of the proposed algorithm for future medical applications with the x-ray phase contrast imaging technique.

Assisted walking in Malagasy dwarf chamaeleons

Chamaeleons are well known for their unique suite of morphological adaptations. Whereas most chamaeleons are arboreal and have long tails, which are used during arboreal acrobatic manoeuvres, Malagasy dwarf chamaeleons (Brookesia) are small terrestrial lizards with relatively short tails. Like other chamaeleons, Brookesia have grasping feet and use these to hold on to narrow substrates. However, in contrast to other chamaeleons, Brookesia place the tail on the substrate when walking on broad substrates, thus improving stability. Using three-dimensional synchrotron X-ray phase-contrast imaging, we demonstrate a set of unique specializations in the tail associated with the use of the tail during locomotion. Additionally, our imaging demonstrates specializations of the inner ear that may allow these animals to detect small accelerations typical of their slow, terrestrial mode of locomotion. These data suggest that the evolution of a terrestrial lifestyle in Brookesia has gone hand-in-hand with the evolution of a unique mode of locomotion and a suite of morphological adaptations allowing for stable locomotion on a wide array of substrates.

2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance

A method of tomographic phase retrieval is developed for multi-material objects whose components each has a distinct complex refractive index. The phase-retrieval algorithm, based on the Transport-of-Intensity equation, utilizes propagation-based X-ray phase contrast images acquired at a single defocus distance for each tomographic projection. The method requires a priori knowledge of the complex refractive index for each material present in the sample, together with the total projected thickness of the object at each orientation. The requirement of only a single defocus distance per projection simplifies the experimental setup and imposes no additional dose compared to conventional tomography. The algorithm was implemented using phase contrast data acquired at the SPring-8 Synchrotron facility in Japan. The three-dimensional (3D) complex refractive index distribution of a multi-material test object was quantitatively reconstructed using a single X-ray phase-contrast image per projection. The technique is robust in the presence of noise, compared to conventional absorption based tomography.

A comparison of iterative algorithms and a mixed approach for in-line x-ray phase retrieval

Previous studies have shown that iterative in-line x-ray phase retrieval algorithms may have higher precision than direct retrieval algorithms. This communication compares three iterative phase retrieval algorithms in terms of accuracy and efficiency using computer simulations. We found the Fourier transformation based algorithm (FT) is of the fastest convergence, while the Poisson-solver based algorithm (PS) has higher precision. The traditional Gerchberg-Saxton algorithm (GS) is very slow and sometimes does not converge in our tests. Then a mixed FT-PS algorithm is presented to achieve both high efficiency and high accuracy. The mixed algorithm is tested using simulated images with different noise level and experimentally obtained images of a piece of chicken breast muscle.

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

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.

Phase retrieval from one single phase contrast x-ray image

Phase retrieval is required for achieving artifact-free x-ray phase-sensitive 3D imaging. A phase-retrieval approach based on the phase-attenuation duality with high energy x-rays can greatly facilitate for phase sensitive imaging by allowing phase retrieval from only one single projection image. The previously derived phase retrieval formula is valid only for small Fresnel propagator phases corresponding to common clinical imaging tasks. In this work we presented a new duality-based phase retrieval formula that can be applied for cases with large Fresnel propagator phases corresponding to high spatial resolution imaging. The computer simulation demonstrated superiority of this new formula over the previous phase retrieval formula in reconstructing the high frequency components of imaged objects. A modified Tikhonov regularization technique has been devised for phase retrieval in cases of very high resolution and large object-detector distance such that some Fresnel propagator phases may be close or greater than pi. This new phase retrieval formula lays the foundation for implementing high-resolution phase-sensitive 3D imaging of soft tissue objects.

Skull and brain of a 300-million-year-old chimaeroid fish revealed by synchrotron holotomography

Living cartilaginous fishes, or chondrichthyans, include numerous elasmobranch (sharks and rays) species but only few chimaeroid (ratfish) species. The early history of chimaeroids, or holocephalans, and the modalities of their divergence from elasmobranchs are much debated. During Carboniferous times, 358-300 million years (Myr) ago, they underwent a remarkable evolutionary radiation, with some odd and poorly understood forms, including the enigmatic iniopterygians that were known until now from poorly informative flattened impressions. Here, we report iniopterygian skulls found preserved in 3 dimensions in approximately 300-Myr-old concretions from Oklahoma and Kansas. The study was performed by using conventional X-ray microtomography (muCT), as well as absorption-based synchrotron microtomography (SR-muCT) [Tafforeau P, et al. (2006) Applications of X-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens. Appl Phys A 83:95-202] and a new holotomographic approach [Guigay P, Langer M, Boistel R, Cloetens P (2007) Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region. Opt Lett 32:1617-1619], which revealed their peculiar anatomy. Iniopterygians also share unique characters with living chimaeroids, suggesting that the key chimaeroid skull features were already established 300 Myr ago. Moreover, SR-muCT of an articulated skull revealed a strikingly brain-shaped structure inside the endocranial cavity, which seems to be an exceptional case of soft-tissue mineralization of the brain, presumably as a result of microbially induced postmortem phosphatization. This was imaged with exceptional accuracy by using holotomography, which demonstrates its great potential to image preserved soft parts in dense fossils.

Feasibility study of the iterative x-ray phase retrieval algorithm

An iterative phase retrieval algorithm was previously investigated for in-line x-ray phase imaging. Through detailed theoretical analysis and computer simulations, we now discuss the limitations, robustness, and efficiency of the algorithm. The iterative algorithm was proved robust against imaging noise but sensitive to the variations of several system parameters. It is also efficient in terms of calculation time. It was shown that the algorithm can be applied to phase retrieval based on one phase-contrast image and one attenuation image, or two phase-contrast images; in both cases, the two images can be obtained either by one detector in two exposures, or by two detectors in only one exposure as in the dual-detector scheme.

Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography

A well-known problem in x-ray microcomputed tomography is low sensitivity. Phase contrast imaging offers an increase of sensitivity of up to a factor of 10(3) in the hard x-ray region, which makes it possible to image soft tissue and small density variations. If a sufficiently coherent x-ray beam, such as that obtained from a third generation synchrotron, is used, phase contrast can be obtained by simply moving the detector downstream of the imaged object. This setup is known as in-line or propagation based phase contrast imaging. A quantitative relationship exists between the phase shift induced by the object and the recorded intensity and inversion of this relationship is called phase retrieval. Since the phase shift is proportional to projections through the three-dimensional refractive index distribution in the object, once the phase is retrieved, the refractive index can be reconstructed by using the phase as input to a tomographic reconstruction algorithm. A comparison between four phase retrieval algorithms is presented. The algorithms are based on the transport of intensity equation (TIE), transport of intensity equation for weak absorption, the contrast transfer function (CTF), and a mixed approach between the CTF and TIE, respectively. The compared methods all rely on linearization of the relationship between phase shift and recorded intensity to yield fast phase retrieval algorithms. The phase retrieval algorithms are compared using both simulated and experimental data, acquired at the European Synchrotron Radiation Facility third generation synchrotron light source. The algorithms are evaluated in terms of two different reconstruction error metrics. While being slightly less computationally effective, the mixed approach shows the best performance in terms of the chosen criteria.

Phase-space evolution of x-ray coherence in phase-sensitive imaging

X-ray coherence evolution in the imaging process plays a key role for x-ray phase-sensitive imaging. In this work we present a phase-space formulation for the phase-sensitive imaging. The theory is reformulated in terms of the cross-spectral density and associated Wigner distribution. The phase-space formulation enables an explicit and quantitative account of partial coherence effects on phase-sensitive imaging. The presented formulas for x-ray spectral density at the detector can be used for performing accurate phase retrieval and optimizing the phase-contrast visibility. The concept of phase-space shearing length derived from this phase-space formulation clarifies the spatial coherence requirement for phase-sensitive imaging with incoherent sources. The theory has been applied to x-ray Talbot interferometric imaging as well. The peak coherence condition derived reveals new insights into three-grating-based Talbot-interferometric imaging and gratings-based x-ray dark-field imaging.

Phase-contrast X-ray tomography: contrast mechanism and roles of phase retrieval

The direct phase-contrast X-ray tomography, which reconstructs tomograms directly from projection phase-contrast images without the phase retrievals, finds many interesting applications for detection of interfaces for different tissues. In this work we explore and clarify the contrast mechanism of the direct phase-contrast tomography, and show that the direct phase-contrast tomogram is a mixture of three components: the 3D map of the imaged object's linear attenuation coefficients, the map of the rescaled 3D Laplacians of its refraction indices, and artifacts related to the global distribution of the attenuation coefficients and refraction indices. The need of phase retrieval for accurate and quantitative tomography for medical applications is pointed out. A phase tomography reconstruction formula for soft tissue imaging, which requires just a single phase-contrast image per projection for reconstruction, is presented.