computer programs
A phaseretrieval toolbox for Xray holography and tomography
^{a}Institut für Röntgenphysik, Universität Göttingen, Germany, and ^{b}Deutsches ElektronenSynchrotron, Hamburg, Germany
^{*}Correspondence email: tsaldit@gwdg.de
Propagationbased phasecontrast Xray imaging is by now a well established imaging technique, which – as a fullfield technique – is particularly useful for tomography applications. Since it can be implemented with synchrotron radiation and at laboratory microfocus sources, it covers a wide range of applications. A limiting factor in its development has been the phaseretrieval step, which was often performed using methods with a limited regime of applicability, typically based on linearization. In this work, a much larger set of algorithms, which covers a wide range of cases (experimental parameters, objects and constraints), is compiled into a single toolbox – the HoloTomoToolbox – which is made publicly available. Importantly, the unified structure of the implemented phaseretrieval functions facilitates their use and performance test on different experimental data.
1. Introduction
For imaging complex samples at high resolution, hard Xrays offer a unique potential because of their small wavelength and high i.e. the phase difference in the Xray wave induced by different phase velocities in a partially coherent wavefront, has to be exploited. Propagationbased phasecontrast Xray imaging (PBI) has proven to be a suitable method to make this phase difference visible via the conversion of phase information into measurable intensity patterns by ways of selfinterference of the beam between object and detector, i.e. by freespace propagation (Wilkins et al., 1996; Cloetens et al., 1999; Paganin, 2006; Nugent, 2011). PBI has also been compared with other phasecontrast methods such as analyser or gratingbased techniques and is particularly well suited for highresolution imaging (Zanette et al., 2013; Lang et al., 2014). Image formation in PBI is well understood. The measurable intensity pattern depends on the length scales of propagation distance z, photon wavelength λ and sample structure A. The dependence on these parameters can be summarized in terms of the Fresnel number F_{(A)} = A^{2}/(zλ) (associated with the object scale A). Depending on the Fresnel numbers F_{min} and F_{max} associated with the smallest and largest relevant image structures (e.g. the size of one pixel and of the full fieldofview, respectively), different imaging regimes can be distinguished: the directcontrast regime (), in which phasecontrast effects are visible as edge enhancements but contrast transfer for low spatial frequencies is weak [see Fig. 1(a)]; the holographic regime (, ), in which multiple interference fringes are visible and coarser image structures exhibit high contrast transfer [see Fig. 1(b)]; and the farfield regime (), where the intensity pattern is essentially given by the squared modulus of the Fourier transform of the wavefield behind the sample (Fraunhofer farfield), without any interference with the primary beam. An important advantage of PBI is the high number of resolution elements which can be imaged in parallel with a single acquisition without any scanning, making it extremely suitable for tomography. Hence in this work, PBI is always associated with primarily 3D applications.
In classical radiography, contrast is based on absorption of the beam when passing through an object. However, in order to resolve the inner structure of weakly absorbing materials and biological specimens at sufficient contrast, in addition to absorption, the phase information,PBI is predominantly carried out with synchrotron radiation (SR) because of the relatively high degree of spatial coherence offered at these sources. High resolution at the nanoscale is achieved by focusing SR to a secondary source and then exploiting the conebeam geometry (Mokso et al., 2007; Bartels et al., 2015) and the Fresnel scaling theorem (Paganin, 2006). A wide range of dedicated beamlines are now available for parallelbeam PBI (see, for example, Weitkamp et al., 2010; Longo et al., 2019; Wang et al., 2001; Schell et al., 2013) as well as – at a somewhat smaller scale and number – for conebeam PBI based on nanofocused undulator radiation (da Silva et al., 2017; Wilde et al., 2016; Lovric et al., 2016; Salditt et al., 2015). In addition, a new generation of microfocus Xray sources, with increased and improved spatial coherence (Hemberg et al., 2003), has enabled the implementation of PBI and computed tomography at the micrometre scale (microCT) in the laboratory (Bartels et al., 2013; Vågberg et al., 2015; Larsson et al., 2016).
As for other coherent Xray imaging techniques, a central challenge for PBI is the reconstruction of the projected phase images from the measured interference patterns, i.e. the phaseretrieval step. There exist efficient algorithms to perform this task in practice, either by a direct singlestep inversion based on various approximations or by iterative techniques based either on minimizing an error metric (Maretzke et al., 2016; Davidoiu et al., 2011) or in the framework of projection algorithms (Luke et al., 2002; Marchesini, 2007). While powerful software packages have been published for farfield coherent imaging, notably for coherent diffractive imaging (Maia, 2008) and ptychography (Marchesini et al., 2016; Enders & Thibault, 2016), publicly available phaseretrieval and reconstruction software for PBI is still sparse (Weitkamp et al., 2011), limited to the directcontrast regime of PBI. Here, the commonly used technique is the single material object algorithm (SMO algorithm) (Paganin et al., 2002). Based on this, different research teams at PBI beamlines have developed elaborate codes which have also resulted in actively maintained opensource frameworks (Weitkamp et al., 2011; da Silva, 2019; Gürsoy et al., 2014). Among them, TomoPy (Gürsoy et al., 2014) is an outstanding example of a framework for tomography that also includes phase retrieval for the directcontrast regime. However, it is neither suitable for the holographic imaging regime with SR nor for the low coherence setting of laboratory microCT, which requires a different set of reconstruction algorithms. Further progress and development of imaging techniques could benefit tremendously from commonly accessible code which is more general with respect to the phaseretrieval approach applicable for different imaging regimes, requirements of probe coherence and object constraints. In addition to processing of experimental data, a framework should also allow simulating experiments beforehand and finding appropriate phaseretrieval settings. In terms of algorithms, many schemes established for the farfield can be adapted to PBI by exchange of the propagators. Yet, since the mathematical structure of the phaseretrieval problem changes its nature when applied to the nearfield setting (see, for example, Maretzke, 2015), it has to be kept in mind that the same algorithm can behave quite differently in different imaging regimes.
In this work, we present a software toolbox containing a collection of MATLAB functions developed by and implemented in our group over several years, which allows for phase retrieval in both the directcontrast and holographic regimes. It originated from the need to facilitate analysis for data obtained at the holotomography endstation GINIX, which is installed at the P10 beamline at PETRA III located at DESY (Salditt et al., 2015), as well as at the emerging laboratorybased setups which are now available for PBI in the directcontrast regime.
The rising number of internal and external users at these setups has raised the demand for a selfcontained toolbox covering the entire dataanalysis process illustrated in Fig. 2, including a collection of phaseretrieval algorithms. For that purpose, the toolbox has recently undergone a thorough revision. The function names and interfaces were harmonized and the code was refactored and partially rewritten from scratch, exploiting more recent features of MATLAB and eliminating redundancy. In addition, detailed documentation was added. The toolbox, called HoloTomoToolbox, is now presented in this work as an overview and at the same time made publicly available under the GPLv3 license. HoloTomoToolbox is available under https://gitlab.gwdg.de/irp/holotomotoolbox including documentation and workedthrough examples. The name accounts for the fact that the applications include in particular the holographic regime, which is not yet well covered, and that we assume that the vast majority of applications will be 3D.
Note that for the 3D tomographic reconstruction, HoloTomoToolbox interfaces the efficient and versatile ASTRA toolbox (van Aarle et al., 2015, 2016) that, in particular, provides a fast implementation of the Feldkamp–Davis–Kress (FDK) algorithm (Feldkamp et al., 1984) for conebeam tomography. Here we do not want to give a comparison of algorithms for specific experimental conditions and parameters, since this has partly been the topic of previous work (see, for example, Burvall et al., 2011; Krenkel et al., 2017). Rather, we want to provide a userfriendly collection of different phaseretrieval approaches suitable for realworld applications. This also accounts for the fact that there is no optimal method covering all cases but that instead each configuration requires a careful consideration of the different alternatives.
2. Propagationbased phasecontrast Xray imaging
To illustrate the various regimes and assumptions made in phaseretrieval algorithms, we will briefly cover the underlying forward models [see also Maretzke (2019) and references therein]. Image formation of fully coherent PBI is modeled by formulating the measured intensity pattern I as a function of projected phase (assuming that the projection approximation holds) and absorption dz where n = 1 − δ + iβ is the z is the optical axis and k is the wave number as
Here, denotes Fresnel propagation of the wavefield in the object plane to the detector, governed by the Fresnel number F. Phase retrieval consists of reconstructing the phase ϕ (and absorption μ) from data of the form (1). To facilitate this task, additional simplifying assumptions can be made, as described in the following.
2.1. Weak object approximation
One of the most widely used approximations in nearfield phase retrieval is the assumption that the sample interacts sufficiently weakly with the Xrays such that the relation (1) can be linearized with respect to the phase and absorption images ϕ and μ. Neglecting higherorder contributions in the components of the and rearranging, this leads to the well known contrasttransfer function (CTF) model (Guigay, 1977),
where s_{F} and c_{F} denote the phase and absorption CTFs, and denotes the Fourier transform. The linearization in (2) has a surprisingly large regime of validity (Turner et al., 2004) but can also fail completely in the presence of large phase gradients (Hagemann et al., 2018).
2.2. Pure phase and singlematerial approximation
Typically, the absorption contrast μ is much weaker than the phase ϕ. Depending on the optical constants and size of the object, it can even be justified to neglect absorption altogether, i.e. to assume pure phase contrast μ = 0, and hence use this constraint for phase retrieval. If the absorption μ cannot be neglected, but is expected to scale with the local electron density, a popular constraint is to set it proportional to the phase, μ/ϕ = β/δ = c_{β/δ} = constant. This singlematerial approximation strictly holds only for objects where the stochiometry of elements (not their density) is homogeneous throughout the object. The obvious algorithmic benefit of either pure phase contrast or singlematerial assumptions is that only one image (ϕ) instead of two (ϕ and μ) have to be recovered from the measured data.
2.3. Linearized transport of intensity – the directcontrast regime
In the directcontrast regime, the governing Fresnel numbers are large enough that the model in equation (1) can be well approximated by linearization with respect to 1/F. This is equivalent to a frequently used model based on the transportofintensity (TIE) equation (see, for example, Teague, 1983),
3. Overview of the toolbox
The HoloTomoToolbox extends the builtin functionalities of MATLAB to enable the entire dataanalysis workflow of a typical holotomography measurement. First, reading and preprocessing the raw data to obtain clean and aligned holograms; second, the phaseretrieval step from holograms to phase maps; third, the tomographic reconstruction from 2D projections to 3D volumes; and, finally, postprocessing of the 3D volume (see Fig. 2).
Besides the main functionalities, the toolbox also contains a large number of convenience functions, e.g. for simulation of PBI data and general image processing. In addition, the toolbox provides exemplary experimental holograms, acquired at laboratory and synchrotron setups by our group, including all the 2D PBI examples presented in Section 4. HoloTomoToolbox is built on top of a number of lowlevel functions to work around various shortcomings of MATLAB and allow for more efficient and clearer code. Prominent examples are the functions fftfreq and fftfreqNormSq, inspired by NumPy, which create a (properly shifted) grid in Fourier frequencies corresponding to the sampling points of an ndimensional fast Fourier transform and its (Euclidean) norm square, respectively. They simplify the definition of Fourierspace filters tremendously and are used in virtually all phaseretrieval functions.
The main aims of the revision were to make the code clearer and easier to read, but also making it faster, avoiding redundancy and using MATLAB's builtin algorithms whenever possible.
In the same spirit, the function interfaces were designed to be as easy to use as possible, while at the same time exposing sufficient control knobs. On the one hand, this requires as few (mandatory) parameters as possible while including optional parameters that have meaningful default values, and on the other hand this requires meaningful and consistent parameter names. Because of MATLAB's syntax for function definitions, it is impossible to have multiple optional parameters and provide default values in a simple userfriendly way. For this reason, many functions in HoloTomoToolbox optionally accept a structure array settings as their last argument to pass extra parameters. All omitted parameters are filled with their default values. The default values can be obtained by calling the function with no arguments and can also be found in the documentation.
All phaseretrieval functions in the toolbox have a unified interface so that different algorithms for a specific task can be easily tested and compared. Every function comes with detailed documentation and simple examples are included to help nonexperts in getting started. An important concept of the toolbox is to use pixel units and the unitless Fresnel number F for all tasks. We note that we can only give a brief overview with references here, while detailed documentation with examples for most functions is available online (https://gitlab.gwdg.de/irp/holotomotoolbox).
3.1. Preprocessing: data input and alignment
Various functions are provided for reading the acquired detector data, complementing MATLAB's builtin imread.
Apart from the obvious corrections by subtraction of dark images or division by the flat field (intensity profile of the illumination) or filtering of lowfrequency artifacts, which can trivially be accomplished using MATLAB's builtin functions, other preprocessing steps are more involved, e.g. the removal of hot pixels/outliers or stripes. The toolbox also provides a collection of other recurring imageprocessing tasks, such as cropping, padding and windowing of 2D or 3D data.
A major part of preprocessing is image alignment. For this reason, the toolbox provides general methods for image registration of shifts and rotations based on GuizarSicairos et al. (2008), and methods to shift, rotate and magnify images. In addition, it provides functions for more specific tasks, such as the automatic alignment of holograms acquired at different defocus/propagation distances or the automatic alignment of the tomographic axis.
3.2. Phase retrieval
A unique feature of the toolbox is the variety of implemented methods for nearfield phase retrieval. All methods require (dark and flatfield corrected) holograms and, depending on the regime, the corresponding Fresnel numbers as input. Freespace propagation is computed assuming a fully coherent planewave illumination. In practice, however, the methods still perform well for partially coherent data (Hagemann & Salditt, 2018) and turn out to be much more robust than farfield methods.
The phaseretrieval algorithms represent the core of the toolbox and are grouped into two major classes, one for the directcontrast regime and the other for the holographic regime. Depending on the problem and particular algorithm, phase retrieval can be performed with or without the different approximations (weak object) and constraints (pure phase or range constraints, singlematerial and support constraints). The methods for the holographic regime encompass both singlestep reconstruction algorithms, which are computationally faster, and iterative ones, which require less assumptions (in particular, no weak object) and provide more flexibility in terms of imposing a priori constraints. By performing computations on a graphics processing unit (GPU), reconstruction with iterative algorithms also becomes feasible for tomography applications.
Table 1 lists the three algorithms implemented for the directcontrast regime [SMO, modified Bronnikov algorithm (MBA), Bronnikov aided correction (BAC)], with corresponding original references and function names. A comparison for laboratory microCT applications of the three approaches has been presented in the work of Töpperwien et al. (2016), which also gives a guideline on how to choose the respective regularization parameters. Further help and example scripts are provided in the documentation.
Table 2 lists the implemented algorithms for phase retrieval in the holographic regime. The CTF implementation follows very closely the seminal work by Cloetens and coworkers (Cloetens et al., 1999; Turner et al., 2004; Zabler et al., 2005). It has been the working horse for holographic phase retrieval for nearly two decades but the trend is turning to iterative methods nowadays to overcome the limitations imposed by the weak object approximation. In our toolbox, the implementation of CTFbased phase retrieval is improved by the possibility to include additional constraints as a support or by the positivity of the reconstructed phase. The extension of the TIE approach to the holographic regime (HoloTIE algorithm) is also based on multipledistance acquisitions as CTFs, but here very closely spaced to approximate the intensity derivative with propagation distance z. It has been introduced in the work of Krenkel et al. (2013, 2017) and offers a suitable alternative for cases where no object constraint is available, i.e. for objects which are extended, not pure phase and not single material. The nonlinear Tikhonov algorithm is based on a fully nonlinear phasecontrast model inverted with Tikhonov regularization (unpublished work by Maretzke) and will be discussed in a future article.

Next, the table lists the iterative projection algorithms [alternating projections (AP), relaxed averaged alternating reflections (RAAR) and modified hybrid input output (mHIO)] again with the corresponding function names and references, referring to their first use for PBI and representing the starting points for the toolbox implementation. These algorithms are based on alternating application of projectionoperators in the object and data space corresponding to constraints in the respective spaces and offer a similarly large regime of applicability and flexibility in terms of constraints. Similar to the implementation in the work of Luke (2012), the toolbox provides the projector classes as well as abstract algorithms that can be freely combined to try out new algorithmconstraint combinations. In addition, the most useful and robust combinations (e.g. the reconstruction of a purephase object and support constraints) are provided ready to use with meaningful default parameter values.
Finally, two iterative algorithms for simultaneous phase retrieval and tomographic reconstruction (3D phase retrieval) are provided: iterative reprojection phase retrieval (IRP) and the more recent 3D CTF method. These fully 3D approaches enable improved reconstruction quality as demonstrated in the work of Ruhlandt & Salditt (2016), yet at the cost of increased computation time.
3.3. Tomographic reconstruction
The toolbox contains easytouse wrappers for GPUbased tomographic reconstruction methods implemented in the ASTRA toolbox (van Aarle et al., 2015, 2016), including filtered back projection, the simulatenous iterative reconstruction technique for parallelbeam tomography and the FDK algorithm for conebeam data. Additionally, wrappers for tomographic (unfiltered) forward and backprojections are provided, which can be used for data simulation or as building blocks for iterative algorithms.
3.4. Postprocessing (ring removal)
Because of the varying sensitivities of detector pixels or dust particles on the sensors, tomographic reconstructions often suffer from socalled ring artifacts. The toolbox offers functions to remove concentric ring structures from the recovered images using established algorithms (Ketcham, 2006; Münch et al., 2009).
4. Examples
This section lists reconstructions obtained with HoloTomoToolbox for typical use cases. As a first example, laboratory microCT applications are presented where phase effects are observed in the directcontrast regime. A particularly well suited approach for PBI at such sources is to use the inverse geometry at low magnification, i.e. a sourcetosample distance much larger than the sampletodetector distance and a highresolution detector. Here, the system resolution is mainly determined by the pointspread function of the detector, and partial spatial coherence is sufficient to observe phase effects. As shown in the work of Töpperwien et al. (2016), a suitable phasereconstruction strategy follows BAC (De Witte et al., 2009), since this approach is robust with respect to the nonideal beam conditions at compact Xray sources, namely low spatial coherence and large bandwidth. A comparison of different phaseretrieval approaches for the directcontrast regime is given in Fig. 3. The example shows a detailed view of an iodine stained cobweb spider leg, with the normalized projection, and the corresponding reconstructions by the MBA (Groso et al., 2006), the SMO (Paganin et al., 2002) and the BAC approach (De Witte et al., 2009). Only BAC provides a reconstruction with sharpness comparable with the raw data. This affects in particular the resolution of the 3D volume data obtained from BACreconstructed projections, as demonstrated in the work of Töpperwien et al. (2016). Note that since BAC is in essence a physically motivated smoothing operator, it enhances the signaltonoise ratio compared with projections recorded in the directcontrast regime while at the same time exploiting the pronounced contrast of the edgeenhancement regime.
Fig. 4 illustrates phase retrieval in the holographic regime for the case of highresolution (conebeam) SR data, showing (a) the flatfieldcorrected hologram of freezedried Deinococcus radiodurans cells [singledistance recording with 13.8 keV at the GINIX setup with a sub20 nm secondary source (Bartels et al., 2015)], and (b)–(d) different phase reconstructions without (left side) and with (right side) support constraint. The source code required to produce the shown reconstructions is listed in Fig. 5.
5. Conclusions
The HoloTomoToolbox, which we hope may prove useful for other groups and provides a reference for further work. To our knowledge, this is the first published and free collection of nearfield phaseretrieval methods for different imaging regimes. It facilitates phaseretrieval applications at current instruments, both with SR and microCT. We also suggest to use HoloTomoToolbox for educational purposes to demonstrate/experiment with nearfield phaseretrieval methods in classroom or workshop tutorials. Finally, it may also inspire further development, in particular when it comes to more efficient codes and more rigorous GPU implementations.
in the nearfield holographic setting has been the subject of intensive research for more than two decades and significant progress has been achieved. However, the range of experimental parameters, the object's optical properties and the variety in prior knowledge require a set of phaseretrieval algorithms instead of a single tool. To this end, we have presentedAcknowledgements
We are grateful to the many contributors, developers and users of holotomography. This project is built on early initial work in phase retrieval and MATLAB code by Klaus Giewekemeyer and Matthias Bartels, who have pioneered holographic imaging in our group. We also thank our colleagues of SFB755, in particular Russel Luke and Thorsten Hohage for collaboration and mathematical treatment of phase retrieval, on which this work is based, as well as Markus Osterhoff and Jan Goemann for continuous support with all numerical challenges. We also acknowledge the team at ID16/ESRF led by Peter Cloetens for fruitful exchange.
Funding information
We thank SFB755 Nanoscale Photonic Imaging and BMBF project 05K19MMG2 NeuroTomo for their financial support.
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