HoToPy: a toolbox for X-ray holo-tomography in Python

We present a Python toolbox for holographic and tomographic X-ray imaging. It comprises a collection of phase retrieval algorithms for the deeply holographic and direct contrast imaging regimes, including non-linear approaches and extended choices of regularization, constraint sets and optimizers, all implemented with a unified and intuitive interface. Moreover, it features auxiliary functions for (tomographic) alignment, image processing and simulation of imaging experiments. The capability of the toolbox is illustrated by an example of a catalytic nanoparticle, imaged in the deeply holographic regime at the `GINIX' instrument of the P10 beamline at the PETRA III storage ring (DESY, Hamburg, Germany). Due to its modular design, the toolbox can be used for algorithmic development and benchmarking in a lean and flexible manner, or be interfaced and integrated in the reconstruction pipeline of other synchrotron or X-ray free-electron laser instruments for phase imaging based on propagation.

1. Introduction

The ability of X-ray radiation to penetrate matter is key to its use as a non-destructive probe for the inner structure of objects, materials and tissues, by way of computed tomography (CT). Penetration relies on weak attenuation and therefore has always been a limitation as much as an enabling property. Contrast vanishes for soft tissues and low-Z materials at scales in the micrometer range and below, when attenuation becomes insufficient. The more recent exploitation of phase contrast based on (partially) coherent beam propagation has helped to overcome these limits, offering sufficient contrast even to unstained soft biological tissue, soft matter materials and/or nanoscale structures in solution. Different experimental techniques exist to transform phase shifts imparted by a sample into measurable intensity patterns. One example is propagation based imaging (PBI), where intensity patterns emerge through self-interference after sufficient (optics-free) free space propagation, as illustrated in Fig. 1. In ptychography this is additionally combined with lateral sample shifts (Rodenburg & Faulkner, 2004; Robisch et al., 2015; Pfeiffer, 2018). Other techniques probe the phase by additional optical elements such as grating-based (Momose et al., 2009; Pfeiffer et al., 2013) or speckle imaging (Zanette et al., 2014). For a detailed introduction to X-ray phase contrast imaging we refer to the following references: Paganin (2006); Salditt et al. (2020); Quenot et al. (2022). Hence, X-ray phase contrast imaging and computed tomography (XPCT) is a unique tool for a wide range of applications. Recent cutting edge examples are diverse, including nanoimaging of neuronal tissue for connectomics (Livingstone et al., 2025; Azevedo et al., 2024), morphological transitions of nanoparticles in solution (Grote et al., 2022; Veselý et al., 2021), or ultrafast imaging of hydrodynamics at X-ray free-electron lasers (XFELs) for cavitation (Hoeppe et al., 2024) and fusion confinement research (Montgomery, 2023). In all cases, efficient and high-quality phase retrieval is a key element in phase contrast imaging, in particular in the high resolution full-field variant of holographic tomography (holo-tomography) (Cloetens et al., 1999).

Figure 1 Schematic for PBI in cone-beam geometry. A diverging X-ray beam illuminates a sample with weak absorption, which is placed at a defocus distance z01. The sample imprints phase shifts onto the X-ray wavefront, which render into a measurable self-interference pattern on a detector, placed at distance z12 downstream of the sample, after sufficient free-space propagation. For tomography the sample is rotated and imaged at multiple angles.

Emerging capabilities of fourth-generation synchrotrons (Chushkin & Zontone, 2025; Li et al., 2022) and XFEL sources, of larger and faster detectors (Correa et al., 2024; Donath et al., 2023; Hatsui & Graafsma, 2015), as well as of optics and instruments, enable higher spatial resolution, dose efficiency, larger fields of view and shorter acquisition times (Vijayakumar et al., 2024; Spiecker et al., 2023; Astolfo et al., 2017). At the other end of the image chain, new paradigms in image analysis are fueled by machine learning, which requires large image libraries, for example as training data (Bellens et al., 2024; Flenner et al., 2022; Hendriksen et al., 2020). To meet both ends, reconstruction software and computing pipelines must keep up with the boost in efficiency and throughput, while at the same time achieving image quality beyond the standard linearized phase retrieval approaches.

A variety of toolboxes already exist for X-ray holography, tomography and ptychography, reflecting the growing demand for advanced reconstruction software. A few examples are PyPhase (Langer et al., 2021), CIL (Jørgensen et al., 2021), TomoPy (Gürsoy et al., 2014), HoloTomoToolbox (Lohse et al., 2020), Holotomocupy (Nikitin, 2023), TOFU/UFO (Faragó et al., 2022; Vogelgesang et al., 2016), PyHST (Mirone et al., 2014), PtyLab (Loetgering et al., 2023), PyNX (Favre-Nicolin et al., 2020) and PtyPy (Enders & Thibault, 2016). Each of these has its own strengths and capabilities. With HoToPy, we contribute implementations of recent advances in phase retrieval (Faragó et al., 2024; Huhn et al., 2022), iterative tomographic alignment techniques and image processing tools, that to our knowledge are not yet available in other frameworks. We pair this with high level implementations of smooth and non-smooth optimization. Emerging from HoloTomoToolbox (Lohse et al., 2020) for MATLAB, HoToPy not only serves as a drop-in replacement but extends it with recent methods in a more flexible and modular framework. By leveraging the Python ecosystem including the `PyTorch' (Ansel et al., 2024) library for GPU acceleration and automatic differentiation, HoToPy seamlessly integrates with other existing frameworks simplifying integration into existing analysis workflows and pipelines. It offers a broad spectrum of functionality for the entire range of data analysis, from image processing, phase retrieval and alignment to tomographic reconstruction, but also the simulation of experiments. The source code is licensed under GNU General Public License and publicly available, ref. HoToPy (2025); up-to-date installation instructions, detailed documentation and examples are provided therein.

Currently, HoToPy is rapidly evolving. In this manuscript, we showcase its current status and demonstrate its capabilities on experimental data taken on the Göttingen instrument for nano-imaging with X-rays (GINIX) operated by our research group at the P10 beamline of the PETRA III storage ring at DESY in Hamburg, Germany.

2. HoToPy toolbox

The HoToPy package is a Python toolbox for holographic and tomographic X-ray imaging. Its primary use is the reconstruction of propagation based X-ray phase contrast tomography—holo-tomography—data recorded at synchrotron or laboratory X-ray sources, but it can also be used for visible light or electron holographic imaging.

HoToPy is implemented using the machine learning library `PyTorch' (Ansel et al., 2024), which provides strong GPU acceleration and flexible automatic differentiation. High-performance tomographic primitives are provided through an interface to the ASTRA toolbox (Palenstijn et al., 2013; van Aarle et al., 2015; van Aarle et al., 2016). HoToPy can either be used as a software library in reconstruction pipelines or it can be used interactively by the user, thanks to its sane defaults and intuitive interface. Furthermore, real experimental datasets for testing and development of novel algorithms are included.

2.1. Numerical concepts

HoToPy's flexibility, extensibility and high numerical performance resides on two core principles: the use of state-of-the-art numerical optimization algorithms and automatic differentiation (AD), whose concepts are briefly introduced in the following. In order to ease adoption of these concepts we provide reusable optimization algorithms through the hotopy.optimize submodule.

State-of-the-art numerical optimization underpins HoToPy's phase retrieval algorithms and enables fast and robust reconstructions with high numerical efficiency. We provide algorithms for smooth and non-smooth optimization, including a robust proximal gradient method (PGM) with backtracking line search and adaptive step sizes described by Goldstein et al. (2014a), an accelerated alternating direction method of multipliers (ADMM) (Goldstein et al., 2014b) and fast iterative shrinkage-thresholding algorithm (FISTA) (Beck & Teboulle, 2009). In particular, the ADMM algorithm is used for the constrained variant of the contrast transfer function (CTF) and intensity transfer function (ICT) phase retrieval algorithms and the PGM with automatic differentiation for Tikhonov and TikhonovTV, as described by Huhn et al. (2022).

Automatic differentiation (AD) is popular for training of (deep) neural networks (Paszke et al., 2017; Baydin et al., 2018), where it is used to dynamically compute the gradient of a scalar-valued objective or loss function. Thus, neither finite-difference approximation nor analytical derivation and explicit implementation of gradients is required. In general, this allows fast prototyping of new algorithms or adaptation of existing ones without the burden of manually deriving and updating explicit gradients with an insignificant overhead in performance.

In HoToPy, AD is automatically used within the gradient-based optimization algorithms, e.g. PGM and FISTA, and thus in the phase retrieval algorithms based on them, such as the Tikhonov algorithm.

Using AD provides a high degree of flexibility. For example, this allows extension or modification of the Tikhonov algorithm with any (smooth) regularization, such as the smoothed L¹-norm for a total variation (TV) regularization (Hansen et al., 2021) used in TikhonovTV. Furthermore, any modification of the forward model can easily be combined with any other constraint or regularization.

2.2. Phase retrieval

Since the phase of an X-ray wavefront cannot be measured with current technology, a central step in holo-tomography is the computational reconstruction of quantitative phase and attenuation images from recorded near-field diffraction patterns, or (inline) holograms. This so-called `phase problem' poses a nonlinear, ill-posed inverse problem. To solve this inverse problem, several algorithms have been developed incorporating different assumptions and priors, such as assuming short propagation distance, a single-material object (Paganin et al., 2002) or an optically weak object (Cloetens et al., 1999). Generally, in PBI two regimes can be distinguished characterized by a dimensionless quantity, the Fresnel number. It is defined by = σ²/zλ, with reference length scale (e.g. sample diameter, structure size or pixel size) σ, propagation distance z between sample and detector, and wavelength λ. One regime is the so-called direct contrast or edge-enhancement regime, where ≅ 1; the other is the holographic regime, where << 1. For each regime a range of different algorithms are available in HoToPy, see Table 1.

Table 1
Overview of available phase retrieval methods for the holographic as well as the direct contrast regime

Name	Class	Reference
Holographic regime
Contrast transfer function (CTF)	CTF	Cloetens et al. (1999)
— constrained CTF	CTF	Huhn et al. (2022)
Intensity transfer function (ICT)	ICT	Faragó et al. (2024)
Nonlinear Tikhonov	Tikhonov	Huhn et al. (2022)
— (smoothed) TV regularization	TikhonovTV	Lucht et al. (unpublished)
Alternating projections (AP)	AP	Hagemann et al. (2018)
Direct contrast regime
Paganin	Paganin	Paganin et al. (2002)
Generalized Paganin	GeneralizedPaganin	Paganin et al. (2020)
Bronnikov-aided correction	BronnikovAidedCorrection	De Witte et al. (2009)
Modified Bronnikov	ModifiedBronnikov	Groso et al. (2006)

In HoToPy, holography related methods (propagation and phase retrieval) are provided through the hotopy.holo submodule. So far, it is focused on propagation based phase contrast in the direct-contrast as well as holographic regime. All algorithms for the holographic regime support imposing priors as object constraints, e.g. pixel-wise non-positivity of the phase or finite supports. An overview of the available phase retrieval algorithms at the time of writing is given in Table 1, while an updated list can be found in the online documentation. All are implemented with GPU computation support. Moreover, they allow for astigmatism, so that the effective propagation distance or equivalently the Fresnel number can be different in the two directions orthogonal to the optical axis. This can, for example, be caused by anisotropic magnifications in the horizontal and vertical directions, see for example the case of Bragg magnifiers (Spiecker et al., 2023). For detailed documentation of the algorithms we refer to the online documentation, example notebooks and their respective literature. In addition, a number of methods for preprocessing are provided. These include the automated removal of faulty pixels as well as principal component analysis based (Nieuwenhove et al., 2015) and curvature based methods for empty-beam division.

2.3. Computed tomography

The tomographic methods are organized in the hotopy.tomo submodule. Through interfaces to the ASTRA toolbox (Palenstijn et al., 2013; van Aarle et al., 2015; van Aarle et al., 2016), efficient (multi-)GPU reconstruction and projection algorithms in two and three dimensions are provided, including, for example, filtered back-projection (FBP), the cone-beam algorithm of Feldkamp, Davis and Kress (FDK), and the simultaneous iterative reconstruction technique (SIRT). For both parallel beam and cone beam geometric models, the source, sample and detector can be positioned freely for each projection image. Thus, any inexactness of the tomographic trajectory can be incorporated directly into the geometric model. This is computationally more efficient than aligning and interpolating the projection images which also degrades image quality. The toolbox contains methods based on image registration (Guizar-Sicairos et al., 2008) for determining deviations from an assumed tomographic trajectory.

The center of rotation (CoR) can be found by registering the shift between either two opposing projection images, or, for scans with an angular range larger than 0° to 180°, two opposite segments of a sinogram—ideally two half rotations—can instead be registered. In the latter, the CoR estimate becomes an average over all acquisition angles, making it more robust for acquisitions with little position precision. The registration can be repeated for sinograms from different detector rows to further increase robustness and also determine small tilts of the rotation axis.

The iterative reprojection alignment algorithm (van Leeuwen et al., 2018) in the toolbox enables reconstructing and correcting rigid sample movement between individual projections during a tomographic scan. In each iteration, the volume is reconstructed based on the current geometry estimate. (Re-)Projection images are generated from the reconstructed volume, registered against the acquired images and the geometric model for the respective projection updated according to the detected shift. In practice it is often advisable to apply pixel binning and a bandpass filter to the projection images prior to the alignment routine to accelerate the computation and improve the registration. Nonlinearities in the detector response, but also illumination, can cause stripes in the sinograms which lead to ring artifacts in the reconstructed volumes. HoToPy contains implementations of additive (Ketcham, 2006) and wavelet based (Münch et al., 2009) ring-removal algorithms to mitigate these artifacts.

3. Reconstruction example: catalytic particle

We demonstrate the HoToPy toolbox with the example of an X-ray holo-tomography dataset of an isolated catalytic particle with a diameter of approximately 33 µm used for olefin polymerization. The particle morphology and fragmentation (`cracks') is of particular interest. For an in-depth description of the sample, and how the inner structure matters for the catalytic function in the application context, we refer to Werny et al. (2022a,b) and Veselý et al. (2021). The particle is attached to the interior wall of a Kapton tube. The data were recorded at the P10 beamline of the PETRA III storage ring at DESY in Hamburg with the GINIX. The GINIX was used in its cone-beam configuration to achieve geometrical magnification and effective pixel sizes down to the nanometer range. To this end, the incident beam is focused with a Kirkpatrick–Baez (KB) mirror system onto an X-ray waveguide acting as quasi-point source for holographic illumination. The waveguide (ID 4743) is a combination of two orthogonally crossed thin-film waveguides with a diameter of 58 nm and depth of 600 µm each, functioning together as a two-dimensional waveguide (Krüger et al., 2010). Two tomograms at two defocus distances were acquired with 1501 projections each, covering an angular range of 180°. The source-to-sample distance z₀₁ was adjusted to 13.53 mm and 16.73 mm, at constant source-to-detector distance z₀₂ = 5110 mm. The photon energy E_ph = 13.8 keV was selected by a Si(111) monochromator. Images were recorded using a Gadox scintillator of 15 µm thickness fiber-coupled to an Andor Zyla sCMOS sensor with a pixel size Δ_x of 6.5 µm and 2160 × 2560 pixels. The exposure time per acquisition was 1 s. The geometry corresponds to geometric magnifications of M = z₀₂/z₀₁ = 378 and 305, resulting in an effective pixel sizes of 17.2 nm and 21.3 nm, respectively. The Fresnel numbers (with respect to the effective pixel size), = , evaluate to 2.44 × 10⁻⁴ and 3.02 × 10⁻⁴, respectively.

3.1. Phase retrieval

First, the phase of the catalytic particle has to be reconstructed by phase retrieval. To this end, the recorded raw intensity patterns are preprocessed by dark current subtraction and divided by interpolated empty beam images, i.e. images taken without a sample. The empty image used for normalization at the tomographic angle θ with range [0, θ_max] is interpolated by a linear combination of the (average) empty images before and after the tomographic scan, = + . The resulting normalized holographic diffraction patterns are also called (inline) holograms. Afterwards, residual low frequency background variations are suppressed by a least curvature inpainting of the background within a compact support of the particles. Prior to phase retrieval the holograms of the second distance are magnified to an effective parallel beam geometry with effective pixel size 17.2 nm and respective Fresnel numbers 2.44 × 10⁻⁴ and 1.98 × 10⁻⁴.

An exemplary empty-beam divided hologram is shown in Fig. 2(a) together with different phase reconstructions in Figs. 2(b) and 2(c). Fig. 2(b) shows a single step phase reconstruction using the contrast transfer function (CTF) method (Cloetens et al., 1999) without the use of any constraints. Figs. 2(c) and 2(d) show a non-linear and constraint-based phase reconstruction using the Tikhonov (Huhn et al., 2022) algorithm with pixel-wise non-positivity, and non-positivity combined with a compact disk-shaped support, respectively. All reconstructions were computed with the input of the recordings of the two distances, assuming a homogeneous object with beta/delta ratio β/δ = 0.035 and applying a two-level frequency regularization using the weights α_low = 2 × 10⁻⁵, α_high = 3 × 10⁻⁵ (Huhn et al., 2022).

Figure 2 Hologram and phase reconstructions of a catalytic particle. (a) Example showing one of the two normalized holographic intensity interference patterns I/I0 (hologram) of a catalytic particle at one tomographic angle. (b–d) Comparison of different phase retrieval methods and constraints. The reconstruction in (b) uses an unconstrained linear contrast transfer function (CTF). The reconstructions (c) and (d) are obtained using the HoToPy–Tikhonov algorithm. For these, a pixel-wise non-positivity constraint is used and for (d) additionally a finite disk-shaped support, indicated by the dashed circle. Scale bars: 5 µm. Effective pixel size: 17.2 nm. Images have 2160 × 2560 pixels.

By comparing the reconstructions, we can directly observe strong background variations in Figs. 2(b) and 2(c). In the lower right corner of Fig. 2(c) we can observe that this also affects the phase reconstruction within the particle. Furthermore, the linear and unconstrained reconstruction in Fig. 2(b) incorrectly contains positive values; due to the convention used here a higher density sample relative to the empty beam should solely have non-positive values (negative phase shift or retarded waves). Thus, using this as prior knowledge for the reconstructions yields more faithful reconstructions [Fig. 2(c)]. Finally, if additionally combined with a disk-shaped support, low frequency background variations are effectively suppressed [Fig. 2(d)].

The corresponding code snippets used for the phase reconstructions are given in Appendix A. A comparison of computation times of two phase retrieval algorithms with different sets of constraints is given in Table 2 in Appendix B.

Table 2
Compute times of two phase retrieval algorithms implemented in HoToPy with different sets of constraints

Computations were performed on a NVIDIA RTX 6000 Ada and a NVIDIA H100 GPU with the same dataset shown in Fig. 2 and the full catalytic particle dataset, respectively.

Algorithm	RTX 6000 Ada	H100
Single tomographic angle, two distances
CTF, no constraints (single-step method)	4.1 ms	3.9 ms
CTF, phase ϕ ≤ 0 (24 iterations)	38.5 ms	30.2 ms
Tikhonov, no constraints (25 iterations)	245.8 ms	217.9 ms
Tikhonov, phase ϕ ≤ 0 (25 iterations)	284.5 ms	233.8 ms
Tikhonov, phase ϕ ≤ 0 and disk-shaped support (30 iterations)	439.5 ms	317.6 ms
Full dataset of catalytic particle
CTF, no constraints	9 s	6 s
CTF, phase ϕ ≤ 0	72 s	47 s
Tikhonov, no constraints	6.1 min	4.4 min
Tikhonov, phase ϕ ≤ 0	7.4 min	5.1 min
Tikhonov, phase ϕ ≤ 0 and disk-shaped support	11.8 min	7.3 min

3.2. Tomography

A three dimensional model of the sample is created from the phase projections by means of tomographic reconstruction. A wavelet-based filter (Münch et al., 2009) is applied to reduce stripes in the sinograms. The corresponding reduction of ring artifacts in the reconstructed slice can be seen in Fig. 3(a). The direct tomographic reconstruction of the sinograms [Fig. 3(b)] suffers from artifacts caused by inexactness of the acquisition trajectory, mainly due to sample movement. With a geometric model which includes the sample movement [Fig. 3(c)], the quality of the reconstruction improves drastically and fragmentation cracks can be traced [Fig. 3(d)]. To extract the sample movement from the acquired projection images, the CoR and drift were first estimated by registration of opposite projections. The reprojection alignment algorithm was then applied to a reduced dataset which was obtained by 8 × 8 pixel binning and high-pass filtering with a Gaussian filter kernel (σ = 40/8). The influence of the Kapton tube's strong edge was reduced by a directional Fourier filter applied to images with observation angles close to the edge surface. The displayed shifts were obtained after 100 iterations, taking about 9.3 min on a machine with 24 CPU cores [Intel(R) Xeon(R) w7-3455] and an NVIDIA RTX 6000 Ada Generation GPU. The FDK reconstruction of the full volume takes about 2 min. The code snippets for the tomography reconstruction are given in Appendix A.

Figure 3 Tomographic reconstruction by the FDK algorithm. (a) Zoom into the center of a reconstructed horizontal slice. Concentric ring artifacts (top left) are mitigated (bottom right) after applying wavelet-based ring removal (l = 4, σ = 1) to the sinogram. (b) Vertical slice through a volume reconstruction assuming an idealized acquisition trajectory. (c) Shifts of the projection images estimated by registration of opposite projections and reprojection registration (8 × 8 pixel binning, high-pass filter with σ = 5, 100 iterations). (d) Virtual slice through the volume reconstruction after applying the shift correction. The effective voxel size is 17.2 nm.

4. Conclusion

We have presented HoToPy, an open source toolbox for holographic and tomographic X-ray imaging in Python. It provides a collection of phase retrieval algorithms for the direct contrast and in-line holographic regime, suitable for propagation-based phase imaging at synchrotron and laboratory µCT instruments alike. Tomographic reconstruction and alignment methods are also included. It is easily extensible without sacrificing speed by leveraging automatic differentiation and GPU computation, facilitating rapid testing of novel algorithms. Compared with our current `workhorse' – the HoloTomoToolbox for MATLAB – HoToPy not only integrates into the extensive ecosystem of scientific computing in Python but also features iterative tomographic reprojection alignment and new phase retrieval algorithms such as ICT (Faragó et al., 2024) and TikhonovTV, extending the approach of Huhn et al. (2022) to TV regularization. We have showcased the toolbox by reconstructing the volume from a deeply holographic and severely misaligned dataset of a catalytic particle.

The toolbox can serve at least four different purposes. First, it can be used for testing and development of algorithms, operated stand-alone on pre-recorded or simulated data, for example using the supplied phantoms. Second, in the same manner, it can serve classroom teaching and visualization. Third, it can be operated with a script-based analysis pipeline for a specific instrument such as the `GINIX' instrument. In this case, wrapper scripts handling the instrument specific metadata and data formats, or default reconstruction parameters, can easily be written and supplied to the user. Fourth, algorithmic implementations and functions of interest can be integrated into entirely different reconstruction pipelines tailored to specific instruments and requirements. For example, integration and interoperability with other phase and tomographic reconstruction platforms such as CIL (Jørgensen et al., 2021) should be possible with little effort. Finally, beyond its primary use as a tool for X-ray phase imaging, we envision it to be useful also for inline optical or electron holography. To this end, the generic formulation in units of pixel size, Fresnel number or real and imaginary part of the refractive index is particularly helpful.

APPENDIX A

Code examples from HoToPy

For the complete reconstruction script, full documentation and further examples, refer to HoToPy (2025). The given code examples in this manuscript are based on HoToPy version v0.21.

A source code snippet for phase reconstruction of a catalytic particle with HoToPy follow in Fig. 4.

Figure 4 Source code snippet for phase reconstruction of a catalytic particle with HoToPy.

A source code snippet for the tomographic reconstruction and reprojection alignment to correct the acquisition trajectory follows in Fig. 5.

Figure 5 Source code snippet for the tomographic reconstruction and reprojection alignment to correct the acquisition trajectory.

APPENDIX B

Computations times for phase retrieval

We performed a comparison of the computation times for the phase retrieval listed in Table 2. Here we compared the CTF and HoToPy–Tikhonov algorithms with and without enforcing pixel-wise non-positive phase. Additionally, for the Tikhonov a disk-shaped object support was also tested. The single-angle reconstructions were performed on the dual distance holograms shown in Fig. 2 (2160 × 2560 pixels), which corresponds to a single angle of the tomogram. All computations were done with single precision floats. We used default settings in HoToPy with homogeneous object assumption with a beta/delta ratio β/δ = 0.0035 and two-level frequency regularization α_low = 2 × 10⁻⁵, α_high = 3 × 10⁻⁵. The iterations were stopped with the default stopping conditions. For the single tomographic angle reconstruction the iterations needed to meet these condition are given. The phase reconstruction for the full dataset were performed sequentially for each tomographic angle on the GPU.

Clearly, the linear CTF algorithm without constraints is the fastest in the comparison, since it is a single-step algorithm without iterations. When constraints are set, like the pixel-wise phase non-positivity used here, ADMM iterations are applied which takes approximately one order of magnitude longer than the single-step method. The HoToPy–Tikhonov algorithm takes again one order of magnitude longer to converge without constraints. If pixel-wise phase non-positivity is added, convergence is reached in a comparable time. This time is approximately doubled if phase non-positivity is combined with a disk-shaped support.

This benchmark gives an impression of the performance of HoToPy's phase retrieval using a real dataset on standard hardware. When considering these results, one should keep in mind that the performance also strongly depends on the dataset, parameters and constraints.