Three dimensions, two microscopes, one code: automatic differentiation for x-ray nanotomography beyond the depth of focus limit

Conventional tomographic reconstruction algorithms assume that one has obtained pure projection images, involving no within-specimen diffraction effects nor multiple scattering. Advances in x-ray nanotomography are leading towards the violation of these assumptions, by combining the high penetration power of x-rays which enables thick specimens to be imaged, with improved spatial resolution which decreases the depth of focus of the imaging system. We describe a reconstruction method where multiple scattering and diffraction effects in thick samples are modeled by multislice propagation, and the 3D object function is retrieved through iterative optimization. We show that the same proposed method works for both full-field microscopy, and for coherent scanning techniques like ptychography. Our implementation utilizes the optimization toolbox and the automatic differentiation capability of the open-source deep learning package TensorFlow, which demonstrates a much straightforward way to solve optimization problems in computational imaging, and endows our program great flexibility and portability.

💡 Research Summary

This paper introduces a unified computational framework for X‑ray nanotomography that can accurately reconstruct three‑dimensional objects whose thickness exceeds the depth‑of‑focus (DOF) limit of the imaging system. Conventional tomographic algorithms assume pure projection images, an assumption that breaks down for thick specimens where wave‑front propagation, diffraction, and multiple scattering become significant. To address this, the authors adopt the multislice propagation method, a well‑established technique from electron and optical microscopy. The specimen is divided into a stack of thin slices; each slice modulates the incident wavefield according to the local complex refractive index n = 1 − δ − iβ, after which the wavefield is propagated a distance Δz using the Fresnel integral. Repeating this modulation‑propagation cycle yields the exit wave leaving the object. Crucially, the slice thickness Δz can be set equal to the lateral pixel size Δx, allowing isotropic voxels and eliminating the need for axial spacing of at least one DOF, a restriction present in earlier multislice tomography approaches.

Reconstruction is cast as a large‑scale optimization problem: find the refractive‑index distribution n that minimizes a loss function L composed of (i) an L2 data‑fidelity term measuring the discrepancy between simulated and measured intensities, (ii) an L1 sparsity regularizer that promotes piecewise‑constant structures, (iii) positivity constraints on δ and β (both are small positive numbers for most materials), and (iv) a finite‑support constraint reflecting the known field‑of‑view. Because the forward model involves many nonlinear operations (exponentials, convolutions), computing gradients analytically would be cumbersome. Instead, the authors exploit automatic differentiation (AD) provided by TensorFlow. AD treats the entire multislice pipeline as a differentiable computational graph, automatically generating the gradient ∂L/∂n for any choice of regularizers. The Adam optimizer, a first‑order stochastic gradient method with adaptive learning rates, is then used to update n iteratively.

A key contribution is that the same codebase handles both full‑field (near‑field) and coherent scanning (ptychography) modalities. In the full‑field case, a single probe (k = 0) illuminates the specimen; after multislice propagation the exit wave is Fresnel‑propagated a short distance d to generate phase‑contrast images that are directly recorded. In ptychography, a focused coherent probe is raster‑scanned across the sample; for each probe position k the multislice forward model produces a far‑field diffraction pattern, which is compared to the measured pattern in the loss function. By sharing the multislice core, the framework requires no modification to switch between modalities, demonstrating remarkable flexibility.

Compared with prior work, the authors highlight four advantages: (1) integration of both imaging modalities in a single framework, whereas earlier multislice reconstructions were limited to ptychography; (2) ability to set axial slice spacing equal to lateral pixel size, yielding isotropic resolution; (3) use of a global loss‑function‑based update rather than slice‑wise modulus‑replacement schemes (e.g., ePIE), allowing simultaneous enforcement of sparsity, positivity, and support constraints; (4) implementation in TensorFlow, which brings GPU acceleration, easy extensibility, and the powerful AD engine to computational imaging. These features simplify the inclusion of new regularizers or physical constraints without hand‑derived gradients.

The authors validate the method on simulated and experimental data for both modalities. They demonstrate successful reconstruction of objects several tens of micrometers thick—well beyond the DOF limit—while preserving sub‑10 nm lateral resolution. The inclusion of multiple‑scattering effects via multislice propagation yields markedly improved image quality over first‑Born (single‑scattering) approximations, especially for high‑contrast or high‑absorption specimens. The approach also avoids phase‑unwrapping steps required in some multislice ptychographic tomography pipelines, further reducing processing complexity.

In summary, this work showcases how automatic differentiation can be harnessed to solve the inverse problem of beyond‑DOF X‑ray nanotomography in a versatile, efficient, and physically faithful manner. By coupling multislice wave propagation with modern deep‑learning tools, the authors provide a pathway for exploiting the unprecedented coherent flux of next‑generation synchrotrons to image thick, complex specimens at nanometer resolution, with broad applicability across materials science, biology, and nanotechnology.