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REVIEW 3 major objections 6 minor 6 cited by

TorchRadon: Fast Differentiable Routines for Computed Tomography

T0 review · 3 major / 6 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read TorchRadon makes CT forward and back projections differentiable, GPU-resident, and up to 125× faster than the comparison toolbox, so they can act as trainable layers inside neural networks and power iterative reconstruction.

desk verdict TorchRadon is a genuinely useful, openly released PyTorch library for differentiable CT operators, but the headline 125x speed claim is under-specified without Astra's GPU configuration and the gradient correctness is asserted rather than tested. read the letter →

arxiv 2009.14788 v1 pith:DJIBQQ6U submitted 2020-09-29 eess.IV cs.CVcs.LG

Matteo Ronchetti This is my paper · ORCID
classification eess.IVcs.CVcs.LG
keywords computedtomographydifferentiableprogrammingRadontransformbackprojectionGPUaccelerationCUDAkernelsshearletiterativereconstruction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

TorchRadon is an open-source GPU library that implements computed-tomography operations as differentiable routines, so Radon forward projection, backprojection, and shearlet transforms can sit inside a deep network and receive gradients through the ordinary backward pass. The paper's central claim is that these operators are both fast and accurate: up to 125 times faster than the comparison toolbox for projection and backprojection, more than 2.5 times faster again in half precision, and close enough numerically that filtered-backprojection reconstruction error is essentially unchanged. The paper supports this with numerical comparisons of projections, filtered backprojection, and shearlet coefficients, and with a shearlet-regularized limited-angle reconstruction that runs in seconds rather than minutes. If the claims hold, researchers can combine deep learning with model-based CT reconstruction without writing custom gradient code or shuttling data between CPU and GPU.

What carries the argument

The load-bearing mechanism is a set of custom CUDA kernels for Radon transforms whose inputs are stored in CUDA textures, a read-only memory layout that caches two-dimensional neighborhoods and improves hit rates for line integrals. When given half-precision input, the kernels pack four images into the color channels of a single texture, cutting memory reads by a factor of four, and upconvert to single precision for all arithmetic. These kernels are wrapped as automatic-differentiation-aware operations, so the same code path that computes a projection also computes its gradient via the framework's backward pass, and batched execution keeps the GPU fully occupied. The same GPU-resident, batched design carries the shearlet transform and the iterative solvers, which is what lets an ADMM reconstruction loop run without CPU-GPU copies.

What would settle it

Re-run the paper's batch forward and backprojection benchmarks with the comparison toolbox using its GPU backend and GPU-resident data; if the speedup over that baseline does not approach 40× in single precision or 125× in half precision, the central speed claim is not supported. A second decisive check would compare TorchRadon's backward pass to finite-difference gradients of its forward operator, since the paper does not report such a check.

Watch

Extended reading notes

Core claim

The central contribution is a set of differentiable CUDA routines for computed tomography: parallel-beam and fan-beam Radon forward projection, backprojection, filtered backprojection with standard filters, shearlet transforms, and the iterative solvers Landweber, CGNE, and conjugate gradient. The author's claim is that because these routines are registered with the automatic differentiation system, gradients can be computed automatically through them, and because input data stays resident on the GPU, is processed in batches, and is read through texture-cached memory, they run up to 125 times faster than the comparison toolbox under the paper's benchmarks. Numerically, the paper reports relative errors of about $8\times10^{-4}$ against the comparison toolbox for forward projections and $4\times10^{-5}$ for backprojections, nearly identical filtered-backprojection reconstruction error ($2.22\times10^{-4}$ versus $2.02\times10^{-4}$ mean squared error), and shearlet coefficients matching to $3.86\times10^{-7}$ in single precision. Half-precision storage more than doubles speed while leaving the filtered-backprojection reconstruction error essentially unchanged, at $2.0567\times10^{-4}$ versus $2.0566\times10^{-4}$ in single precision.

Load-bearing premise

The load-bearing premise is that the comparison toolbox was benchmarked in its best-performing GPU configuration, since the paper never states whether its GPU backend was used when the 40× to 125× speedups were measured.

Editorial extensions

If this is right

  • Radon forward projection, backprojection, and shearlet transforms become drop-in layers for end-to-end training, with gradients obtained by the framework's ordinary backward pass rather than hand-derived code.
  • Iterative reconstruction algorithms such as Landweber, CGNE, and conjugate gradient run entirely on GPU and support batched data, so a model-based reconstruction can be called within a training loop without CPU-GPU copies.
  • A shearlet-regularized limited-angle CT pipeline reimplemented with the library runs in about 1.6 seconds on a server GPU versus 2.5 minutes reported on CPU for the original implementation, and batch processing brings the average to about 1.2 seconds per image.
  • Half-precision storage gives more than 2.5 times the speed of single precision for memory-bound Radon transforms while keeping filtered-backprojection reconstruction quality effectively identical, so precision can be traded for speed in large runs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Since the underlying kernels are differentiable and exposed to the autograd system, the same design could be extended to make acquisition geometry (angles, detector positions, source trajectories) learnable parameters for limited-view CT.
  • The half-precision texture-packing trick is a general recipe for memory-bound integral operators; analogous packing could speed up line-integral transforms in electron microscopy, seismic imaging, or synchrotron tomography.
  • The paper does not report a finite-difference check of its backward pass, so a direct numerical gradient comparison would settle whether the differentiability claim holds in floating-point practice.
  • If the measured speedups generalize to other geometries and problem sizes, unrolled optimization networks could afford many more iterations per forward pass, shifting the practical balance between learned and model-based reconstruction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper introduces TorchRadon, an open-source CUDA-based PyTorch extension providing differentiable Radon forward/backprojection, fan-beam projection, shearlet transforms, and iterative solvers for CT reconstruction. The main claims are that the library is up to 125× faster than Astra Toolbox for Radon projections, that its operators integrate with PyTorch autograd so gradients can be computed via backward(), and that half-precision storage gives large speedups with negligible accuracy loss. Numerical comparisons are reported against Astra Toolbox for parallel-beam and fan-beam projections and FBP, and against AlphaTransforms for shearlets. A use case reproduces a shearlet-regularized ADMM reconstruction, reporting a 125× speedup over the runtime reported in the original paper.

Significance. If the claims are substantiated, TorchRadon would be a practically valuable tool for researchers combining deep learning with model-based CT reconstruction, since differentiable GPU-resident forward operators are a bottleneck for such work. The paper makes concrete empirical comparisons against external libraries (Astra Toolbox and AlphaTransforms), the source code is publicly available, and the reported numerical errors are small. However, two load-bearing aspects are under-supported: the headline speedup benchmark is not specified enough to be reproduced or interpreted, and the differentiability claim is asserted without any numerical or analytical verification of the backward pass. The central idea is plausible and the reported comparisons are a good start, but the manuscript needs additional methodological detail before the main claims can be accepted.

major comments (3)
  1. [Section 4.1, Figure 13] The benchmark underlying the headline speedup is under-specified. The text states that TorchRadon processes a batch of 32 continuous GPU arrays, but it never states Astra Toolbox's version, which projector kernel was used (e.g., 'cuda', 'linear', or 'strip'), whether Astra data were GPU-resident through its own GPU/OpenCL backend, how batching was emulated for Astra, how many timing repetitions were performed, or whether error bars were computed. Without these details, the reported 40× and 125× speedups could reflect differences in API overhead or CPU-GPU copies rather than algorithmic performance. Figure 13 also appears to show only single timing points without error bars, making it impossible to assess the stability of the claimed ratios.
  2. [Section 2 and Section 3] The differentiability claim is not verified. The paper states that TorchRadon operators allow gradient computation via PyTorch backward(), and this is a central advertised feature, but no numerical gradient check or adjoint-test is reported for the Radon forward/backprojection or shearlet transforms. A finite-difference comparison (e.g., ||g_analytic - g_numeric|| / ||g_numeric|| for random inputs) or a consistency check of the backward pass against the adjoint of the forward operator should be added; without this, the reader cannot confirm that the autograd integration is correct.
  3. [Section 4.3] The second 125× claim compares TorchRadon's runtime to the 2.5-minute runtime reported in the original paper on an Intel i7 CPU, not to a directly benchmarked implementation of the same algorithm on the same hardware. The manuscript says 'Compared to an implementation made using Astra and AlphaTransforms' but such an implementation is not actually timed. This comparison is suggestive but not a controlled benchmark; the authors should either run the reference implementation on the same machine or clearly label the comparison as a cross-hardware, cross-implementation estimate rather than a measured speedup.
minor comments (6)
  1. [Abstract and Section 2] The abstract says 'up to 125 faster' without the multiplication sign, and Section 2 likewise states 'up to 125 × faster'; the distinction between the half-precision 125× figure and the single-precision 40× figure should be stated in the abstract as well, since the current phrasing could mislead a reader into attributing 125× to the default single-precision mode.
  2. [Section 4.3 code listing] The code listing has apparent typos: 'clamp min' should be 'clamp_min', and the variable 'n angles' appears to be undefined, likely intended to be 'n_angles'. Please correct the listing so that the example is runnable as printed.
  3. [Section 3.1 heading] The heading reads 'Filtered Bacprojetion'; this should be 'Filtered Backprojection'.
  4. [Section 4.3 bullet list] The second bullet reads 'There are are no CPU-GPU memory copies'; the duplicated word 'are' should be removed.
  5. [Reference [4]] Reference [4] is incomplete: the journal name is missing (it should be Inverse Problems).
  6. [Section 4.1, Figure 13] Figure 13 would benefit from explicit axis labels and units (e.g., milliseconds or seconds), and from a statement of the number of trials and hardware/software versions in the caption or text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's claims are direct measurements against external tools and implementations, with no fitted parameters or self-referential derivation.

full rationale

TorchRadon's central claims are empirical: speedups of 40x-125x over Astra Toolbox, numerical agreement with Astra and AlphaTransforms, and reconstruction quality metrics. All of these are measured comparisons against external, independently developed libraries, not quantities derived from the paper's own assumptions. The accuracy figures (relative errors, MSE) are direct outputs of running both implementations. The speed claims are benchmark measurements, and while the Astra configuration is under-specified in the paper, that is a reproducibility or measurement-fairness concern, not a circular one. The differentiability claim (autograd integration) is asserted but not numerically verified; again, this is a missing validation, not an input-output equivalence. There are no self-citations, no fitted parameters masquerading as predictions, and no equations where the conclusion is equivalent to the premises. The paper is self-contained against external benchmarks, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claims are empirical: speed and correctness. They rest on external reference implementations and the correctness of the CUDA kernels, not on a mathematical derivation. No new entities or fitted parameters are introduced.

assumptions (3)
  • domain assumption Astra Toolbox outputs are treated as ground truth for validating Radon and FBP implementations.
    Section 3.1 computes relative errors between TorchRadon and Astra; this presumes Astra is correct enough to serve as reference.
  • domain assumption AlphaTransforms coefficients are treated as ground truth for shearlet correctness.
    Section 3.2 compares shearlet coefficients and reconstructions against AlphaTransforms, assuming the reference library is correct.
  • domain assumption PyTorch autograd correctly composes custom backward functions.
    The library relies on PyTorch's automatic differentiation machinery, but no gradient checks against finite differences are provided.

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Cite this review

Pith. "Pith review of TorchRadon: Fast Differentiable Routines for Computed Tomography." pith.science (2026). https://pith.science/paper/DJIBQQ6U

@misc{pith2026200914788,
  author       = {Pith},
  title        = {Pith review of: TorchRadon: Fast Differentiable Routines for Computed Tomography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJIBQQ6U}},
  note         = {Machine review of arXiv:2009.14788}
}
read the original abstract

This work presents TorchRadon -- an open source CUDA library which contains a set of differentiable routines for solving computed tomography (CT) reconstruction problems. The library is designed to help researchers working on CT problems to combine deep learning and model-based approaches. The package is developed as a PyTorch extension and can be seamlessly integrated into existing deep learning training code. Compared to the existing Astra Toolbox, TorchRadon is up to 125 faster. The operators implemented by TorchRadon allow the computation of gradients using PyTorch backward(), and can therefore be easily inserted inside existing neural networks architectures. Because of its speed and GPU support, TorchRadon can also be effectively used as a fast backend for the implementation of iterative algorithms. This paper presents the main functionalities of the library, compares results with existing libraries and provides examples of usage.

Figures

Figures reproduced from arXiv: 2009.14788 by the authors.

Figure 1
Figure 1. The classic Shepp-Logan phantom projections with a very small accuracy loss. Refer to section 3.3 for more details about the loss of numerical accuracy. 2.1 Package Structure The library is divided into submodules grouped by functionality: • torch radon: main module containing Radon projections. Currently parallel beam and fan-beam projections are implemented. • torch radon.shearlets: GPU implementa￾tion of Alpha Sh… view at source ↗
Figure 2
Figure 2. Visualization of parallel beam projec￾tion. Results of forward projection on the phantom are visualized in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Comparison of forward Radon transform with parallel beam projection. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Comparison of backward Radon transform with parallel beam projection. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: Visualization of fan-beam projection. sino = radon.forward(x) # Radon backward projection bp = radon.backprojection(sino) In the above code we made use of TorchRadon’s RadonFanbeam default values: det distance by default is assumed equal to source distance and det spac…
Figure 6
Figure 6. Figure 6: Comparison of forward Radon transform with fan-beam projection. [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: Comparison of backward Radon transform with fan-beam projection. [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 8
Figure 8. Figure 8: Comparison of reconstructions using FBP with Ram-Lak filter. [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]
Figure 9
Figure 9. Figure 9: 12 coefficients of the shearlet transform of the Shepp-Logan phantom. [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: Shearlet reconstructions obtained by AlphaTransforms and [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 11
Figure 11. Figure 11: Comparison between sinograms obtained using single precision (32bits) and half precision [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]
Figure 12
Figure 12. Figure 12: Comparison between FBP reconstructions obtained using single precision (32bits) and half [PITH_FULL_IMAGE:figures/full_fig_p007_12.png]
Figure 13
Figure 13. Figure 13: Comparison of the speed of Torch Radon and Astra Toolbox on a modern server GPU (Tesla [PITH_FULL_IMAGE:figures/full_fig_p009_13.png]
Figure 14
Figure 14. Figure 14: Comparison between the reconstructions obtained using Filtered Backprojection and the [PITH_FULL_IMAGE:figures/full_fig_p010_14.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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