Method

CTO is an end-to-end model following an unrolled network design. The input sinogram $\mathbf{p}$ is first processed by a sinogram-space neural operator $\mathrm{NO}_s$, which performs function-space learning that unifies different sampling rates along the view-angle axis $\theta$. The intermediate result is then fed to an unrolled network with 3 cascades, each consisting of an image-space neural operator $\mathrm{NO}_i$ and a data-consistency (physical update) term, which together denoise and reconstruct the final image.

Both operators are built on Discrete–Continuous (DISCO) convolutions, which parametrize kernels in continuous function space and discretize them to match the input or output resolution — making the whole framework inherently resolution-agnostic. DISCO is discretization-convergent: the approximation error is bounded across resolutions and converges to zero as resolution increases.

Sinogram-space operator $\mathrm{NO}_s$

$\mathrm{NO}_s$ acts on the function space of $\mathbf{p}(\theta, r)$, with detector position $r$ and view angle $\theta$ as the two sinogram axes. DISCO convolutions are applied directly in this $(\theta, r)$ coordinate system — without reparameterizing to $(x, y)$ — so the model learns in the intrinsic measurement domain and avoids projection error. Because the sinogram's local support corresponds, by Radon duality, to direction-selective global support in image space, $\mathrm{NO}_s$ learns global structural information, complementing the local detail captured by $\mathrm{NO}_i$.

Joint spatial–frequency learning. $\mathrm{NO}_s$ comprises two parallel U-shaped DISCO branches. $\mathrm{NO}_{s,\text{spatial}}$ operates on $\mathbf{p}(\theta, r)$ to capture local, geometry-aware correlations. $\mathrm{NO}_{s,\text{freq}}$ operates on the sinogram's frequency representation, obtained by a 1D Fourier transform along the detector axis, to model long-range angular–radial dependencies; the outputs are averaged. This is motivated by the Fourier slice theorem,

$\mathcal{F}_r\{\mathbf{p}(\theta, r)\}(\omega) = \hat{f}(\omega\cos\theta, \omega\sin\theta),$

which says the 1D Fourier transform of the sinogram along $r$ encodes the object's frequency spectrum in polar coordinates — so localized operations there correspond to global modifications of the reconstructed image. It is also a data-driven generalisation of filtered back projection (FBP): rather than imposing a hand-crafted ramp filter, $\mathrm{NO}_{s,\text{freq}}$ learns a frequency-domain prior directly from data.

Rotation equivariance. Parallel-beam CT induces two symmetries on the sinogram: rotating the image by $\phi$ is a shift in view angle, $(\mathbf{p} \circ R_\phi)(\theta, r) = \mathbf{p}(\theta - \phi, r)$; and $\mathbf{p}(\theta + \pi, r) = \mathbf{p}(\theta, -r)$ — the $\pi$-periodicity-with-flip identity. We honour both with a modified circular padding along $\theta$ that first flips the boundary data along the detector axis $r$ and then wraps it around $\theta$ (standard reflective padding is used along $r$). This provides physically consistent boundary conditions, preventing discontinuities at $\theta = 0^\circ$ and $180^\circ$, and aligns the network's inductive bias with the sinogram's geometry.

Image-space operator $\mathrm{NO}_i$

$\mathrm{NO}_i$ is a U-shaped DISCO network operating in the spatial domain of image space, which lets it train and infer at different image resolutions. It refines and denoises the intermediate reconstruction from $\mathrm{NO}_s$, recovering high-frequency local information and fine structural detail, so that final reconstructions preserve both large-scale structural consistency and small-scale detail.

Results

We evaluate on the AAPM Low-Dose Abdominal CT dataset and the C4KC-KiTS Kidney CT dataset, at 9-, 18-, 36- and 72-view acquisition settings. For all methods a single model is trained across all sampling rates, with the rate chosen uniformly at random per batch, so every method sees the same amount of data at each rate. We report RMSE after conversion to Hounsfield Units (clinical accuracy) alongside PSNR and SSIM in the original reconstruction space (image similarity).

CTO outperforms learning-free methods, diffusion methods, and CNN and transformer architectures. Averaged over view counts, CTO is 3 dB PSNR better than the unrolled CNN baseline, 6 dB better than DPS, and 6 dB better than DuDoTrans on the abdomen dataset; on the kidney dataset the corresponding gains are 4 dB, 3 dB and 5 dB. Note that some baselines score below their published numbers — expected, since those works train a separate model per sampling rate.

Table 1: 18-view sparse-view CT reconstruction on the AAPM dataset. Lower is better for RMSE, higher for PSNR/SSIM. All learning methods are a single model trained jointly on $N_v = \{9, 18, 36, 72\}$ views. Full results at 36 and 72 views, with standard deviations, are in the paper.

Zero-shot super-resolution

Because CTO's kernels are defined as continuous functions, they maintain the same relative receptive field when the discretization changes — a CNN's fixed-size kernel covers half the relative field when resolution doubles. We test this without any fine-tuning.

Image domain ($\mathrm{NO}_i$). All models are trained to produce $256 \times 256$ reconstructions from 18-view sinograms; at inference the intermediate output is bilinearly upsampled to $512 \times 512$ before image-space processing. CTO gains 3 dB PSNR over the CNN baseline and 4.8 dB over LEARN. The CNN baseline shows substantial blurring and aliasing at the higher resolution, while CTO preserves much sharper detail.

Sinogram domain ($\mathrm{NO}_s$). Models are trained on sinograms with 72 views and 672 detectors, then evaluated at larger sinogram sizes. At 144 views / 672 detectors, CTO outperforms the unrolled CNN by 3.5 dB, LEARN by 3.2 dB and RegFormer by 2.8 dB; at 144 views / 1344 detectors, by 3.7 dB, 2.7 dB and 2.4 dB respectively.

Analysis and ablations

Overfitting to a single rate. A model trained only at 72 views collapses when tested elsewhere: LEARN reaches 32.32 dB at 72 views but 4.16 dB at 18 views, and RegFormer 35.11 dB versus 5.04 dB. Multi-rate co-training mitigates this (LEARN 25.51 / 27.53 / 29.90 dB and RegFormer 25.67 / 28.00 / 30.43 dB at 18 / 36 / 72 views), but CTO remains well ahead at 31.57 / 35.06 / 37.88 dB.

Inference cost. On an NVIDIA A100, CTO runs in 0.065 s per reconstruction with no per-rate tuning, against 51.58 s for DPS and 32.72 s for ALD — over $500\times$ faster than diffusion, which additionally requires hyperparameter tuning for each sampling pattern. SART (0.417 s) also needs per-rate tuning.

Component ablations. Averaged over view counts, removing $\mathrm{NO}_i$ costs 3.86 dB PSNR and removing $\mathrm{NO}_s$ costs 4.82 dB. Within $\mathrm{NO}_s$, removing the spatial branch costs 3.1 dB and the frequency branch 2.7 dB. The rotation-equivariant circular padding is worth 0.83 dB at 24-view subsampling (33.00 dB with, 32.17 dB without).

Conclusion

We present CTO, a unified neural-operator framework for sparse-view CT reconstruction that generalises across sampling rates of the sensory data, overcoming the limitation of existing deep learning methods that are tied to a fixed acquisition setting. By combining rotation-equivariant DISCO convolutions with joint spatial–frequency learning in the sinogram domain, CTO achieves robust, resolution-agnostic reconstructions and consistently outperforms state-of-the-art methods across downsampling regimes — a scalable and flexible alternative to traditional CT reconstruction.

Future work could extend CTO to more clinically relevant datasets, to 3D CT and other computational imaging tasks, add uncertainty analysis, and evaluate clinical applicability across broader scanner protocols.

Acknowledgements

This work is supported in part by ONR (MURI grant N000142312654 and N000142012786). A.D. is supported in part by the Summer Undergraduate Research Fellowships (SURF) at Caltech. J.W. is supported in part by the Pritzker AI+Science initiative and Schmidt Sciences. A.A. is supported in part by the Bren endowed chair and the AI2050 senior fellow program at Schmidt Sciences.

Citation

If you find this work useful, please cite:

@inproceedings{datta2026cto,
    title={Resolution-Agnostic Neural Operators for Multi-Rate Sparse-View CT},
    author={Datta, Aujasvit and Wang, Jiayun and Aali, Asad and Anandkumar, Anima},
    booktitle={European Conference on Computer Vision (ECCV)},
    year={2026}
}