A collection of large-scale inverse problems

A benchmark should not be tied to a single imaging modality. Different communities work with different forward models, acquisition geometries and data formats, yet many reconstruction algorithms share the same mathematical structure and face the same computational challenges when images become too large for a single accelerator.

For this reason, the benchmark integrates several representative inverse problems within a common interface. Each use case provides an imaging task together with its corresponding forward model, datasets and evaluation protocol, while relying on the same distributed reconstruction framework.

The current benchmark includes multi-frame super-resolution, X-ray tomography (in both 2D and 3D), and radio interferometric imaging. Additional applications can be incorporated following the same design, allowing the benchmark to evolve towards a broader collection of large-scale inverse problems.

1. Multi-frame super-resolution

Multi-frame super-resolution provides a simple yet representative inverse problem that is easy to scale to very large image sizes. Starting from a high-resolution image, the benchmark generates several complementary low-resolution observations by applying different blur operators, followed by downsampling and additive noise.

The acquisition model is

where $x$ is the unknown high-resolution image, $B_k$ denotes a frame-dependent blur, $S$ is the downsampling operator and $n_k$ is additive Gaussian noise.

Each observation contains only part of the spatial information. The goal is to recover a single high-resolution image by jointly exploiting all available frames.

The benchmark also provides a three-dimensional variant following the same multi-frame philosophy. In this case, several noisy observations of a synthetic volume are generated,

allowing the benchmark to evaluate reconstruction methods on very large 3D volumes while keeping the acquisition model intentionally simple. This makes it possible to focus on the scalability of the reconstruction algorithm without introducing additional complexity from the forward model.

2. X-ray tomography

Tomography is one of the most widely used imaging techniques for recovering the interior of an object from a series of X-ray measurements acquired at different viewing angles. It plays a central role in medical imaging, industrial inspection, materials science and many other scientific applications, making it a natural benchmark for large-scale inverse problems.

In the benchmark, the measurements are modeled as

where $x$ denotes the unknown object, $P_{\Theta}$ is the X-ray projection operator associated with the acquisition geometry, and $n$ represents measurement noise. Reconstructing the object therefore amounts to recovering $x$ from a finite collection of projection images.

The benchmark currently provides two tomography problems covering both two- and three-dimensional reconstruction.

The 2D benchmark is based on a resizable Shepp–Logan phantom, allowing image size and acquisition parameters to be varied while keeping a fully controlled experimental setting. It is well suited for reproducible performance studies and algorithmic comparisons.

The 3D benchmark is based on the Walnut cone-beam CT dataset, a widely used reference dataset for large-scale tomographic reconstruction, with volumes of 500³ voxels. It combines a realistic acquisition geometry with high-resolution volumetric data, providing a representative large-scale reconstruction problem while remaining fully reproducible.

Both variants share the same reconstruction interface, making it possible to evaluate distributed algorithms on synthetic and experimental tomography data without changing the reconstruction pipeline.

3. Radio interferometric imaging

Radio interferometry is a key imaging technique in modern astronomy, enabling high-resolution observations by combining the signals recorded by many individual antennas. Rather than measuring an image directly, an interferometer samples the Fourier transform of the sky brightness at a collection of irregularly distributed spatial frequencies. Reconstructing an image therefore requires solving an inverse problem from incomplete and noisy Fourier measurements.

In the benchmark, the acquisition model is represented as

where $x$ is the unknown sky image, $\Omega$ denotes the irregular Fourier sampling locations, $\mathcal{F}_{\Omega}$ is the corresponding non-uniform Fourier transform, $W$ contains measurement weights, and $n$ models the measurement noise.

The benchmark relies on realistic radio astronomy datasets and simulations to provide representative reconstruction problems. Sky images are stored as FITS files, while interferometric observations are generated using the Karabo simulation framework for the MeerKAT telescope. The resulting Measurement Sets are converted into weighted non-uniform Fourier operators through the DeepInverse physics framework, allowing reconstruction algorithms to be evaluated under realistic acquisition conditions.

Two image scales are currently provided, with native resolutions of 1,024 × 1,024 and 10,000 × 10,000 pixels. The latter corresponds to one hundred million image pixels, making it representative of the large-scale imaging problems targeted by distributed reconstruction methods.

As for the other benchmark problems, the radio interferometry pipeline follows the same reconstruction interface, making it possible to evaluate algorithms consistently across very different imaging modalities while keeping the underlying implementation transparent to the user.