Methods in LION

A map of what LION has: the kinds of learned reconstruction, the models of each kind, how each is trained and how it reconstructs, and where each comes from. Every method from a paper cites it: Class.cite() gives its BibTeX, Class.cite("MLA") the MLA form. The tutorials show them in use (Tutorials); the API reference has every argument (API reference).

The kinds of learned reconstruction

A learned reconstruction is a model (a network), trained by a solver, and used to reconstruct, by itself or inside an algorithm. They differ in where the network sits:

Kind	What the network does	Trained by	Reconstructs with
Post-processing	Removes the artefacts of a classical reconstruction (e.g. FDK): image to image.	SupervisedSolver, or a self-supervised solver	ForwardReconstructor
Unrolled iterative	An iterative algorithm unrolled into a fixed number of steps, each with learned parts, the operator between them: takes the measurements.	SupervisedSolver	ForwardReconstructor
Learned FBP	Filtered backprojection with learned parts (filter, interpolation, fusion): takes the measurements.	SupervisedSolver	ForwardReconstructor
Plug-and-Play (PnP)	A denoiser, trained on its own, used as the prior of an iterative algorithm (ADMM, HQS, FBS).	GaussianDenoiserSolver	PnP
Learned regulariser	A function of the image (low for good images) used as the regulariser of a variational problem.	ARSolver	LearnedRegularizerReconstructor

Self-supervision is a way of training, not a kind of model: the self-supervised solvers train image models (post-processing, usually) from noisy measurements alone (see Training).

Models

All in LION.models, each a LIONmodel. “Takes” is what forward takes: measurements (a model that uses the operator, made with its parameters: Model(geometry)) or images (a solver or reconstructor gives it the operator’s default reconstruction, FDK for CT).

Post-processing

Model	Takes	Paper
FBPConvNet	measurements (FDK, then a U-Net)	Jin et al., “Deep convolutional neural network for inverse problems in imaging”, IEEE Trans. Image Processing, 2017
FBPConvNetImage	images	the same, with the reconstruction made outside the model
cFBPConvNet	images	Runkel, Biguri and Schönlieb, “Continuous Learned Primal Dual”, MLSP 2024 (its continuous blocks in FBPConvNet)

Any image-to-image network below (CNNs) is a post-processing model too.

Unrolled iterative

Model	Takes	Paper
LG (Learned Gradient)	measurements	Adler and Öktem, “Solving ill-posed inverse problems using iterative deep neural networks”, Inverse Problems, 2017
LPD (Learned Primal-Dual)	measurements	Adler and Öktem, “Learned primal-dual reconstruction”, IEEE Trans. Medical Imaging, 2018
cLPD	measurements	Runkel, Biguri and Schönlieb, “Continuous Learned Primal Dual”, MLSP 2024
ItNet	measurements	Genzel et al., “Near-exact recovery for tomographic inverse problems via deep learning”, ICML 2022

Learned FBP

Model	Takes	Paper
DeepFBPNetwork	measurements	Tan et al., “Deep Filtered Back Projection for CT Reconstruction”, IEEE Access, 2024
FusionFBPNetwork, DeepFusionBPNetwork	measurements	Sainz Bear, “Deep Filtered Back Projection for CT Reconstruction”, MPhil thesis, 2025

Image networks (CNNs), denoisers

Image to image: post-processing networks, denoisers for PnP, or the models of self-supervised training.

Model	Notes	Paper
UNet		Ronneberger, Fischer and Brox, “U-Net: Convolutional Networks for Biomedical Image Segmentation”, MICCAI 2015
MSDNet	few parameters, no downsampling	Pelt and Sethian, “A mixed-scale dense convolutional neural network for image analysis”, PNAS, 2018
DnCNN	denoiser	Zhang et al., “Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising”, IEEE Trans. Image Processing, 2017
DRUNet	denoiser, takes the noise level; DRUNet.pretrained(): the authors’ weights	Zhang et al., “Plug-and-Play Image Restoration With Deep Denoiser Prior”, IEEE TPAMI, 2022
GSD (gradient-step denoiser)	wraps any denoiser N as x - ∇g(x), g(x) = ½‖x - N(x)‖²	Hurault, Leclaire and Papadakis, “Gradient Step Denoiser for Convergent Plug-and-Play”, ICLR 2022
ResUNet3D	3D, for segmentation	Yu et al., “Design of lung nodules segmentation and recognition algorithm based on deep learning”, BMC Bioinformatics, 2021

Learned regularisers

Model	Notes	Paper
AR (adversarial regulariser)		Lunz, Öktem and Schönlieb, “Adversarial regularizers in inverse problems”, NeurIPS 2018
ACR (adversarial convex regulariser)	convex in the image	Mukherjee et al., “Learned convex regularizers for inverse problems”, arXiv:2008.02839, 2020

Your own

Any torch module is a LION model with TorchModel(YourClass, arguments); a model that uses the operator derives from LIONmodel (Extending LION).

Training

All in LION.optimizers, each a LIONsolver.

Solver	Trains	Paper
SupervisedSolver	any model, on (measurements, image) pairs
GaussianDenoiserSolver	a denoiser, on clean images with Gaussian noise added (noise ranges, patches, augmentation, as in the literature)	as DnCNN and DRUNet’s training (Zhang et al.)
ARSolver	an adversarial regulariser (WGAN loss, gradient penalty)	Lunz et al., NeurIPS 2018
Noise2InverseSolver	self-supervised, from splits of the measurements	Hendriksen, Pelt and Batenburg, “Noise2Inverse: Self-supervised deep convolutional denoising for tomography”, IEEE TCI, 2020
Sparse2InverseSolver	self-supervised, sparse view	Gruber et al., “Sparse2Inverse: Self-supervised inversion of sparse-view CT data”, arXiv:2402.16921, 2024
Noisier2InverseSolver	self-supervised, correlated noise	Gruber et al., “Noisier2Inverse: Self-Supervised Learning for Image Reconstruction with Correlated Noise”, arXiv:2503.19468, 2025
Proj2ProjSolver	self-supervised, masked projections	Unal, Ertas and Yildirim, “Proj2Proj: self-supervised low-dose CT reconstruction”, PeerJ Computer Science, 2024
EquivariantSolver	self-supervised, equivariance to rotations	Chen, Tachella and Davies, “Equivariant imaging: Learning beyond the range space”, ICCV 2021
Equivariance2InverseSolver	self-supervised, equivariance and projection-wise cross-validation	Schut et al., “Equivariance2Inverse: A Practical Self-Supervised CT Reconstruction Method Benchmarked on Real, Limited-Angle, and Blurred Data”, arXiv:2510.23317, 2025
SelfSupervisedSolver	any loss of the model and its input (e.g. SURE)

Penalties on the model (a solver’s model_regularization, LION.losses.jacobian_penalty), for PnP denoisers that converge:

• FirmlyNonexpansivePenalty: Pesquet et al., “Learning maximally monotone operators for image recovery”, SIAM J. Imaging Sciences, 2021;

• ProximalDenoiserPenalty: Hurault, Leclaire and Papadakis, “Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization”, ICML 2022.

Losses (LION.losses): SURE (Metzler et al., “Unsupervised learning with Stein’s unbiased risk estimator”, arXiv:1805.10531, 2018; not usable yet for CT, issue #144) and SUREpgLoss (Poisson-Gaussian SURE, Le Montagner, Angelini and Olivo-Marin, IEEE TIP, 2014).

Reconstruction

Reconstructors (LION.reconstructors, each a LIONReconstructor) use a trained model; Reconstructors puts them side by side.

Reconstructor	Does	Paper
ForwardReconstructor	the model alone, as its solver ran it
PnP	ADMM, HQS or FBS with a denoiser as the prior	Venkatakrishnan, Bouman and Wohlberg, “Plug-and-Play priors for model based reconstruction”, GlobalSIP 2013; HQS as in Zhang et al., TPAMI 2022
LearnedRegularizerReconstructor	gradient descent on ½‖Ax - y‖² + α R(x), for any regulariser R	Lunz et al., NeurIPS 2018

Classical reconstructions, for any operator (LION.classical_algorithms): conjugate gradient, FISTA (Beck and Teboulle, 2009), SPGL1 (van den Berg and Friedlander, 2008); for CT (LION.operators.ct): FDK/FBP, SIRT and TV minimisation (tomosipo’s algorithms).

Data and experiments

Dataset	What it is	Paper
LIDC_IDRI	lung CT of 1010 patients with annotated nodules; LION simulates the sinograms	Armato et al., Medical Physics, 2011
deteCT	2DeteCT: measured sinograms of 109 objects in three modes (5,000 slices)	Kiss et al., “2DeteCT”, Scientific Data, 2023

Experiments (LION.experiments) fix a problem on them: LION’s simulated ones on LIDC-IDRI (ct_experiments: clinical, low and extremely low dose; full, sparse-view and limited-angle sampling), which also run on 2DeteCT’s measured data, and the 2DeteCT benchmark paper’s (deteCT_benchmark: Kiss et al., “Benchmarking learned algorithms for computed tomography image reconstruction tasks”, Applied Mathematics for Modern Challenges, 2025).

Operators

LION.operators: CT (ct: the geometry, the projector, noise, FDK), photocurrent mapping (pcm: Koutsourakis, Thompson and Blakesley, Solar RRL, 2021), wavelets, and composite and debiasing operators for sparse recovery. Everything above that uses an operator works with any of them that defines what it needs (e.g. autograd() to train through it).