One tree, four decades, thirty papers. CT reconstruction did not advance along a single line — it forked: into networks that repair the sinogram as well as the image, into coordinate networks that represent one scan at a time, into explicit Gaussian primitives, into generative priors, and finally into foundation models that try to hold all the branches at once. This hand-curated tree starts at FDK, the analytic root, and follows every fork. Each card is a paper in the library; each short label names the step that was taken; and the reading list below the canvas tells every step at full length.
FDK turns filtered back-projection into a practical 3D cone-beam algorithm — one filtering pass and one back-projection reconstruct a volume directly from cone-beam projections, exact on the midplane and close enough almost everywhere else.
The compressed-sensing turn for CT: ASD-POCS reconstructs by minimizing total variation under data-consistency and non-negativity constraints — the first convincing proof that a cone-beam volume can be recovered from far fewer views than FDK needs.
FDK is exact only when the data are complete. Cut the view count or the angular range and the analytic formula has nothing to say — the missing measurements must be filled by assumptions about the image. Sidky and Pan's assumption: real images are nearly piecewise-constant, so their gradient is sparse. Reconstruction becomes an optimization problem — minimize TV subject to staying consistent with the measured data — and the hand-written prior enters CT for good.
Sparsity, second draft: instead of assuming the image gradient is sparse, learn a redundant dictionary of image patches and assume each patch is a sparse combination of its atoms — the sparse-representation era's canonical CT method, set inside a statistical iterative reconstruction.
The prior becomes a module: plug-and-play replaces the proximal operator inside a model-based iteration with any off-the-shelf denoiser, so the denoiser itself acts as an implicit prior.
TV works, but it is one fixed guess about images, and every new hand-crafted prior demands its own bespoke solver. Venkatakrishnan, Bouman and Wohlberg notice that the prior's step inside these iterations is, mechanically, a denoising step — so they let you plug in any denoiser, including ones nobody can write a formula for. The prior stops being an equation and becomes a replaceable component. In 2013 the denoisers being plugged in are still classical — but the socket is ready for what comes next.
Learning enters through the back door: RED-CNN takes the FBP image as given and learns to clean it — a residual encoder-decoder combining autoencoder, deconvolution and shortcut connections that suppresses low-dose noise while preserving structure and lesions.
PnP proved a denoiser can carry the prior — which raises the obvious question: why hand-craft the denoiser when it can be learned from data? The first wave of deep learning in CT answers in the simplest possible place: after reconstruction. Reconstruct with FBP as always, then let a trained CNN remove the noise and streaks. It works strikingly well — and its limits are just as clear: the network never sees the measurements, so it can smooth away real structure or hallucinate plausible texture, and nothing checks its output against the physics.
Post-processing grows a theory: WavResNet denoises low-dose CT with a wavelet residual network built on framelet theory, pairing the framelet algorithm's performance guarantees with a CNN's expressive power.
The radical end-to-end answer: AUTOMAP recasts reconstruction itself as supervised learning, letting the mapping from sensor domain to image domain emerge from training data instead of being derived from geometry.
If a network can clean the image, why keep the analytic reconstruction at all? AUTOMAP, published in Nature, takes the opposite extreme from post-processing: learn the entire domain transform, projection data in, image out, with the manifold structure of real images doing the work that formulas used to do. It is a landmark of ambition — and a demonstration of the cost: the learned transform is welded to the geometry and distribution it was trained on, and it needs enormous training corpora to learn what FDK encodes in a page of math.
The synthesis: LPD unrolls a proximal primal-dual optimization and replaces only the proximal operators with CNNs, keeping the (possibly non-linear) forward operator inside the network — physics stays in the loop, learning fills the steps nobody could hand-write.
The field's answer to AUTOMAP's brittleness came the same year. Adler and Öktem keep the classical solver's skeleton — a primal-dual iteration that alternates between measurement space and image space — and learn only the proximal steps, as small CNNs. The forward operator is evaluated exactly at every iteration, so the reconstruction can never drift away from the measured data; the learned parts stay local and the training data requirement collapses. Unrolling becomes the era's dominant design pattern.
Unrolling, codified: MoDL alternates a CNN regularization prior with an exact data-consistency solve, sharing weights across iterations — a small network suffices, and the template transfers across inverse problems.
LPD proved the pattern on tomography; MoDL writes it down as a general architecture for inverse problems. One CNN prior block, one exact data-consistency step, repeated — with the same network weights at every iteration, so the model stays small and trains on modest data. Where LPD learned iteration-specific operators, MoDL's discipline is economy: the physics does most of the work, the network only supplies the prior. Model-based deep learning now has a reference design that a decade of papers will instantiate.
The unrolled template, made recurrent: LEARN++ revisits learned iterative reconstruction for compressed-sensing CT as a recurrent dual-domain network, carrying the LEARN design of 2018 into the dual-domain era.
The object itself becomes the model — as a coordinate field, or as explicit primitives.
MoDL's template assumes reconstruction is a mapping to be learned once and reused. Around 2021 a different instinct appears: stop learning mappings altogether, and learn a representation of this one object instead. A small coordinate network, optimised on the scan's own measurements, can stand in for the measurement field or the image itself — no training set, no distribution to overfit; the per-scan optimisation of the DIP era returns with better machinery. And once the object is a representation, it need not be a neural field: it can be a cloud of explicit Gaussian primitives, fitted by differentiable rendering. The two children of this box are those two answers — implicit and explicit — and they grow into the busiest branches of the 2020s tree.
Internal learning: CoIL trains a coordinate MLP on the scan itself — the network encodes the object's complete measurement field from its own sparse measurements, and the completed field then feeds ordinary reconstruction.
CoIL is the implicit line's cleanest early statement: an MLP that maps measurement coordinates to responses, trained only on the sparse measurements of the object on the table. The network is the completed measurement field; hand it to any ordinary reconstruction and sparse-view results improve. Nothing about the object class is learned in advance — the prior is the continuity of the field itself, a hand-written assumption wearing neural clothes, which makes this branch the tree's quiet echo of its 2008 root.
The image as a continuous function: NeRP parameterises the image itself as an MLP, embeds a prior image as its initialisation, and optimises it against the measurement physics — no training set, and the same recipe crosses between CT and MRI.
NeRF, made structure-aware for X-rays: SAX-NeRF builds a neural radiodensity field whose Lineformer backbone reasons over ray segments, reconstructing 3D volumes from sparse views.
The explicit turn: X-Gaussian represents the object as a cloud of 3D Gaussian primitives and renders X-ray projections by differentiable radiative rasterisation — reconstruction becomes fitting the primitives until their renderings match the scan.
The explicit sibling: where the implicit line hides the object inside a network's weights, X-Gaussian writes it out as primitives — a cloud of 3D Gaussians whose projections are rendered by a differentiable radiative rasteriser, the X-ray analogue of 3D Gaussian splatting. Fitting the scan becomes fitting the primitives. The representation is inspectable, editable and fast to render, and within two years it supports a full reconstruction framework (R2-Gaussian), a cross-modal unification (X-Splat) and a change of imaging geometry (LamiGauss).
Gaussians as a feature field: DIF-Gaussian lets 3D Gaussians carry feature distributions that supply extra spatial information for estimating the attenuation coefficients, targeting extremely sparse-view cone-beam CT.
Splatting, made honest for tomography: R2-Gaussian rectifies how 3D Gaussians project to 2D, giving the first 3DGS framework built for sparse-view tomographic reconstruction rather than view synthesis.
One representation, two modalities: X-Splat's Gaussian primitives share a single geometry across RGB and X-ray, with only surface-proximal primitives carrying appearance — splatting grows from a CT technique into a unified 3D representation.
Gaussians leave the cone-beam nest: LamiGauss pitches radiative rasterisation into laminography geometry — a detector-to-world transform for the tilt angle plus an initialisation that filters laminographic artifacts — reconstructing plate-like objects directly from sparse projections.
The dual-domain turn: DuDoNet restores the sinogram and enhances the image inside one end-to-end network, joined by a Radon inversion layer that carries gradients between the two domains — metal artifacts are fought where they start, not only where they land.
Post-processing has a structural blind spot: it only ever sees the image, so whatever FBP discarded in the sinogram is gone for good — and for metal artifacts the corruption is in the sinogram, in the measurements themselves. DuDoNet's move is to work both sides of the Radon transform at once: restore the sinogram where the damage starts, reconstruct, then enhance the image, with a Radon inversion layer carrying gradients between the domains so the two halves train as one network. Single-domain gives way to dual-domain — and the idea proves durable enough to return, years later, as a pretraining strategy for foundation models (CTPretrain, in the foundation group below).
Dual-domain, generalised: DuDoUFNet progressively restores under-complete sinograms into fully complete images in two stages, taking on metal artifacts and low dose as one joint problem rather than two separate ones.
The two lines braid: InDuDoNet+ unfolds a joint sinogram–image optimisation into learned proximal operators with CT geometry constraints embedded — dual-domain learning written in the unrolled, interpretable form.
The canonical post-processing entry: FBPConvNet applies direct inversion first, then a multi-resolution U-Net with residual learning regresses the FBP image to the ground truth — the network only ever sees FBP images, never the raw measurements.
RED-CNN showed a learned denoiser could clean an FBP image. Jin and colleagues ask what that denoiser should look like when the corruption is not just noise but the inverse problem itself: their answer, FBPConvNet, keeps the analytic step and learns the rest. A direct inversion — FBP — produces a first image, and a multi-resolution U-Net with residual learning maps that image to the ground truth, trained to correct exactly the artifacts the inversion leaves behind. The design is deliberately conservative: the geometry stays analytic — the network never touches the measurements — so the learning problem is a pure image-to-image mapping from a corrupted image to a clean one. It is the post-processing idea at its most explicit — inversion first, learning second — and the template RED-CNN and FBPConvNet share: reconstruct, then repair.
The borrowed prior: 3DIP — the paper the field cites as DiffusionMBIR — reconstructs 3D volumes with a pre-trained 2D diffusion model, coupled across the remaining dimension by a model-based prior at test time; no CT-specific training anywhere.
This is the paper the field cites as DiffusionMBIR: a pre-trained 2D diffusion model, applied slice by slice with a model-based coupling across the third dimension, solves 3D inverse problems — CT included — with no task-specific training. It proved a generative prior did not have to be trained on CT to reconstruct CT. Its sibling here, DOLCE, is the opposite answer: train one on CT anyway.
The prior becomes a generative model of CT images themselves: DOLCE uses a conditional diffusion model as the image prior inside a model-based framework, and one pretrained model reaches state-of-the-art results across drastically different image types on real limited-angle data.
The unrolled networks of 2018–19 still carry a discriminative CNN prior trained for one task and one degradation; change the protocol and you retrain. The next relocation of the prior: learn what CT images look like, full stop — a diffusion model trained as a generative prior — then couple it to the measurement model at reconstruction time. DOLCE does this for the hardest geometry, limited-angle CT, and the payoff is the one PnP promised in 2013 at a new scale: a single pretrained prior, plugged into the physics, generalizing across image types it was never fine-tuned on. The plug-in denoiser has grown into a generative model of the image manifold.
Full circle, one level up: DC-PnPDP restores the classical dual variable inside plug-and-play ADMM and puts a diffusion model in the prior seat, using integral feedback to force the data-consistency and prior steps into agreement.
Thirteen years after the socket was built, the plug comes home. DC-PnPDP puts a diffusion prior inside plug-and-play ADMM — and, dissatisfied with how loosely the prior step and the data step usually hold each other, restores the classical dual variable so an integral feedback term progressively forces the two into agreement. Every thread of this branch is in it: PnP's modularity, the generative prior, and the optimisation discipline of the unrolled era.
The prior at foundation scale: pretrained once, pointed at many degradations.
DOLCE showed a pretrained generative prior can serve many images, but it remained a framework for one problem family, trained and tuned per setting. FORCE asks the foundation-model question directly: can the prior-plus-physics package be made task-agnostic — one unsupervised engine that conditions on whatever measurements arrive, across low-dose, sparse-view and metal-artifact CT, without paired examples for any of them? Its answer replaces the diffusion prior with a Poisson-flow generative model (PFGM++) and presents the result as what its title claims: a tomographic foundation model.
The generative prior, packaged as a foundation model: FORCE unifies CT data fidelity with a PFGM++ generative prior for unsupervised reconstruction across low-dose, sparse-view and metal-artifact scenarios — no paired training data required.
One supervised model, many geometries: ProCT's view-aware prompting lets a single transformer handle diverse sparse-view and limited-angle settings, with artifact-aware contextual learning that senses artifact patterns from in-context pairs.
FORCE reaches "one model for everything" through an unsupervised generative prior, sampled iteratively. ProCT is the supervised counter-answer: keep one transformer, and tell it what it is looking at. A view-aware prompt encodes the acquisition geometry, so the same weights serve many sparse-view and limited-angle protocols; in-context pairs teach it the artifact pattern of the current scan. The interface idea is borrowed from foundation models in language and vision — the model is not retrained per task, it is prompted.
The foundation-model move in full: TAMP pretrains on 10.8M physics-driven simulated non-ideal-measurement images, then adapts with a parameter-efficient fine-tune on a few slices — one backbone for low-dose, sparse-view and limited-angle CT at once.
ProCT adapts one model to many settings, but the model is still trained at the scale of a single task family. TAMP completes the trajectory this chain has been tracing: pretrain at foundation scale — 10.8 million images simulated with the physics of non-ideal measurement — so that the prior is no longer a filter, a formula, a denoiser or even a sampler, but the accumulated statistics of CT itself, stored in a backbone that a few slices of fine-tuning can steer to a new protocol. In 1984 the knowledge lived in the filter design; in 2026 it lives in the pretraining corpus. The chain ends where the field now stands.
Pretraining, in both domains at once: CTPretrain couples a sinogram-domain restoration transformer with an image-domain one pre-trained through learnable Gaussian bandpass decomposition, so a single backbone adapts across low-dose, sparse-view and limited-angle degradations.
CTPretrain is where the dual-domain idea — born with DuDoNet in 2019 as a way to fight metal artifacts — returns as a foundation strategy: pre-train a sinogram-domain and an image-domain transformer together, and one backbone adapts across low-dose, sparse-view and limited-angle work. On the tree it sits in the foundation group, but its edge back to 2019 is written here, because a tree can only draw one parent.
The all-in-one endpoint: AMIFound trains one shared encoder with a dynamically routed expert pool on 3.76M images across seven modalities, covering seven degradation types — CT metal-artifact removal among them, on 37 CT datasets.