Atlases Are Already Inside: Recovering Population Templates from Pretrained Diffusion Models

1 INTRODUCTION

• The sampler converges from every random start to one sharp image, with cross-seed SSIM ≥0.99, across brains, faces, and 3D shapes.The paper calls this behavior canonical convergence.Source: §1 · p. 2
• A population is represented as one template and the transformations that carry it to each member; in medical imaging, that template is an atlas.Existing atlas construction commonly aligns thousands of individuals into a common frame under a chosen deformation model.Source: §1 · pp. 1–2
• The hypothesis is that the denoiser carries shared population structure while injected noise selects individual variation, so removing noise traces the central instance of a coherent population.This proposed correspondence links the diffusion reverse process to pattern theory’s template-and-transformations account of a population.Source: §1 · p. 2
• The intrinsic atlas is best or second-best on every dataset and is the most central and most regular template on held-out brain cohorts.Evaluation spans brain MRI, faces, chest X-ray, and 3D shapes under a standard registration protocol.Source: §1 · p. 2
• Age conditioning yields age-specific atlases on demand that reproduce healthy-aging CSF expansion, while populations with several centers or none yield multiple templates or a blurry image.The sampler is intended to return one image per template when a population has several templates, and no stable image when it has none.Source: §1 · p. 2

2.1 ATLAS AND TEMPLATE CONSTRUCTION

• Classical and learned atlas methods use atlas-specific construction objectives, whereas this approach obtains an atlas from a generator trained for synthesis alone.Classical methods align and average under an explicit deformation model; learned methods amortize alignment or train a network to output the atlas.Source: §2.1 · pp. 2–3
• Vision and 3D-shape methods train templates and evaluate them under their own aligners, while this work trains no template and tests it under an unseen registration protocol.Examples include congealing and GANgealing for images, and DIT and DIF-Net for 3D shapes.Source: §2.1 · p. 3

2.2 DETERMINISTIC SAMPLING AND THE ROLE OF NOISE

• Unlike DDIM and probability-flow ODE sampling, which preserve marginals and yield distinct samples, this sampler discards injected noise; the authors report no prior work with this exact convergence or atlas use.Prior work edits injected noise to edit samples, exploiting its role in carrying individual variation.Source: §2.2 · p. 3

Canonical convergence

• 0.66 vs. 0.9999 cross-seed correlation distinguishes stochastic sampling from posterior-mean convergence in Figure 3.Each row uses an independent seed; the stochastic samples remain distinct while posterior-mean trajectories converge to the same image.Source: Figure 3 · p. 4

The posterior-mean iteration

• Replacing each stochastic reverse transition with its conditional mean removes injected noise and lets the denoiser pull the state toward the population center.The posterior-mean iteration re-inserts no noise, unlike DDIM, which re-inserts predicted noise at the schedule’s amplitude.Source: §3 · p. 4

Canonical convergence

• On a 2D Gaussian population with a known score, all 300 seeds reach the center with an offset of 2×10−4, while stochastic sampling spreads across the population.This demonstrates that convergence to the center is a property of the operator, not of a trained network.Source: §3 · p. 4

Generality

• Intrinsic atlases span medical and non-medical 2D and 3D populations, retaining shared structure and omitting features the population does not share.Examples include a clean sans-serif “A” from mixed typefaces and a chair atlas without legs.Source: Figure 5 · p. 5, §3 · p. 5

Evaluation scope

• Brain MRI evaluation uses IBSR18, Mindboggle101, and ABIDE-I; other tested domains include faces, chest X-rays, and 3D shapes.Neither intrinsic-atlas model was trained on the brain MRI test cohorts.Source: §4 · pp. 5–6

Brain MRI results

• On brain MRI, MultiMorph’s Dice is higher by at most 0.004 on Mindboggle101 and ABIDE, and by three points on IBSR18.The intrinsic atlas remains the most central and regular template on every cohort.Source: §4 · p. 5

Registration utility

• Table 1 reports the intrinsic atlas as best or second-best on every dataset across domains, and the most central and regular template on every brain MRI cohort.Registration uses the same protocol for every template within each domain, and each table cell averages at least three repeated runs.Source: §4 · p. 5, Table 1 · p. 6

Other domains

• Across non-brain comparisons, VoxelMorph leads on faces but is fourth on chairs, while Aladdin leads on chest X-ray but ranks last on faces.The intrinsic atlas is stronger than DIF-Net and DIT in PCK on both 3D shape datasets, and best on both PCK@0.05 and part-label IoU on 3D cars.Source: §4 · p. 6

On-demand atlas families

• One continuously age-conditioned brain-T1 model recovers an atlas at a fixed age with the same deterministic procedure and no per-age optimization.The model was fine-tuned on 899 cognitively normal IXI and OASIS-1 scans spanning ages 18–94.Source: §5 · p. 6

Biological validity

• The recovered atlas family reproduces monotonic, accelerating CSF expansion across age, with per-seed correlation r=+0.88 and spread below 0.2 percentage points at every age.The comparison to Yamada et al. (2023) removes one constant shift because the studies use different volumetry pipelines.Source: §5 · p. 7

Scope

• The displayed age family covers ages 22–80, excluding sparsely populated extremes of the model’s 18–94 training range.The figure varies only the age condition across columns.Source: §5 · p. 7, Figure 6 · p. 7

Registration utility

• −0.05 mm mean displacement favors age-matched over unconditioned atlases in every decade, with lower displacement for 53/78 subjects.The reported comparison has p=4×10−5; the VoxelMorph template lies a further 0.4–0.5 mm above the age-matched atlas.Source: §5 · pp. 7–8

Age specificity

• Age-matched atlases yield lower displacement than distant-age atlases for 57/78 subjects, with p=5×10−7.The mean displacement is also lower for the age-matched atlas in every decade.Source: §5 · p. 8

• 60 Pieridae seeds resolve to one of three similar pattern variants, revealing multiple population centers.The variants appear as distinct generated intrinsic atlases.Source: §6 · p. 8, Figure 8 · p. 8
• OCT consensus depends on modeling scale: the 32 × 32 atlas shows a clear fovea, while coherence degrades at higher resolutions.Pixel-space consensus reflects only the structures represented at a given scale.Source: §6 · p. 8, Figure 8 · p. 8
• FFHQ frames with varying hair, background, and clothing yield a featureless consensus, but cropping the same subjects to the face oval restores a sharp atlas.This is an example of a population whose shared structure becomes clear after removing varying content.Source: Figure 8 · p. 8, §6 · p. 9

• The intrinsic atlas follows the generator’s density, unlike a Fréchet mean defined by summed deformation under a chosen model.The paper attributes its centrality by construction to Tweedie’s formula and reports it as the most central under registration in Table 1.Source: §7 · p. 9
• When a population has multiple centers, unconditioned convergence pulls recovered centers toward one another; conditioning separates them.For the age family, the unconditioned atlas resembles age 45, near the mean training age of 47; conditioning gains increase with distance from that mean, most for subjects in their 70s.Source: §7 · p. 9, Figure 9 · p. 9
• Seed splits can diagnose multiple centers without labels, but a mode-detection tool with a principled agreement threshold remains future work.Source: §7 · p. 9
• The intrinsic atlas is a registration target, but deformation onto it is still computed outside the generator.If correspondence also requires no training, congealing could reduce to inference on a generator built only to synthesize.Source: §7 · p. 9

• Removing injected noise from diffusion inference yields the population template, called the intrinsic atlas, without retraining.Source: §8 · p. 9
• The intrinsic atlas is best or second-best as a registration target across brain MRI, faces, chest X-ray, and 3D shapes.Conditioning yields atlases matched to generator covariates; populations with several templates or none can yield multiple images or a blur.Source: §8 · p. 9
• Where a population coheres and a generator has learned it, atlas construction is reframed as a byproduct of generative modeling.Source: §8 · p. 9

Contraction rate

• For Gaussian populations, every noise-free step contracts deviation toward the center (0 < ρ_t(λ) < 1), multiplying the initial distance by the product of contraction factors.The rate is direction-dependent; directions with greater population variance shrink more slowly at low noise.Source: A MECHANISM: CONTRACTION RATE AND… · p. 14
• 3.8×10−3 measured endpoint offset matches the 4.1×10−3 prediction; at T=400, measured wide-axis spread is 0.079 versus 0.095 predicted.For the shorter schedule, the measured spread is elongated along the wide axis, as predicted.Source: A MECHANISM: CONTRACTION RATE AND… · p. 14

Several templates

• 0.20 endpoint displacement remains from each mixture template because high-noise responsibilities first pull seeds toward the global mean.Later contraction within each component is too small to undo that earlier pull.Source: A MECHANISM: CONTRACTION RATE AND… · p. 15

Controls

• Exact-score substitution preserves collapse, reintroduced noise degrades it, and deterministic DDIM does not converge, isolating noise removal rather than learning or determinism.Table 2 describes endpoint spread and center offset, cross-seed NCC under noise reinjection, and cross-seed NCC across samplers.Source: Table 2 · p. 14, A MECHANISM: CONTRACTION RATE AND… · p. 15

An instance, not an average

• 0.020 variance of the Laplacian within the brain for the atlas versus 0.0024 for the mean of 50 samples makes the atlas about 8× sharper.The atlas-to-mean NCC is 0.84; stochastic samples have pairwise NCC 0.47 over 50 seeds.Source: A MECHANISM: CONTRACTION RATE AND… · p. 15, Figure 10 · p. 15

B.1 DATA, GENERATORS, AND BASELINES

• The benchmark uses held-out subjects and domain-specific generators and baselines for brain MRI, faces, chest X-rays, and 3D chairs and cars.Brain MRI includes off-the-shelf and age-conditioned generators; all atlases are read out at t*=99.Source: §B.1 · pp. 15–16

B.2 REGISTRATION AND METRICS

• Registration uses ANTs defaults, with SyN variants selected by domain, and measures deformation fields on each template’s foreground mask.The protocol uses antsRegistrationSyNQuick[b] for brain volumes, SyNCC for 2D images, and SyN for upsampled occupancy volumes.Source: §B.2 · p. 16

B.2.1 METRICS FOR T1 BRAIN MRI

• Brain MRI metrics include Dice, centrality, mean displacement, and SDlogJ; centrality vanishes when opposing cohort deformations cancel.Absolute centrality values are not comparable with Abulnaga et al. (2025), but within-table comparisons use the same code and cohort.Source: §B.2.1 · p. 16

B.2.2 METRICS FOR OTHER MODALITIES

• Other-modality evaluation uses landmark PCK for images and shapes, part-label IoU for shapes, and pairwise cross-seed SSIM or NCC.Shape PCK follows a 5-shot protocol using five fixed source shapes; IoU averages over parts present per shape.Source: §B.2.2 · pp. 16–17

B.3 VERIFICATION AND REPEATED RUNS

• Metric implementations were checked against upstream code and analytic cases; repeated-run variation is negligible for brain MRI and faces but substantial for chest X-rays and chairs.On chairs, the intrinsic atlas and DIF-Net are separated from other templates but not from each other; on chest X-rays, only Aladdin’s lead exceeds run-to-run variation.Source: §B.3 · p. 17, Table 3 · p. 17

C.1 STOPPING TIME

• Across t* from 99 to 499, Dice stays nearly flat and Table 1 rankings remain unchanged; below t*=49 it falls sharply as subject-specific detail leads to under-registration.The chosen operating point is t*=99, selected before the runs rather than at the Dice maximum.Source: §C.1 · p. 18

C.2 QUALITATIVE COMPARISON

• A clinician judged the intrinsic atlas anatomically plausible, with no missing or duplicated structures, in a comparison beside established and learned brain atlases.Figure 13 places the intrinsic atlas beside three comparison templates.Source: Figure 13 · p. 18, §C.2 · p. 18

D.1 AGE-CONDITIONED FAMILY

• The age-conditioned evaluation compares decade-midpoint, distant-age, null-age, and VoxelMorph targets; OASIS-1’s Talairach-space scans require more deformation than IXI scans because the sources occupy different spaces.The training population has two centers, and the intrinsic atlas settles at the majority IXI center; the authors identify source-conditioned or space-matched generation as appropriate for Talairach-space subjects.Source: §D.1 · p. 19, Table 4 · p. 19

D.2 CONTRAST-CONDITIONED FAMILY

• Switching the anatomy-class token produces separate T1- and T2-weighted atlases from the same pretrained model, each identical across 20 seeds.Cross-seed SSIM is 0.9997 and 0.9992, respectively.Source: §D.2 · p. 19, Figure 14 · p. 20

E OCCUPANCY AS CONSENSUS CONFIDENCE

• Soft occupancy grades how consistently structures are shared: chair legs appear only below occupancy 0.2, while the airplane atlas does not change across levels.Figure 5 shows the level whose occupied voxel count matches a median training shape.Source: E OCCUPANCY AS CONSENSUS CONFIDENCE · p. 20, Figure 15 · p. 20

F REPRODUCIBILITY

• The 3D medical generator is a pretrained class-conditional latent diffusion model, and one atlas at 32×64×64 latent resolution takes about two minutes on an A100.The 2D and 3D generators are pixel- and voxel-space DDPMs trained from scratch.Source: F REPRODUCIBILITY · p. 20