MUMINS: Metadata-conditioned Uncertainty-aware Medical Image Next-state Synthesis Anna Oliveras1,2 , Roger Marı́1 , Rafael Redondo1 , Oriol Guardià-Olivella1 , Cynthia Ifeyinwa Ugwu1 , Ana Tost1 , Bhalaji Nagarajan3 , Carolina Migliorelli1 , Vicent Ribas1 , Petia Radeva2,4
arXiv:2609.17169v1 [cs.CV] 15 Sep 2026
1
2
Eurecat, Centre Tecnològic de Catalunya, Barcelona, Spain Dept. de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain 3 Barcelona Supercomputing Center (BSC), Barcelona, Spain 4 Institut de Neurociències, Universitat de Barcelona, Barcelona, Spain
[email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]
𝒛𝒕
Abstract
MUMINS
Forecasting anatomical changes such as tumor growth and neurodegeneration is a challenging generative vision task. Morphological evolution is subtle relative to static anatomy, highly patient-specific, and inherently stochastic. Existing methods struggle with several issues: deterministic networks ignore biological stochasticity, while standard diffusion models require computationally prohibitive multi-pass sampling to quantify uncertainty. We propose MUMINS (Metadata-conditioned Uncertainty-aware Medical Image Next-state Synthesis), an efficient diffusion framework that jointly diffuses a baseline scan and its follow-up residual, summed to synthesize the follow-up scan, while concurrently predicting a spatial uncertainty map, in a single reverse diffusion process. Conditioned on the time interval and relevant metadata, it preserves fine-grained anatomy by dynamically re-injecting the baseline as a soft anchor at every denoising step, and a negative-log-likelihood head learns the uncertainty map to explicitly flag error-prone regions. Designed without organ-specific heuristics, the same architecture is reused across anatomies via separate, dataset-specific retraining. Extensive evaluations demonstrate that dataset-specific retraining of MUMINS matches or outperforms dedicated, domain-specific state-of-the-art methods on lung CT (PNG) and brain MRI (OASIS-3). Project page: https://github.com/aolivtous/MUMINS.
Noise Head
𝑳𝒓𝒆𝒄𝒐𝒏
Variance Head
𝑳𝑵𝑳𝑳
𝒚
𝒙
𝑩𝒂𝒔𝒆𝒍𝒊𝒏𝒆 𝑭𝒐𝒍𝒍𝒐𝒘 − 𝒖𝒑 𝒓𝒆𝒔𝒊𝒅𝒖𝒂𝒍
+
𝐜
↑ TRAINING ↑ ↓INFERENCE↓ Add noise
3D U-Net 𝑓𝜃 (𝑧𝑡 , 𝑡, 𝑐)
𝐜
𝒖𝒏𝒄𝒆𝒓𝒕𝒂𝒊𝒏𝒕𝒚
𝒙(𝒇𝒖𝒕𝒖𝒓𝒆) +
Guide
Recon
𝒙𝒕
𝒙
𝒚 = (𝒙(𝒇𝒖𝒕𝒖𝒓𝒆) − 𝒙) Baseline to Follow-up residual
𝒚𝒕
𝒚 Predicted Baseline to Follow-up residual
𝒙(𝒇𝒖𝒕𝒖𝒓𝒆) Predicted Follow-up
𝒙𝒕−𝟏
𝒚𝒕−𝟏
𝒙
𝒚
𝒛 = (𝒙,𝒚) Two-channel volumetric tensor
𝒄 𝒕
Conditional embedding Diffusion timestep
Figure 1. MUMINS overview. Training: the model learns to jointly generate the baseline image, the baseline-to-follow-up residual, and a voxel-wise uncertainty map via denoising diffusion. Inference: the real baseline is re-injected at every denoising step to guide residual generation. Joint state (baseline-residual) representation enhances consistency between scans, while predicted uncertainty highlights error-prone regions of the synthetic follow-up.
1. Introduction Predicting how anatomy evolves over time is a clinical necessity. In lung cancer, Computed Tomography (CT) screening identifies millions of indeterminate pulmonary
nodules yearly, and survival hinges on resolving the “waitand-watch” dilemma between benign, indolent lesions and early-stage malignancies [31]. In neurodegeneration, longitudinal brain Magnetic Resonance Imaging (MRI) underpins the tracking of Alzheimer’s disease progression, where patient-specific forecasts could accelerate trial stratification and treatment decisions [1]. In both modalities, forecasting a future scan (follow-up) from a starting scan (baseline) at an arbitrary interval is a prognostic necessity. From a computer-vision standpoint, this is a hard conditional generation problem: the morphological change between scans is small relative to the static anatomy, highly patient-specific, and inherently stochastic [2, 11]. A generative model must therefore preserve fine baseline scan details while generating plausible changes, and, since a single deterministic output cannot capture the range of biologically plausible futures, should at least flag where its own prediction is most likely to diverge from the true outcome. Progression modeling has evolved from classical analytical models [12, 22] to discriminative deep-learning predictors [8, 18, 36] and, recently, generative models [19, 29, 39, 42]. However, three limitations persist. First, determinism: the dominant deterministic and kinetic-model methods yield a single future for each input. Second, organ specificity: existing methods embed anatomy- or modalityspecific components (e.g., brain-age regularizers, growthkinetic priors), so each new setting demands a custom architecture [7, 9]. Third, uncertainty: recent diffusion models offer probabilistic synthesis [19, 29, 42], but extracting a per-voxel uncertainty map requires costly Monte Carlo aggregation over many reverse-diffusion passes [20, 41, 43]. We propose MUMINS (Metadata-conditioned Uncertainty-aware Medical Image Next-state Synthesis), a diffusion framework that forecasts a patient’s follow-up scan (next state) at an arbitrary future interval ∆τ from their current one (baseline), along with a spatial uncertainty map, addressing all three limitations above. Fig. 1 gives an overview of the method. Our contributions are: • Joint baseline-residual representation. MUMINS jointly denoises the baseline and baseline-to-follow-up residual, with baseline re-injection at each inference step to preserve patient-specific anatomy while synthesizing plausible morphological change. • Single-pass per-voxel uncertainty prediction. We embed likelihood-based uncertainty learning to predict voxel-wise uncertainty from a single reverse-diffusion process. This bypasses the prohibitive cost of Monte Carlo sampling, which requires multiple inferences per sample, yielding uncertainty maps that correlate with reconstruction error. • Domain-agnostic design. MUMINS uses a universal architecture with generic metadata conditioning, free of organ- or modality-specific components. After dataset-
specific retraining, it matches or outperforms domainspecific state-of-the-art methods on PNG lung CT [35] and OASIS-3 brain MRI [17].
2. Related Work Longitudinal medical image progression aims to forecast patient-specific anatomical change from baseline scans, with growing attention to the uncertainty associated with such forecasts [46]. In line with the experimental benchmarks considered in this work, we first discuss existing methods for pulmonary nodule evolution and neurodegenerative brain progression, then review recent diffusion-based approaches to uncertainty quantification.
2.1. Longitudinal Medical Image Progression Deterministic and registration-based methods. Early progression models are deterministic regressors that learn a fixed mapping from baseline to follow-up. For lung nodules, NoFoNet [18] learns voxel-wise displacement fields to warp the baseline scan, refining intensities via a dedicated TextureNet branch, while Tao et al. [36] similarly warp a prior scan conditioned on the follow-up interval. LCTformer [21] instead couples a longitudinal Transformer with a ConvLSTM to capture spatiotemporal growth dynamics. A parallel line of work couples clinical kinetic growth models (e.g., Gompertz functions) with deep generative networks: GM-AE [8], LNGNet [38], and STGNet [10] all follow a two-stage paradigm, first predicting future nodule mass or volume via kinetic equations, then conditioning a downstream decoder to synthesize the final shape and texture. Recently, NGP-Net [35] introduced a lightweight growth architecture and released the longitudinal Pulmonary Nodule Growth (PNG) dataset, which, unlike fixed-interval datasets such as NLST [24], features irregular scan intervals handled via a dedicated spatiotemporal encoding module. Despite their differing mechanisms, deterministic approaches yield only a single predicted future without progression uncertainty, and kineticmodel variants are additionally vulnerable to error propagation, as downstream shape and texture generation depend on the accuracy of upstream growth predictions. Adversarial and Flow Matching methods. GP-WGAN [39] proposed adversarial training for nodule progression by generating one-year follow-up scans, demonstrating that synthesized images could improve malignancy classification. To model continuous dynamics, CorrFlowNet [42] maps baseline CT embeddings to follow-up embeddings via a latent Neural ODE trained with flow matching, using a correlational autoencoder to isolate nodule progression from static lung anatomy. Similarly, IMMFM [14] employs piecewise-quadratic interpolation paths as smooth targets for flow matching in longitudinal neuroimaging. While these methods can model progression over continuous time,
they mainly learn transitions from one state to another rather than capturing dense distributions of plausible follow-up scans, motivating diffusion-based approaches. Diffusion-based progression. Denoising diffusion probabilistic models (DDPMs) have become a dominant paradigm for high-fidelity image generation [6, 23, 26, 27]. Recent methods leverage DDPMs for follow-up synthesis, but remain deeply anatomy-, disease-, or modality-specific. In pulmonary imaging, CIResDiff [15] predicts idiopathic pulmonary fibrosis progression by learning residual differences between scans, conditioned on Contrastive LanguageImage Pretraining (CLIP) [30] embeddings. For brain MRI, TADM-3D [19] models volumetric progression as residual diffusion over intensity differences, enforcing temporal coherence via Brain-Age Estimator conditioning and BackIn-Time Regularization, while TaDiff [20] integrates treatment awareness into the diffusion process for diffuse glioma growth prediction, guided by sequential multi-parametric MRI and treatment plans. BrLP [29] operates in a compact latent space using a latent diffusion model, a ControlNet [45], an auxiliary volumetric network, and a patch discriminator for adversarial training, averaging multiple generation paths to improve spatiotemporal consistency and volumetric accuracy in Alzheimer’s disease.
2.2. Uncertainty Quantification in Diffusion Models Quantifying uncertainty in diffusion-based synthesis is an active, clinically relevant area. The most direct approach is Monte Carlo (MC) sampling: running the reverse process N times and taking the per-voxel variance [20, 29, 41, 43]. This scales as N × T network evaluations per volume and is sensitive to N [32]. BayesDiff [16] reduces this cost via a post-hoc Last-Layer Laplace Approximation but still resamples weights every step, leaving the multi-pass bottleneck intact. A complementary line, from sports trajectory forecasting, instead learns uncertainty in a single forward pass: U2Diff [3] and its heteroscedastic extension U2Diffine [4] add a bivariate NLL term to the DDPM objective, jointly predicting the noise’s mean and variance. U2Diffine propagates this variance to the output via a firstorder Taylor expansion of the reverse update, linearizing the denoiser via its Jacobian; U2Diff forgoes this for speed.
3. Methodology In this section, we describe MUMINS in four stages. First, follow-up synthesis is formulated as conditional generation of a joint baseline-residual state, separating static anatomy from longitudinal change. Second, a 3D temporally conditioned diffusion model learns this joint state and its associated uncertainty. Third, during sampling, the baseline scan is re-injected at every reverse step to anchor the generated residual to patient-specific anatomy. Finally, uncertainty estimates from the reverse steps are accumulated into
a voxel-wise map of where the synthesized follow-up is less reliable. Fig. 1 gives an overview of the method.
3.1. Joint Baseline-Residual Representation Let a patient’s baseline anatomical state be an observation x ∈ X ⊂ RD×H×W acquired at time τ0 , and the follow-up state be x(future) acquired at τf uture , with temporal interval ∆τ = τf uture − τ0 . Directly modeling p(x(future) | x, ∆τ ) often leads to shortcut learning, where the network ignores the temporal condition or fails to preserve patient-specific anatomy [19, 20]. We instead target the longitudinal residual y = x(future) − x, isolating disease progression and anatomical evolution from the static anatomy, and synthesize it jointly with the baseline as z = (x, y). This formulation is inspired by DiffAtlas [44], a segmentation method that jointly diffuses an image and its mask and, at inference, substitutes the known image at each reverse step to guide mask generation. We adapt this idea to progression along two axes. First, rather than an image-mask pair, our joint state pairs the static baseline x with the dynamic residual y, so that substituting the known baseline at inference (Sec. 3.5) guides the generated residual. Second, we condition on clinical and acquisition variables c (Sec. 3.3) to further encourage patient- and time-specific synthesis. We therefore learn the conditional joint distribution pθ (z | c); modeling baseline and residual jointly keeps the generated progression structurally consistent with the baseline.
3.2. Forward Process and Noise Prediction We construct a discrete-time Markovian diffusion process that progressively adds Gaussian noise to the joint data distribution z0 ∼ pdata (z) [13]. Following [25], the noise is scheduled with a cosine variance profile β1 , . . . , βT over T steps, defining the forward transitions as q(zt | zt−1 ) = √ N (zt ;Q 1 − βt zt−1 , βt I). Using the reparameterization t ᾱt = s=1 (1 − βs ), the marginal distribution at any diffusion timestep t is given in closed form: √ q(zt | z0 ) = N (zt ; ᾱt z0 , (1 − ᾱt )I). (1) The reverse process is parameterized by a 3D U-Net [5] fθ (zt , t, c) outputting four channels, split into a noiseprediction tensor ϵ̂θ (zt , t, c) (the noise-mean head) and a second tensor ŝθ (zt , t, c) passed through a sigmoid to produce a per-voxel standard deviation (the variance head): σ̂ θ (zt , t, c) = sigmoid(ŝθ (zt , t, c)) + ε,
(2)
with ε = 10−6 for numerical stability. Because zt is a joint variable, both outputs factorize into baseline and residual components, ϵ̂θ = (ϵ̂xθ , ϵ̂yθ ) and σ̂ θ = (σ̂ xθ , σ̂ yθ ). For brevity, we henceforth write ϵ̂θ ≡ ϵ̂θ (zt , t, c) and σ̂ θ ≡ σ̂ θ (zt , t, c), with the same convention applied to their perchannel components ϵ̂kθ and σ̂ kθ for k ∈ {x, y}, suppressing dependence on (zt , t, c) unless required for clarity.
3.3. Temporal and Metadata Conditioning The conditioning vector c = (∆τ, a, ∆bx , ∆by , ∆bz ) aggregates clinical and acquisition signals relevant to longitudinal modeling: the inter-scan interval ∆τ , the patient’s age at τ0 (a), and a triplet of acquisition-quality descriptors (∆bx , ∆by , ∆bz ) encoding the per-axis change in image blur between scans, computed during training as the difference in gradient magnitude along axis i between paired scans. At inference, we report two settings: ∆b = 0, which preserves the baseline’s own sharpness and requires no knowledge of the unseen follow-up, and ∆b = GT, an oracle upper bound computed directly from the true followup. The former is the only information-respecting choice in a genuine forecasting deployment; the latter is reported purely as a diagnostic ceiling (Table 1). Follow-up scans can otherwise exhibit global sharpness differences from acquisition variability rather than anatomical progression, as in the PNG benchmark [35]; blur conditioning disentangles these acquisition-related appearance changes from true anatomical change (See Supp. Sec. G). Each scalar is independently mapped through a sinusoidal positional embedding [37] and a two-layer MLP, then concatenated with the diffusion timestep embedding. This combined embedding modulates the network’s intermediate representations via the time-conditioned residual blocks, letting the noise estimator adapt its transition dynamics to the current noise scale, target longitudinal horizon, and acquisition context.
3.4. Hybrid Training Objective with Voxel-wise Uncertainty We adapt the NLL-based uncertainty learning of U2Diff [3], originally developed for trajectory forecasting, to volumetric follow-up synthesis. The model predicts a per-voxel variance alongside each diffusion noise estimate, enabling uncertainty maps to be obtained from a single reversediffusion process. Training uses a hybrid objective that combines standard noise prediction with the NLL term. Joint Noise Prediction. The primary objective trains the network to denoise the joint state. For each sample, we draw ϵ ∼ N (0, I) and t ∼ U(1, T ), and minimize a channel-wise reconstruction loss on the predicted noise mean: Lrecon = Ez0 ,ϵ,t,c ∥ϵx − ϵ̂xθ ∥1 + ∥ϵy − ϵ̂yθ ∥1 , (3) x and y channels are weighted equally so uncertainty in either branch is not absorbed by the other. Uncertainty Loss. To learn predictive variances without destabilizing noise prediction, we add an NLL term where the predicted mean acts only as a target for the variance head (Sec. 3.2), whose per-voxel standard deviation σ̂ kθ (Eq. 2) is the predicted noise scale in the Gaussian NLL of Eq. (4). Following [3], we apply a stop-gradient sg[·] to the noise mean so gradients from the NLL flow only through the un-
certainty head: LN LL = Ez0 ,ϵ,t,c
X
log
√
2π σ̂ kθ
k∈{x,y}
h i 2 ϵk − sg ϵ̂kθ 2 + , 2 (σ̂ kθ )2
(4)
This decomposition decouples optimizing the deterministic noise estimate (Eq. 3) from calibrating its uncertainty, preventing the network from trivially reducing LN LL by inflating σ̂ θ at the cost of Lrecon . The total objective is: Ltotal = Lrecon + λN LL LN LL ,
(5)
with λN LL = 0.01 set empirically so the uncertainty term does not dominate denoising.
3.5. Baseline-Guided Conditional Synthesis During inference, the objective is to sample from the conditional distribution pθ (x, y | c) given an observed baseline x and the conditioning vector c. Because our model is trained on this joint distribution, we perform conditional sampling via baseline injection. Starting from pure Gaussian noise zT ∼ N (0, I), at each reverse step t the network predicts the joint posterior parameters. To ground the generative process in the patient’s actual anatomy, prior to each network evaluation we overwrite the x-component of zt with an exact sample from the forward marginal of the known baseline: √ (6) xt ∼ N ᾱt x, (1 − ᾱt )I . We realize this by drawing a single noise realization ϵ ∼ N (0, I) once per sample and √ it, unchanged, at ev√ reusing ery reverse step, i.e. xt = ᾱt x + 1 − ᾱt ϵ for all t: this satisfies the marginal of Eq. (6) at every step while tracing one coherent forward-diffusion trajectory of the known baseline, rather than resampling ϵ independently at each step. By iteratively projecting the baseline channel onto this fixed trajectory, the residual yt is continually guided by accurate, temporally consistent anatomical context. We ablate this choice against per-step noise resampling in Supp. Sec. F Table F.5, finding the single-trajectory injection preferable. At the end of the reverse process, we get the denoised joint state ẑ0 = (x̂0 , ŷ0 ). We retain its residual channel and add it to the baseline to obtain the synthesized follow-up x̂(future) = x + ŷ0 .
3.6. Uncertainty Propagation at Inference Beyond a single volume, our framework yields a per-voxel uncertainty map by propagating the network’s predicted variance through the reverse process. Two approximations make this tractable at volumetric resolution. First, we adopt a diagonal predictive covariance, so the network outputs a
per-voxel variance σ̂ 2θ . Second, as in U2Diff [3], we set the Jacobian Jt = ∇z ϵ̂θ to zero, since retaining it requires a backward pass through the 3D denoiser at every reverse step, infeasible at our scale even under the diagonal approximation [4]. Our sampling uses the stochastic DDIM [33] update with skip interval ζ and stochasticity η, with perstep noise scale σt . Let Vt = Var(zt ) be the accumulated per-voxel variance (VT = 0). Assuming zt , ϵ̂θ , and the injected noise are mutually independent (diagonal assumption), a first-order variance decomposition yields: Vt−ζ = rt2 Vt + c2t σ̂ 2θ (zt , t, c) + σt2 , p
(7)
p
ᾱt−ζ /ᾱt and ct = 1 − ᾱt−ζ − σt2 − with √ rt = rt 1 − ᾱt . The three terms propagate accumulated uncertainty along the trajectory, add the network’s predicted uncertainty at step t, and inject the sampler’s stochastic noise, respectively; the full derivation is in Supp. Sec. A. Setting Jt = 0 removes the stabilizing cross-term of the full formulation [4], leaving an unopposed homogeneous map with per-step amplification rt2 > 1. Propagating from the start of the chain thus accumulates variance monotonically, yielding over-dispersed, poorly calibrated maps by t = 0 (shown to be structural, not numerical, in Supp. Sec. B). We therefore adopt the delayed variance propagation of U2Diff [3], activating the recursion only for t ≤ ŝ and holding Vt = 0 earlier, selecting ŝ by minimizing the NLL. Unlike U2Diff and U2Diffine, which sample deterministically, our baseline-anchored synthesis requires stochastic sampling: the injected term σt2 enters the recursion only for η > 0, and is needed both to calibrate uncertainty and, as our ablation shows, to maintain reconstruction quality (Supp. Sec. F, Table F.4). From residual variance to follow-up uncertainty. The recursion above propagates variance over the two-channel joint state, so the final map V0 contains a baseline component V0x and a residual component V0y . Since the synthesized follow-up is x̂(future) = x + ŷ0 with the baseline x injected as a known measurement, its predictive variance reduces exactly to the residual component: Var x̂(future) = Var(x + ŷ0 ) = Var(x) | {z } =0
+ Var(ŷ0 ) + 2 Cov(x, ŷ0 ) = V0y , | {z }
position of Eq. (7) is unaffected by the single-trajectory-vsresampling choice.V0y is thus the per-voxel predictive variance of the synthesized follow-up, discarding the baseline component. We report its element-wise square root as the p final uncertainty map, σ ≜ V0y , distinguishing it from the two quantities it is built from: the per-step network output σ̂ θ (Eq. 2), and the accumulated variance Vt it propagates into via the recursion of Eq. (7). The reverse chain concludes with a standard DDPM [13] step from t1 to t = 0; following U2Diff [3], the accumulated variance is carried through unchanged, V0 = Vt1 .
(8)
=0
where Var(x) = 0 because x is deterministic and fully observed, and Cov(x, ŷ0 ) = 0 follows from the diagonal predictive covariance of Sec. 3.6. This holds regardless of which guide-noise scheme injects the baseline (Sec. 3.5): x itself is the same deterministic, fully-observed measurement under either injection variant, so the variance decom-
4. Experiments and Results 4.1. Datasets and Preprocessing Lung Nodule Progression. We use the Pulmonary Nodule Growth (PNG) dataset [35]: 378 chest CT scans from 103 patients, 226 longitudinal nodules with at least three time points (2–64 month intervals), radiologist-verified. We follow NGP’s [35] preprocessing and splits, converting longitudinal trios into pairs while keeping patients within a single split (full pipeline in Supp. Sec. C). Alzheimer’s Disease Progression. We use OASIS-3 [17]: 1352 T1-weighted scans from 492 subjects (ages 42–97; CN/MCI/AD), spanning ∼10 years with a mean baselineto-follow-up interval of 3.7 years. We follow the Turboprep [28] preprocessing of [29] and use 1283 volumes as in [19], split 70/15/15% with intra-patient pairs kept within a single split (full pipeline in Supp. Sec. C).
4.2. Implementation Details MUMINS uses a 3D U-Net [5] with per-resolution attention factorized into in-plane and rotary-embedding depthaxis terms [34], adapted from the spatial/temporal split used in video diffusion models. We train with Adam (10−4 ) and EMA (decay 0.995) for 150k iterations, batch 2 (300k for brain at batch 1; Supp. D) on 2 NVIDIA H100 (64 GB) GPUs, using T = 300 diffusion steps with a cosine schedule [25] and λNLL = 0.01. At inference we use stochastic DDIM [33] with skip interval ζ = 5 (60 steps), η = 1, and variance start-step ŝ = 15 (selected on held-out validation; ablated in Supp. F). All baselines are retrained and evaluated on identical splits; extended details in Supp. D.
4.3. Evaluation metrics As described in Sec. 4.1, samples are organized as longitudinal pairs; all metrics are averaged over pairs rather than subjects or nodules. Synthesis quality. We compare each generated follow-up x̂(future) against the ground truth x(future) using three complementary metrics: mean absolute error (MAE ↓), the average voxel-wise intensity error; peak signal-to-noise ratio (PSNR ↑), the reconstruction quality to the maximum sig-
Baseline GT Follow-up GT
age: 53
dt: 6m
MUMINS
NGPNet
TADM-3D
GM-AE
age: 51
dt: 43m
age: 52
dt: 6m
Absolute Error (CT)
0.4 0.3 0.2 0.1
Baseline GT Follow-up GT
age: 68
dt: 2.86y
age: 52
dt: 3.42y
MUMINS
BrLP
0.0
TADM-3D 0.175
Absolute Error (MRI)
0.150 0.125 0.100 0.075 age: 66
dt: 2.87y
0.050 0.025 0.000
Figure 2. Qualitative next-state synthesis on lung CT (top) and brain MRI (bottom): baseline (age, green), ground-truth follow-up (∆τ , blue; in years y or months m), and per-method predictions with absolute error maps. MUMINS inference with ∆b = 0.
nal; and structural similarity (SSIM ↑) [40], the perceptual and structural agreement beyond intensity. Uncertainty evaluation. MUMINS outputs a per-voxel uncertainty map σ in a single reverse-diffusion pass. We assess σ against the absolute error map e, with ei = |xi − x̂i |, along two axes: localization, whether σ is high where error is large, and calibration, whether the predicted uncertainty magnitude matches the observed error scale. For localization, we report Spearman (ρ) correlation between σ and e, and the Area Under the Sparsification Error curve (AUSE); lower is better. For calibration, we report the 95% prediction-interval coverage probability (PICP95 , ideal 0.95) and the mean prediction-interval width (MPIW), computed in normalized image-intensity units (Supp. Sec. D). Coverage at additional nominal levels is given in Supp. Sec. E. As a widely used uncertainty estimate in diffusion models [20, 29, 41, 43], we compute a Monte Carlo (MC) reference: for each input we draw K = 20 stochastic generations from the same trained model and take their voxel-wise standard deviation. Since MC and our single-pass σ capture different uncertainty sources, we do not treat MC as ground truth; both are scored independently against the true error, and we report their cross-agreement (ρ). We discuss the compromise of this K=20 choice, including MC’s own convergence behavior, in Supp. Sec. E.
4.4. Results We evaluate MUMINS against GM-AE [8], NGP-Net [35], and TADM-3D [19] on PNG [35] lung CT, and BrLP [29] and TADM-3D on OASIS-3 [17] brain MRI, plus a nochange copy baseline (x̂(future) = x, i.e. ŷ0 = 0).
Table 1 reports both blur-conditioning settings from Sec. 3.3. MUMINS faithfully follows whatever ∆b it is given (r = 0.974–0.980 in Supp. Fig. G.4), including ∆b = 0, which preserves the baseline’s own blur. These retrospective benchmarks, however, contain real inter-visit blur drift that ∆b = 0 does not chase, which is why it trails ∆b = GT below (See Supp. Sec. G). Unless stated otherwise, MUMINS refers to the deployable ∆b = 0 setting, for parity with baselines that lack access to the true follow-up. On PNG lung CT, MUMINS (∆b = 0) attains the best aggregate whole-image MAE, PSNR, and SSIM among lung baselines, narrowly edging out NGP-Net (23.53 vs. 22.85 dB PSNR) despite NGP-Net’s extra scan, and the best aggregate nodule-ROI PSNR (24.89 dB), with ROI MAE essentially tied with GM-AE. The copy baseline trails all methods but TADM-3D on MAE/PSNR, confirming change is rewarded over copying. Whole-image gains mostly hold across subgroups at ∆b = 0: SSIM trails NGP-Net in Stable/Shrink, MAE trails GM-AE in Stable. ROI-level accuracy is more sensitive: MUMINS clearly leads NGP-Net in Grow (26.12 vs. 23.38 dB PSNRROI ) but trails it in Stable and, more narrowly, Shrink, where GM-AE’s kineticgrowth model tops both. In Shrink, this reflects acquisitionblur mismatch, fully resolved by ∆b = GT (25.45 vs. 24.56 dB PSNRROI ); in Stable, the gap persists even with the true blur, plausibly reflecting that NGP-Net’s extra scan helps on near-static nodules. Aggregated over all nodules, supplying the true drift (∆b = GT, an oracle diagnostic) recovers a uniform margin over NGP-Net (aggregate PSNR +2.65 dB; nodule-ROI PSNR +2.45 dB).
Table 1. Synthesis results on the PNG lung CT benchmark and OASIS-3 brain MRI; subgroups indicate nodule growth trend (lung) and cognitive status (brain). Metrics reported over the whole image and the ROI (nodule/brain). Copy is a no-change reference.
OASIS-3 Brain MRI [17]
PNG Lung CT [35]
Subgroup Method
MAE (×10−2 ) ↓ PSNR (dB) ↑
SSIM (%) ↑
MAEROI (×10−2 ) ↓ PSNRROI (dB) ↑
Grow
GM-AE [8] TADM-3D [19] NGP-Net [35] MUMINS ∆b = 0 MUMINS ∆b = GT
4.03 ± 1.32 4.93 ± 1.57 4.74 ± 1.70 3.87 ± 1.60 2.95 ± 1.29
22.83 ± 2.61 21.49 ± 2.41 21.87 ± 2.75 23.65 ± 3.87 26.22 ± 4.41
65.29 ± 11.39 68.33 ± 10.10 68.48 ± 13.64 71.66 ± 12.36 80.82 ± 11.21
4.47 ± 2.51 5.92 ± 4.37 6.05 ± 3.80 4.23 ± 2.71 3.42 ± 2.16
25.18 ± 4.11 23.87 ± 5.28 23.38 ± 5.11 26.12 ± 4.81 28.39 ± 5.29
Stable
GM-AE TADM-3D NGP-Net MUMINS ∆b = 0 MUMINS ∆b = GT
3.71 ± 1.45 4.46 ± 1.20 3.95 ± 1.24 3.79 ± 1.90 3.06 ± 1.71
23.75 ± 2.93 67.34 ± 15.02 22.12 ± 2.14 67.53 ± 11.35 23.22 ± 2.56 72.81 ± 9.57 24.36 ± 4.35 72.41 ± 15.06 26.16 ± 4.94 76.91 ± 15.83
7.02 ± 5.47 8.34 ± 6.40 4.66 ± 2.88 7.52 ± 5.67 6.00 ± 5.80
22.46 ± 6.03 20.91 ± 5.60 25.15 ± 4.41 22.43 ± 7.43 24.85 ± 7.98
Shrink
GM-AE TADM-3D NGP-Net MUMINS ∆b = 0 MUMINS ∆b = GT
4.18 ± 1.46 5.03 ± 1.50 4.13 ± 1.58 4.09 ± 1.75 3.47 ± 1.73
22.67 ± 3.59 21.03 ± 2.72 22.85 ± 3.72 23.12 ± 3.95 24.43 ± 4.50
65.61 ± 13.55 64.69 ± 12.31 73.18 ± 12.19 70.76 ± 14.38 75.08 ± 14.90
5.40 ± 3.71 6.60 ± 4.08 5.34 ± 4.93 5.73 ± 4.68 4.82 ± 3.52
24.75 ± 5.96 22.24 ± 4.89 24.56 ± 4.96 24.26 ± 5.75 25.45 ± 5.97
All above
Copy (x̂(future) = x) GM-AE TADM-3D NGP-Net MUMINS ∆b = 0 MUMINS ∆b = GT
4.54 ± 1.56 4.05 ± 1.38 4.91 ± 1.50 4.13 ± 1.46 3.95 ± 1.69 3.18 ± 1.55
21.91 ± 3.07 22.88 ± 3.06 21.39 ± 2.52 22.85 ± 2.94 23.53 ± 3.96 25.50 ± 4.56
65.21 ± 12.66 65.69 ± 12.65 66.77 ± 11.24 71.34 ± 11.54 71.40 ± 13.47 78.01 ± 13.59
5.82 ± 4.02 5.17 ± 3.56 6.51 ± 4.60 5.37 ± 3.74 5.26 ± 4.16 4.32 ± 3.48
23.63 ± 5.21 24.65 ± 5.19 22.83 ± 5.24 24.30 ± 5.08 24.89 ± 5.69 26.75 ± 6.12
CN
BrLP [29] TADM-3D [19] MUMINS ∆b = 0 MUMINS ∆b = GT
1.57 ± 0.24 1.21 ± 0.24 1.15 ± 0.46 1.06 ± 0.34
27.16 ± 1.31 30.12 ± 1.73 30.78 ± 2.87 31.34 ± 2.40
94.15 ± 1.19 95.00 ± 1.59 95.21 ± 2.82 95.52 ± 2.37
6.90 ± 1.11 3.82 ± 0.80 4.16 ± 1.84 3.69 ± 1.14
20.59 ± 1.34 26.06 ± 1.67 25.66 ± 3.22 26.37 ± 2.47
MCI
BrLP TADM-3D MUMINS ∆b = 0 MUMINS ∆b = GT
1.55 ± 0.20 1.19 ± 0.32 0.98 ± 0.33 0.95 ± 0.31
27.17 ± 1.10 30.19 ± 1.95 31.72 ± 2.17 31.82 ± 2.17
94.24 ± 1.12 95.02 ± 1.69 96.25 ± 1.82 96.61 ± 1.41
6.88 ± 0.91 3.69 ± 1.02 3.42 ± 1.21 3.43 ± 1.21
20.58 ± 1.09 26.47 ± 1.92 27.01 ± 2.39 27.00 ± 2.44
AD
BrLP TADM-3D MUMINS ∆b = 0 MUMINS ∆b = GT
1.73 ± 0.25 1.17 ± 0.21 1.16 ± 0.29 1.17 ± 0.31
26.17 ± 1.34 30.01 ± 1.60 30.44 ± 1.97 30.44 ± 2.16
93.47 ± 1.36 94.77 ± 2.15 95.38 ± 2.03 95.52 ± 1.74
7.74 ± 1.19 3.79 ± 0.73 4.18 ± 1.05 4.27 ± 1.21
19.52 ± 1.39 26.01 ± 1.56 25.35 ± 2.06 25.26 ± 2.31
All above
Copy (x̂(future) = x) BrLP TADM-3D MUMINS ∆b = 0 MUMINS ∆b = GT
1.14 ± 0.56 1.57 ± 0.24 1.21 ± 0.25 1.13 ± 0.44 1.05 ± 0.34
30.91 ± 3.60 27.12 ± 1.30 30.12 ± 1.75 30.87 ± 2.77 31.35 ± 2.37
95.55 ± 3.37 94.12 ± 1.20 94.99 ± 1.62 95.34 ± 2.71 95.64 ± 2.28
4.34 ± 2.25 6.94 ± 1.10 3.80 ± 0.82 4.08 ± 1.76 3.69 ± 1.15
25.61 ± 3.98 20.54 ± 1.33 26.10 ± 1.69 25.80 ± 3.12 26.38 ± 2.47
On OASIS-3 brain MRI, MUMINS (∆b = 0) is essentially matched with the near-saturated copy baseline on whole-image metrics since baseline and follow-up differ subtly in most pairs, and ahead of TADM-3D and BrLP on all three (Table 1). At the ROI level, MUMINS and TADM3D are closely matched: MUMINS edges TADM-3D on MCI (3.42 vs. 3.69 × 10−2 MAEROI ) but trails narrowly on CN (4.16 vs. 3.82 × 10−2 MAEROI ). Supplying ∆b = GT lifts MUMINS’s aggregate ROI to its best reported scores, matching or leading TADM-3D on CN, MCI, and aggregate, with TADM-3D remaining stronger on AD (3.79 vs. 4.27 × 10−2 MAEROI ), where atrophy produces the largest anatomical change. Fig. 2 shows qualitative results of all methods.
On
lung CT, MUMINS yields the sparsest, lowest-amplitude error maps and shows little error over the nodule; TADM3D spreads high error beyond the localized errors of GMAE and NGP-Net. On brain MRI, all methods reconstruct well, so error maps use a much finer scale: MUMINS produces the cleanest maps overall, BrLP yields visibly blurrier follow-ups (likely from its latent-space compression) with error concentrated around the central ventricles and outer cortical boundary, and TADM-3D distributes error more diffusely. Supp. Sec. I shows failure cases, such as abrupt nodule growth that MUMINS under-predicts. Uncertainty. Table 2 scores the single-pass predicted σ against voxel-wise error, over the image and the ROI (nodule/brain). The meaningful measure differs by dataset. For
Lung CT samples
Pred σ
0.26±0.07 0.03±0.01 0.97±0.04 0.47±0.01
MC std
ρ↑ AUSE↓ PICP95 MPIW
0.70±0.14 0.01±0.01 0.78±0.14 0.22±0.08
0.17±0.29 0.05±0.05 0.61±0.23 0.30±0.20
0.56±0.06 0.001±0.001 0.20±0.05 0.04±0.00
0.32±0.08 0.02±0.01 0.59±0.14 0.12±0.01
ρ↑
0.79±0.08
0.50±0.26
0.73±0.02
0.65±0.03
age: 54
age: 68
age: 81
Δτ: 2m
Δτ: 43m
Δτ: 2m
Δτ: 1.7y
Δτ: 1.5y
Δτ: 2.8y
GT Baseline GT Follow-up
Brain MRIROI
0.54±0.05 0.002±0.001 0.98±0.02 0.44±0.00
age: 47
Predicted Follow-up
Brain MRIImg
0.13±0.28 0.06±0.05 0.80±0.22 0.46±0.02
age: 51
Predicted Uncertainty
Lung CTROI
0.61±0.15 0.02±0.01 0.91±0.06 0.45±0.01
Brain MRI samples
age: 48
Absolute Error
Lung CTImg ρ↑ AUSE↓ PICP95 MPIW
X
Table 2. Uncertainty evaluation on both test sets, for the image and the ROI (nodule for CT, brain for MRI). Rows compare singlepass σ and 20-seed MC std, each against voxel-wise error, and their agreement (X). ρ: Spearman; PICP95 → 0.95; mean ± std.
lung, the 643 volumes are cropped around the nodule, so whole-image scores are anatomically meaningful: σ localizes error well (ρ = 0.61) and is reasonably calibrated (PICP95 0.91), far cheaper than the 20×-costlier MC reference (ρ = 0.70, PICP95 0.78). The nodule ROI is a small sub-region, so its correlations are unstable for both σ (ρ = 0.13±0.28) and MC (ρ = 0.17±0.29) and not meaningfully interpretable. For the brain, volumes are skullstripped, inflating whole-image scores with the large zero background, so uncertainty inside the ROI is the meaningful measure. There, reconstruction is near-saturated, and error is small and diffuse, leaving little structure to localize: MC’s repeated sampling gives a modest localization edge (ρ = 0.32 vs. our 0.26), but calibration is the discriminating axis, where the single diffusion process uncertainty σ remains better calibrated (PICP95 0.97) while MC severely under-covers (0.59). Across both datasets, both estimates are correlated (cross-ρ 0.79 lung, 0.65 brain ROI), so a single pass recovers much of the uncertainty structure MC obtains only by repeated sampling. This structure could stem from the learned variance head or the sampler’s injected stochasticity; setting η = 0 removes the latter, yet σ remains error-correlated, confirming the variance head learns meaningful uncertainty on its own (Supp. Sec. F). Figure 3 shows predicted uncertainty and absolute-error maps for representative cases (axial, sagittal, and coronal views in Supp. Fig. H.6). Sparsification curves (Fig. 4) show both estimators’ MAE dropping monotonically toward the oracle as high-uncertainty voxels are removed first; the single-pass curve sits above the 20-seed MC reference, consistent with its higher AUSE (Table 2).
Figure 3. MUMINS follow-up and uncertainty predictions (test samples). ∆τ denotes inter-scan interval (months m, years y).
4.5. Ablation Studies
JD RP BC UP BR MAE (×10−2 ) ↓ PSNR (dB) ↑ SSIM (%) ↑
We ablate MUMINS on the PNG lung validation split [35]; the 8×-larger brain (1283 ) configuration precludes exhaustive sweeps. Table 3 isolates five components on a shared 3D U-Net backbone [5] and training iterations: joint diffusion (JD), residual prediction (RP), blur conditioning (BC), uncertainty prediction (UP), and baseline re-injection (BR),
Mean abs error of remaining
ROI ROI
0.175 0.150 0.125 0.100 0.075 0.050 0.025 0.000
0.0
0.2
pred (20 seeds) Oracle
0.4
0.6
ROI MC std Image pred (20 seeds)
0.8
0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00
Fraction of voxels removed
(a)
0.0
0.2
Image Image
Oracle MC std
0.4
0.6
0.8
Fraction of voxels removed
(b)
Figure 4. Sparsification curves on PNG lung CT (a) and OASIS brain MRI (b) test sets; ROI is the nodule (a) and brain (b).
starting from a single-channel diffusion (JD-free) as in typical conditional DDPM models [19, 20, 29]. Predicting the image directly (rows 1–2) yields high variance but JD outperforms in all metrics the baseline; RP sharply tightens this spread (PSNR std 4.09 → 1.65 dB), and BC then lifts fidelity (PSNR 17.40 → 20.69 dB), with UP adding a further, essentially cost-free gain (21.00 vs. 20.69 dB). BR, a parameter-free inference-time mechanism, gives the largest gains (MAE 5.55 → 3.86 × 10−2 , PSNR 21.00 → 23.73 dB, SSIM 56.27% → 72.38%). Full analysis, qualitative examples, and sampler sweeps (ζ, ŝ, η) are in Supp. Sec. F. Table 3. Ablation study of MUMINS variants on the PNG validation set [35]. Starting from an image-only conditional baseline (top row), we add joint diffusion (JD), residual prediction (RP), acquisition-blur difference conditioning (BC) with ∆b = 0 inference, uncertainty prediction (UP) and baseline reinjection (BR).
✗ ✓ ✓ ✓ ✓ ✓
✗ ✗ ✓ ✓ ✓ ✓
✗ ✗ ✗ ✓ ✓ ✓
✗ ✗ ✗ ✗ ✓ ✓
✗ ✗ ✗ ✗ ✗ ✓
19.76 ± 10.82 11.54 ± 5.33 8.62 ± 1.68 5.75 ± 1.60 5.55 ± 1.70 3.86 ± 1.69
14.19 ± 5.43 41.84 ± 26.55 16.98 ± 4.09 54.28 ± 19.84 17.40 ± 1.65 45.45 ± 10.15 20.69 ± 2.29 54.64 ± 11.42 21.00 ± 2.59 56.27 ± 12.49 23.73 ± 3.98 72.38 ± 13.65
5. Conclusions We presented MUMINS, a diffusion framework that synthesizes a follow-up scan and a voxel-wise uncertainty map in a single reverse diffusion process. Modeling a joint baseline-residual state with per-step baseline re-injection preserves fine anatomy while generating plausible change; a likelihood-based variance head yields uncertainty without Monte Carlo’s costlier repeated sampling. With no organ-specific components, MUMINS matches or outperforms domain-specific baselines on lung CT and brain MRI, with single-pass uncertainty correlating cheaply with the MC reference. Two limitations remain: localization trails MC on near-saturated brain MRI targets, and predictions queried at different ∆τ from the same baseline are generated independently, with no explicit mechanism enforcing cross-horizon consistency. Efficient computation of the denoiser’s volumetric Jacobian for uncertainty estimation is a promising direction, as is extending MUMINS to further organs and modalities toward a general progression-synthesis model.
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