Spherical Brownian Bridge Diffusion Model (SBDM)

• SBDM is a bidirectional conditional Brownian bridge diffusion model that predicts subject-specific cortical thickness trajectories from a single baseline MRI scan.
• It integrates a discrete Gaussian bridge with a conditional spherical U-Net employing spherical convolutions and dense cross-attention to merge cortical features with covariate data.
• SBDM achieves state-of-the-art forecasting accuracy with significantly lower mean absolute error compared to previous methods on ADNI and OASIS datasets, enabling both factual and counterfactual trajectory sampling.

Searching arXiv for the specified paper and closely related Brownian bridge diffusion work. arXiv search query: (Stoyanov et al., 10 Sep 2025) The Spherical Brownian Bridge Diffusion Model (SBDM) is a bidirectional conditional Brownian bridge diffusion model for individualized, vertex-level forecasting of cortical thickness (CTh) trajectories from a single baseline MRI scan on registered spherical cortical surfaces. It is designed to predict future CTh maps $\tau_t$ at each cortical mesh vertex, given baseline thickness $\tau_0$, a time horizon $t$, and subject-specific covariates $c$ such as age, sex, and diagnosis. In the formulation reported for conditional cortical thickness forecasting, SBDM combines a discrete Gaussian Brownian bridge with a denoising architecture called the conditional spherical U-Net (CoS-UNet), which uses spherical convolutions and dense cross-attention to integrate cortical surfaces and tabular conditions. The model is evaluated on ADNI and OASIS longitudinal datasets, where it achieves the lowest reported mean absolute error among the compared methods and is also used to generate factual and counterfactual CTh trajectories (Stoyanov et al., 10 Sep 2025).

1. Problem setting and spherical formulation

SBDM addresses forecasting of individualized cortical thickness trajectories at the resolution of vertices on a registered cortical mesh. The target quantity is the future cortical thickness map $\tau_t$, expressed through the relation

$\tau_t = \tau_0 + \Delta \tau_t,$

where $\tau_0$ is the baseline thickness and $\Delta \tau_t$ is the future relative change. The stated goal is to support discrimination between healthy aging and neurodegenerative change, including transitions such as MCI to AD, from a single baseline MRI scan and subject-specific covariates (Stoyanov et al., 10 Sep 2025).

The spherical formulation arises from the geometric properties of the cerebral cortex. The cortex is described as a highly folded 2D manifold embedded in 3D. After surface reconstruction, for example with the FreeSurfer longitudinal stream, cortical meshes are registered to a population template with spherical topology, specifically the FsAverage icosphere. This yields a consistent vertex set $V$ and geodesic neighborhood structure across subjects. In this setting, spherical representation makes convolutions and pooling well-defined across subjects, preserves local topology, and avoids distortions inherent in Euclidean gridding of a folded sheet.

A common misunderstanding is to equate the adjective spherical with stochastic evolution directly constrained to the sphere $S^2$. In SBDM, the spherical aspect refers primarily to the cortical surface representation and to the neural operators acting on that representation. The diffusion variables are scalar thickness values indexed by the vertices of a fixed spherical mesh, not coordinates moving on the sphere. This distinction is central to the model’s construction and to its relation to intrinsic manifold diffusion.

2. Brownian bridge mechanism and denoising objective

The core stochastic object in SBDM is a discrete-time conditional Brownian bridge between two structured endpoints. The bridge distribution is

$\tau_0$0

where $\tau_0$1 is the bridge step, $\tau_0$2 is a monotone schedule, and

$\tau_0$3

In SBDM, $\tau_0$4, $\tau_0$5, and $\tau_0$6 is the vector of per-vertex thickness values. The bridge index $\tau_0$7 is not the physical time; the physical interval $\tau_0$8 enters as a condition (Stoyanov et al., 10 Sep 2025).

During training, the forward or noising process samples intermediate states as

$\tau_0$9

with $t$0. This construction differs from standard denoising diffusion probabilistic models that replace the signal by pure Gaussian noise. Here, the bridge always interpolates between the baseline endpoint and the future-change endpoint while adding time-varying noise. The stated motivation is that this preserves structure and facilitates multimodal conditioning.

At inference, $t$1 is unavailable, so SBDM uses a learned recursive sampler conditioned on baseline thickness, bridge step, physical time, and covariates:

$t$2

instantiated through the non-Markovian recursion

$t$3

where $t$4, $t$5 is the denoiser, and $t$6 are the BBDM coefficients computed from $t$7. After the final step, $t$8 and $t$9.

The denoiser is trained to recover the bridge noise term at step $c$0 with the objective

$c$1

This is a denoising-style regression in the native vertex space $c$2 and does not use a separate autoencoder or latent space. The formulation therefore remains directly tied to vertex-level cortical thickness values.

3. Extrinsic geometry, discretization, and numerical structure

SBDM is explicitly described as extrinsic rather than intrinsic. The stochastic process lives in the Euclidean feature space $c$3 of scalar thickness values defined on a fixed spherical mesh. Spherical geometry is incorporated by the denoising network, which uses spherical convolutions, rather than by constraining diffusion trajectories to lie on $c$4 (Stoyanov et al., 10 Sep 2025).

For comparison, the exposition contrasts SBDM with an intrinsic Brownian motion or Brownian bridge on $c$5, whose generator is $c$6 and whose drift toward a terminal state can be written using the Riemannian logarithm:

$c$7

where $c$8 projects Euclidean increments to the tangent space $c$9. SBDM does not implement this intrinsic stochastic differential equation. This clarification addresses a second recurring misconception: the model is not a manifold Brownian bridge in the strict Riemannian sense.

The cortical surfaces themselves are represented as a fixed triangular mesh mapped to the FsAverage template. The mesh is a geodesic polyhedron derived by recursively subdividing an icosahedron, and spherical pooling and upsampling invert subdivision between encoder and decoder levels. Because the stochastic updates $\tau_t$0 are thickness values per vertex, no geodesic computation or projection onto $\tau_t$1 is required inside the diffusion loop. Sphericality is instead respected through mesh-topology-aware convolutions.

The reported numerical regime uses $\tau_t$2 bridge steps for training and non-Markovian sampling with 200 steps for inference. The process is a discrete Gaussian bridge; neither Stratonovich nor Itô calculus is invoked. Stability is attributed to the BBDM coefficientization and the conditioning mechanism in the denoiser, with DDIM-style subsampling used to reduce inference cost.

4. CoS-UNet architecture and multimodal conditioning

The denoising model in SBDM is the conditional spherical U-Net (CoS-UNet), an encoder-decoder with residual links tailored to spherical meshes. In each encoder stage, two residual blocks are used; each block applies one-hop spherical graph convolution, group normalization, and SiLU activation. Spherical pooling reduces resolution by inverting icosphere subdivision. The decoder mirrors the encoder, uses transposed spherical convolutions for upsampling, and includes skip connections to preserve high-resolution detail. In the reported experiments, the channel width is set to $\tau_t$3 (Stoyanov et al., 10 Sep 2025).

Conditioning enters through dense cross-attention after the second residual block in each stage. Let $\tau_t$4 denote spherical features at a stage, and let the conditions be embedded as $\tau_t$5. Queries, keys, and values are formed as

$\tau_t$6

where $\tau_t$7 and $\tau_t$8, with the condition features replicated across vertices. The resulting attention and fusion are

$\tau_t$9

The bridge step $\tau_t = \tau_0 + \Delta \tau_t,$0 is embedded separately and added to latent feature maps to disentangle it from $\tau_t = \tau_0 + \Delta \tau_t,$1.

The conditioning variables are specified as

$\tau_t = \tau_0 + \Delta \tau_t,$2

together with the real-valued time horizon $\tau_t = \tau_0 + \Delta \tau_t,$3. In factual generation, $\tau_t = \tau_0 + \Delta \tau_t,$4 is unspecified. In counterfactual generation, $\tau_t = \tau_0 + \Delta \tau_t,$5 is set to a target diagnosis. This design yields dense, stage-wise guidance from tabular conditions to spatial spherical features and is the mechanism by which SBDM integrates multimodal information for subject-specific forecasting.

5. Forecasting, factual sampling, and counterfactual trajectories

The factual sampling procedure begins from the baseline cortical thickness map. The inputs are $\tau_t = \tau_0 + \Delta \tau_t,$6, $\tau_t = \tau_0 + \Delta \tau_t,$7, and $\tau_t = \tau_0 + \Delta \tau_t,$8, and the initialization is $\tau_t = \tau_0 + \Delta \tau_t,$9. For each selected bridge step, the model samples $\tau_0$0, computes the coefficients $\tau_0$1, and updates

$\tau_0$2

After the final step, the output is $\tau_0$3 and $\tau_0$4 (Stoyanov et al., 10 Sep 2025).

Counterfactual sampling uses the same procedure but changes the conditioning tuple to

$\tau_0$5

thereby steering the generated trajectory toward a chosen diagnostic state. The model is therefore not restricted to predicting the most likely factual trajectory under observed conditions; it can also be used to explore “what-if” scenarios at the level of subject-specific cortical thinning patterns.

Uncertainty in SBDM is stochastic rather than analytically calibrated. Because inference depends on $\tau_0$6, multiple independent samples produce an empirical distribution over $\tau_0$7 and $\tau_0$8. The reported work demonstrates trajectory generation but does not report formal calibration metrics for SBDM. The exposition notes that uncertainty could be summarized by per-vertex sample variance, credible bands, or coverage, if desired. A plausible implication is that the model provides a natural sampling-based uncertainty mechanism, but not yet a validated uncertainty quantification pipeline.

6. Empirical performance, baselines, and reported limitations

The reported experiments use ADNI for training, validation, and test, and OASIS as an external test set. For ADNI, the subject-level splits are 921 train subjects (4,112 scans), 306 validation subjects (1,412 scans), and 306 test subjects (1,387 scans), with age 55–97, CN/MCI/AD, left hemisphere evaluation, and time horizon up to 168 months. For OASIS, the external test set contains 590 subjects and 1,750 scans, age 42–95, CN and AD, with the medial region connecting hemispheres ignored unless stated otherwise. Baselines are linear regression, Spherical U-Net, Surface Vision Transformer (SiT), CTh-DDPM, and DDPM with CoS-UNet denoiser (DDPM/CoS-UNet). All methods were adapted to receive the same conditions and were trained on ADNI (Stoyanov et al., 10 Sep 2025).

Method	ADNI All MAE (mm)	OASIS All MAE (mm)
Linear regression	0.135 ± 0.031	0.146 ± 0.154
Spherical U-Net	0.111 ± 0.029	0.115 ± 0.027
SiT	0.108 ± 0.028	0.112 ± 0.025
CTh-DDPM	0.208 ± 0.019	0.210 ± 0.017
DDPM/CoS-UNet	0.119 ± 0.029	0.121 ± 0.030
SBDM	0.097 ± 0.031	0.100 ± 0.030

The subgroup results are also reported. On ADNI, SBDM attains $\tau_0$9 for CN, $\Delta \tau_t$0 for MCI, and $\Delta \tau_t$1 for AD. On OASIS, it attains $\Delta \tau_t$2 for CN and $\Delta \tau_t$3 for AD. The paper states that SBDM consistently achieves the lowest MAE, with $\Delta \tau_t$4 improvement versus the second-best method, SiT, and strong generalization to OASIS, including CN error increase $\Delta \tau_t$5 mm and AD decrease $\Delta \tau_t$6 mm. Paired two-sided Wilcoxon signed-rank tests yield $\Delta \tau_t$7 against all four pre-existing baselines on both datasets.

Qualitative evaluation reports vertex-wise MAE maps with uniformly lower errors for SBDM across cortex. The precentral gyrus is identified as a hard region for all methods, but SBDM is reported to avoid the dotted heterogeneities seen in Spherical U-Net and SiT. In denoiser ablations on ADNI validation, the reported MAE values are $\Delta \tau_t$8 for a Spherical U-Net-based denoiser, $\Delta \tau_t$9 for an MLP-based denoiser, $V$0 for an SiT-based denoiser, and $V$1 for CoS-UNet, with CoS-UNet yielding the best accuracy.

The practical implementation uses PyTorch 2.0.1 and SphericalUNet 1.2.2, trained on a single Nvidia Titan RTX (24 GB). The optimizer is AdamW with initial learning rate $V$2 and Reduce-on-Plateau after 100 epochs; an exponential moving average of weights is used; training runs for up to 2000 epochs; the bridge uses $V$3 training steps; inference uses DDIM-like 200-step sampling; and the channel width is $V$4. Reported limitations include dependency on FreeSurfer reconstruction and spherical alignment, sensitivity to the schedules $V$5 and $V$6, diagnostic heterogeneity in AD and MCI, the need for broader validation across populations and pathologies, and the absence of formal uncertainty calibration metrics. Suggested future directions include intrinsic manifold bridges based on Log/Exp maps, multiscale temporal modeling such as Neural ODEs over $V$7, multi-visit conditioning, and clinical integration with biomarkers, normative modeling, and trial selection.

In relation to prior work, SBDM is positioned against Euclidean DDPMs, prior Brownian bridge models for 2D images, and spherical CNNs or transformers. The paper states that SBDM is the first to adapt a Brownian bridge to vertex-level cortical surfaces with spherical neural operators and dense cross-attention to tabular conditions, while keeping the diffusion process extrinsic in vertex-value space. It also states that Spherical U-Net and SiT operate on spherical meshes but lack flexible conditioning and generative trajectory sampling. This suggests that SBDM’s main contribution is not only improved error but also the unification of spherical surface processing, structured conditional diffusion, and subject-specific factual and counterfactual trajectory generation.