AnoBFN: Bayesian Flow Network for PET Analysis

• AnoBFN is an unsupervised anomaly detection method for brain FDG PET that reconstructs pseudo-healthy images from healthy training data.
• It integrates conditional image generation with simplex noise and a tailored accuracy schedule to maintain anatomical fidelity.
• The method employs recursive Bayesian feedback from the original input to balance anomaly removal with subject-specific detail.

Searching arXiv for AnoBFN and related Bayesian Flow Networks to ground the article. AnoBFN denotes an unsupervised anomaly detection method based on Bayesian Flow Networks (BFNs) for brain FDG PET in the context of Alzheimer’s disease. It is trained on healthy subjects only, generates a pseudo-healthy reconstruction for a test scan, and identifies anomalies from discrepancies between the input and reconstruction. Its defining modifications are conditional image generation under high levels of spatially correlated noise and recursive feedback from the original input image throughout the generative process, with the stated purpose of preserving subject specificity while reducing false positive rates (Roy et al., 23 Jul 2025).

1. Terminology and scope

In the supplied literature, the label “AnoBFN” is used in more than one sense. The primary usage is the neuroimaging method introduced as “AnoBFN” for unsupervised anomaly detection in brain FDG PET (Roy et al., 23 Jul 2025). A second usage appears in the antibody-design literature, where “AnoBFN” refers to antibody-oriented BFN frameworks such as AntibodyDesignBFN, AbBFN, and IgCraft (Hu et al., 9 Jan 2026). A third, separate interpretation treats the label as analog-only beamforming for near-field multiuser MIMO communications (Wang et al., 2024). This multiplicity is terminological rather than methodological: the three usages share the acronymic surface form and, in two cases, a connection to BFNs, but they address different technical problems.

Usage	Domain	Description
AnoBFN	Neuroimaging	Unsupervised anomaly detection for brain FDG PET
“AnoBFN” in antibody-design context	Protein design	Antibody-oriented BFN frameworks such as AntibodyDesignBFN, AbBFN, and IgCraft
Separate interpretation	Wireless communications	Analog-only beamforming for near-field multiuser MIMO

For the term as a named method, the neuroimaging usage is the most specific: it denotes a concrete reconstruction-based UAD system for Alzheimer’s disease-related anomaly detection in FDG PET. The antibody and beamforming usages are best understood as adjacent or contextual interpretations rather than the same method.

2. Neuroimaging problem formulation

AnoBFN is situated in unsupervised anomaly detection (UAD), where a generative model is trained only on healthy data and then used to reconstruct a pseudo-healthy version of a possibly abnormal test image (Roy et al., 23 Jul 2025). In brain FDG PET for Alzheimer’s disease, the target anomalies are regional hypometabolism patterns in characteristic networks, including posterior cingulate and temporo-parietal cortex. The operational premise is straightforward: when a model has learned the healthy distribution, deviations between a scan and its pseudo-healthy reconstruction can be interpreted as anomalies.

The method addresses a setting in which abnormalities are diffuse, subtle, and ill-defined rather than sharply bounded. The supplied description emphasizes three difficulties. First, AD-related hypometabolism lacks hard lesion boundaries. Second, voxel-wise ground-truth annotations of abnormal metabolism do not exist in real data. Third, classical UAD systems based on VAEs or GANs can encode abnormal inputs poorly, over-smooth normal regions, and generate high false-positive rates while losing subject-specific anatomical details.

AnoBFN is designed around these constraints. It aims not merely to reconstruct healthy-looking scans, but to do so without erasing individual anatomy. This dual requirement explains the two design goals stated in the abstract: conditional image generation under high levels of spatially correlated noise, and recursive feedback from the input image throughout generation (Roy et al., 23 Jul 2025). A plausible implication is that the method is intended to balance anomaly removal against anatomical fidelity rather than optimize either objective in isolation.

3. Bayesian Flow Network foundation

AnoBFN inherits the BFN formulation in which the generative process evolves parameters of a distribution rather than noisy data directly (Roy et al., 23 Jul 2025). The latent state is written as

$\vtheta_t=\{\vmu_t,\rho_t\},$

where these parameters define a Gaussian input distribution

$p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$

This parameter-space construction is the central distinction from standard DDPM-style formulations.

During training, noisy observations are produced by a sender distribution

$p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$

with $\alpha_t=\frac{d\beta(t)}{dt}$. Given current parameters $\vtheta_{t+1}$ and a noisy observation $\vx_t$, the Bayesian update uses Gaussian conjugacy:

$\vmu_t=\frac{\rho_{t+1}\vmu_{t+1}+\alpha_t\vx_t}{\rho_{t+1}+\alpha_t}, \qquad \rho_t=\rho_{t+1}+\alpha_t.$

At inference, the model no longer observes real $\vx_t$ and instead samples from a receiver distribution whose mean is predicted by a neural network $\Psi$:

$p_R(\hat{\vx}_t \mid \vtheta_{t+1};\alpha_t) = \mathcal{N}(\hat{\vx}_t;\Psi(\vtheta_{t+1}),\alpha_t^{-1}\mathbf{I}).$

The same Bayesian update is then applied with $p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$0 in place of $p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$1.

Training minimizes the KL divergence between sender and receiver across timesteps:

$p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$2

The paper states that there is no extra explicit ELBO or KL term beyond this; image fidelity follows from matching $p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$3 to $p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$4 across timesteps, while anomaly sensitivity arises from training only on healthy data together with AnoBFN’s modified noise schedule and inference-time guidance.

4. AnoBFN-specific modifications

AnoBFN introduces two explicit contributions, denoted [C1] and C2. The first is conditional image generation under high spatially correlated noise. The second is recursive Bayesian feedback of the original input image at every inference step.

For [C1], AnoBFN replaces i.i.d. Gaussian noise with simplex noise, described as a gradient-based noise with spatial continuity. This follows AnoDDPM and is motivated by the spatial coherence of medical-image anomalies. The method also changes the accuracy schedule $p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$5 so that the prior at $p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$6 remains input-dependent while having large variance. Assuming prior $p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$7, the schedule is built through

$p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$8

with the cosine-based choice

$p_I(\vx_t)=\mathcal{N}(\vx_t;\vmu_t,\rho_t^{-1}\mathbf{I}).$9

The reported properties are that $p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$0 is maximal, the variance term $p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$1 is large, and the corresponding mean at $p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$2 becomes roughly $p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$3. This construction is intended to make priors for healthy and abnormal scans overlap while still retaining information about the specific input.

For [C2], AnoBFN injects the original input $p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$4 into every Bayesian update during inference. The modified update is

$p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$5

with

$p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$6

Here $p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$7 is an auxiliary weight controlling how strongly the original input influences each voxel. It is defined by

$p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$8

The residual-based factor preserves input information where prediction and input agree, while downweighting the input where they diverge. The temporal logistic factor reduces input guidance early in the chain and increases it later. This suggests that AnoBFN uses the input image not as a fixed condition alone, but as a recursive, region-dependent source of evidence.

5. Data, evaluation protocol, and empirical behavior

The experiments use FDG PET from ADNI, preprocessed with Clinica through co-registered frame averaging, standardization to uniform resolution, linear registration to MNI space, intensity normalization using mean uptake in cerebellum and pons, cropping, resampling to a $p_S(\vx_t \mid \vx_0, \alpha_t)=\mathcal{N}(\vx_t; \vx_0, \alpha_t^{-1}\mathbf{I}),$9 grid, and rescaling to $\alpha_t=\frac{d\beta(t)}{dt}$0 (Roy et al., 23 Jul 2025). From each 3D volume, 20 axial slices centered around the midplane are extracted. The cohort contains 733 scans from 301 cognitively normal subjects. The split is subject-level and stratified by age and sex: 540 training scans, 57 validation scans, and 80 test scans, each from a distinct cognitively normal subject. For each test CN scan, a synthetic abnormal scan with approximately 30% AD-like hypometabolism is generated, yielding test_CN for reconstruction evaluation and test_sAD for anomaly detection evaluation.

AnoBFN and AnoDDPM share a U-Net backbone with 3 downsampling and 3 upsampling stages, channel widths $\alpha_t=\frac{d\beta(t)}{dt}$1 with $\alpha_t=\frac{d\beta(t)}{dt}$2, attention layers with 4 heads and head dimension 16, and about 10M parameters. Optimization uses AdamW with learning rate $\alpha_t=\frac{d\beta(t)}{dt}$3, weight decay $\alpha_t=\frac{d\beta(t)}{dt}$4, $\alpha_t=\frac{d\beta(t)}{dt}$5, EMA decay 0.9999, batch size 30, and gradient clipping at 1. Baselines are $\alpha_t=\frac{d\beta(t)}{dt}$6-VAE with $\alpha_t=\frac{d\beta(t)}{dt}$7 and latent dimension 128, f-AnoGAN, AnoDDPM with simplex noise, the original BFN, and an ablation denoted AnoBFN w/o [C2].

Evaluation is reconstruction-based. On test_CN, metrics are MSE, PSNR, and SSIM. On test_sAD, anomaly localization is measured by pixel-level IoU with threshold 0.05 and pixel-level AP. Statistical testing uses a paired Wilcoxon signed-rank test with Bonferroni correction, and differences $\alpha_t=\frac{d\beta(t)}{dt}$8 are marked with *.

The reported quantitative pattern is specific. AnoDDPM achieves the best MSE and PSNR on healthy reconstructions, with $\alpha_t=\frac{d\beta(t)}{dt}$9 and $\vtheta_{t+1}$0, while AnoBFN is close at $\vtheta_{t+1}$1 and $\vtheta_{t+1}$2, and obtains the best SSIM at $\vtheta_{t+1}$3. For anomaly detection, AnoBFN is the strongest method in both IoU and AP, reaching $\vtheta_{t+1}$4 and $\vtheta_{t+1}$5, compared with $\vtheta_{t+1}$6 and $\vtheta_{t+1}$7 for $\vtheta_{t+1}$8-VAE, $\vtheta_{t+1}$9 and $\vx_t$0 for f-AnoGAN, $\vx_t$1 and $\vx_t$2 for AnoDDPM (simplex), $\vx_t$3 and $\vx_t$4 for the original BFN, and $\vx_t$5 and $\vx_t$6 for AnoBFN w/o [C2].

The ablation pattern is also explicit. Moving from the original BFN to AnoBFN w/o [C2], the addition of simplex noise and the new accuracy schedule improves both reconstruction and detection. Moving from AnoBFN w/o [C2] to the full model, the input-guided Bayesian update yields further gains. Qualitatively, the paper describes AnoBFN reconstructions as sharp and anatomically realistic, with normal regions remaining close to the input while hypometabolic regions are filled in to healthy intensities. By contrast, AnoDDPM is said to reconstruct the abnormal input too faithfully, which weakens residual-based anomaly maps.

6. Limitations, misconceptions, and related uses of the name

The paper states several limitations for the neuroimaging AnoBFN (Roy et al., 23 Jul 2025). Evaluation is based on synthetic hypometabolism rather than real patient pathology; generalization beyond ADNI and beyond FDG PET is untested; explicit uncertainty quantification is not yet implemented despite the Bayesian formulation; computation remains heavier than simple autoencoders; and the method depends on accurate preprocessing, including registration, intensity normalization, and resampling. Future directions mentioned include uncertainty-aware scoring, alternative accuracy schedules and scaling metrics for Bayesian updates, extension to other pathologies and modalities, and robustness studies across scanners and domains.

A common misconception is to treat “AnoBFN” as a generic synonym for BFNs in any application area. The supplied literature does not support that interpretation. In antibody design, the relevant model is AntibodyDesignBFN, a fixed-backbone inverse-folding framework that uses a discrete BFN conditioned on full 3D backbone geometry through a Geometric Transformer with Invariant Point Attention. In that context, “AnoBFN” denotes antibody-oriented BFN frameworks rather than the neuroimaging method, and the reported benchmark is amino acid recovery on a 2025 temporal test set, with AntibodyDesignBFN reaching 49.9% average AAR versus 44.2% for ProteinMPNN (Hu et al., 9 Jan 2026). This is a distinct problem class involving sequence design on the probability simplex rather than medical-image anomaly localization.

A further separate interpretation appears in near-field wireless communications, where the term is read as analog-only beamforming for near-field multiuser MIMO. There the central objects are not generative models but beam focusing, beam nulling, SLNR optimization, and MM-based phase updates under constant-modulus constraints. The paper reports that two analog-only beamforming schemes can approach the sum rate of HBF schemes while outperforming HBF schemes in energy efficiency (Wang et al., 2024). This usage is methodologically unrelated to the neuroimaging AnoBFN except for the surface form of the label.

Taken together, these distinctions matter because the neuroimaging AnoBFN is a specific BFN-based UAD method defined by pseudo-healthy reconstruction, simplex-noise conditioning, and recursive Bayesian feedback from the input image. The same string can name or suggest different systems in antibody design and communications, but those systems operate on different state spaces, optimize different objectives, and should not be conflated.