TEMDiff: 3D Diffusion in Electron Tomography

• The paper introduces TEMDiff, which integrates a learned volumetric diffusion prior with a projector-based data-consistency correction to address the missing-wedge problem in limited-angle electron tomography.
• TEMDiff operates directly in 3D volume space, enforcing cross-slice consistency by conditioning on an initial inverse-Radon reconstruction and acquisition geometry.
• Trained on simulated FIB-SEM data, TEMDiff demonstrates superior reconstruction metrics (RMSE, PSNR, SSIM) compared to classical and 2D diffusion methods under extreme angular sparsity.

TEMDiff is a 3D diffusion-based iterative reconstruction framework for limited-angle transmission/scanning transmission electron tomography. It addresses the reconstruction of a volume $x \in \mathbb{R}^{D\times H\times W}$ from a small number of TEM/STEM tilt projections acquired over a narrow angular span, a regime dominated by the missing-wedge problem and by severe ambiguity in the recovered morphology. The method combines a learned volumetric diffusion prior, conditioning on an initial inverse-Radon reconstruction, and a projector-based data-consistency correction within the reverse diffusion process. Its defining claim is that operating directly in 3D volume space allows the model to learn realistic structural priors while implicitly enforcing consistency across slices, and that a model trained on simulated tilt series derived from volumetric FIB-SEM data can generalize to real TEM/STEM tilt series without retraining or fine-tuning (Deng et al., 7 Oct 2025).

1. Problem domain and scope

TEMDiff is formulated for limited-angle electron tomography, where the measured tilt series $y$ is modeled as a Radon transform of the unknown volume,

$y = R(x) = Ax ,$

with $A$ determined by the tilt range $\theta$ and the angular increment $\Delta\theta$. In this setting, reconstruction quality degrades because practical acquisition cannot provide dense coverage close to $180^\circ$. The resulting missing wedge removes a wedge of frequencies in Fourier space and produces elongation, streaking, blurred membranes, structural deformation, and loss of fidelity. The paper emphasizes that the problem becomes extreme when the tilt range drops to around $10^\circ$ or less, because many distinct 3D volumes can explain the observed projections nearly equally well (Deng et al., 7 Oct 2025).

The method is positioned against both classical and learned reconstructions. Classical filtered backprojection and algebraic schemes such as SART are dominated by missing-wedge artifacts in the narrow-angle regime. Prior diffusion-based tomography methods are described as slice-by-slice 2D approaches, including DiffusionMBIR and DOLCE. TEMDiff is introduced specifically to replace slice-wise reconstruction with full-volume reconstruction, on the premise that very narrow-angle tomography requires a volumetric prior rather than only explicit or implicit cross-slice coupling (Deng et al., 7 Oct 2025).

A second part of the problem definition concerns supervision. Clean paired data of TEM tilt series and artifact-free 3D ground truth are characterized as essentially unavailable, because real electron-tomography reconstructions are themselves corrupted by the missing wedge. TEMDiff therefore adopts a synthetic-supervision strategy based on FIB-SEM volumes rather than paired TEM ground truth (Deng et al., 7 Oct 2025).

2. Projector-guided 3D diffusion formulation

TEMDiff is described as a projector-guided conditional 3D diffusion reconstruction framework. The diffusion prior supplies a learned distribution over plausible 3D ultrastructure, while a projection-consistency correction constrains the reverse process toward agreement with the measured tilt series. The idealized projector $P$ is defined by

$A\,P(x)=y.$

In practical implementation, this projector is not realized as an explicit pseudoinverse. Instead, it is approximated by successive gradient-descent steps minimizing the projection mismatch

$y$0

with update direction

$y$1

This makes TEMDiff a predictor-corrector-style reconstruction procedure in which a diffusion denoiser proposes a cleaner 3D volume and the projector step restores measurement consistency (Deng et al., 7 Oct 2025).

The paper does not present the method as pure end-to-end regression from projections to volume, nor as explicit posterior sampling with a derived Bayesian score decomposition. It instead adopts a practical DDIM-style reverse process in which the denoising step and the projector correction alternate inside the sampling loop. This distinction is important: the learned component is a generative prior over volumes, while the forward operator enters explicitly during inference rather than only during training (Deng et al., 7 Oct 2025).

The conditioning signal is a coarse reconstruction

$y$2

obtained by inverse Radon transform or filtered backprojection from the measured tilt series. TEMDiff uses this coarse volume as an anchor rather than reconstructing only from noise and geometry. The paper frames the objective in MAP-like terms: because $y$3 admits many solutions in limited-angle tomography, the target is the most likely $y$4 satisfying the measurements, with plausibility supplied by the diffusion prior and feasibility supplied by the projector (Deng et al., 7 Oct 2025).

3. Conditional volumetric prior and architecture

The denoising model is trained over full 3D patches rather than isolated slices. Architecturally, the network treats depth slices as channels of a 2D U-Net with self-attention layers. This is not a full 3D convolutional U-Net in the strict sense, but it processes the entire volume patch jointly, which the paper identifies as the main distinction from prior 2D diffusion methods. The stated effect is that each slice is reconstructed while seeing all other slices and their shared volumetric context, so cross-slice consistency is enforced implicitly rather than by an additional regularizer such as $y$5-direction TV (Deng et al., 7 Oct 2025).

Conditioning is twofold. First, the network is conditioned on the initial reconstruction $y$6. Second, it is conditioned on the acquisition geometry $y$7, injected at the bottleneck after Fourier positional encoding and a 2-layer MLP. The conditioning on $y$8 is implemented with a classifier-free-guidance-like mechanism. At sampling time the model predicts both unconditional and conditional noise estimates,

$y$9

$y = R(x) = Ax ,$0

and combines them as

$y = R(x) = Ax ,$1

where $y = R(x) = Ax ,$2 is the guidance scale. The null condition $y = R(x) = Ax ,$3 is an all-zero conditioning input. This design is analogous to classifier-free guidance, except that the condition is a coarse tomographic reconstruction plus acquisition geometry rather than a class or text prompt (Deng et al., 7 Oct 2025).

The paper further states that TEMDiff is not latent diffusion and not score-SDE/ODE. It is framed as a DDIM-style noise-prediction model trained with a mean squared error objective on the predicted noise. This places the method in the family of conditional iterative diffusion reconstructions, but with a domain-specific volumetric prior and an explicit projector-based corrector (Deng et al., 7 Oct 2025).

4. Training-data strategy based on FIB-SEM simulation

A central technical feature of TEMDiff is its replacement of unavailable TEM ground truth with simulated tilt series derived from volumetric FIB-SEM data. Four FIB-SEM volumes are used for training: three containing mitochondria and one containing synapses, drawn from brain and HeLa-cell datasets. From these volumes, the method extracts patches of size

$y = R(x) = Ax ,$4

The spatial crop is chosen so that a patch typically covers one mitochondrion or synapse, and depth $y = R(x) = Ax ,$5 is selected to approximate physical specimen thickness in STEM biological imaging. The training pipeline uses 3D augmentations including flipping, resizing, affine, elastic, and perspective transforms (Deng et al., 7 Oct 2025).

The simulator begins from a contrast model for FIB-SEM intensity,

$y = R(x) = Ax ,$6

where $y = R(x) = Ax ,$7 and $y = R(x) = Ax ,$8 denotes local heavy-atom content. For TEM/STEM attenuation, the paper uses

$y = R(x) = Ax ,$9

which yields

$A$0

Approximating HAADF contrast by the inelastic component produces

$A$1

and substituting the FIB-SEM heavy-atom surrogate gives the simulation mapping

$A$2

The parameters $A$3 and $A$4 are fitted so that synthetic HAADF tilt histograms and dynamic range match real STEM data (Deng et al., 7 Oct 2025).

The paper is explicit that this simulator is approximate rather than fully physical. Its function is to bridge the domain gap statistically: FIB-SEM intensity is treated as a proxy for heavy-atom density, and HAADF contrast is approximated through exponential attenuation of a transformed density under the Radon projector. This approximation is what makes large-scale supervised training of the 3D prior feasible (Deng et al., 7 Oct 2025).

5. Reverse-diffusion update and uncertainty weighting

TEMDiff inserts a data-consistency corrector into every reverse diffusion step. After the conditional and unconditional noise predictions are combined, the method forms a denoised estimate, applies the projector, and then fuses the two estimates using a voxel-wise uncertainty map. The uncertainty is derived from the variance across tilt-wise backprojected voxel estimates: $A$5 Here $A$6 is the value assigned to voxel $A$7 by backprojecting the $A$8-th tilt, $A$9 is the mean over tilts, and $\theta$0 is the number of projections. The normalization constrains $\theta$1 (Deng et al., 7 Oct 2025).

Given the denoised estimate $\theta$2 and its projector-corrected version $\theta$3, the reverse step becomes

$\theta$4

The intended interpretation is explicit: low-uncertainty voxels rely more heavily on the projection-corrected estimate, whereas high-uncertainty voxels rely more heavily on the learned prior. This mechanism is motivated by the fact that strict consistency with $\theta$5 is only idealized; real tilt series contain distortions and alignment errors, and those errors become especially harmful when the number of views is very small (Deng et al., 7 Oct 2025).

The full inference procedure initializes from Gaussian noise, computes the coarse condition $\theta$6, estimates $\theta$7, and then iterates the classifier-free-guided denoising step, the projector correction, and the uncertainty-weighted fusion until $\theta$8 is obtained. The stopping criterion is simply completion of the reverse schedule rather than convergence of a separate optimization loop (Deng et al., 7 Oct 2025).

6. Experimental evidence and operating regimes

TEMDiff is evaluated on both synthetic and real data. For synthetic testing from FIB-SEM volumes, the tilt geometry uses total angular ranges of $\theta$9, $\Delta\theta$0, and $\Delta\theta$1, symmetric around $\Delta\theta$2, with $\Delta\theta$3 increments. Training randomizes geometry over $\Delta\theta$4 and $\Delta\theta$5. For real-data-derived tests, six real tilt-series reconstructions are used as references: three single-cell STEM mitochondria volumes and three brain-cell TEM synapse volumes, each reconstructed from $\Delta\theta$6–$\Delta\theta$7 projections over $\Delta\theta$8–$\Delta\theta$9 with $180^\circ$0 increment and $180^\circ$1 image resolution, then cropped and downsampled to $180^\circ$2 (Deng et al., 7 Oct 2025).

On simulated FIB-SEM datasets, the method is reported to be strongest in the most underdetermined setting, especially at $180^\circ$3. On the brain dataset at $180^\circ$4, TEMDiff achieves RMSE $180^\circ$5, PSNR $180^\circ$6, and SSIM $180^\circ$7, compared with FBP RMSE $180^\circ$8, SSIM $180^\circ$9, SART RMSE $10^\circ$0, SSIM $10^\circ$1, DiffusionMBIR RMSE $10^\circ$2, SSIM $10^\circ$3, and DOLCE RMSE $10^\circ$4, SSIM $10^\circ$5. On HeLa2 at $10^\circ$6, TEMDiff reaches RMSE $10^\circ$7, PSNR $10^\circ$8, and SSIM $10^\circ$9. On the synapse dataset at $P$0, TEMDiff ties DiffusionMBIR in RMSE at approximately $P$1 while increasing SSIM from $P$2 to $P$3 (Deng et al., 7 Oct 2025).

On simulated datasets derived from real tilt reconstructions, TEMDiff remains competitive even though the references still contain residual missing-wedge artifacts. For mitochondria at $P$4, it reports RMSE $P$5, PSNR $P$6, and SSIM $P$7, outperforming all baselines. For mitochondria at $P$8, it reaches RMSE $P$9 and PSNR $A\,P(x)=y.$0, although the paper notes that FBP can obtain slightly higher SSIM by reproducing directional artifacts present in the AreTomo reference rather than removing them. For synapse at $A\,P(x)=y.$1, TEMDiff records RMSE $A\,P(x)=y.$2 and PSNR $A\,P(x)=y.$3, ahead of FBP, SART, DiffusionMBIR, and DOLCE in RMSE and PSNR (Deng et al., 7 Oct 2025).

The most practically important claim concerns raw real limited-angle tilt series. The method is applied to biological TEM/STEM data from samples of approximately $A\,P(x)=y.$4 nm, $A\,P(x)=y.$5 nm, and $A\,P(x)=y.$6 nm thickness, acquired over only $A\,P(x)=y.$7 total range with $A\,P(x)=y.$8 or $A\,P(x)=y.$9 increments. The paper reports that a model trained only on simulated FIB-SEM-derived data generalizes to these real tilt series without retraining or fine-tuning, and that it produces clearer and more plausible 3D reconstructions than FBP and SART in this regime (Deng et al., 7 Oct 2025).

7. Relation to prior approaches, limitations, and interpretation

TEMDiff differs from classical reconstruction and earlier learned approaches in three linked respects. Relative to FBP and SART, it inserts a learned structural prior into an otherwise ill-posed inverse problem. Relative to DiffusionMBIR and DOLCE, it reconstructs directly in 3D rather than slice by slice in 2D. Relative to purely learned inverse mappings, it retains explicit measurement consistency through the projector-corrected reverse process. The paper repeatedly identifies this combination—3D conditional diffusion, conditioning on $y$00, and projector guidance—as the reason it remains effective under extreme angular sparsity (Deng et al., 7 Oct 2025).

The limitations are equally explicit. The FIB-SEM-to-HAADF simulator is only approximate, so a residual domain gap remains between the training distribution and real microscope data. Severe microscope misalignment or geometric distortion can still harm reconstruction quality even with uncertainty weighting. Quantitative evaluation on real-data-derived test sets is imperfect because the large-angle AreTomo references retain missing-wedge artifacts and therefore do not constitute artifact-free ground truth. The projection and contrast model is simplified and may not capture all specimen-specific imaging physics (Deng et al., 7 Oct 2025).

Within those constraints, TEMDiff occupies a specific place in the literature of electron-tomography reconstruction: it is a volumetric generative prior coupled to a forward-model corrector, designed for regimes where angular coverage is so narrow that slice-wise priors and classical inversion become inadequate. Its central contribution is therefore not merely the use of diffusion, but the use of a projector-guided 3D diffusion prior trained from simulated FIB-SEM supervision to recover coherent ultrastructure from tilt ranges as narrow as $y$01 without retraining (Deng et al., 7 Oct 2025).