EqDiff-CT: Equivariant Diffusion for CT Synthesis

• The paper introduces a slice-wise conditional DDPM that integrates C4-equivariant steerable layers in a U-Net backbone to synthesize HU-consistent CT from CBCT.
• It employs a hybrid training objective combining MSE and SSIM losses to improve structural fidelity, yielding significant SSIM and PSNR gains over non-equivariant models.
• Clinical benefits include enhanced dose calculation and adaptive planning through improved anatomical detail, though limitations involve 2D context and iterative sampling speed.

EqDiff-CT is a conditional denoising diffusion probabilistic model (DDPM) for CBCT-to-CT synthesis in image-guided radiotherapy (IGRT). It generates anatomically consistent, HU-preserving synthetic CT (sCT) from cone-beam CT (CBCT) by combining a slice-wise DDPM with a group-equivariant conditional U-Net backbone implemented with e2cnn steerable layers and cyclic $C_4$ rotational symmetry. In the reported study, the model was trained and validated on the SynthRAD2025 Head-and-Neck cohort and compared with CycleGAN and a baseline DDPM, with improvements in structural fidelity, HU accuracy, and quantitative metrics (Altalib et al., 26 Sep 2025).

1. Clinical setting and problem formulation

CBCT is widely used for daily image guidance because it is integrated into linear accelerators and acquired with a single rotation. However, scatter, truncated projections, and low-dose acquisition cause noise and streaking artifacts, and CBCT lacks standardized HU calibration. CT, conversely, offers accurate HU and high SNR but is often acquired offline during planning, so it cannot reflect day-to-day anatomical changes. Synthetic CT from CBCT is intended to bridge this gap and enable dose recalculation and anatomical monitoring on the daily anatomy without repeat CT scans (Altalib et al., 26 Sep 2025).

The artifact profile emphasized for CBCT includes increased scatter and beam hardening, truncated projections and non-uniformity, rotationally oriented streaks emanating from high-contrast structures such as bone and dental fillings, and reduced soft-tissue contrast and signal-to-noise ratio. Within adaptive radiotherapy workflows, these degradations are clinically relevant because HU inaccuracies translate into erroneous electron density, affecting path length and attenuation modeling in photon or proton dose engines. The HU definition is given as

$$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$

where $\mu(x)$ is the linear attenuation coefficient at voxel $x$ and $\mu_{water}$ is the water attenuation coefficient. The reported motivation states that percent-level HU errors can produce several percent dose discrepancies that jeopardize target coverage or increase OAR dose (Altalib et al., 26 Sep 2025).

EqDiff-CT is framed specifically as a method for closing the imaging-quality gap between daily CBCT and planning CT. The stated objective is accurate CBCT-to-CT synthesis for dose calculation, adaptive planning, and anatomical monitoring on intra-treatment anatomy.

2. Conditional diffusion formulation

Let $x_0$ denote a ground-truth CT slice and $y$ the paired CBCT slice. EqDiff-CT learns $p_\theta(x_0 \mid y)$ via the DDPM framework. The forward noising process uses a variance schedule $\{\beta_t\}_{t=1}^T$, with $\alpha_t = 1-\beta_t$ and $$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$0, and is written as

$$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$1

with the closed form

$$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$2

so that during training

$$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$3

The reverse denoising step is

$$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$4

with

$$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$5

and

$$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$6

Sampling uses

$$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$7

with $$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$8 for $$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$9 (Altalib et al., 26 Sep 2025).

Conditioning is implemented by concatenation conditioning: the CBCT slice $\mu(x)$0 is concatenated channel-wise to $\mu(x)$1 at every scale $\mu(x)$2. No classifier-free guidance or conditioning dropout is used. The training objective is a hybrid noise-prediction loss,

$\mu(x)$3

$\mu(x)$4

$\mu(x)$5

with $\mu(x)$6. The reported rationale is that the use of SSIM on noise stabilizes perceptual structure in reconstructions (Altalib et al., 26 Sep 2025).

The paper states three contributions in this formulation: a slice-wise conditional DDPM framework with volume-wise analysis for CBCT$\mu(x)$7CT synthesis; a group-equivariant, attention-enhanced, time-conditioned U-Net backbone implemented with e2cnn $\mu(x)$8 symmetry), with CBCT concatenated at every scale; and a stable hybrid training objective combining MSE and SSIM on the predicted noise, coupled with an HU-preserving normalization/denormalization pipeline.

3. Group-equivariant denoiser and architectural design

The denoiser is a group-equivariant U-Net in which equivariance is defined by

$\mu(x)$9

where $x$0 and $x$1 are input and output representations, and $x$2 corresponds to in-plane rotations by $x$3, $x$4, $x$5, and $x$6. The stated reason for enforcing rotational equivariance is that CBCT acquisition and its artifacts have strong rotational structure; enforcing equivariance reduces the need to learn multiple rotated copies of filters and helps preserve orientation-consistent details such as bone edges and airway (Altalib et al., 26 Sep 2025).

The steerable group convolution is summarized as

$x$7

with kernel constraint

$x$8

for all $x$9. The implementation uses e2cnn with $\mu_{water}$0, $\mu_{water}$1, and $\mu_{water}$2-$\mu_{water}$3. The nonlinearity is written as

$\mu_{water}$4

where $\mu_{water}$5 is a gating function; dependence only on invariant norms preserves equivariance (Altalib et al., 26 Sep 2025).

Architecturally, the network is a 2D U-Net with three scales and channel multipliers $\mu_{water}$6. The encoder consists of three double G-equivalent conv blocks $\mu_{water}$7, each followed by $\mu_{water}$8 and ReLU $\mu_{water}$9-$x_0$0, with equivariant down-sampling. The decoder is the mirror of the encoder with up-sampling and skip connections. The residual topology is standard U-Net skip connections, with no internal residual addition within blocks. CBCT $x_0$1 is concatenated channel-wise at every resolution scale. Self-attention blocks capture long-range dependencies, and the attention mechanism is implemented with equivariant self-attention in which $x_0$2 are computed via equivariant $x_0$3 intertwiners and attention weights use a $x_0$4-invariant inner product to remain equivariant (Altalib et al., 26 Sep 2025).

Time conditioning uses sinusoidal encoding,

$x_0$5

followed by a two-layer MLP that maps $x_0$6 to $x_0$7, injected at the bottleneck via a linear projection and broadcast addition. In combination, rotational equivariance, self-attention, and sinusoidal time embeddings constitute the core inductive bias of EqDiff-CT.

4. Dataset, preprocessing, and optimization protocol

The reported experiments use SynthRAD2025 (Head-and-Neck cohort), comprising 325 patients and approximately 23,927 axial slices, with paired CBCT and planning CT volumes curated for CBCT$x_0$8CT synthesis research (Altalib et al., 26 Sep 2025).

Preprocessing consists of cropping to a non-zero bounding box for anatomical centering; HU windowing to $x_0$9 HU; normalization to $y$0 using

$y$1

2D axial slice extraction; zero-padding to $y$2 to preserve aspect ratio and network compatibility; slice pairing by axial index across modalities; and random horizontal and vertical flips. The split is patient-wise 80/20, with 259 train and 66 test patients. Training uses paired CBCT-CT slices, and testing uses CBCT as input and CT for evaluation (Altalib et al., 26 Sep 2025).

The diffusion schedule uses linear $y$3 with $y$4 steps and $y$5. Optimization uses Adam with $y$6, $y$7, learning rate $y$8, batch size $y$9, and $p_\theta(x_0 \mid y)$0 epochs. The loss is the hybrid noise-prediction objective with equal weights $p_\theta(x_0 \mid y)$1 for $p_\theta(x_0 \mid y)$2 and $p_\theta(x_0 \mid y)$3. Multi-GPU training is noted; inference was measured on an NVIDIA RTX A5000 with FP32. For a 61-slice case and 250 sampling steps, EqDiff-CT averages $p_\theta(x_0 \mid y)$4 s/slice (Altalib et al., 26 Sep 2025).

These design choices place EqDiff-CT in a slice-wise regime with volume-wise analysis, rather than a full volumetric 3D diffusion regime. The paper explicitly identifies this as a modeling decision rather than a claim of full 3D contextual reasoning.

5. Quantitative performance, ablations, and runtime

Evaluation uses SSIM, MSE, and PSNR, with

$p_\theta(x_0 \mid y)$5

and

$p_\theta(x_0 \mid y)$6

On the test set, the reported CT-versus-sCT metrics are as follows (Altalib et al., 26 Sep 2025).

Method	SSIM	PSNR
EqDiff-CT	$p_\theta(x_0 \mid y)$7	$p_\theta(x_0 \mid y)$8 dB
Baseline DDPM (non-equivariant)	$p_\theta(x_0 \mid y)$9	$\{\beta_t\}_{t=1}^T$0 dB
CycleGAN	$\{\beta_t\}_{t=1}^T$1	$\{\beta_t\}_{t=1}^T$2 dB
Baseline CBCT vs CT	$\{\beta_t\}_{t=1}^T$3	$\{\beta_t\}_{t=1}^T$4 dB

The relative improvements reported for EqDiff-CT are $\{\beta_t\}_{t=1}^T$5 SSIM and $\{\beta_t\}_{t=1}^T$6 dB PSNR versus the non-equivariant DDPM, and $\{\beta_t\}_{t=1}^T$7 SSIM and $\{\beta_t\}_{t=1}^T$8 dB PSNR versus CycleGAN. The baseline CBCT-versus-CT gap is included to highlight the synthesis problem that the method is attempting to close (Altalib et al., 26 Sep 2025).

The ablation on equivariance compares the model with a non-equivariant attention U-Net denoiser. The equivariant model improved average SSIM by approximately $\{\beta_t\}_{t=1}^T$9 and PSNR by approximately $\alpha_t = 1-\beta_t$0 dB, with reduced loss variability. Training loss converged more stably, and PSNR and SSIM increased more smoothly over epochs. Qualitatively, EqDiff-CT produces sharper soft-tissue boundaries, realistic bone reconstructions in the mandible and cervical spine, and suppresses rotationally patterned streaks. Difference maps show lower residual intensity near sudden HU transitions and interfaces such as airway walls and jawbone (Altalib et al., 26 Sep 2025).

Runtime measurements on a 61-slice test case with 250 sampling steps report $\alpha_t = 1-\beta_t$1 min for the non-equivariant DDPM $\alpha_t = 1-\beta_t$2 s/slice; $\alpha_t = 1-\beta_t$3 slices/min$\alpha_t = 1-\beta_t$4 and $\alpha_t = 1-\beta_t$5 min for EqDiff-CT $\alpha_t = 1-\beta_t$6 s/slice; $\alpha_t = 1-\beta_t$7 slices/min$\alpha_t = 1-\beta_t$8, indicating negligible overhead for $\alpha_t = 1-\beta_t$9 equivariance and attention (Altalib et al., 26 Sep 2025).

6. Clinical role, limitations, and later developments

The clinical argument for EqDiff-CT is that by learning a CBCT-conditioned CT distribution and enforcing rotational equivariance, the model reduces artifacts and yields HU-consistent sCT volumes. Better HU correspondences reduce electron density errors, which improves dose calculation accuracy for adaptive planning. The paper further states that daily sCT enables rapid dose recalculation on the current anatomy, supports adaptive re-planning criteria, and improves confidence in target and OAR delineations. Enhanced bone detail and soft-tissue boundary recovery are described as supporting robust auto-segmentation and reducing inter-observer variability, while artifact suppression yields more reliable CBCT guidance without additional imaging dose (Altalib et al., 26 Sep 2025).

The stated limitations are equally explicit. The current design is 2D and cannot leverage through-plane 3D context. Training and validation were limited to the Head-and-Neck SynthRAD2025 domain, so generalization to other anatomies or scanners requires validation and potentially domain-adaptive fine-tuning. End-to-end dose recalculation comparisons were not reported, so clinical physics validation across photon and proton pipelines remains needed. DDPM sampling remains iterative, and accelerating sampling through fewer steps or distillation is identified as a route to reduced latency in online settings. Future work is described in terms of richer symmetry groups such as $$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$00 or approximate $$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$01, multi-view conditioning such as neighboring slices or projections, uncertainty quantification, and patient-specific fine-tuning strategies (Altalib et al., 26 Sep 2025).

A later related framework, "EPC-3D-Diff: Equivariant Physics Consistent Conditional 3D Latent Diffusion for CBCT to CT Synthesis" (Altalib et al., 19 May 2026), describes an equivariant, physics-consistent conditional 3D latent diffusion framework that introduces a projection-domain equivariance loss derived from acquisition physics, operates in a compact 3D latent space, and reports $$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$02 dB PSNR on phantom data and $$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$03 dB on clinical data compared to state-of-the-art methods. This suggests a subsequent design trajectory in which slice-wise $$HU(x) = 1000 \cdot \frac{\mu(x)-\mu_{water}}{\mu_{water}},$$04-equivariant denoising is extended toward volumetric latent diffusion and projection-domain physics consistency rather than remaining confined to image-domain rotational equivariance (Altalib et al., 19 May 2026).

Within CBCT-to-CT synthesis, EqDiff-CT is therefore characterized by the conjunction of three elements: conditional diffusion synthesis, rotational equivariance implemented with e2cnn steerable layers, and an HU-preserving training pipeline. In the source description, this combination is presented as delivering anatomically faithful, HU-consistent sCT suitable for adaptive radiotherapy workflows (Altalib et al., 26 Sep 2025).