Denoising Diffusion Gamma Model

• Denoising Diffusion Gamma Model is a diffusion framework that substitutes Gaussian noise with Gamma-distributed noise while preserving the noise variance schedule via centered noise increments.
• It employs closed-form Gamma noise accumulation to enable efficient one-shot forward sampling and supports both unconditional generation and posterior-sampling denoising techniques.
• Adaptations like accelerated reverse sampling and OCT-specific modifications (e.g., NRFT in GARD) demonstrate improved image quality and reduced computational cost in practical applications.

Denoising Diffusion Gamma Model (DDGM) denotes a class of diffusion models in which the Gaussian perturbation used in standard DDPM/DDIM-style processes is replaced by Gamma-distributed noise, typically centered to zero mean so that the variance schedule is preserved while the forward noising process remains analytically tractable. In its canonical form, DDGM retains the core diffusion architecture—a forward corruption chain, a learned reverse denoiser $\varepsilon_\theta(x_t,t)$, and iterative sampling—but changes the underlying noise law from Gaussian to Gamma. Subsequent work extended Gamma-based diffusion beyond unconditional generation to posterior-sampling denoising and to OCT despeckling, where the statistical mismatch between Gaussian assumptions and observed corruption is especially consequential (Nachmani et al., 2021, Xie et al., 2023, Fazekas et al., 12 Sep 2025).

1. Origins and rationale

The original motivation for DDGM is the claim that Gaussian noise is mathematically convenient but overly restrictive. Standard DDPMs exploit the fact that sums of Gaussians remain Gaussian, which yields a closed-form latent $x_t$ at arbitrary timesteps. DDGM argues that Gamma noise offers a comparable practical benefit because sums of Gamma random variables with the same scale remain Gamma distributed, while also introducing a two-parameter family $\Gamma(k,\theta)$ with more degrees of freedom than the single-distribution Gaussian setting (Nachmani et al., 2021).

Related non-Gaussian diffusion work broadened this argument by considering both Gamma noise and mixtures of Gaussians. That line of work reported that, after multiple diffusion steps, Gamma and Gaussian mixtures fit residual histograms better than a single Gaussian, and that the added flexibility can improve generation quality and sometimes convergence speed (Nachmani et al., 2021).

A domain-specific rationale appears in retinal OCT. There, the dominant corruption is speckle noise arising from coherent interference of backscattered light. Speckle is described as positive-only and skewed, and as matching the intensity fluctuations created by coherent imaging more closely than a Gaussian. In GARD, this motivates replacing the Gaussian diffusion assumption with a Gamma-based diffusion process adapted to retinal OCT (Fazekas et al., 12 Sep 2025).

2. Centered-Gamma forward diffusion

The defining DDGM forward step replaces Gaussian perturbation with centered Gamma noise:

$$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$

where

$$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$

This parameterization is chosen so that the centered increment has zero mean and variance $\beta_t$:

$$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$

In this sense, DDGM preserves the variance-schedule role played by Gaussian increments in standard diffusion while changing the higher-order distributional structure (Nachmani et al., 2021).

The key analytical property is the closed-form accumulated noising relation:

$$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$

with

$$\bar g_t\sim \Gamma(\bar k_t,\theta_t), \qquad \bar k_t=\sum_{i=1}^t k_i.$$

This is the Gamma analogue of the Gaussian DDPM identity $$x_t=\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\,\varepsilon$$. It permits direct sampling of $x_t$0 from $x_t$1 without simulating all intermediate steps, which is the main reason Gamma diffusion remains computationally practical (Nachmani et al., 2021).

The same construction was transferred to OCT despeckling in GARD, with the paper emphasizing that the additive Gamma formulation is applied to post-processed display-ready images, often log-compressed or fourth-root transformed, where the noise becomes approximately additive even though speckle is physically multiplicative in raw OCT. This suggests that the mathematical convenience of additive centered-Gamma diffusion is being used in an operating domain chosen to reduce the mismatch with the underlying acquisition physics (Fazekas et al., 12 Sep 2025).

3. Reverse process, learning objective, and accelerated sampling

As in Gaussian diffusion, DDGM uses a neural network $x_t$2 to predict the noise component required for reverse-time denoising. The reverse update is written in DDPM/DDIM-style form, but with Gamma-based noise terms in place of Gaussian perturbations. The training derivation proceeds through the usual diffusion-model ELBO decomposition, with $x_t$3 treated as constant and $x_t$4 reported as empirically negligible in the Gamma setup. The paper concludes that minimizing the resulting variational objective is equivalent to minimizing an $x_t$5-style noise-prediction objective (Nachmani et al., 2021).

The broader non-Gaussian diffusion formulation preserves the same training recipe in spirit as DDPM: sample a random timestep $x_t$6, generate $x_t$7 from $x_t$8 in one shot using the closed-form forward process, train a network to predict the injected noise, and use a learned reverse update at inference. The practical significance is that non-Gaussian noise does not force explicit simulation of every intermediate training step (Nachmani et al., 2021).

Fast sampling is particularly important because Gamma-based reverse chains remain iterative. GARD adapts DDIM-style deterministic sampling to the Gamma model by setting $x_t$9, removing the stochastic part of the reverse transition, and allowing skipped timesteps. In the reported OCT implementation, inference does not use all $\Gamma(k,\theta)$0 timesteps: denoising starts at $\Gamma(k,\theta)$1 and samples every $\Gamma(k,\theta)$2th timestep. The paper reports that this greatly reduces compute without sacrificing image quality (Fazekas et al., 12 Sep 2025).

4. Alternative Gamma-denoising formulations

A distinct line of work formulates Gamma denoising not as unconditional generation from centered Gamma noise, but as posterior sampling under an explicit multiplicative Gamma corruption model. In that setting,

$\Gamma(k,\theta)$3

and denoising is framed as sampling from $\Gamma(k,\theta)$4 rather than estimating a single deterministic output (Xie et al., 2023).

To preserve Gamma structure across intermediate states, that paper defines a multiplicative Markov chain with a decreasing sequence of shape parameters $\Gamma(k,\theta)$5, using Beta-distributed transition factors so that every state $\Gamma(k,\theta)$6 remains a Gamma-corrupted version of the clean image. The reverse chain replaces the unknown clean image with a neural estimate $\Gamma(k,\theta)$7, and the theoretical result is that the optimal predictor is the posterior mean:

$\Gamma(k,\theta)$8

The corresponding training problem reduces to an $\Gamma(k,\theta)$9 regression objective, even though the reverse model itself is probabilistic (Xie et al., 2023).

These two Gamma-diffusion families are related but not identical. The centered additive DDGM preserves the DDPM/DDIM structure while altering the forward noise law. The posterior-sampling denoiser instead redesigns the diffusion chain so that the entire corruption family remains Gamma-consistent under multiplicative noise. A plausible implication is that “Gamma diffusion” is best understood as a design space rather than a single equation set: the common principle is alignment between the diffusion process and the assumed corruption statistics.

5. OCT-specific instantiation: GARD

GARD, short for Gamma-based Anatomical Restoration and Denoising, is an OCT despeckling system that combines three elements: a Gamma diffusion model for OCT-speckle-aware denoising, a Noise-Reduced Fidelity Term (NRFT), and DDIM-style accelerated inference (Fazekas et al., 12 Sep 2025).

The NRFT addresses a specific failure mode of diffusion-based restoration. A fidelity term tied directly to the noisy input can preserve unwanted high-frequency speckle. GARD therefore constructs a less-noisy guidance image

$$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$0

where NLM is Non-Local Means filtering. The NLM output is described as retaining low-frequency structure, major edges, and anatomical boundaries while suppressing much of the speckle. After each diffusion reverse step, the estimate is refined by solving a fidelity optimization problem involving the current reverse-step output, the NLM-filtered image, and a fidelity weight $$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$1; the paper uses Newton’s method for this optimization. The intended role of NRFT is structure preservation without reintroducing speckle (Fazekas et al., 12 Sep 2025).

The implementation uses a U-Net backbone for both Gaussian and Gamma diffusion models, a linear $$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$2 schedule from $$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$3 to $$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$4, $$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$5 diffusion steps, and $$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$6 for the Gamma model. Training uses AdamW with learning rate $$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$7, batch size $$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$8, and $$x_t = \sqrt{1-\beta_t}\,x_{t-1} + \bigl(g_t-\mathbb{E}(g_t)\bigr),$$9 iterations, with random horizontal flip as data augmentation. The training set contains $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$0 OCT volumes and $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$1 B-scans (Fazekas et al., 12 Sep 2025).

Quantitative evaluation uses $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$2 OCT volumes acquired with a Spectralis device, each containing paired single-sweep noisy B-scans and less-noisy ART-averaged B-scans from $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$3 registered scans. Compared methods include NLM, SCUNet, Speckle2Speckle, N2V2, standard DDPM, CPDM, DDGM, and variants with NRFT and/or CPDM. GARD reports the best overall metrics: SSIM $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$4, PSNR $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$5 dB, and MSE $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$6. The paper highlights PSNR gains of $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$7 dB over SCUNet, $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$8 dB over standard DDPM, and $$g_t \sim \Gamma(k_t,\theta_t), \qquad \theta_t=\sqrt{\bar\alpha_t}\,\theta_0, \qquad k_t=\frac{\beta_t}{\alpha_t\theta_0^2}.$$9 dB over vanilla DDGM, with statistically significant improvement over all methods at $\beta_t$0. Qualitatively, GARD is reported to produce sharper edges, better preservation of thin anatomical structures, improved layer boundaries, and better maintenance of subtle retinal details. Qualitative examples on Cirrus, Topcon, and Spectralis scans further suggest generalization across vendors (Fazekas et al., 12 Sep 2025).

6. Empirical behavior, misconceptions, and limitations

The empirical record for DDGM is strongest in low- and mid-step sampling regimes. On LJ Speech, the original DDGM paper reports improvements over WaveGrad at all reported sampling budgets. At $\beta_t$1 steps, PESQ increases from $\beta_t$2 to $\beta_t$3 and STOI from $\beta_t$4 to $\beta_t$5; at $\beta_t$6 steps, PESQ increases from $\beta_t$7 to $\beta_t$8 and STOI from $\beta_t$9 to $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$0 (Nachmani et al., 2021).

For image generation, Gamma diffusion improves FID substantially under fast sampling. On CelebA $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$1, DDGM with DDIM inference reports $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$2, $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$3, $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$4, $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$5, and $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$6 FID at $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$7, $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$8, $$\mathbb{E}(g_t-\mathbb{E}(g_t))=0, \qquad \mathrm{Var}(g_t-\mathbb{E}(g_t))=\beta_t.$$9, $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$0, and $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$1 steps, versus $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$2, $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$3, $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$4, $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$5, and $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$6 for the Gaussian DDIM baseline. On LSUN Church $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$7, Gamma improves over the Gaussian baseline at all reported step counts for both DDPM and DDIM inference (Nachmani et al., 2021).

In supervised denoising under Gamma corruption, the posterior-sampling formulation exhibits a different metric profile. On Kodak and CSet9, the mean of $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$8 diffusion samples nearly matches supervised learning PSNR and SSIM, while individual generated samples are visually richer and more detailed but can score lower on pixelwise metrics. For $$x_t=\sqrt{\bar\alpha_t}x_0+(\bar g_t-\bar k_t\theta_t),$$9, the mean of $$\bar g_t\sim \Gamma(\bar k_t,\theta_t), \qquad \bar k_t=\sum_{i=1}^t k_i.$$0 samples attains $$\bar g_t\sim \Gamma(\bar k_t,\theta_t), \qquad \bar k_t=\sum_{i=1}^t k_i.$$1 on Kodak and $$\bar g_t\sim \Gamma(\bar k_t,\theta_t), \qquad \bar k_t=\sum_{i=1}^t k_i.$$2 on CSet9, compared with $$\bar g_t\sim \Gamma(\bar k_t,\theta_t), \qquad \bar k_t=\sum_{i=1}^t k_i.$$3 and $$\bar g_t\sim \Gamma(\bar k_t,\theta_t), \qquad \bar k_t=\sum_{i=1}^t k_i.$$4 for the supervised baseline. This behavior is consistent with the paper’s framing of denoising as posterior sampling rather than posterior-mean regression (Xie et al., 2023).

Several misconceptions are explicitly countered by the literature. First, a Gamma diffusion model is not merely a Gaussian DDPM with a cosmetic reparameterization; its defining change is the underlying noise law together with a parameterization that preserves zero mean, the desired variance schedule, and closed-form accumulation (Nachmani et al., 2021). Second, Gamma noise is not universally better than Gaussian noise. The non-Gaussian diffusion work states that it does not establish universal criteria for when Gamma beats Gaussian or vice versa, and it reports that Gamma is worse than the baseline on MCD for speech, which the authors attribute to extra noise and less sharp spectrograms, possibly due to the bounded nature of the Gamma distribution (Nachmani et al., 2021). Third, in OCT despeckling, the use of additive Gamma diffusion does not deny that raw speckle is multiplicative; rather, the model is applied to post-processed images where the noise is approximately additive (Fazekas et al., 12 Sep 2025).

The main limitations reported across the literature are therefore methodological rather than merely empirical. Gamma closed-form accumulation depends on a shared scale parameter across summed Gamma variables. Some derivations and algorithm blocks are typographically imperfect. Posterior-sampling denoising requires a specified noise model and remains more complex than direct regression because reverse sampling is iterative. Even in generation, DDGM is not uniformly dominant at very long sampling chains; on CelebA with $$\bar g_t\sim \Gamma(\bar k_t,\theta_t), \qquad \bar k_t=\sum_{i=1}^t k_i.$$5 DDPM-style steps, the Gaussian baseline slightly outperforms DDGM in FID (Nachmani et al., 2021, Xie et al., 2023, Nachmani et al., 2021).