Blur-Noise Mixture Diffusion Model (BNMD)

• BNMD is a generative diffusion framework that combines frequency-dependent blurring with isotropic Gaussian noise to interpolate between hot and cold diffusion regimes.
• It employs a parameterizable Blur-to-Noise Ratio schedule and a divide-and-conquer design to achieve enhanced control and improved performance on standard image generation benchmarks.
• The model leverages DCT-based blurring with noise addition to preserve natural-image spectral statistics, enabling robust zero-shot factorized sampling and prompt-controlled inverse problem solutions.

The Blur-Noise Mixture Diffusion Model (BNMD) is a theoretical and algorithmic framework for generative modeling that unifies and generalizes standard Gaussian ("hot") diffusion, pure blurring ("cold") diffusion, and their intermediate regimes. BNMD introduces explicit coupling of frequency-dependent blurring with isotropic Gaussian noise at each diffusion step, dictated by a parameterizable Blur-to-Noise Ratio (BNR) schedule. This framework improves inductive bias towards natural-image spectral statistics, enables enhanced controllability and compositional generation, and achieves strong empirical results across canonical image generation benchmarks. Key instantiations of BNMD include the "Warm Diffusion" method and its generalized counterparts, as well as a family of zero-shot factorized diffusion modifications enabling fine-grained, component-wise prompt control.

1. Model Foundations and Mathematical Formulation

The Blur-Noise Mixture Diffusion Model modifies the classic Markovian forward process of denoising diffusion probabilistic models (DDPM) by introducing an additional frequency-selective blurring operator, typically implemented as a diagonal mask in the Discrete Cosine Transform (DCT) domain. For a clean image $x_0$, the degraded image at time $t$ is formulated as

$$q\left(x_{α_t,β_t} \mid x_0\right) = \mathcal{N}\left(V M_{α_t} V^T x_0, \; β_t^2 I\right),$$

where $V, V^T$ are the (inverse) DCT transforms, and $M_{α_t}$ is a frequency-wise blur mask that attenuates high frequencies as $\alpha_t$ increases. The parameter $\beta_t$ controls the additive white noise level.

This generalizes:

• Hot diffusion: $\alpha_t = 0$ and $\beta_t > 0$ (pure noise).
• Cold diffusion: $\beta_t = 0$ and $\alpha_t > 0$ (pure blur).
• BNMD: intermediate settings, $\alpha_t, \beta_t > 0$.

The critical control variable is the Blur-to-Noise Ratio,

$BNR_t = \frac{\alpha_t}{\beta_t},$

which places the model on a continuum between noisy and blurry degradation (Hsueh et al., 21 Nov 2025, Hoogeboom et al., 2022).

2. Forward and Reverse Diffusion Processes

In the BNMD paradigm, both the forward (noising/blurring) and reverse (generative) processes are analytically tractable in closed form due to the preservation of Gaussianity under linear transforms. The forward kernel at step $t$ is defined via DCT-domain attenuation and isotropic noise as above, resulting in a Markov chain both in pixel and transform domains.

The reverse process is constructed as a learned Markovian transition, parameterized either as direct mean prediction or via latent variable inference. For generative modeling, maximizing the standard evidence lower bound (ELBO) or minimizing the mean-squared error (MSE) between predicted and true means suffices, exploiting closed-form expressions for the reverse transition mean and variance at each step. This preserves the flexibility of classic score-based frameworks but with enhanced spectral structure (Hsueh et al., 21 Nov 2025, Hoogeboom et al., 2022).

3. Divide-and-Conquer Parameterization and Training

BNMD capitalizes on the structure of natural images, which exhibit strong spectral dependencies. The "divide-and-conquer" parameterization splits the reverse mapping into parallel branches for denoising and deblurring:

• Denoiser $D_\theta$: predicts the blurred, denoised image ($V M_{α_t} V^T x_0$).
• Deblurrer $R_\theta$: predicts the high-frequency residual ($x_0 - V M_{α_t} V^T x_0$).

The generative mean is assembled according to

$$μ_\theta = x_{α_t,β_t} + V (M_{α_{t-1}} - M_{α_t}) (I - M_{α_t})^{-1} V^T R_\theta + \left(β_t - \sqrt{β_{t-1}^2 - σ_t^2}\right) \frac{D_\theta - x_{α_t,β_t}}{β_t}.$$

Training objectives decompose into separate MSE losses for denoising and deblurring targets, enabling fast and stable optimization. Empirical ablation studies confirm that the two-branch (denoise+residual) split yields superior FID scores compared to monolithic or naive parameterizations (Hsueh et al., 21 Nov 2025).

4. Spectral Perspective and Data Manifold Geometry

The fundamental innovation in BNMD is control over the spectral signature of the forward process. Spectral analysis reveals that natural-image power spectra $S_x(f)$ typically decay as $1/\|f\|^2$, whereas noise is spectrally flat. Blurring suppresses high-frequency components in a frequency-selective manner, while noise destroys signal indiscriminately.

A moderate $BNR \approx 0.5$ (i.e., balanced blur and noise) is optimal, as it simplifies the learning problem for the denoising branch without excessively collapsing the data manifold. Excessive blurring (high BNR) risks the forward distribution becoming too narrow, causing the reverse process to stray off-manifold unless a large number of sampling steps is used (Hsueh et al., 21 Nov 2025). The model therefore interpolates between the tractable spectral bias of blurring and the expressiveness of noise-driven diffusion.

5. Algorithmic Structure and Pseudocode

Training and inference in BNMD follow standard procedures for diffusion models, with the following modifications:

• Training: For each batch, apply the forward process by first blurring in DCT space then adding noise, then compute denoise and deblur targets, predict with parallel branches, and backpropagate via combined MSE.
• Sampling (reverse): Initialize from isotropic Gaussian, alternate between DCT blurring, adding noise, and updating via predicted denoise and deblur streams until $x_0$ is recovered.

Pseudocode is detailed in (Hsueh et al., 21 Nov 2025) and summarized below:

for x0 in dataset: t = random timestep xt = DCT_blur(x0, alpha_t) + beta_t * epsilon D_target = DCT_blur(x0, alpha_t) R_target = x0 - D_target D_pred = D_theta(xt, alpha_t, beta_t) R_pred = R_theta(xt, alpha_t, beta_t) loss = MSE(D_pred, D_target) + MSE(R_pred, R_target) update_theta(loss)

6. Extensions: Factorized Diffusion, Perceptual Illusions, and Inverse Problems

BNMD supports explicit factorization of image space into linear components $f_i(x)$, permitting per-component prompt conditioning in text-to-image setups. For $N$ components (e.g., low/high frequency, color/grayscale, motion-blur/residual), it performs $N$ noise predictions per step, reassembling them according to the specified decomposition:

$\tilde{\epsilon} = \sum_{i=1}^N f_i(\epsilon_i),$

where each $\epsilon_i$ is predicted under its prescribed conditioning $y_i$ (Geng et al., 17 Apr 2024).

This enables hybrid images exhibiting distance-dependent, color-dependent, or motion-dependent illusions, and allows "inverse problem" extensions wherein a subset of components is held fixed (e.g., from a reference image) while others are regenerated to match target semantics. Empirically, factorized BNMD sampling achieves higher subjective and objective alignment with target prompts and can reconstruct missing components robustly (Geng et al., 17 Apr 2024).

7. Empirical Results and Practical Considerations

Experimental results validate the advantage of BNMD across canonical datasets. On CIFAR-10 (NFE=35, BNR=0.5), BNMD achieves FID=1.85 compared to EDM’s 1.97, and Inception Score IS=10.02 vs. 9.78. Similar improvements are reported for conditional settings, human perceptual studies, and on larger, higher-resolution datasets such as FFHQ and LSUN-Church (Hsueh et al., 21 Nov 2025, Geng et al., 17 Apr 2024). Performance degrades for excessively high BNR due to manifold collapse, while low BNR recovers classic DDPM results.

Algorithmically, the additional overhead of DCT-based blurring is minor compared to the main network forward/backward passes. Standard U-Net architectures may be adopted, with minor adaptions to incorporate blur/noise schedules and component decompositions. The underlying framework is compatible with DDPM, DDIM, and related score-based methods, and generalizes to per-component prompt control, compositional generation, and robust inverse problem formulations.

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References:

• "Warm Diffusion: Recipe for Blur-Noise Mixture Diffusion Models" (Hsueh et al., 21 Nov 2025)
• "Blurring Diffusion Models" (Hoogeboom et al., 2022)
• "Factorized Diffusion: Perceptual Illusions by Noise Decomposition" (Geng et al., 17 Apr 2024)