Blur-to-Noise Ratio (BNR)

• Blur-to-Noise Ratio (BNR) is a dimensionless metric defined as the ratio of blur level to noise level, crucial in both generative diffusion models and imaging reconstruction.
• In diffusion models, BNR modulates spectral trade-offs by enhancing low-frequency details while high BNR values may induce unstable generation due to out-of-manifold effects.
• Optimal BNR tuning, typically around 0.5 for diffusion models and CT, balances noise reduction with spatial resolution, improving overall algorithm performance.

The blur-to-noise ratio (BNR) is a dimensionless figure-of-merit quantifying the trade-off or relative dominance between blur and noise in imaging and generative modeling processes. Its formal definition and utility have emerged in both inverse problems (e.g., CT reconstruction) and generative diffusion models, where it provides an interpretable metric for analyzing and optimizing algorithms that jointly or competitively modulate blurring and noise.

1. Formal Definitions of Blur-to-Noise Ratio (BNR)

In diffusion-based generative modeling, BNR is defined as the ratio of the blur level $\alpha$ to the noise level $\beta$ at a given step $t$ in the forward corrupting process:

$\mathrm{BNR}_t = \frac{\alpha_t}{\beta_t}$

as introduced in Equation 13 of "Warm Diffusion: Recipe for Blur-Noise Mixture Diffusion Models" (Hsueh et al., 21 Nov 2025). If the forward process is given by

$$x_{\alpha_t,\beta_t} = V M_{\alpha_t} V^T x_0 + \beta_t \epsilon$$

then BNR is $\alpha_t/\beta_t$.

In model-based iterative reconstruction for flat-panel cone-beam CT, BNR is operationalized as the ratio of spatial blur to image noise variance:

$\mathrm{BNR} = \frac{\mathrm{FWHM}}{\sigma}$

where FWHM is the system’s spatial resolution (full-width at half-maximum, in mm) and $\sigma$ is the standard deviation of noise in a uniform region (in mm$^{-1}$) (II et al., 2017). This renders BNR dimensionless.

2. Spectral and Statistical Rationale

Diffusion Models

BNR encapsulates the spectral trade-off between blurring and noise injection. In the discrete cosine transform (DCT) domain, at frequency $j$, the corrupted image component is:

$$\hat{x}_{\alpha,\beta}(j) = m_j \hat{x}_0(j) + \beta \epsilon_j$$

where $m_j$ is an entry of the spectral blur mask $M_\alpha$ (monotonically decreasing with frequency), and $\epsilon_j \sim \mathcal{N}(0,1)$. The frequency-resolved signal-to-noise ratio is

$$\mathrm{SNR}(j) = \frac{(m_j \cdot [\mathrm{PS}(x_0)]^{1/2}(j))^2}{\beta^2}$$

Given empirical $\mathrm{PS}(x_0)(j) \approx C/f_j^2$ for natural images, increasing BNR enhances low-frequency spectral dominance by blur, while high frequencies are dominated by noise. The boundary (“corner frequency” $f^*$) where noise overtakes signal is controlled by BNR, thus shaping which frequencies are learned or synthesized in which stage of the diffusion process (Hsueh et al., 21 Nov 2025).

Imaging Inverse Problems

For CT reconstruction, BNR provides a single axis for assessing the spatially-dependent interplay between physical (or algorithmic) blurring and measurement noise. By matching noise ($\sigma$) and comparing FWHM between reconstruction algorithms, relative BNR reductions quantify the gain in resolution (sharper images) at fixed noise, or vice versa (II et al., 2017).

3. Impact on Learning Dynamics and Data Manifold

In blur-noise mixture diffusion models (“Warm Diffusion”), BNR directly modulates generative learning dynamics and the topology of the data manifold traversed by the reverse process. Two key mechanisms are observed:

• Learning difficulty: Low BNR (noise-dominated) obliges the denoiser $D_\theta$ to reconstruct all spectral bands from strong noise, increasing regression difficulty. High BNR (blur-dominated) concentrates denoising on low-frequency content, relegating high-frequency detail recovery to a subsequent deblurrer $R_\theta$. This “divide-and-conquer” approach empirically simplifies network training and shifts training targets towards lower-frequency bands at early diffusion steps (see Fig. 3 in (Hsueh et al., 21 Nov 2025)).
• Manifold integrity: With increasing BNR, the effective thickness of the corrupted data distribution (“noise-covering”) shrinks. If BNR is excessive, reverse steps are likely to explore parts of latent space not consistent with the true data manifold, leading to unstable or degraded generation (“out-of-manifold” inputs; see Fig. 4 in (Hsueh et al., 21 Nov 2025)).

A plausible implication is that there exists an optimal intermediate BNR which balances ease of learning with maintenance of data manifold integrity.

4. Empirical Characterization and Optimization

Diffusion Models

Sweeping BNR on unconditional CIFAR-10 (NFE=35) reveals a U-shaped performance curve in FID and Inception Score (IS):

BNR	FID (↓)	IS (↑)
0.0	1.97	9.78
0.1	1.97	9.96
0.3	1.90	10.02
0.5	1.85	10.02
0.65	1.91	10.00
1.0	2.01	9.96
2.0	2.57	9.89
10.0	11.97	8.51

Performance improves as BNR increases from 0 to $\approx$0.5 (best FID), then degrades for higher BNR as reverse sampling becomes less stable and out-of-manifold effects dominate (Hsueh et al., 21 Nov 2025). For high BNR, increasing the number of sampling steps (NFE) partially recovers quality, but manifold mismatch remains limiting for short schedules.

Flat-Panel CBCT Reconstruction

A set of five systems, varying in focal-spot vs. detector blur but with matched total blur, shows up to 42% reduction in blur (i.e., BNR) at matched noise for the correlated-noise model:

System	$B_d$ FWHM (mm)	$B_s$ FWHM (mm)	FWHMuncorr (mm)	FWHMcorr (mm)	$\Delta_{\%}$BNR
a	0.77	0.00	0.270	0.270	0%
b	0.70	0.34	0.370	0.340	8%
c	0.55	0.55	0.360	0.320	11%
d	0.34	0.70	0.400	0.230	42%
e	0.00	0.77	0.400	0.400	0%

For system (d), a 42% BNR reduction corresponds to achieving 0.23 mm FWHM versus 0.40 mm at identical noise variance ($6.9 \times 10^{-8}$ mm$^{-2}$) (II et al., 2017).

5. Practical Guidelines for BNR Selection

Guidelines for optimal BNR tuning depend on the application context:

• In diffusion models, an intermediate BNR of approximately 0.5 is empirically optimal: high-frequency bands become noise-dominated, maintaining manifold coverage and leveraging spectral structure. Very large BNR ($\gg 1$) regimes should be avoided unless sampling schedules are greatly extended, as cold-diffusion–like processes become unstable and out-of-manifold effects dominate.
• For imaging reconstruction, algorithmic or system design should aim for minimizing BNR (i.e., maximizing spatial resolution at fixed noise) through accurate noise modeling and deblurring, especially in regimes where detector blur and noise correlation are significant (II et al., 2017).

6. Interconnections and Broader Context

BNR as a unifying concept bridges the logic of forward process design in generative modeling (trading blur and noise for controllable manifold traversals and tractable learning) with the long-standing statistical trade-offs in imaging inverse problems (resolution vs. noise). In both cases, BNR provides a parameterization or figure-of-merit capturing the efficiency of information transfer and the fidelity achievable in estimation, sampling, or reconstruction pipelines.

Empirical and theoretical results demonstrate that, while minimizing either blur or noise in isolation is suboptimal, joint modulation via BNR enables improved generation quality and more robust, higher-resolution reconstruction. This suggests broader applicability of BNR analyses in any domain where blurring and noise compete or cooperate in shaping data distributions and learning objectives.

7. Summary Table: BNR Definitions across Domains

Domain	BNR Formula	Interpretation
Diffusion Models	$\alpha/\beta$	Ratio of blur to noise per step
Imaging (CT)	$\mathrm{FWHM}/\sigma$	Spatial blur per unit noise variance

BNR provides a principled, empirical, and theoretically interpretable axis for optimizing and understanding algorithms that balance denoising, deblurring, and the preservation of data manifold structure in both synthesis and reconstruction contexts (Hsueh et al., 21 Nov 2025, II et al., 2017).