Unified Denoising Objective Framework

• Unified denoising objective is a mathematical framework that minimizes an error functional combining data fidelity with model-specific regularization across diverse domains.
• It leverages principled techniques such as unbiased risk estimators, variational bounds, and score matching to unify classical, deep, and hybrid denoisers.
• The framework supports cross-domain implementations and systematic benchmarking, advancing applications in inverse problems like super-resolution and beamforming.

A unified denoising objective is a mathematical and algorithmic framework that formulates denoising—across signal, image, graph, and sequence domains—as the explicit minimization of a risk or discrepancy functional that simultaneously accounts for data fidelity and model-dependent structure. These objectives typically admit closed-form or principled variational/score-matching proxies and unify a broad range of classical, deep, probabilistic, and hybrid denoisers under a single theoretical lens. They often enable the design of new denoising schemes, systematic benchmarking, and practical cross-domain implementations.

1. General Structure and Theory

At the core, unified denoising objectives formalize denoising as the minimization of an error functional $\mathcal{L}$ between a denoised estimate $f_\theta(y)$ (parameterized by $\theta$) and the true (unknown) signal $x$, often regularized for structure. The typical form is

$$\mathcal{L}(\theta) = \mathbb{E}_{x,y} \left[ \| f_\theta(y) - x \|_2^2 \right] + \lambda \cdot\,\text{Reg}(f_\theta)$$

where $y$ is a noisy observation of $x$, and $\text{Reg}(\cdot)$ encodes domain-specific regularization (e.g., patch similarity, graph smoothness, or segmentation constraints).

When $x$ is unknown, unbiased risk estimators (e.g., SURE or URE) or variational bounds (ELBO, KL divergence) provide tractable surrogates. In probabilistic frameworks, the setting may leverage time-reversal or backward generators, Doob's $h$‐transform, or Markovian variational objectives to directly minimize path-space discrepancies or measure-transport costs (Ren et al., 2 Apr 2025). This mechanism encompasses continuous, discrete, and mixed-processes, such as classical diffusions, Lévy processes, or discrete jump dynamics.

In graph signal settings, the unified denoising objective is the minimizer of

$$\mathcal{L}(F) = \| F - S \|_F^2 + c \operatorname{Tr}(F^\top L F)$$

with $L$ the graph Laplacian, $S$ the noisy signal, and $c>0$ a smoothness parameter, directly linking classical and neural GNN architectures to denoising variational principles (Ma et al., 2020).

2. Classical and Patch-Based Image Denoising

Patch-averaging, PCA-Wiener, NL-Bayes, BM3D, and related non-local methods have been rigorously unified via quadratic-risk minimization over patch groups: $$\Theta^* = \arg \min_\Theta\, \mathbb{E} \left\| f_\Theta(Y) - X \right\|_F^2$$ with $Y \in \mathbb{R}^{n \times m}$ a matrix of grouped noisy patches, $X$ the clean patches, and $f_\Theta(Y) = Y\Theta$ a (possibly constrained) linear estimator. Because $X$ is unknown, a two-step strategy is employed: first, SURE- or URE-based minimization with respect to $Y$; second, internal adaptation using the pilot estimate $\hat{X}^{(1)}$ as a stand-in for $X$ to solve a regularized objective (ridge or affine constraint), yielding closed-form weights (Herbreteau et al., 2022, Herbreteau et al., 21 Feb 2024).

This framework recovers NL-Bayes via group-wise mean/covariance estimation and BM3D via shrinkage in the transformed domain, and encompasses new methods (e.g., NL-Ridge) that outperform both in challenging benchmarks while retaining complete interpretability. It extends to Poisson, Poisson–Gaussian, and other noise families via corresponding unbiased risk estimators (Herbreteau et al., 21 Feb 2024).

3. Probabilistic and Markovian Unification

Modern generative denoising models are formalized using Markovian stochastic dynamics, wherein a forward process transforms a target distribution to an easy-to-sample reference, and the backward process (learned or constructed) transports probability mass back to the target. Unified objectives minimize the KL divergence between the true and model backward path-laws: $$\mathrm{KL}(P\,\|\,Q^\theta) = E_P\Biggl[\int_0^T \bigl( L_t^\theta\,h_t^{-1}(x_t) + L_t[\log h_t](x_t) \bigr)\,dt\Biggr]$$ where $L_t^\theta$ is the parameterized backward generator derived via generalized Doob–$h$ transform, and $h_t$ links the path and reference distributions (Ren et al., 2 Apr 2025).

Specializations yield classical diffusion (e.g., Anderson reverse SDE, discrete-jump processes) and score-matching losses; for continuous diffusions: $$\mathfrak L[s^\theta] = E\left[\int_0^T \frac12 \| \sigma_t (s_t^\theta(x_t) - \nabla\log p_t(x_t)) \|^2 dt \right]$$ thus providing a precise foundation for continuous- and discrete-time denoising diffusion models, as well as for models driven by Lévy, compound-Poisson, or geometric Brownian motion forward processes.

4. Extension to Structured, Multi-Modal, and Task-Coupled Denoising

Unified denoising objectives have been adapted to architectures coupling denoising with auxiliary tasks, e.g., segmentation or action prediction. The AMDiff model (Xia et al., 17 Mar 2025) defines

$$\mathcal{L}_{\mathrm{total}}(\theta) = \mathcal{L}_{\mathrm{diff}} + \lambda_{\mathrm{warm}}(e)\big(\mathcal{L}_{\mathrm{lor}} + \mathcal{L}_{\mathrm{rev}} + \mathcal{L}_{\mathrm{seg}}\big)$$

where a diffusion-based denoising term, lesion-organ-specific regularizer, revision loss, and segmentation loss are combined, with a warm-up schedule to stabilize multi-task interaction and enable direct inference of clinical metrics (e.g., lesion glycolysis). This design demonstrates significant reduction in quantification bias and improved Dice/NRMSE compared to ablated objectives.

For joint vision-language-action, unified denoising of token sequences representing images and actions is achieved via discrete diffusion, cross-entropy (mask-predict) loss, and hybrid-attention transformers, enabling tightly coupled generation and inference in embodied multi-modal models (Chen et al., 3 Nov 2025).

5. Unified Denoising Objectives in Modern Diffusion and Generative Models

Score-based, directly denoising, and hybrid diffusion models leverage unified objectives for both variance-preserving (VP) and variance-exploding (VE) processes. In uDDDM (Wang et al., 31 May 2024), the shared form

$$f(x_0, x_t, t) = a(\sigma_t)x_t + b(\sigma_t)F(x_0, x_t, t)$$

with an adaptive Pseudo-Huber metric,

$d(x, y) = \sqrt{\|x-y\|^2 + c^2} - c$

supports training for both VP and VE regimes. Existence, uniqueness, and the non-intersection of solution paths are formally guaranteed, with state-of-the-art FID scores achieved in one-step and multi-step settings. Equivalent structures arise in unconditional and conditional denoising diffusion autoencoders, where the same quadratic objective both trains the generator and scaffolds linearly-separable representations for self-supervised learning (Xiang et al., 2023).

A further bridge is established via conditional diffusion for generative denoising across Gaussian, Gamma, and Poisson noise: the unified training loss

$$L(\theta) = \mathbb{E}_{t, q} \| f_\theta(x_t, [x_N?], t) - x_0 \|_2^2$$

proves, under mild conditions, to minimize the KL at each step and to yield the exact posterior mean for each noise family (Xie et al., 2023).

6. Denoising as the Foundation for Diverse Inverse and Learning Problems

Unified denoising objectives are also deployed within general inverse problems such as super-resolution and beamforming. For super-resolution, the plug-and-play prior (PPP) and regularization-by-denoising (RED) frameworks define MAP objectives—using arbitrary denoisers as black-box regularizers—and leverage ADMM solvers that decouple data fidelity and denoising (Brifman et al., 2018). For audio, convolutional beamformers optimize a single quadratic form that simultaneously accounts for dereverberation and denoising, unifying classical weighted prediction error and minimum-power distortionless response objectives into a single closed-form solution (Nakatani et al., 2018).

7. Design Principles, Variational Regularity, and Future Extensions

General conditions for well-posed unified objectives include Feller evolution of the forward process, existence/smoothness of path densities, and practical computability of adjoint operators (Ren et al., 2 Apr 2025). The framework is agnostic to process type—diffusive, jump, or hybrid—enables the explicit construction of backward generators, and naturally yields score-matching and variational objectives for learning.

This generality supports the development of new classes of denoising Markov models, potentially utilizing geometric, heavy-tailed (Lévy), or manifold-anchored dynamics as forward processes, provided the path-space variational criterion remains tractable and the backward process can be synthesized.

A plausible implication is that future unified denoising designs could flexibly integrate domain priors, learned structure, complex noise models, and even task-specific auxiliary constraints (e.g., segmentation, abstention) while remaining grounded in mathematically rigorous risk functionals, as evidenced by the breadth of recent developments across image, audio, graph, and multimodal AI (Herbreteau et al., 2022, Herbreteau et al., 21 Feb 2024, Ma et al., 2020, Ren et al., 2 Apr 2025, Chen et al., 3 Nov 2025, Xia et al., 17 Mar 2025).