Material-Agnostic Denoising (MAD) Methods

• Material-Agnostic Denoising (MAD) is a strategy that decouples noise removal from material-specific parameters by operating on intermediate representations in both rendering and diffusion contexts.
• In real-time rendering, MAD denoises projected irradiance before material shading, mitigating BSDF-dependent noise and preserving fine visual details.
• In diffusion models, MAD adapts sampling dynamics to attract outputs toward low-dimensional manifolds, effectively suppressing off-manifold noise.

Searching arXiv for the cited MAD-related papers to ground the article. Material-Agnostic Denoising (MAD) denotes a class of denoising strategies that seek to remove noise without embedding assumptions about a specific material class, object category, or handcrafted domain prior. In the literature represented here, the term appears in two closely related but non-identical senses. In real-time rendering, MAD is introduced explicitly as Material-Agnostic Denoising, where Monte Carlo noise is removed before material shading by denoising a representation of incident radiance that is independent of the scene’s actual materials (Schied et al., 23 Jul 2025). In score-based generative modeling, MAD stands for Manifold Attracted Diffusion, a modified inference procedure for diffusion models trained on noisy data; it is described as a form of material-agnostic denoising because it uses the geometry of the learned score field to suppress off-manifold noise without specifying what clean images or materials should look like (Elbrächter et al., 29 Sep 2025). Across these usages, the unifying idea is to separate signal structure from nuisance variation by exploiting an intermediate representation—either projected irradiance or manifold geometry—rather than relying on material-specific priors.

1. Terminological scope and research contexts

The expression Material-Agnostic Denoising is used directly in the rendering paper “Parametric Integration with Neural Integral Operators” (Schied et al., 23 Jul 2025). There, MAD is the core of a broader “neural shading pipeline” that denoises Monte Carlo estimates in a pre-shading representation of incident radiance, rather than in the final shaded RGB image. The denoiser is therefore agnostic to the BSDF parameters at a pixel, while material dependence is handled later by a small neural integral operator (Schied et al., 23 Jul 2025).

A second, partially overlapping usage appears in “MAD: Manifold Attracted Diffusion” (Elbrächter et al., 29 Sep 2025). In that work, MAD denotes Manifold Attracted Diffusion, not Material-Agnostic Denoising as an acronym. However, the method is explicitly characterized as a form of material-agnostic denoising because it uses a model trained directly on noisy data and alters the reverse diffusion dynamics so that generated samples are “attracted” to a lower-dimensional data manifold and have less noise than the training data, without specifying what clean images or materials look like (Elbrächter et al., 29 Sep 2025).

These two lines of work share a structural commitment to domain-independence at the denoising stage. The rendering formulation removes BSDF dependence from the denoised quantity; the diffusion formulation removes dependence on explicit clean-data supervision or known corruption models. This suggests that “material-agnostic” functions less as a narrow application label than as a design principle: denoising should operate on a representation in which nuisance variability is geometrically or physically separable from the content of interest.

2. Pre-shading MAD in real-time rendering

In real-time rendering, MAD arises from the observation that the conventional pipeline first estimates the reflected radiance

$$L_r(x, \omega_r) \;=\; \int_{\mathcal S^2_+(x)} f(x,\omega,\omega_r)\,L_i(x,\omega)\,\cos\vartheta\,d\omega,$$

and then denoises the resulting shaded image (Schied et al., 23 Jul 2025). This directly filters the BSDF-weighted signal $f \cdot L_i$, making noise statistics strongly material-dependent and increasing the risk of blurring high-frequency albedo, normal maps, and sharp specular structure (Schied et al., 23 Jul 2025).

The MAD alternative is to move denoising before material shading. The method projects incident radiance $L_i(x,\omega)$ onto a fixed, material-independent set of directional functions, yielding a vector of projected irradiance coefficients $\mu$. It then denoises $\mu$ with a U-Net using geometry buffers such as normals and depth, but no material parameters, and finally reconstructs shaded radiance with a small material decoder $M_\theta$ (Schied et al., 23 Jul 2025).

The projected irradiance is defined as

$$\mu^* (x,\omega_r) \;=\; \int_{\mathcal S^2_+(x)} (\omega,\omega_r)\,L_i(x,\omega)\,\cos\vartheta\,d\omega \;\in \mathbb{R}^{3d},$$

with a Monte Carlo estimator

$$\mu := \left[ \frac{1}{n} \sum_{i=0}^{n-1} (\omega_i,\omega_r)\,R_i \;\Big|\; \frac{1}{n} \sum_{i=0}^{n-1} (\omega_i,\omega_r)\,G_i \;\Big|\; \frac{1}{n} \sum_{i=0}^{n-1} (\omega_i,\omega_r)\,B_i \right] \in \mathbb{R}^{3d}.$$

A crucial design choice is BSDF stealing: the Monte Carlo estimator omits the BSDF factor $f$, so $\mu$ contains incident radiance shaped by geometry, visibility, and next event estimation, but not by material reflectance (Schied et al., 23 Jul 2025).

This separation is the basis of material agnosticism in the rendering sense. The denoiser operates on a field that is meant to be universal across materials, while the BSDF-dependent stage is deferred to a later, inexpensive decoder. The practical significance is that fine material detail and textures are applied after denoising, rather than being subjected to spatial filtering in the already-shaded image (Schied et al., 23 Jul 2025).

3. Neural integral operators and the rendering architecture

The rendering formulation interprets light transport as a parametric integral and approximates the corresponding operator with a neural network (Schied et al., 23 Jul 2025). Instead of directly regressing shaded RGB from noisy samples, the method approximates

$f \cdot L_i$0

where $f \cdot L_i$1 is a neural integral operator acting on the projected irradiance representation (Schied et al., 23 Jul 2025).

The full pipeline has three components: irradiance projection, denoising in projected-irradiance space, and material decoding. The denoiser $f \cdot L_i$2 is a U-Net variant with approximately 9M weights, using only one convolution per layer, decoder-side $f \cdot L_i$3 convolutions implemented as ResNet blocks with 16 neurons each, and ReLU activations. The paper states that the denoiser U-Net can be implemented without bias terms and normalizations, which makes it invariant to positive input scaling (Schied et al., 23 Jul 2025).

Because projected irradiance is high dynamic range, the method applies two invertible transformations before denoising: channel-wise normalization and local-average-based log compression. The normalized red-channel input is

$f \cdot L_i$4

and the compressed red channel is

$f \cdot L_i$5

with analogous formulas for green and blue (Schied et al., 23 Jul 2025). The blurred irradiance $f \cdot L_i$6 is obtained by applying an a-trous wavelet with a $f \cdot L_i$7 Gaussian kernel for 6 iterations, intentionally without edge stopping (Schied et al., 23 Jul 2025).

The denoised projected irradiance is restored as

$f \cdot L_i$8

where $f \cdot L_i$9 is the view-space normal and $L_i(x,\omega)$0 is a range-compressed depth cue (Schied et al., 23 Jul 2025).

Material dependence is then reintroduced by a very small MLP-like decoder. For $L_i(x,\omega)$1, the material network takes input dimension $L_i(x,\omega)$2, uses 6 residual blocks with 16 neurons each and ELU activations, and has about 3,754 weights (Schied et al., 23 Jul 2025). It outputs unnormalized per-channel weights and a scalar intensity:

$L_i(x,\omega)$3

followed by a softmax on each color channel and a convex-combination shading rule such as

$L_i(x,\omega)$4

with analogous expressions for green and blue (Schied et al., 23 Jul 2025).

The rendering paper makes the material-agnostic property explicit in three ways: the denoiser does not consume material parameters, the projected irradiance is computed with BSDF stealing, and denoiser training randomizes material parameters per pixel while passing supervision through the material decoder (Schied et al., 23 Jul 2025).

4. Manifold Attracted Diffusion as material-agnostic denoising

“MAD: Manifold Attracted Diffusion” addresses a different setting: score-based diffusion models trained on noisy data (Elbrächter et al., 29 Sep 2025). Standard score-based diffusion assumes a forward process

$L_i(x,\omega)$5

with densities

$L_i(x,\omega)$6

and sampling via the probability flow ODE

$L_i(x,\omega)$7

where $L_i(x,\omega)$8 is the score (Elbrächter et al., 29 Sep 2025).

The problem considered is that the observed data come from a noisy distribution $L_i(x,\omega)$9, where $\mu$0 is an unknown clean data distribution. A diffusion model trained on samples from $\mu$1 will reproduce the noisy distribution, not the clean one. MAD therefore modifies inference so that sampling moves from the noisy manifold back toward the clean manifold, despite training only on noisy data (Elbrächter et al., 29 Sep 2025).

The geometric motivation is the manifold hypothesis: clean data lie near a low-dimensional manifold $\mu$2, and noise is primarily off-manifold, in directions normal to $\mu$3, with small magnitude. On-manifold directions correspond to meaningful variation and typically exhibit larger variance, whereas off-manifold directions correspond mostly to noise and have small variance (Elbrächter et al., 29 Sep 2025). The method’s central claim is that a denoising dynamic can be obtained if small-variance directions are treated differently from large-variance directions.

This is realized through the extended score. For densities with smooth positive density, the extended score coincides with the standard score. For singular measures such as Dirac deltas or lower-dimensional manifolds, it remains well-defined and behaves as if small-variance distributions had been collapsed toward point masses (Elbrächter et al., 29 Sep 2025). The paper states that for the Dirac delta at 0,

$\mu$4

and for a degenerate Gaussian supported on a subspace one obtains

$\mu$5

The interpretation given is that along manifold directions one gets the usual score, while orthogonal directions are pulled linearly back toward the manifold “like a spring” (Elbrächter et al., 29 Sep 2025).

In toy examples, this yields a soft-thresholding effect: small-variance Gaussians behave almost like Diracs and are aggressively shrunk, while large-variance Gaussians behave similarly to the standard score and preserve their variance (Elbrächter et al., 29 Sep 2025). In high-dimensional data, the intended consequence is that off-manifold noise is suppressed while on-manifold content is retained.

5. Inference dynamics and empirical behavior in diffusion MAD

The diffusion paper derives a modified Euler update for sampling. Standard probability-flow sampling uses

$\mu$6

MAD instead introduces an effective noise level

$\mu$7

constrains $\mu$8, and parameterizes

$\mu$9

so that $\mu$0 vanishes together with $\mu$1 (Elbrächter et al., 29 Sep 2025).

The discrete-time update is

$\mu$2

where $\mu$3, $\mu$4 is a finite-difference approximation of $\mu$5,

$\mu$6

and

$\mu$7

is a correction factor chosen so that the update reproduces the standard Dirac behavior in a special case (Elbrächter et al., 29 Sep 2025). The practical approximation of the extended score requires one extra forward pass per time step and therefore has roughly 2× the cost of standard sampling (Elbrächter et al., 29 Sep 2025).

The method is described as “material-agnostic” because it uses no domain-specific prior, works with any pretrained score model that supports a continuous range of $\mu$8, and does not assume Gaussian corruption in the original data, a known noise level of the corruption, or a known manifold dimension or parametrization (Elbrächter et al., 29 Sep 2025). The stated assumptions are instead geometric: the manifold hypothesis should approximately hold, the corruption should amplify off-manifold variance without destroying manifold structure, and the score model should accurately approximate the score field of the noisy distribution (Elbrächter et al., 29 Sep 2025).

The experiments span toy distributions, pretrained EDM models on FFHQ, AFHQv2, and ImageNet, a synthetic denoising task, and cryo-EM data from EMPIAR-11618 (Elbrächter et al., 29 Sep 2025). On synthetic shapes, a DDPM++ model trained only on noisy samples reproduced corruption under standard sampling but produced clean, fully visible shapes under MAD sampling (Elbrächter et al., 29 Sep 2025). On cryo-EM data, standard sampling generated images resembling the noisy training data, while MAD sampling produced images with shapes closely resembling 3D structures reconstructed in prior specialized cryo-EM pipelines, although no clean references were used in training (Elbrächter et al., 29 Sep 2025).

A plausible implication is that diffusion-based MAD and rendering-based MAD instantiate the same abstract strategy in different computational regimes: both alter the operator applied at inference time so that denoising occurs in a representation where nuisance variation is more separable from intrinsic structure.

6. Training protocols, assumptions, and computational trade-offs

The rendering and diffusion formulations differ sharply in training protocol. The rendering pipeline separates training into two phases: an offline supervised training stage for the material decoder $\mu$9, and a Noise2Noise-style training stage for the denoiser $M_\theta$0 with $M_\theta$1 fixed (Schied et al., 23 Jul 2025). The material decoder is trained procedurally by sampling BSDF parameters uniformly, sampling 4 incident directions and one outgoing direction, assigning random RGB radiances in $M_\theta$2, and evaluating the reference Disney BSDF without Monte Carlo noise (Schied et al., 23 Jul 2025). To improve coverage of highlights, incident directions are sampled uniformly with 95% probability and via BSDF importance sampling with 5% probability (Schied et al., 23 Jul 2025).

The denoiser is trained on real scene geometry using two independent noisy projected irradiances $M_\theta$3 and $M_\theta$4 generated from the same geometry and the same randomly assigned BSDF parameters, but with different Monte Carlo sampling. The primary loss is a relative MSE defined on shaded RGB after passing through the material decoder, with a detached denominator used for normalization (Schied et al., 23 Jul 2025). The paper also describes an alternative training mode defined directly in projected space, but reports that training with the material decoder in the loop yields better results, while a purely projected-space loss “did not yield usable results” (Schied et al., 23 Jul 2025).

By contrast, the diffusion formulation requires no modification to training: a standard EDM-style score-based model, such as DDPM++, is trained on the available noisy data with the standard diffusion loss, and MAD affects only the sampling dynamics (Elbrächter et al., 29 Sep 2025). This separation between training and inference is central to the diffusion paper’s claim that denoising beyond the training corruption can be achieved at sampling time alone (Elbrächter et al., 29 Sep 2025).

The two approaches also differ in computational profile. Rendering MAD is designed for real-time operation, requires data from only a single frame, and is described as complementary to existing denoisers and temporal anti-aliasing (Schied et al., 23 Jul 2025). Diffusion MAD, in contrast, incurs about double the cost of standard probability-flow sampling because of the extra score evaluation used to approximate the derivative with respect to noise scale (Elbrächter et al., 29 Sep 2025).

7. Limitations, related directions, and conceptual significance

Both MAD formulations are explicit about failure modes. In rendering, the limitations include scintillation and temporal artifacts under extremely low sample counts and aggressive temporal reuse, the absence of support for depth of field or motion blur, restriction to isotropic BSDFs, and reliance on a single BSDF model, namely the Disney principled BSDF as implemented in UE4’s DefaultLit material (Schied et al., 23 Jul 2025). The finite-dimensional projected representation also limits the accuracy of the neural integral operator for high-frequency angular phenomena such as very sharp specular lobes or caustics (Schied et al., 23 Jul 2025).

In diffusion, the primary limitations are hyperparameter sensitivity, incomplete theoretical understanding outside simplified settings, dependence on the quality of the base score model, and the absence of explicit estimation of the clean versus noisy distributions (Elbrächter et al., 29 Sep 2025). The paper notes that overly aggressive parameters can overshrink images, remove important details, or even change semantic content, whereas weak parameters may leave the corruption largely untouched (Elbrächter et al., 29 Sep 2025).

A related but distinct line of work is “Learning to Recorrupt: Noise Distribution Agnostic Self-Supervised Image Denoising” (Monroy et al., 26 Mar 2026). That paper presents a noise distribution-agnostic denoising technique that eliminates the need for knowledge of the noise distribution and introduces a learnable monotonic neural network trained through a min-max saddle-point objective (Monroy et al., 26 Mar 2026). Although it is not presented as MAD, it occupies adjacent conceptual territory: denoising without explicit access to a parametric corruption model. This suggests that “agnostic” denoising has become a broader research theme extending across rendering, diffusion, and self-supervised image restoration.

Taken together, the literature shows that Material-Agnostic Denoising is not a single algorithm but a family of strategies for restructuring the denoising problem around an intermediate representation. In rendering, that representation is projected irradiance prior to BSDF application (Schied et al., 23 Jul 2025). In diffusion, it is the geometry of the learned score field near a low-dimensional manifold (Elbrächter et al., 29 Sep 2025). In adjacent self-supervised denoising work, it is a learned recorruption process that avoids committing to a known noise distribution (Monroy et al., 26 Mar 2026). The common methodological claim is that denoising becomes more robust when the operator acts on a representation where material dependence, off-manifold variation, or unknown corruption structure has been factored out rather than directly filtered in the final observation space.