Wavelet-Based Denoising Diffusion

• Wavelet-based denoising diffusion is a multiresolution method that integrates wavelet transforms with diffusion processes to clearly separate low- and high-frequency details.
• It employs various formulations—including direct wavelet-space, wavelet-residual, and hybrid spectral diffusion—to boost efficiency and fidelity in image restoration tasks.
• Architectural innovations use wavelet information as conditioning or supervisory signals, resulting in faster sampling and improved performance in applications like super-resolution and medical imaging.

Wavelet-based denoising diffusion denotes a family of methods that couple wavelet representations with denoising diffusion processes or diffusion-inspired denoising operators. In contemporary generative modeling, the diffusion state may be the full wavelet coefficient tensor, a wavelet-domain residual, a selected low-frequency subspace, or a hybrid spectral state that mixes wavelet and Fourier components; in other variants, wavelets act as conditioning priors, architectural side information, or loss-space constraints rather than as the diffusion state itself (Moser et al., 2023, Huang et al., 2023, Yuan et al., 2023, Kiruluta et al., 4 Apr 2025). Across these variants, the recurring premise is that wavelet decompositions make low- and high-frequency structure explicit, preserve spatial localization, and often reduce the spatial size of the tensors processed at each denoising step. Earlier literature also uses related terminology in a different sense, where wavelet denoising is applied to statistical diffusion processes or is derived from nonlinear diffusion equations rather than from DDPM-style generative models (Hoffmann et al., 2010, Alt et al., 2019).

1. Historical and conceptual lineage

Wavelet-based denoising diffusion has two distinct lineages. One lineage predates denoising diffusion probabilistic models and arises from the connection between nonlinear diffusion PDEs and wavelet shrinkage. In "Learning a Generic Adaptive Wavelet Shrinkage Function for Denoising," one explicit time step of nonlinear diffusion is treated as equivalent to a coupled Haar wavelet shrinkage rule, with the shrinkage written as

$S_\theta\!\left(\!\!\colvec{3}{w_x}{w_y}{w_{xy}\!\!\right) = \Big(1 - g\big(w_x^2 + w_y^2 + 2\, w_{xy}^2\big)\Big) \colvec{3}{w_x}{w_y}{w_{xy},$

and with a Forward-and-Backward diffusivity that can both shrink and amplify coefficients depending on scale and noise level (Alt et al., 2019). In that formulation, diffusion is not a generative Markov chain; it is a nonlinear smoothing-and-enhancement mechanism expressed in wavelet coordinates.

A second pre-DDPM lineage comes from statistical estimation of stochastic diffusion processes under noisy observations. In "Adaptive wavelet estimation of the diffusion coefficient under additive error measurements," the observed process is

$Z_{j,n} = X_{j/n} + \epsilon_{j,n},$

and the method combines pre-averaging, bias correction, and hard thresholding of estimated wavelet coefficients to reconstruct the diffusion coefficient path $\sigma_t^2$ under Besov smoothness assumptions (Hoffmann et al., 2010). Here the phrase concerns wavelet denoising for a diffusion coefficient, not a denoising diffusion model.

A persistent source of confusion is therefore terminological. In the classical literature, “diffusion” often refers to a stochastic process or a nonlinear diffusion PDE; in recent generative modeling, it refers to a forward noising chain and learned reverse denoising chain. The modern literature inherits the older intuition that wavelets isolate sparse, localized high-frequency structure, but repurposes that intuition for DDPM- and DDIM-style generative inference.

2. Core formulations in wavelet and spectral domains

Modern wavelet-based denoising diffusion methods differ mainly in what variable is diffused. A practical taxonomy is shown below.

Formulation	Representative papers	Diffusion variable
Direct wavelet-space diffusion	SFUNet, WDNO	Full wavelet coefficient tensor
Wavelet-residual diffusion	DiWa	Residual between target coefficients and an initial predictor
Low-frequency-only diffusion	WaveDM, DiffLL	Low-frequency or LL coefficients only
Hybrid spectral diffusion	Wavelet-Fourier-Diffusion	Fourier low band plus wavelet high bands
Wavelet-conditioned latent diffusion	ResQu, TIR-Diffusion	Latent variable; wavelets enter as condition or loss

The most literal construction replaces pixels with wavelet coefficients and applies an otherwise standard DDPM. In "Spatial-Frequency U-Net for Denoising Diffusion Probabilistic Models," the image $x_0$ is transformed to $u_0=W(x_0)$, the forward process is

$$u_t = \sqrt{\bar{\alpha}_t}\, u_0 + \sqrt{1 - \bar{\alpha}_t} \, \epsilon,$$

and reverse diffusion predicts noise on the 5D tensor $$u_t \in \mathbb{R}^{B \times C \times F \times H/2 \times W/2}$$, where the explicit frequency axis $F=4$ corresponds to the LL, LH, HL, and HH subbands (Yuan et al., 2023). WDNO extends the same idea to PDE trajectories, applying diffusion to wavelet coefficients of whole trajectories rather than to images, with DDPM/DDIM mechanics operating directly in wavelet space (Hu et al., 2024).

A second formulation diffuses a wavelet-domain residual rather than the coefficients themselves. DiWa computes

$\check{\mathbf{y}_\text{init} = g_\theta(\check{\mathbf{x}), \qquad \Delta\check{\mathbf{y} = \check{\mathbf{y} - g_\theta(\check{\mathbf{x}),$

and then applies the forward process to that residual: $\mathbf{z}_t = \sqrt{\gamma_t} \left(\check{\mathbf{y} - g_\theta(\check{\mathbf{x})\right) + \sqrt{1-\gamma_t}\,\varepsilon_t.$ The final reconstruction is

$Z_{j,n} = X_{j/n} + \epsilon_{j,n},$0

This residualization makes the diffusion target closer to missing high-frequency detail and correction terms than to the entire HR image (Moser et al., 2023).

A third formulation diffuses only selected low-frequency content and delegates high-frequency synthesis to separate modules. WaveDM applies a 2-level 2D Full Wavelet Packet Transform, separates the first three low-frequency bands from the remaining forty-five high-frequency bands, diffuses only the low-frequency spectrum $Z_{j,n} = X_{j/n} + \epsilon_{j,n},$1, and reconstructs the high-frequency spectrum with a High-Frequency Refinement Module (Huang et al., 2023). DiffLL adopts an analogous division in low-light enhancement: diffusion is performed on the coarsest LL coefficient $Z_{j,n} = X_{j/n} + \epsilon_{j,n},$2, while scale-wise high-frequency subbands are reconstructed by dedicated high-frequency modules (Jiang et al., 2023).

A fourth formulation is hybrid spectral diffusion. "A Hybrid Wavelet-Fourier Method for Next-Generation Conditional Diffusion Models" decomposes an image into a wavelet low-frequency band and wavelet high-frequency bands, then applies a Fourier transform only to the low-frequency band. The state is $Z_{j,n} = X_{j/n} + \epsilon_{j,n},$3, the forward process uses radial masking in the Fourier low band and Gaussian noise in wavelet detail bands, and the reverse process reconstructs both before applying inverse Fourier and inverse wavelet transforms (Kiruluta et al., 4 Apr 2025). This suggests that “wavelet-based” diffusion need not be purely wavelet-domain; it can denote a broader class of frequency-aware denoising chains.

3. Architectural patterns and conditioning mechanisms

Once diffusion is moved into wavelet-aware representations, architecture becomes decisive. SFUNet shows that simply concatenating wavelet subbands as ordinary channels is inadequate: its direct wavelet baseline with standard spatial convolutions performs poorly, whereas explicit spatial-frequency convolution and separable spatial/frequency attention substantially improve FID on high-resolution datasets (Yuan et al., 2023). The central architectural lesson is that the representation’s frequency semantics must be made visible to the denoiser.

WaveDM and DiffLL instead encode a strong asymmetry between low and high frequencies. WaveDM uses a noise estimation network for low-frequency diffusion, an HFRM for high-frequency prediction, and an Efficient Conditional Sampling procedure that stops at an intermediate timestep $Z_{j,n} = X_{j/n} + \epsilon_{j,n},$4 and directly reconstructs the clean low-frequency spectrum from $Z_{j,n} = X_{j/n} + \epsilon_{j,n},$5 (Huang et al., 2023). DiffLL likewise trains both forward diffusion and denoising in the training phase, adding a content-consistency term on the restored low-frequency band to suppress “chaotic content” and reduce inference-time randomness (Jiang et al., 2023). In both systems, wavelets are not merely an efficiency device; they induce a model factorization in which global illumination or coarse structure and fine detail are treated by different mechanisms.

Other systems keep diffusion in pixel or latent space but inject wavelet information through conditioning paths. DDRF introduces wavelet modulation and style transfer modulation into a conditional diffusion U-Net for remote sensing fusion, using wavelet features as fine-grained guidance in the decoder (Cao et al., 2023). ResQu conditions Stable Diffusion with quaternion wavelet embeddings produced by QUAVE and a quaternion wavelet- and time-aware encoder whose features modulate latent U-Net activations via Spatial Feature Transform layers (Sigillo et al., 1 May 2025). WCC-Net uses a frozen pretrained 3D DDPM backbone for PET denoising and injects wavelet-based structural guidance through a ControlNet-style branch and zero-initialized convolutions, with low-frequency wavelet coefficients acting as structural priors (Jing et al., 11 Jan 2026).

A related but distinct design uses wavelets in the objective rather than in the diffusion state or condition. TIR-Diffusion fine-tunes a pretrained Stable Diffusion v1-4 U-Net for thermal infrared denoising with a combined latent-space and wavelet-domain objective, where a 2-level DWT or DTCWT is applied to reconstructed and clean images and the loss is imposed on high-frequency subbands (Rhee et al., 30 Jul 2025). A common misconception is therefore that wavelet-based denoising diffusion always means “diffusion on wavelet coefficients.” The literature shows at least three legitimate interpretations: wavelets as the state space, wavelets as a conditioning prior, and wavelets as a supervisory domain.

4. Task domains and representative applications

Single-image super-resolution is the most extensively developed use case. DiWa applies wavelet-residual diffusion to SISR and explicitly reports stronger PSNR, SSIM, and LPIPS than SR3 and SRDiff while using fewer parameters (Moser et al., 2023). ResQu uses quaternion wavelet conditioning within latent diffusion for super-resolution, emphasizing phase- and orientation-aware structure in the conditioning path rather than direct diffusion on wavelet coefficients (Sigillo et al., 1 May 2025).

General image restoration supplies a second major application cluster. WaveDM addresses raindrop removal, rain streak removal, dehazing, defocus deblurring, demoiréing, and denoising with a common wavelet-based framework (Huang et al., 2023). DiffLL specializes the same general principle to low-light image enhancement, treating the coarse wavelet band as the diffusion target and using separate high-frequency restoration modules (Jiang et al., 2023). TIR-Diffusion adapts pretrained latent diffusion to thermal infrared denoising by supervising the reconstruction in DWT or DTCWT space (Rhee et al., 30 Jul 2025).

Medical imaging has motivated explicitly structural variants. WCC-Net applies a single-level 3D Haar DWT to low-dose PET patches, uses low-frequency wavelet priors to guide a frozen 3D diffusion backbone, and frames the method as decoupling anatomical structure from noise (Jing et al., 11 Jan 2026). The same motif appears in wavelet-conditioned frameworks for other high-frequency-sensitive domains, where low-pass bands provide stable anatomy and high-pass bands are treated cautiously because they may be noise-dominated.

Compression and generative coding provide a different interpretation of denoising diffusion. UGDiff performs wavelet conditional diffusion only on high-frequency coefficients, predicts those coefficients from reconstructed low-frequency content, and then compresses the prediction residual with an uncertainty-weighted rate-distortion objective (Song et al., 2024). A plausible implication is that wavelet-based denoising diffusion can function as a generative prior within a larger transmission or inverse-problem pipeline rather than as a standalone restorer.

The paradigm is not confined to 2D images. WDNO performs diffusion-based generative modeling in the wavelet domain for PDE trajectories and control signals, explicitly targeting abrupt changes and multi-resolution generalization (Hu et al., 2024). WaveletDiff trains diffusion models directly on multilevel wavelet coefficients for multivariate time-series generation and combines dedicated transformers for each decomposition level with cross-level attention mechanisms and energy preservation constraints (Wang et al., 13 Oct 2025). These examples suggest that wavelet-based denoising diffusion is best understood as a multiresolution generative principle rather than as a purely image-specific technique.

5. Empirical behavior: fidelity, perceptual quality, and efficiency

Empirically, the literature repeatedly reports gains in either efficiency, fidelity, or both, but the gains are task-dependent. DiWa reports, on face SR, PSNR 23.34 versus 23.04 and SSIM 0.67 versus 0.65 relative to SR3, while using approximately 92M parameters rather than 550M; on general SR, it reports PSNR 28.09 and LPIPS 0.104 on DIV2K validation, outperforming SRDiff in PSNR and LPIPS while using 9.3M parameters rather than 12M (Moser et al., 2023). The ablation in the same paper also shows that DWT and the initial predictor individually improve performance, with their combination giving the highest PSNR on Set5.

WaveDM emphasizes that wavelet-domain modeling changes not only quality but sampling regime. It reports that its Efficient Conditional Sampling reduces the number of total sampling steps to around 5 and that the method is over $Z_{j,n} = X_{j/n} + \epsilon_{j,n},$6 faster than existing image restoration methods using vanilla diffusion models, while retaining state-of-the-art performance across twelve restoration benchmarks (Huang et al., 2023). On the Raindrop dataset, the paper gives 32.19 dB for ECS versus 30.17 dB for DDIM at a 4-step trajectory, and for patch-based diffusion it reports approximately 301 seconds per image versus approximately 0.30 seconds for WaveDM.

For unconditional generation, SFUNet demonstrates that wavelet-space diffusion is not uniformly beneficial but can be highly effective at higher resolutions. On FFHQ-256 it reports FID 7.12 for SFUNet versus 13.53 for the pixel-space DDPM U-Net, and on LSUN-Bedroom-256 it reports FID 3.88 versus 4.89, with better precision and recall as well (Yuan et al., 2023). The same paper also reports that naive wavelet concatenation is markedly inferior, reinforcing that the gain is architectural rather than merely representational.

Hybrid and conditional spectral models report related gains. Wavelet-Fourier-Diffusion gives FID 2.9 versus 3.4 on CIFAR-10, 4.8 versus 7.6 on CelebA-HQ 64×64, and 16.7 versus 18.2 on an ImageNet subset 128×128 relative to pixel diffusion baselines, with corresponding improvements in Inception Score (Kiruluta et al., 4 Apr 2025). WCC-Net reports, on the internal 1/20-dose PET test set, PSNR 43.594 versus 42.380, SSIM 0.984 versus 0.976, GMSD 0.011 versus 0.014, and NMAE 0.111 versus 0.117 relative to a 3D DDPM baseline, with similarly improved performance on unseen 1/50-dose and 1/4-dose settings (Jing et al., 11 Jan 2026).

Wavelet-based supervision in latent diffusion also yields strong task-specific results. TIR-Diffusion reports PSNR 27.97 dB and SSIM 0.8382 for its DTCWT variant, 26.49 dB and 0.8594 for its Biorthogonal DWT variant, and for the cascaded DTCWT model LPIPS 0.1529 and FID 38.13 (Rhee et al., 30 Jul 2025). UGDiff, in compression, reports BD-rate improvements of $Z_{j,n} = X_{j/n} + \epsilon_{j,n},$7 on Kodak and $Z_{j,n} = X_{j/n} + \epsilon_{j,n},$8 on Tecnick relative to VVC, while attributing part of the gain to wavelet conditional diffusion on high-frequency coefficients (Song et al., 2024).

6. Limitations, misconceptions, and open directions

The literature is explicit that wavelet-based denoising diffusion is not uniformly superior. SFUNet reports that on CIFAR-10 at $Z_{j,n} = X_{j/n} + \epsilon_{j,n},$9, the pixel-space DDPM remains best, with FID 3.53 for the pixel model versus 4.88 for SFUNet and 9.29 for naive wavelet concatenation (Yuan et al., 2023). This directly refutes the common assumption that wavelet transforms are automatically beneficial for all resolutions and all architectures.

Another recurring limitation is incomplete recovery of the finest details. DiWa notes that very fine details such as pores, pimples, or tiny objects may remain imperfect, and also reports slight localized Gaussian noise artifacts in some general-SR samples (Moser et al., 2023). WaveDM states that training cost remains large-scale and that wavelet choice and decomposition level materially affect the balance between efficiency and fidelity (Huang et al., 2023). WCC-Net observes that a fixed Haar wavelet and single-level decomposition may not fully capture complex anatomical structures or fine directional textures (Jing et al., 11 Jan 2026). TIR-Diffusion likewise notes that balancing the latent and wavelet losses is crucial and that directional transforms such as DTCWT introduce additional implementation complexity (Rhee et al., 30 Jul 2025).

A further limitation is theoretical heterogeneity. Some models retain the full DDPM formalism in wavelet space, as in SFUNet and DiWa; others adopt procedural spectral corruption without a closed-form Gaussian Markov chain, as in Wavelet-Fourier-Diffusion (Kiruluta et al., 4 Apr 2025). This suggests that “wavelet-based denoising diffusion” is a methodological family rather than a single mathematically unified model class.

The current research directions are correspondingly diverse. DiWa explicitly points to multi-level DWT, latent diffusion, more powerful wavelet CNN initial predictors, EMA, improved noise schedules, DDIM, progressive distillation, and band-specific attention mechanisms (Moser et al., 2023). Wavelet-Fourier-Diffusion proposes adaptive masking schedules, more advanced wavelets such as Daubechies, directional transforms such as curvelets or shearlets, and latent-space variants (Kiruluta et al., 4 Apr 2025). WCC-Net highlights other modalities, especially CT, MRI, and ultrasound, as natural extensions of wavelet-conditioned diffusion priors (Jing et al., 11 Jan 2026). WaveletDiff points toward conditional generation, denoising, forecasting, and imputation directly in multilevel wavelet spaces with cross-level attention and energy-preserving constraints (Wang et al., 13 Oct 2025).

Taken together, these works indicate that wavelet-based denoising diffusion is best viewed as a multiresolution design doctrine: encode coarse and fine structure explicitly, let the denoiser reason in a domain where localization and frequency separation are native, and decide case by case whether wavelets should define the stochastic state, the conditioning signal, the supervisory loss, or all three simultaneously.