Wavelet Diffusion Models Overview

• Wavelet Diffusion Models are generative frameworks that perform diffusion in the wavelet domain by separating low-frequency structure from high-frequency detail.
• They use discrete wavelet transforms to decompose signals into multiple subbands, enabling task-specific processing and significantly faster inference.
• These models are versatile, achieving impressive efficiency and quality in applications ranging from image restoration and generation to speech synthesis and 3D medical imaging.

Searching arXiv for recent and foundational papers on Wavelet Diffusion Models. Wavelet Diffusion Model (WDM) denotes a family of diffusion-based generative frameworks that operate in a wavelet-domain representation rather than directly in the original signal domain. Across image generation, image restoration, speech synthesis, medical image synthesis, time-series generation, geoscience downscaling, and motion prediction, the defining idea is to apply a discrete wavelet transform or a wavelet packet transform, perform diffusion and denoising on the resulting multiscale coefficients, and reconstruct the final sample with the corresponding inverse transform (Phung et al., 2022, Friedrich et al., 2024, Wang et al., 13 Oct 2025). In the broad literature, WDMs are motivated by the observation that wavelet decompositions separate low-frequency structure from high-frequency detail while reducing spatial resolution and preserving invertibility; this enables lower computational cost, explicit multiscale control, and task-specific handling of distinct frequency bands (Phung et al., 2022, Huang et al., 2023, Zhang et al., 2024).

1. Conceptual definition and scope

The core principle of a WDM is that the diffusion process is transferred from the native data representation to a wavelet representation. In two-dimensional image settings, a discrete wavelet transform decomposes an image into low-frequency and high-frequency subbands such as $LL$, $LH$, $HL$, and $HH$, each at reduced spatial size (Phung et al., 2022, Yi et al., 2 Jul 2025). In three-dimensional medical imaging, a 3D discrete wavelet transform produces eight sub-band coefficient volumes, and the diffusion model operates on their concatenation rather than on the full-resolution voxel grid (Friedrich et al., 2024). In one-dimensional speech and time-series settings, the transformed representation may consist of approximation and detail coefficients or a multilevel wavelet hierarchy processed by dedicated denoisers (Zhang et al., 2024, Wang et al., 13 Oct 2025).

Within this general pattern, different papers instantiate the concept differently. Some models apply diffusion to all wavelet bands jointly and reconstruct the sample by inverse wavelet transform (Phung et al., 2022, Friedrich et al., 2024). Others diffuse only selected components, such as low-frequency bands in image restoration or average coefficients in low-light enhancement, while estimating high-frequency components with dedicated auxiliary modules (Huang et al., 2023, Jiang et al., 2023). Still others impose conditioning in wavelet space, for example for super-resolution, image fusion, or ground-motion generation, where wavelet coefficients are combined with cross-attention, conditional encoders, or posterior sampling mechanisms (Yi et al., 2 Jul 2025, Cao et al., 2023, Ding et al., 7 Feb 2026).

The term also appears in narrower, application-specific senses. In speech synthesis, “Wavelet Diffusion Model” refers to a method that redirects the generative target from waveform space to the wavelet domain while keeping the core DDPM architecture essentially unchanged (Zhang et al., 2024). In intellectual-property protection, WDM denotes a “Watermark Diffusion Model,” where the acronym refers not to wavelet processing but to a watermarking framework built around a Watermark Diffusion Process (Peng et al., 2023). This is a terminological ambiguity rather than a methodological continuity. In the wavelet-processing literature itself, however, WDM usually denotes a diffusion model whose state variables are wavelet coefficients or closely related multiscale wavelet representations.

2. Wavelet-domain representations and reconstruction

A recurring property of WDMs is the use of fixed, invertible wavelet operators as a front-end and back-end around the diffusion model. In the 2D image-generation formulation, an image $x \in \mathbb{R}^{3 \times H \times W}$ is transformed by a discrete wavelet transform into four subbands and concatenated into $$y \in \mathbb{R}^{12 \times \frac{H}{2} \times \frac{W}{2}}$$, after which diffusion is applied to $y$ rather than to $x$ (Phung et al., 2022). The final sample is reconstructed as $x_0 = \text{IWT}(y_0)$ (Phung et al., 2022). For 3D medical images, the volume $y$ is mapped to

$LH$0

and the synthesized image is recovered as $LH$1 (Friedrich et al., 2024).

Several formulations emphasize exact or near-exact reconstruction. The ground-motion model based on wavelet packets uses a 7-level Daubechies-6 wavelet packet transform, mapping each record $LH$2 of length 16,384 to a $LH$3 matrix and reconstructing the waveform through orthogonal filter banks with near machine-precision reconstruction and no phase retrieval requirement (Ding et al., 7 Feb 2026). The image restoration model WaveDM uses a 2-level 2D Full Wavelet Packet Transform with Haar basis, yielding 48 frequency bands at one-sixteenth the spatial size of the input (Huang et al., 2023). The speech WDM applies a discrete wavelet transform to convert a raw signal $LH$4 into concatenated low- and high-frequency coefficients $LH$5, effectively halving the temporal length while doubling the channel count (Zhang et al., 2024).

In multilevel time-series generation, WaveletDiff uses a multilevel discrete wavelet transform to produce detail coefficients $LH$6 and a coarsest approximation $LH$7, with the number of levels set by

$LH$8

(Wang et al., 13 Oct 2025). In human motion prediction, MotionWavelet applies a 2D discrete wavelet transform over temporal and joint axes, forming a “Wavelet Manifold” by concatenating low- and high-frequency subbands, followed by inverse DWT during generation (Feng et al., 2024).

A plausible implication is that invertible wavelet front-ends give WDMs a different trade-off from latent diffusion models based on learned autoencoders. This suggestion is explicitly supported in WaveDM, which contrasts wavelet transforms with VAE-based latent diffusion by noting the imperfect reconstruction upper bound of the latter and the lossless invertibility of the former (Huang et al., 2023).

3. Diffusion formulations in wavelet space

Although the wavelet representation changes the state space, most WDMs preserve standard diffusion-model training principles. In unconditional or conditional DDPM-style formulations, a noisy wavelet representation is sampled as

$LH$9

or equivalently through a Markov chain with Gaussian transitions (Friedrich et al., 2024, Zhang et al., 2024, Jiang et al., 2023). The denoising network is trained either to predict the added noise or to predict the clean wavelet coefficients themselves, depending on the application (Zhang et al., 2024, Friedrich et al., 2024).

The original image-generation WDM frames the model in the DDGAN setting, combining an adversarial loss with a wavelet-space reconstruction loss

$HL$0

and uses the total generator objective

$HL$1

(Phung et al., 2022). That reconstruction term is described as boosting training convergence and frequency fidelity (Phung et al., 2022). By contrast, the 3D medical-image WDM employs a pure mean-squared error on denoised wavelet coefficients,

$HL$2

after the authors observed that predicting noise led to checkerboard artifacts in that setting (Friedrich et al., 2024).

A more elaborate diffusion protocol appears in the ground-motion model, which adopts the Elucidating Diffusion Model (EDM) with preconditioning

$HL$3

and second-order Heun sampling with 25 noise levels, corresponding to 50 network evaluations (Ding et al., 7 Feb 2026). The same work defines the forward noising as

$HL$4

and trains the denoiser using the EDM-weighted loss over normalized wavelet packet coefficients (Ding et al., 7 Feb 2026).

Conditional score-based formulations also appear. The precipitation-downscaling WDM uses a variance-preserving SDE

$HL$5

with conditional score matching in wavelet space, plus a total variation regularizer applied only to the high-frequency detail channels (Yi et al., 2 Jul 2025). The PET reconstruction model places diffusion in wavelet coefficient space and combines it with diffusion posterior sampling, where denoised wavelet coefficients are mapped back to image space to evaluate a measurement-model likelihood gradient, and the resulting gradient is then propagated back to the coefficient representation (Phung-Ngoc et al., 24 May 2025).

This diversity of formulations indicates that “WDM” is not a single diffusion algorithm but a representational strategy: the forward and reverse stochastic processes are defined over wavelet coefficients, while the specific training objective may be adversarial, MSE-based, score-based, posterior-guided, or EDM-preconditioned depending on the task.

4. Architectural patterns

Despite application diversity, several architectural motifs recur.

A first motif is the use of conventional U-Net or U-Net-like denoisers on wavelet tensors. The image-generation WDM applies diffusion on image-level wavelet coefficients and further introduces wavelet-aware downsampling, upsampling, bottleneck, and residual blocks within the network itself (Phung et al., 2022). The 3D medical-image WDM likewise uses a 3D diffusion backbone that simply receives wavelet coefficients as input, with only the first layer adjusted to map eight channels to the base channel count (Friedrich et al., 2024). WaveDM for image restoration conditions a noise-estimation U-Net on a degraded image’s wavelet spectrum and on a high-frequency estimate supplied by a separate refinement module (Huang et al., 2023).

A second motif is architectural separation of low- and high-frequency pathways. WaveDM learns the lowest three spectral bands through diffusion while a High-Frequency Refinement Module directly estimates the remaining 45 bands (Huang et al., 2023). DiffLL performs diffusion only on the highest-scale average coefficient $HL$6, while a High-Frequency Restoration Module reconstructs $HL$7, $HL$8, and $HL$9 subbands using cross-attention between vertical, horizontal, and diagonal details (Jiang et al., 2023). In sparse-view CT reconstruction, SWORD explicitly stages the process as low-frequency generation, high-frequency refinement, and domain transform (Xu et al., 2023). In video deblurring, VD-Diff separates low-frequency restoration through a Wavelet-Aware Dynamic Transformer and high-frequency prior generation through a latent diffusion model (Rao et al., 2024).

A third motif is multilevel or cross-scale interaction. WaveletDiff assigns a dedicated transformer to each wavelet decomposition level and introduces cross-level attention with adaptive gating,

$HH$0

to regulate information exchange across scales (Wang et al., 13 Oct 2025). The ground-motion model uses a Transformer-based conditional encoder with cross-attention injection into a U-Net operating in wavelet packet space, allowing scalar and vector conditions such as spectral ordinates, Arias intensity, temporal parameters, and Husid curves to influence denoising at multiple scales (Ding et al., 7 Feb 2026). The remote-sensing fusion model DDRF injects wavelet coefficients into decoder stages through cross-attention, using low-frequency coefficients from low-resolution multispectral imagery and high-frequency coefficients from panchromatic imagery (Cao et al., 2023).

A fourth motif is the combination of fixed wavelet transforms with learned modules in adjacent spaces. 3D-WLDM for MR-to-CT translation places the main diffusion process in a learned latent space, but augments the encoder-decoder with a Wavelet Residual Module and a Dual Skip Connection Attention mechanism to improve fine-scale reconstruction and preserve anatomical integrity (Zheng et al., 14 Jul 2025). MotionWavelet uses a wavelet latent manifold as the substrate for a diffusion model and adds Wavelet Space Shaping Guidance and Temporal Attention-Based Guidance during denoising (Feng et al., 2024). These models suggest that wavelet-domain processing can be integrated not only at the diffusion state level but also as a structural bias within broader latent or conditional pipelines.

5. Efficiency, scalability, and fidelity

One of the most consistent empirical themes in WDM research is improved efficiency relative to pixel- or waveform-domain diffusion.

The original image-generation WDM reports that on CIFAR-10, CelebA-HQ, LSUN-Church, and STL-10, wavelet decomposition reduces spatial size and enables sampling in as few as 2 or 4 denoising steps, yielding up to 2.5× faster inference than DDGAN and generation times approaching GANs while maintaining competitive FID (Phung et al., 2022). In speech synthesis, redirecting the generative target to wavelet space “double[s] the training and inference speed” of speech DDPMs, with comparable or superior performance and only an input-layer modification (Zhang et al., 2024). WaveDM for image restoration combines wavelet-space processing with Efficient Conditional Sampling, reducing total sampling steps to around 5 and achieving efficiency comparable to traditional one-pass methods and over 100× faster than vanilla diffusion restorers (Huang et al., 2023).

In low-light enhancement, DiffLL reports that wavelet-domain diffusion on average coefficients, together with denoising during training, yields up to 70× faster inference and 3×–5× lower GPU memory than previous diffusion-based methods (Jiang et al., 2023). The precipitation-downscaling WDM reports a 3.5–9× inference speedup over pixel-based diffusion models, with a two-level DWT variant reducing the time for 512 samples to 0m 53.4s compared with 9m 16.5s for EDM and 12m 41.9s for FourierDM (Yi et al., 2 Jul 2025). In 3D medical image synthesis, WDM is described as the only compared method capable of generating high-quality $HH$1 images on a single 40GB GPU, with inference memory of about 7.27GB at that resolution (Friedrich et al., 2024).

Fidelity and quality are also frequent points of comparison. The unconditional 3D medical WDM reports state-of-the-art FID and MS-SSIM relative to recent GANs, diffusion models, and latent diffusion models on BraTS and LIDC-IDRI, and is the only one among the compared methods to reach $HH$2 synthesis (Friedrich et al., 2024). WaveletDiff reports discriminative scores and Context-FID scores that are $HH$3 smaller on average than the second-best baseline across six real-world datasets (Wang et al., 13 Oct 2025). The precipitation-downscaling WDM reports the best MAE, RMSE, PSNR, SSIM, HI-RMSE, and CSI among the stated baselines, together with markedly lower extreme-quantile errors (Yi et al., 2 Jul 2025). In image restoration, WaveDM reports state-of-the-art or competitive PSNR, SSIM, and FID across twelve benchmarks, while remaining orders of magnitude faster than patch-based diffusion (Huang et al., 2023).

These results do not establish a single universal ranking of WDM against all other efficient diffusion paradigms, because the surveyed papers evaluate different tasks, datasets, and baselines. However, taken together, they consistently support the narrower claim that wavelet-domain diffusion offers a favorable quality-efficiency trade-off in settings where multiscale structure and high resolution are central.

6. Domain-specific variants

The WDM label encompasses a heterogeneous set of domain-specific models, each emphasizing a different aspect of wavelet-domain diffusion.

In image generation, the early WDM literature focuses on accelerating full-image synthesis by decomposing both images and intermediate features into frequency bands and handling them adaptively (Phung et al., 2022). In 3D medical synthesis, the same principle is extended to volumetric data, enabling direct whole-volume generation at $HH$4 and $HH$5 resolutions without patching or slicing (Friedrich et al., 2024).

In image restoration, wavelet diffusion is frequently used to separate difficult high-frequency detail recovery from lower-frequency content. WaveDM learns only the low-frequency bands through diffusion, estimates high-frequency bands directly, and introduces Efficient Conditional Sampling that halts DDIM early and directly computes the clean spectrum at an intermediate step (Huang et al., 2023). DiffLL enhances low-light images by diffusing only the highest-level average coefficient and reconstructing directional details through the High-Frequency Restoration Module (Jiang et al., 2023). VD-Diff for video deblurring likewise assigns low-frequency recovery to a transformer and high-frequency prior synthesis to a latent diffusion model (Rao et al., 2024).

In speech and audio-like sequence modeling, the principal benefit reported is computational compression with minimal architectural disruption. The speech WDM simply applies DDPM in the wavelet domain and keeps the rest of the diffusion vocoder intact, thereby halving feature length and approximately doubling training and inference speed (Zhang et al., 2024).

In structured scientific signals, the emphasis shifts from pure generation quality to controllability and uncertainty. The ground-motion framework uses wavelet packet coefficients together with transformer conditioning and EDM-Heun sampling to match response spectra and temporal energy evolution simultaneously, reporting $HH$6 for almost all response-spectrum matches and $HH$7 for all Husid curves in the fully conditioned case (Ding et al., 7 Feb 2026). It also uses conditional diversity sampling for uncertainty quantification (Ding et al., 7 Feb 2026). In precipitation downscaling, WDM is conditioned on coarse-scale fields and uses wavelet detail regularization to recover sharp kilometer-scale precipitation structures while suppressing artifacts (Yi et al., 2 Jul 2025). In PET reconstruction, WDM serves as a prior model inside diffusion posterior sampling to reconstruct activity and attenuation from emission data alone (Phung-Ngoc et al., 24 May 2025).

In time-series and motion modeling, wavelet diffusion is used to capture multi-resolution temporal structure. WaveletDiff decomposes time series across levels and couples level-specific transformers with cross-level attention and Parseval-based energy preservation (Wang et al., 13 Oct 2025). MotionWavelet forms a wavelet manifold over human motion trajectories and constrains denoising through repeated $HH$8 projections, supplemented by Temporal Attention-Based Guidance (Feng et al., 2024).

This breadth suggests that WDM is less a single architecture than a reusable multiscale prior. The wavelet transform supplies a fixed, interpretable decomposition, while the diffusion component is adapted to the statistical and operational needs of the target domain.

7. Interpretive issues, limitations, and future directions

Several limitations and points of caution recur in the literature. First, the acronym “WDM” is not semantically unique. The Watermark Diffusion Model uses WDM to denote a watermarking framework without wavelet-domain processing (Peng et al., 2023). Consequently, not every “WDM” paper contributes to wavelet-based generative modeling, and the term must be interpreted contextually.

Second, WDMs are not uniformly superior on every axis. The 2025 review of efficient diffusion models in medical imaging characterizes WDM as having high efficiency, moderate stability, high convergence, moderate quality, low sampling cost, and high large-image support relative to DDPM and LDM (Abdullah et al., 9 May 2025). The same review notes that WDM’s absolute image quality can be slightly below the best DDPM results on some benchmarks and that implementation requires careful management of subbands and high-frequency details (Abdullah et al., 9 May 2025). This suggests that efficiency gains do not eliminate the usual generative trade-offs involving fidelity, stability, and architectural complexity.

Third, several application papers explicitly report task-specific failure modes. The PET reconstruction framework performs strongly when time-of-flight information is available, but degrades substantially in non-TOF and low-count conditions (Phung-Ngoc et al., 24 May 2025). The watermarking WDM is robust to quantization and mild fine-tuning, but large-scale fine-tuning can remove the watermark through catastrophic forgetting (Peng et al., 2023). DiffLL introduces an unusual training strategy that includes both forward diffusion and denoising during training specifically to reduce chaotic content and diversity in low-light enhancement; this indicates that vanilla conditional diffusion may be too stochastic for some restoration tasks unless additional stabilization is introduced (Jiang et al., 2023).

Future directions are explicit in both surveys and application papers. The medical-imaging review highlights adaptive sampling, better representation learning, automated selection of wavelet bases, hybrid models that combine WDM with GANs, transformers, or latent encoders, better handling of 3D and 4D data, and improved interpretability (Abdullah et al., 9 May 2025). Other papers propose concrete hybrids already moving in that direction: Wavelet-Fourier-Diffusion combines wavelet sub-band decomposition with partial Fourier steps for conditional image generation (Kiruluta et al., 4 Apr 2025); 3D-WLDM combines wavelet residual learning with latent diffusion and structural anchoring (Zheng et al., 14 Jul 2025); and WaveletDiff augments level-wise wavelet diffusion with cross-level transformer attention and Parseval-based constraints (Wang et al., 13 Oct 2025).

A plausible implication is that the historical role of WDM is shifting from simple acceleration to structured inductive bias. Early work emphasizes runtime reduction and dimensionality compression (Phung et al., 2022, Zhang et al., 2024). Later work increasingly uses wavelet-domain diffusion to encode domain knowledge about energy localization, anatomical structure, high-frequency detail, uncertainty, or multi-resolution dependencies (Ding et al., 7 Feb 2026, Zheng et al., 14 Jul 2025, Wang et al., 13 Oct 2025). In that sense, the wavelet transform functions not merely as a computational convenience but as a way to expose a physically or perceptually meaningful state space for diffusion-based generation and reconstruction.