Frequency-Domain Denoising (FDD)

• Frequency-Domain Denoising (FDD) is a family of techniques that uses spectral transforms to isolate structured signals from broadly distributed noise.
• It employs methods such as coefficient thresholding, band truncation, and neural network integration across various domains including graphs, biomedical signals, and turbulent flows.
• Strengths include adaptive, hybrid approaches that leverage domain-specific spectral properties, though performance hinges on matching the transform to the signal's inherent structure.

Searching arXiv for recent and foundational papers on frequency-domain denoising to ground the article in published work. Frequency-Domain Denoising (FDD) denotes a family of denoising strategies that transform data into a frequency-like representation in which the underlying signal is more compact, structured, or coherent than the corruption, suppress the components identified as noise-dominated, and then reconstruct the signal in the original domain. In the literature, the relevant “frequency domain” is not restricted to the Euclidean Fourier transform: it includes the 2D discrete Fourier domain for network adjacency matrices, the Discrete Cosine Transform (DCT) domain for electrocardiograms, the spectral domain of a graph Laplacian for manifold data, spectral graph wavelet bands, wavelet and curvelet subbands, and Spectral Proper Orthogonal Decomposition (SPOD) modes for turbulent flow data (Frost, 31 Jan 2026, Deutsch et al., 2016, Li et al., 20 Nov 2025, Nekkanti et al., 2020). The central recurring premise is that clean structure is localized or energetically concentrated in selected spectral components, whereas noise is more broadly distributed, less coherent, or less stable under the chosen transform (Yu et al., 4 Aug 2025, Zhao et al., 19 Jun 2025, Kim et al., 2021).

1. Definition and scope

FDD is defined by transform-domain separation rather than by a single algorithm. In one class of methods, denoising is performed by thresholding coefficients in a transform domain and inverting the transform. In another, only selected low-frequency or task-specific bands are retained. In learned systems, the transform coefficients become the representation on which a neural network, adversarial loss, or diffusion model operates. In still other formulations, denoising is cast as estimation from a power spectrum or as modal truncation in a frequency-resolved basis (Frost, 2022, Kim et al., 2021, Li et al., 20 Nov 2025, Civas et al., 2023).

The term therefore covers several distinct but related operations. IterativeFT alternates between real-domain and 2D Fourier-domain sparsification on adjacency matrices until the real-domain support pattern stabilizes (Frost, 31 Jan 2026). Manifold Frequency Denoising (MFD) constructs a graph from noisy samples, computes Spectral Graph Wavelets, and discards high-frequency bands (Deutsch et al., 2016). A fluorescence-imaging variant extracts only the spectral component at a known modulation frequency from each pixel’s time trace (Yu et al., 4 Aug 2025). TFCDiff performs conditional diffusion in a truncated DCT domain for raw 10-second ECG segments and uses a Temporal Feature Enhancement Mechanism (TFEM) to preserve waveform detail (Li et al., 20 Nov 2025).

A common misconception is that FDD is equivalent to a fixed low-pass filter. The surveyed literature does not support that reduction. Some methods are explicitly iterative and bidomain (Frost, 31 Jan 2026, Frost, 2022); some are adaptive and multi-scale (Zhao et al., 19 Jun 2025, Li et al., 27 Jul 2025); some are frequency-aware training strategies rather than standalone test-time filters (Helou et al., 2020, Kim et al., 2021); and some preserve a task-specific narrowband component rather than merely attenuating high frequencies (Yu et al., 4 Aug 2025, Civas et al., 2023).

2. Mathematical principles

The most general mathematical principle behind FDD is transform-domain asymmetry between signal and noise. MFD states this directly for smooth manifolds: each coordinate function behaves as a smooth graph signal, so its energy is concentrated in low graph frequencies, whereas noise affects all frequency bands in a similar way (Deutsch et al., 2016). In TFCDiff, the justification is that ECG energy is concentrated in low frequencies, especially for diagnostic content around $0.5\!-\!50$ Hz, making truncated DCT coefficients a compact latent representation for denoising (Li et al., 20 Nov 2025). In fluorescence imaging, useful signal is made periodic in time by modulated excitation, so it appears as a sharp spectral peak at the modulation frequency $\omega_0$, while background and noise are broadly distributed (Yu et al., 4 Aug 2025).

Several formulations make the transform-domain prior explicit. IterativeFT models a noisy observed adjacency matrix as

$\mathbf{X} = \mathbf{A} * \mathbf{P} + \mathbf{E},$

where $\mathbf{A}$ is the true network, $\mathbf{P}$ is a binary pruning mask, and $\mathbf{E}$ has iid Gaussian entries $e_{i,j}\sim \mathcal{N}(0,\epsilon)$. Its denoising mechanism is repeated thresholding of $\mathbf{X}$ and of its 2D discrete Fourier transform until the real-domain sparsity pattern no longer changes (Frost, 31 Jan 2026). The underlying interpretation is linked to the Fourier uncertainty principle: sparsity in the real domain and sparsity in the frequency domain cannot both be arbitrarily large, so alternating sparsification drives the estimate toward a stable nontrivial compromise rather than the all-zero solution (Frost, 2022).

Graph-spectral methods use a different frequency notion. MFD builds a $k$-nearest-neighbor graph with Gaussian weights

$$W_{ij}=\exp\!\left(\frac{-\|\mathbf{x}_i-\mathbf{x}_j\|_2^2}{2\sigma_D^2}\right),$$

forms the Laplacian $\omega_0$0, and defines graph frequencies through the Laplacian eigenvalues $\omega_0$1. Spectral Graph Wavelet coefficients

$\omega_0$2

are then truncated by retaining low-frequency scaling and wavelet bands and setting higher-frequency bands to zero (Deutsch et al., 2016).

Other FDD formulations are narrowband rather than low-pass. In fluorescence imaging,

$\omega_0$3

so denoising is the extraction of a known coherent component rather than generic attenuation of a high-frequency tail (Yu et al., 4 Aug 2025). In molecular communication, the relevant object is the power spectral density of receptor-binding fluctuations; frequency-domain detection estimates the desired ligand concentration by fitting a theoretical PSD model, exploiting the fact that different ligand-receptor kinetics yield different characteristic frequencies (Civas et al., 2023).

3. Methodological families

The literature presents several recurring FDD designs.

Method	Domain	Core denoising mechanism
IterativeFT	Noisy deterministic networks	Alternating real/Fourier hard thresholding until support stabilization
MFD	Noisy manifold samples	Spectral Graph Wavelet low-band retention
TFCDiff	ECG	Conditional diffusion in truncated DCT domain with TFEM
FDD fluorescence imaging	NIR-II imaging	Retain modulation-frequency component per pixel
SPOD denoising	Turbulent flows	Hard-threshold SPOD eigenvalues above noise floor

The simplest family uses hard thresholding. IterativeFT applies a mean-absolute-value threshold in the real domain, then in the frequency domain, and repeats until the real-domain zero/nonzero pattern stabilizes (Frost, 31 Jan 2026). The earlier signal-denoising version of IterativeFT follows the same logic for 1D periodic spike signals, with convergence defined by unchanged zero indices in the output of the real-domain sparsifier (Frost, 2022). FDNet likewise performs feature-space Fourier denoising by transforming intermediate feature embeddings to frequency space, applying a binary high-pass gate with threshold $\omega_0$4, and inverting the transform before final segmentation refinement (Li et al., 2024).

A second family uses band truncation or subband selection. MFD is explicitly non-iterative: it computes SGW coefficients for each coordinate dimension, retains low-frequency scaling and low-frequency wavelet bands, discards high-frequency bands, and reconstructs denoised coordinates (Deutsch et al., 2016). SPOD-based denoising also follows a hard truncation rule, but in a frequency-resolved modal basis: SPOD eigenpairs below a chosen threshold are discarded, with the threshold selected slightly above the noise floor (Nekkanti et al., 2020).

A third family embeds FDD in learning architectures. MADNet performs adaptive dual-domain denoising by transforming features with FFT, splitting them into learnable high- and low-frequency branches, refining each branch separately, and supervising the result with both spatial and frequency-domain losses (Zhao et al., 19 Jun 2025). HDST uses FFT preprocessing, multi-band convolution in frequency space, a learnable gate that blends original spatial features with an inverse-transformed frequency reconstruction, and cross-domain attention between spatial and frequency representations (Li et al., 27 Jul 2025). In unsupervised image denoising, frequency knowledge can also enter through supervision rather than representation: a spectral discriminator operating on high-pass Fourier features and a Fourier-domain reconstruction loss bias a GAN generator toward clean spectral statistics (Kim et al., 2021).

A fourth family uses diffusion in frequency space. TFCDiff applies type-II DCT to an ECG segment,

$\omega_0$5

keeps only the first $\omega_0$6 coefficients, and performs DDPM-style conditional denoising in that truncated DCT domain (Li et al., 20 Nov 2025). The editing method FDS is not a conventional denoiser, but it extends the FDD idea by decomposing the latent into wavelet subbands and applying diffusion gradients only to selected bands, using stop-gradient on the others to preserve detail or color (Ren et al., 24 Mar 2025).

4. Representative applications and reported behavior

FDD has been used on graph-structured, biomedical, physical, and visual data, with performance strongly tied to whether the transform domain matches the signal class.

For deterministic network denoising, IterativeFT is evaluated on Kautz, lattice, tree, full bipartite, and preferential attachment graphs. It gives the best overall performance on the Kautz and lattice networks, is competitive on tree and full bipartite graphs, has the best overall MSE and second-best F1 on the tree model, and performs poorly on preferential attachment networks, which the authors attribute to the lack of strong regular structure in a stochastic graph (Frost, 31 Jan 2026).

For manifold denoising, MFD is compared with diffusion-based manifold denoising (MD) and locally linear denoising (LLD). The method significantly outperforms MD and LLD, and on real datasets it reports RMSE values of 3.42 for MFD versus 7.35 for LLD and 11.84 for MD on CMU motion capture, and 51.2 for MFD versus 62.2 for LLD and 109.1 for MD on Frey faces (Deutsch et al., 2016).

For ambulatory ECG denoising, TFCDiff targets baseline wander, muscle artifact, electrode motion artifact, and flexible random mixed noise. On the synthesized dataset, the best configuration, TFCDiff-10, reports SSD $\omega_0$7, MAD $\omega_0$8, PRD $\omega_0$9, CosSim $\mathbf{X} = \mathbf{A} * \mathbf{P} + \mathbf{E},$0, and ImSNR $\mathbf{X} = \mathbf{A} * \mathbf{P} + \mathbf{E},$1 dB, and this is the best overall across all five metrics. On the unseen SimEMG database, TFCDiff-10 reports SSD $\mathbf{X} = \mathbf{A} * \mathbf{P} + \mathbf{E},$2, MAD $\mathbf{X} = \mathbf{A} * \mathbf{P} + \mathbf{E},$3, PRD $\mathbf{X} = \mathbf{A} * \mathbf{P} + \mathbf{E},$4, CosSim $\mathbf{X} = \mathbf{A} * \mathbf{P} + \mathbf{E},$5, and ImSNR $\mathbf{X} = \mathbf{A} * \mathbf{P} + \mathbf{E},$6 dB, again the best across all metrics (Li et al., 20 Nov 2025).

For in vivo fluorescence imaging, FDD is used as a temporal-coherence-based post-processing method. The reported gains include more than 2,500-fold improvement in SBR, more than 300-fold improvement in SNR, doubled penetration depth, a 95% reduction in contrast agent dosage or excitation light intensity for mouse vascular imaging, and 600 Hz real-time imaging (Yu et al., 4 Aug 2025). The same work reports SNR values of 45 for tumor imaging and 65 for vascular imaging with ICG, both far above the Rose criterion threshold of 4 used in the study (Yu et al., 4 Aug 2025).

For image and hyperspectral denoising, the evidence favors hybrid and adaptive frequency processing rather than frequency processing alone. MADNet achieves 39.78 dB / 0.959 on SIDD and 39.98 dB / 0.956 on DND, while HDST improves a SERT baseline from 29.68 dB PSNR, 0.9533 SSIM, 2.536 SAM to 30.62 dB PSNR, 0.9555 SSIM, 2.417 SAM on the Realistic hyperspectral dataset (Zhao et al., 19 Jun 2025, Li et al., 27 Jul 2025).

5. Strengths, failure modes, and limits

The strongest empirical pattern is domain dependence. FDD is most effective when the true signal has structured, repetitive, periodic, coherent, or smooth transform-domain organization. This is stated explicitly for deterministic graphs with regular adjacency patterns, for smooth manifolds whose coordinates are low-frequency graph signals, for ECG segments whose diagnostic energy is concentrated below 50 Hz, and for modulated fluorescence signals concentrated at a known temporal frequency (Frost, 31 Jan 2026, Deutsch et al., 2016, Li et al., 20 Nov 2025, Yu et al., 4 Aug 2025).

The same literature also documents limits. IterativeFT can create artifactual network structures, including false edge bands and corner artifacts (Frost, 31 Jan 2026). SPOD denoising depends on choosing a threshold slightly above the noise floor; if the threshold is too high, physically relevant structures are removed (Nekkanti et al., 2020). Fluorescence FDD depends on sufficient acquisition frames and synchronization, and its gains may be smaller when fewer frames are available (Yu et al., 4 Aug 2025). The unsupervised GAN-based image denoiser assumes that noisy and clean images differ meaningfully in the frequency domain and that Fourier high-frequency differences are informative for denoising (Kim et al., 2021).

Several results caution against interpreting frequency processing as sufficient by itself. In TFCDiff, DCT diffusion alone performs poorly, whereas DCT diffusion plus TFEM is best, indicating that frequency-domain modeling and temporal-detail restoration are complementary (Li et al., 20 Nov 2025). In HDST ablations, frequency processing alone yields only a slight gain over baseline, while the best performance arises from combining frequency-domain decoupling with multiscale spatial modeling and dynamic fusion (Li et al., 27 Jul 2025). This suggests that modern FDD often functions most effectively as a component within a hybrid-domain system rather than as an isolated filter.

Another recurrent limit is mismatch between the transform and the data geometry. IterativeFT does not work well on preferential attachment graphs (Frost, 31 Jan 2026). DFT and DCT masking in latent image editing lack spatial localization and can introduce artifacts, whereas wavelets are preferred because they localize both frequency and space (Ren et al., 24 Mar 2025). In microscopy segmentation, feature-space high-pass filtering is helpful because interference is treated as removable low-frequency content, but the full benefit appears only when this step is integrated with contextual fusion and attention (Li et al., 2024).

6. Relation to adjacent methods and current directions

FDD intersects with, but is not identical to, classical filtering, sparse approximation, inverse problems, and deep restoration. Some methods are essentially sparsification analogues of Fourier-based signal recovery (Frost, 31 Jan 2026). Others operate as graph-spectral generalizations of wavelet denoising (Deutsch et al., 2016). Still others treat denoising as conditional generation in a compact transform domain (Li et al., 20 Nov 2025), as frequency-aware adversarial learning (Kim et al., 2021), or as robust training through stochastic masking of selected DCT bands (Helou et al., 2020).

A notable development is the shift from uniform spectral treatment to adaptive decomposition. MADNet argues that previous FFT-based denoisers often treat the frequency domain uniformly and introduces learnable separation into high- and low-frequency branches (Zhao et al., 19 Jun 2025). HDST similarly uses frequency priors as guidance for cross-domain fusion rather than as an end in themselves (Li et al., 27 Jul 2025). In FDS, selective optimization of wavelet subbands is used to preserve specific aspects of an image, such as local detail or color, by freezing irrelevant bands (Ren et al., 24 Mar 2025). These methods indicate a broader movement toward frequency-aware control rather than indiscriminate transform-domain suppression.

Another extension is the replacement of Euclidean Fourier coordinates by geometry-aware or anisotropic representations. MFD uses the graph Laplacian spectrum and Spectral Graph Wavelets for irregular manifold data (Deutsch et al., 2016). De-SwinCANet performs denoising in the curvelet domain, where low-frequency coefficients capture global CSI structure and high-frequency coefficients capture directional detail; a Sigmoid threshold function then suppresses small noisy coefficients before inverse reconstruction (Tao et al., 16 Jun 2026). This suggests that “frequency-domain” in FDD is increasingly understood as task-specific spectral structure rather than as a single canonical transform.

Overall, FDD has evolved from transform-threshold-invert pipelines into a broad class of signal, image, graph, and scientific-data restoration methods. Across the surveyed work, its defining feature is not the presence of a Fourier transform alone, but the deliberate use of a spectral representation in which signal and corruption become more separable, whether by sparsity, coherence, localization, modulation, or modal energy concentration (Frost, 2022, Yu et al., 4 Aug 2025, Nekkanti et al., 2020).