WEGE: Wavelet-Energy Guided Encoding

• Wavelet-Energy-Guided Encoding (WEGE) is a method that explicitly integrates local energy estimates from discrete wavelet transforms into implicit neural representations to mitigate spectral bias.
• It computes and normalizes pixel-wise energy maps using guided filtering, providing spatially consistent frequency information for adaptive reconstruction.
• WEGE enhances tasks like image restoration, super-resolution, and 3D reconstruction by enabling region-adaptive frequency guidance, thereby improving quality and efficiency.

Wavelet-Energy-Guided Encoding (WEGE) is a wavelet-informed encoding strategy in which energy estimates derived from wavelet decompositions are made explicit to a learning system, rather than being left implicit in learned features alone. In the formulation introduced in FLAIR, WEGE uses the discrete wavelet transform (DWT) to compute a spatially varying energy score and concatenates that score with coordinate inputs to an implicit neural representation (INR), thereby providing explicit local frequency information intended to mitigate spectral bias and the lack of region-specific frequency awareness (Ko et al., 19 Aug 2025). Across adjacent literature, the same phrase or closely related descriptions also refer to sparsity-promoting latent regularization, low-frequency-first entropy coding, and wavelet-driven latent pathways, indicating that WEGE functions both as a specific method and, more broadly, as a recurring design principle in wavelet-aware neural encoding.

1. Definition, motivation, and scope

In FLAIR, WEGE is introduced to address two limitations of existing INRs: spectral bias, defined as the tendency to learn low-frequency content early while struggling with fine high-frequency details, and the lack of region-specific frequency awareness, meaning that existing activations or positional encodings cannot distinguish whether a region is high- or low-frequency dominated (Ko et al., 19 Aug 2025). The stated purpose is to inject local frequency information explicitly into the neural representation so that the INR can process frequencies according to spatial context; the paper characterizes this as “region-adaptive frequency guidance” (Ko et al., 19 Aug 2025).

The FLAIR formulation is specific: WEGE is not merely a wavelet transform applied as preprocessing, nor is it a fixed basis expansion. It computes a pixel-wise energy score from wavelet sub-bands, normalizes that score, filters it for spatial consistency, and then appends it to the coordinate input of the INR (Ko et al., 19 Aug 2025). Within that architecture, WEGE complements RC-GAUSS, the companion activation proposed in FLAIR for explicit frequency selection and spatial localization under the time-frequency uncertainty principle (TFUP) (Ko et al., 19 Aug 2025).

The terminology is broader in neighboring work. In “Wavelet-based Variational Autoencoders for High-Resolution Image Generation,” the details explicitly describe a wavelet-energy-guided sparsity penalty on detail coefficients and also label that mechanism “Wavelet-Energy-Guided Encoding (WEGE)” (Kiruluta, 16 Apr 2025). In “WeConvene,” the term is not explicitly used in the paper, but the details identify the low-frequency-first and high-frequency-conditioned entropy coding schedule as energy-guided encoding in the wavelet domain (Fu et al., 2024). “WF-VAE” similarly presents a wavelet-driven “energy flow” that routes low-frequency content into the latent representation (Li et al., 2024). This suggests that WEGE, in current usage, spans a family of architectures in which wavelet-domain energy distributions are made operational in the encoding path.

2. Canonical formulation in FLAIR

The canonical WEGE pipeline described in FLAIR begins with a DWT applied to an input image $I_k$, yielding a low-frequency sub-band $D_k^L$ and high-frequency sub-bands $D_k^H$, where the latter correspond to HL, LH, and HH detail coefficients and the former to the LL approximation coefficients (Ko et al., 19 Aug 2025). From these components, the method defines a pixel-wise energy score map: $$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$ The resulting $E_b$ is described as highlighting regions of high frequency through large values and low frequency through small values (Ko et al., 19 Aug 2025).

To stabilize the signal and place it on a comparable scale across images, the energy map is normalized: $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$ Here $E_{\min}$ and $E_{\max}$ are the minimum and maximum of the energy map, and $\epsilon$ is a small constant for numerical stability (Ko et al., 19 Aug 2025).

FLAIR further states that the raw pixel-wise $w_b$ exhibits discontinuities and can introduce noise or artifacts if supplied directly to an INR. To address this, a guided filter $D_k^L$0 is applied: $D_k^L$1 where $D_k^L$2 is the grayscale guidance image, $D_k^L$3 is the window radius, and $D_k^L$4 is the regularization parameter (Ko et al., 19 Aug 2025). The filtered map $D_k^L$5 is intended to suppress artificial discontinuities while preserving edges.

This sequence yields a concrete seven-step algorithm in the paper’s detailed description: compute DWT; compute $D_k^L$6; normalize to $D_k^L$7; guided-filter to $D_k^L$8; form the input vector $D_k^L$9; feed it to the INR; and allow the network to use the local frequency score for adaptive frequency selection in signal reconstruction (Ko et al., 19 Aug 2025).

3. Integration into implicit neural representations

After filtering, FLAIR concatenates the processed energy score with the original spatial coordinates and feeds the result into the network: $D_k^H$0 Here $D_k^H$1 denotes the INR, which in FLAIR uses RC-GAUSS internally (Ko et al., 19 Aug 2025). The significance of this formulation is explicit in the paper: the input no longer encodes only spatial position, but also the local frequency content at each coordinate.

Within this arrangement, WEGE acts as a conditioning signal for frequency selection. FLAIR states that RC-GAUSS, or any compatible layer, can modulate its frequency selection based on the explicit local frequency signature provided by WEGE (Ko et al., 19 Aug 2025). The method is also characterized as parameter-efficient, increasing parameter count by approximately $D_k^H$2k, and as plug-and-play, although the reported benefits are strongest when combined with RC-GAUSS (Ko et al., 19 Aug 2025).

A key architectural implication is that WEGE localizes frequency guidance spatially. Unlike encodings that supply a fixed global collection of bases, the appended $D_k^H$3 varies by coordinate and is intended to tell the INR where high-frequency allocation is warranted and where smooth reconstruction is preferable (Ko et al., 19 Aug 2025). In the language of the paper, this region-adaptive behavior is meant to reduce artifacts, avoid overfitting to noise in smooth areas, and improve sharp detail recovery where needed (Ko et al., 19 Aug 2025).

4. Reported empirical behavior

FLAIR reports that the full method, combining RC-GAUSS with WEGE, consistently outperforms existing INRs in 2D image representation and restoration as well as in 3D reconstruction (Ko et al., 19 Aug 2025). On the Kodak dataset for image representation, the reported metrics are as follows.

Method	PSNR / SSIM	LPIPS
FINER	30.11 / 0.8227	0.300
WIRE	28.80 / 0.7928	0.333
SIREN	28.47 / 0.7709	0.382
GAUSS	27.30 / 0.7278	0.444
ReLU+PE	21.80 / 0.5447	0.737
FLAIR	30.13 / 0.8506	0.225

The main reported distinction on Kodak is that FLAIR attains the best performance across PSNR, SSIM, and LPIPS, with the largest margin appearing in LPIPS (Ko et al., 19 Aug 2025). In the ablation study summarized in the detailed description, adding WEGE to RC-GAUSS with guided filtering improves PSNR from $D_k^H$4 to $D_k^H$5 and LPIPS from $D_k^H$6 to $D_k^H$7 (Ko et al., 19 Aug 2025). The same source states that WEGE can also be applied to FINER, but that the synergistic gain is larger for RC-GAUSS/FLAIR (Ko et al., 19 Aug 2025).

Outside static image fitting, FLAIR reports that on DIV2K super-resolution, particularly at upscaling factors $D_k^H$8 and $D_k^H$9, the method outperforms competitors in both objective and perceptual metrics (Ko et al., 19 Aug 2025). The denoising experiments are described as showing more effective suppression of noise artifacts while retaining details than baseline methods, and the 3D occupancy reconstruction experiments are described as achieving higher IoU and PSNR while preserving more details than other INR approaches (Ko et al., 19 Aug 2025).

Qualitatively, the paper attributes these outcomes to the fact that WEGE guides the network to allocate high frequencies only where needed, such as edges and textures, while suppressing unwanted artifacts in smooth regions (Ko et al., 19 Aug 2025). This is presented as visually confirmed in reconstructions and ablation figures.

5. Related formulations in adjacent wavelet literature

The most closely related alternate use of the term appears in “Wavelet-based Variational Autoencoders for High-Resolution Image Generation,” where images are encoded as multi-scale Haar wavelet coefficients rather than as Gaussian latent vectors (Kiruluta, 16 Apr 2025). In that model, stochasticity is preserved by adding noise to the coefficients,

$$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$0

and the objective replaces the conventional KL term with a reconstruction term plus an explicit $$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$1 penalty on detail coefficients,

$$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$2

The details explicitly identify this as a wavelet-energy-guided sparsity penalty and also connect it to WEGE (Kiruluta, 16 Apr 2025). Reported metrics show reconstruction loss improving from $$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$3 to $$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$4, SSIM from $$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$5 to $$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$6, and FID from $$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$7 to $$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$8; the ablation further states that fixed noise increases error by $$D_k^H,\; D_k^L = \mathrm{DWT}(I_k), \qquad E_b = \mathrm{IDWT}(D_k^H) - \mathrm{IDWT}(D_k^L).$$9 (Kiruluta, 16 Apr 2025).

In learned image compression, “WeConvene” introduces DWT into both convolutional processing and entropy modeling (Fu et al., 2024). Its WeConv module applies convolution after DWT and then returns to the spatial domain via inverse DWT, while its WeChARM entropy model first codes low-frequency coefficients and then codes high-frequency coefficients conditioned on the reconstructed low-frequency part (Fu et al., 2024). The paper reports that, for the Kodak dataset and a baseline with $E_b$0 BD-Rate saving over H.266/VVC, adding Haar-based WeConv improves the saving to $E_b$1, and enabling Haar-based WeChARM further improves it to $E_b$2 (Fu et al., 2024). The detailed description interprets this low-frequency-first schedule as energy-guided encoding, although it notes that WEGE is not explicitly named in the paper (Fu et al., 2024).

In video autoencoding, “WF-VAE” uses multi-level wavelet transform to route low-frequency energy directly into latent representation while leaving the backbone to process mainly high-frequency information (Li et al., 2024). The paper defines a wavelet-loss term,

$E_b$3

to align encoded and decoded low-frequency coefficients (Li et al., 2024). Reported results include $E_b$4 higher throughput and $E_b$5 lower memory consumption than state-of-the-art video VAEs while maintaining competitive reconstruction quality; the detailed comparison lists, for example, WF-VAE-L with latent channels $E_b$6 at $E_b$7 GB memory, PSNR $E_b$8, LPIPS $E_b$9, throughput $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$0, and FVD $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$1 (Li et al., 2024).

Earlier wavelet coding work provides a classical antecedent. “Wavelet Video Coding Algorithm Based on Energy Weighted Significance Probability Balancing Tree” weights subbands by the square root of their reconstruction energy,

$$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$2

and organizes coefficients into a significance probability balancing tree so that coefficients with similar significance probabilities occupy the same layer (Song et al., 2018). Relative to the asymmetric 3D orientation tree, it reports average PSNR gains of $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$3 dB, $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$4 dB, and $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$5 dB for luminance $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$6 and chrominance $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$7, respectively; relative to the temporal-spatial orientation tree, the gains are $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$8 dB, $$w_b = \frac{E_b - E_{\min}}{E_{\max} - E_{\min} + \epsilon}.$$9 dB, and $E_{\min}$0 dB (Song et al., 2018).

Wavelet-guided dense prediction offers another extension. “WaveSeg” decomposes features into $E_{\min}$1, $E_{\min}$2, $E_{\min}$3, and $E_{\min}$4 bands, constructs a high-frequency prior

$E_{\min}$5

and uses it to modulate decoder features via

$E_{\min}$6

The detailed description connects this explicitly learned high-frequency prior to broader WEGE approaches and states that ablations show consistent mIoU increases of approximately $E_{\min}$7 to $E_{\min}$8 from high-frequency prior guidance alone, with a further $E_{\min}$9 to $E_{\max}$0 gain from the SDA block (Xu et al., 24 Oct 2025).

6. Conceptual interpretation, misconceptions, and limits

A recurrent source of confusion is that WEGE is not identical to “using wavelets” in a generic sense. In FLAIR, the contrast is explicit: other INR encodings, including Fourier and wavelet encodings, are described as providing a fixed global set of bases, which limits adaptability and local detail recovery, whereas WEGE provides per-coordinate frequency guidance derived from an energy map (Ko et al., 19 Aug 2025). The distinction is therefore between fixed basis provision and explicit, spatially varying conditioning.

Another misconception is that wavelet-energy guidance always means prioritizing low frequencies. The surveyed papers show a more task-dependent pattern. FLAIR uses a map that highlights where high-frequency detail should be reconstructed (Ko et al., 19 Aug 2025). WaveSeg constructs an explicit high-frequency prior to reinforce boundaries (Xu et al., 24 Oct 2025). By contrast, WF-VAE emphasizes that most video energy is concentrated in the low-frequency $E_{\max}$1 subband and routes that content into latent space (Li et al., 2024), and WeConvene codes low-frequency coefficients first before high-frequency coefficients (Fu et al., 2024). This suggests that the “guided” quantity may be high-frequency detail, low-frequency energy, or both, depending on the target task.

The term itself is also not standardized across papers. In FLAIR it denotes a DWT-derived coordinate augmentation for INRs (Ko et al., 19 Aug 2025). In Wavelet-VAE it denotes a sparsity-promoting regularization on wavelet detail coefficients (Kiruluta, 16 Apr 2025). In WeConvene and WF-VAE, the details describe closely related energy-guided mechanisms without making the term equally canonical (Fu et al., 2024, Li et al., 2024). A plausible implication is that WEGE is best regarded as an architectural principle—explicitly exposing wavelet-domain energy structure to the encoder, latent model, entropy model, or decoder—rather than as a single invariant algorithm.

The current evidence in the cited works is empirical rather than universal. The reported gains are task-specific: Kodak image representation and restoration for FLAIR, high-resolution image generation for Wavelet-VAE, learned image compression for WeConvene, latent video diffusion support for WF-VAE, and segmentation refinement for WaveSeg (Ko et al., 19 Aug 2025, Kiruluta, 16 Apr 2025, Fu et al., 2024, Li et al., 2024, Xu et al., 24 Oct 2025). What is consistent across them is the claim that making wavelet energy explicit—through coordinate conditioning, latent regularization, entropy scheduling, or feature priors—improves the allocation of representational capacity relative to purely spatial or globally fixed alternatives.