High-Frequency Rectified Diffusion Loss (HR-Loss)

• High-Frequency Rectified Diffusion Loss (HR-Loss) is a framework that integrates frequency constraints into rectified diffusion models to emphasize fine structural details.
• It combines standard reconstruction with wavelet decomposition and HOG feature matching to counteract low-frequency bias and mitigate overly smooth outputs.
• Empirical results in applications like video super-resolution show notable improvements in PSNR, SSIM, and perceptual quality, validating its practical impact.

Searching arXiv for the cited papers on HR-Loss and related rectified diffusion work. High-Frequency Rectified Diffusion Loss (HR-Loss) denotes a class of loss constructions that augment rectified-flow or diffusion objectives with explicit constraints on high-frequency content. In the literature provided here, the term is used explicitly in RealisVSR, where it is defined as a latent-space objective combining a rectified-flow reconstruction term with wavelet and HOG constraints for video super-resolution (Zhao et al., 25 Jul 2025). Closely related ideas also appear under different names in other generative settings: FrePolad introduces a frequency-rectified loss on spherical-harmonic coefficients for point-cloud VAE training (Zhou et al., 2023), and DriveFlow uses high-frequency alignment and dual-frequency background objectives on rectified-flow velocity fields for training-free image editing in autonomous driving (Lin et al., 24 Nov 2025). By contrast, Rectified Diffusion formalizes rectification as matched-pair retraining toward first-order approximate ODE paths, but does not define any high-frequency rectified loss (Wang et al., 2024). Taken together, these works situate HR-Loss as a frequency-aware modification of otherwise standard generative training or adaptation objectives, with the common purpose of counteracting low-frequency bias and improving recovery or preservation of fine structure.

1. Terminological scope and provenance

The expression “High-Frequency Rectified Diffusion Loss” is used explicitly in “RealisVSR: Detail-enhanced Diffusion for Real-World 4K Video Super-Resolution” (Zhao et al., 25 Jul 2025). There, HR-Loss is introduced in Section 3.2, referenced in the introduction as “High-Frequency Rectified Diffusion Loss (HR-Loss) combining wavelet decomposition and HOG feature constraints for texture restoration,” and evaluated through ablations that separate wavelet and HOG contributions (Zhao et al., 25 Jul 2025).

A distinct but related lineage appears in “FrePolad: Frequency-Rectified Point Latent Diffusion for Point Cloud Generation” (Zhou et al., 2023). That work does not use the name HR-Loss, but it introduces a frequency-rectified distance in the spherical-harmonics domain and a corresponding frequency-rectified loss $\mathcal{L}_{\text{Fre}}$ added to the VAE objective. The supplied reconstruction states that a natural mapping is $$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$, in the sense of a high-frequency rectified reconstruction loss. This suggests that “HR-Loss” can function as a unifying editorial label for losses that penalize high-frequency reconstruction errors more strongly, even when the original paper uses different terminology (Zhou et al., 2023).

A further variant appears in “DriveFlow: Rectified Flow Adaptation for Robust 3D Object Detection in Autonomous Driving” (Lin et al., 24 Nov 2025). DriveFlow does not present a single term called HR-Loss in the paper title or abstract, but it introduces “High-Frequency Foreground Preservation” and “Dual-Frequency Background Optimization” within a rectified-flow editing framework. The provided reconstruction explicitly interprets these mechanisms as a concrete instantiation of a high-frequency rectified diffusion/flow loss, defined directly on the velocity field (Lin et al., 24 Nov 2025).

By contrast, “Rectified Diffusion: Straightness Is Not Your Need in Rectified Flow” (Wang et al., 2024) is relevant primarily as a baseline statement of what rectified diffusion loss is not. It formalizes rectified diffusion as standard $\epsilon$-prediction MSE on matched noise-sample pairs and states that the paper never defines “High-Frequency Rectified Diffusion Loss (HR-Loss)” (Wang et al., 2024). This is important because it distinguishes high-frequency rectification from rectification itself.

2. Canonical formulation in RealisVSR

In RealisVSR, the forward latent process is written as

$$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$

where $x_0$ is a clean latent, $\epsilon$ is standard Gaussian noise, and $\alpha_t,\sigma_t$ are time-dependent coefficients (Zhao et al., 25 Jul 2025). The model predicts a velocity field $\mathbf{v}_\Theta(z_t,t)$, and the baseline rectified-flow loss is given as

$$\mathcal{L}_{\text{REC}} = \mathbb{E}\left[\left\| \mathbf{v}_\Theta(\mathbf{z}_t,t)-\epsilon-\mathbf{x}_0 \right\|^2\right],$$

with a later clarification/simplification

$$\mathcal{L}_{\text{REC}} = \left\|\mathbf{v}_\Theta(\mathbf{z}_t,t)-(\epsilon-\mathbf{x}_0)\right\|^2.$$

The supplied material notes a minor minus/plus sign inconsistency, but the stated intent is that the velocity predictor matches a target derived from $$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$0 and $$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$1 (Zhao et al., 25 Jul 2025).

The wavelet component is

$$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$2

where $$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$3 are Haar wavelet subbands and $$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$4 explicitly upweights the high-frequency bands (Zhao et al., 25 Jul 2025).

The HOG component is

$$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$5

where gradient orientations are discretized into 9 bins spanning $$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$6 and gradient magnitudes are normalized with L2-Hys regularization (Zhao et al., 25 Jul 2025).

The final HR-Loss is defined additively as

$$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$7

No extra scalar weights are shown between these three terms in Eq. (6), so the high-frequency emphasis is concentrated inside the wavelet subband weights and in the structural constraint imposed by HOG matching (Zhao et al., 25 Jul 2025).

This formulation is diffusion-native in the sense that it supervises the same latent velocity field used by rectified-flow training, rather than introducing a separate image-space perceptual criterion after the fact. A plausible implication is that HR-Loss is best understood not as an auxiliary regularizer detached from the generative dynamics, but as a reweighting of what the diffusion model is required to fit at each timestep.

3. Frequency mechanisms and representational domains

The three papers that motivate an extended notion of HR-Loss differ chiefly in how they define “frequency” and where they enforce it.

Work	Domain	High-frequency mechanism
RealisVSR (Zhao et al., 25 Jul 2025)	Latent velocity space	Haar subband weighting + HOG descriptor matching
FrePolad (Zhou et al., 2023)	Spherical-harmonic representation of point clouds	Degree-weighted spectral distance with increasing weight near $$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$8
DriveFlow (Lin et al., 24 Nov 2025)	Rectified-flow velocity field	Gaussian low-pass decomposition and masked high-frequency residual alignment

In FrePolad, a point cloud $$\mathcal{L}_{\text{HR}} \equiv \mathcal{L}_{\text{Fre}}$$9 is interpreted as samples on a star-shaped surface and represented by a radius function $\epsilon$0 expanded in spherical harmonics: $\epsilon$1 The frequency-rectified squared distance between two point clouds is

$\epsilon$2

with

$\epsilon$3

Because the Gaussian rectifier is peaked at the highest degree $\epsilon$4, high-degree spherical-harmonic coefficients receive larger weight, thereby emphasizing high-frequency angular structure (Zhou et al., 2023).

DriveFlow instead defines low- and high-frequency components of a velocity field $\epsilon$5 by Gaussian blurring: $\epsilon$6 where

$\epsilon$7

This decomposition is then used to constrain foreground and background regions differently (Lin et al., 24 Nov 2025).

These examples show that HR-Loss is not tied to a single spectral formalism. In RealisVSR, the relevant operators are wavelets and HOG histograms; in FrePolad, spherical harmonics; in DriveFlow, Gaussian low-pass filtering and residual high-frequency velocity components. The unifying property is selective pressure on rapidly varying structure rather than uniform MSE over the full signal.

4. Integration into rectified-flow and diffusion pipelines

In RealisVSR, HR-Loss is the training objective for the main VSR diffusion model. The paper states that the diffusion model (Wan-1.3B + CPC) is trained with

$\epsilon$8

with no additional perceptual, adversarial, or pixel-space reconstruction loss, and with application at all sampled timesteps during training (Zhao et al., 25 Jul 2025). The operators are spatial, applied per latent frame and per latent channel, while temporal consistency is attributed mainly to Wan2.1 and the Consistency Preserved ControlNet (CPC) (Zhao et al., 25 Jul 2025).

FrePolad integrates its frequency-aware term differently. There, the frequency mechanism lives on the VAE side rather than in the DDPM objective. The baseline VAE maximizes the ELBO,

$\epsilon$9

and the frequency-rectified loss

$$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$0

is added through a Lagrangian formulation, yielding the FreELBO

$$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$1

Equivalently, the VAE loss becomes

$$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$2

The subsequent latent DDPM still uses the standard Ho et al. noise-prediction MSE, and “no extra frequency rectification term is added to $$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$3” (Zhou et al., 2023). This is a notable architectural distinction: HR-style frequency rectification shapes the latent space upstream rather than modifying diffusion fitting directly.

DriveFlow presents yet another integration pattern. The pre-trained text-to-image flow model is frozen; instead, the target velocity field is optimized through inner iterations at each time step. With source and target velocities decomposed into low- and high-frequency parts, the total loss is

$$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$4

with typical settings $$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$5, $$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$6, $$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$7 (Lin et al., 24 Nov 2025). Thus, the loss acts as a trajectory-level adaptation signal rather than a parameter-training loss.

Rectified Diffusion provides the control case. Its loss is simply

$$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$8

with matched noise-sample pairs pre-collected from a pretrained model (Wang et al., 2024). The paper explicitly states that there is no frequency weighting and no HR-Loss. This clarifies that high-frequency rectification is an optional augmentation to rectified diffusion, not an intrinsic component.

5. Motivation: low-frequency bias and detail preservation

Across the supplied works, the central motivation is that standard reconstruction or trajectory-matching losses privilege low-frequency structure.

RealisVSR states that standard rectified-flow or diffusion losses are low-frequency biased: they are effective for global structure and pixel-level reconstruction, but “do not explicitly distinguish low vs. high frequency,” with the result that the network allocates more capacity to low-frequency components and produces overly smooth outputs with insufficient recovery of edges, textures, and fine patterns (Zhao et al., 25 Jul 2025). The motivation is particularly strong in real-world 4K VSR, where the paper emphasizes rich textures such as hair strands, fabric, poolside clutter, reflections, eye glasses, skin texture, and eyebrows (Zhao et al., 25 Jul 2025).

FrePolad frames the issue as spectral bias in VAEs. According to the supplied text, VAEs tend to reconstruct low-frequency structure well while losing high-frequency details such as sharp edges, small parts, and fine local geometry, which manifests as overly smooth shapes in point-cloud generation (Zhou et al., 2023). The frequency-rectified loss is meant to force the encoder-decoder pair to preserve those details in the latent representation.

DriveFlow’s motivation is preservation rather than restoration. Rectified-flow-based image editing in autonomous driving can fail to preserve foreground objects with accurate 3D geometry. DriveFlow therefore aligns high-frequency velocity components in foreground regions so that edited scenes maintain precise object contours and geometry, while background low-frequency content remains editable (Lin et al., 24 Nov 2025).

A common misconception is that any rectified-flow or rectified-diffusion objective is already detail-aware because it supervises the full trajectory signal. The supplied evidence does not support that claim. Rectified Diffusion explicitly focuses on first-order approximate ODE paths and matched-pair retraining, not on spectral selectivity (Wang et al., 2024). RealisVSR and DriveFlow both arise precisely because a standard rectified objective is considered insufficient for high-frequency fidelity (Zhao et al., 25 Jul 2025, Lin et al., 24 Nov 2025).

6. Empirical behavior and ablation evidence

The clearest direct empirical evidence for HR-Loss comes from RealisVSR’s ablations on SPMCS (Zhao et al., 25 Jul 2025).

Setting	PSNR ↑	SSIM ↑	LPIPS ↓
Baseline	26.54	0.7843	0.1807
CPC only	26.58	0.8099	0.1520
CPC + Wavelet	27.24	0.8114	0.1549
CPC + HOG	27.33	0.8141	0.1521
CPC + Wavelet + HOG (full)	27.36	0.8169	0.1388

The full table in the supplied text also includes DISTS, where the full model attains the lowest value, $$z_t = \alpha_t \cdot x_0 + \sigma_t \cdot \epsilon,$$9 (Zhao et al., 25 Jul 2025). The reported interpretation is that wavelet loss gives a large PSNR boost and lowers DISTS, HOG provides stronger perceptual gains, and the full HR-Loss yields the best overall distortion and perceptual metrics (Zhao et al., 25 Jul 2025).

The supplementary wavelet-weight ablation further indicates that moderate upweighting is preferable. With $x_0$0, the paper reports the best trade-off, whereas pushing weights to $x_0$1 degrades PSNR, SSIM, LPIPS, and DISTS (Zhao et al., 25 Jul 2025). This suggests that high-frequency emphasis must remain balanced against structural fidelity.

FrePolad’s ablation compares Po, FrePo, Polad, and FrePolad. For airplanes, 1-NNA-CD/EMD are reported as $x_0$2 for Po, $x_0$3 for FrePo, $x_0$4 for Polad, and $x_0$5 for FrePolad (Zhou et al., 2023). The supplied interpretation is that frequency rectification alone improves quality and diversity significantly, and the combination of frequency rectification with latent DDPM performs best (Zhou et al., 2023).

DriveFlow’s evidence is more qualitative in the supplied material. Figure 1 and related discussion reportedly show that $x_0$6 preserves object geometry, $x_0$7 strengthens background editing, and $x_0$8 restores semantic consistency and improves temporal consistency. Quantitatively, the supplied text gives KITTI-C Car mAP examples: DriveGEN (6× aug.) at $x_0$9 average, DriveFlow (Snow only) at $\epsilon$0, and DriveFlow (6× aug.) at $\epsilon$1 (Lin et al., 24 Nov 2025). A plausible implication is that high-frequency preservation is especially consequential for smaller or geometrically delicate targets, since the same text notes pronounced gains for Pedestrian and Cyclist.

7. Implementation patterns and conceptual implications

RealisVSR provides the most explicit implementation recipe. HR-Loss operates in latent space on Wan-VAE latents with channel dimension $\epsilon$2 and spatial resolution downsampled by 8. Wavelet supervision uses a Haar 2D transform per channel per frame with filters

$\epsilon$3

and subband weights $\epsilon$4 and $\epsilon$5 (Zhao et al., 25 Jul 2025). HOG supervision is also applied in latent space, per channel, with 9 orientation bins over $\epsilon$6 and L2-Hys normalization (Zhao et al., 25 Jul 2025). The paper does not describe a schedule; HR-Loss is applied from the beginning of training and at all timesteps (Zhao et al., 25 Jul 2025).

FrePolad’s implementation is more specialized. The loss is applied only during VAE training, computed between the input point cloud $\epsilon$7 and a reconstruction sample $\epsilon$8, where $\epsilon$9. Spherical representations are obtained via KNN interpolation on the sphere, spherical-harmonic coefficients are computed with the differentiable torch_harmonics library, and the weighted coefficient mismatch is combined with the ELBO (Zhou et al., 2023). Reported hyperparameters include latent dimension $\alpha_t,\sigma_t$0, $\alpha_t,\sigma_t$1, $\alpha_t,\sigma_t$2, $\alpha_t,\sigma_t$3, $\alpha_t,\sigma_t$4, and $\alpha_t,\sigma_t$5 (Zhou et al., 2023).

DriveFlow uses Gaussian blur with kernel size $\alpha_t,\sigma_t$6 and $\alpha_t,\sigma_t$7 for channel-wise low-pass filtering, with high-frequency residual obtained by subtraction (Lin et al., 24 Nov 2025). Foreground and background masks are derived from 2D bounding boxes and broadcast across channels. Because the model parameters remain frozen, gradients update the target velocity used along the path rather than the pretrained transformer weights (Lin et al., 24 Nov 2025).

Conceptually, these patterns support three distinct uses of HR-style losses. First, they can modify the primary generative objective itself, as in RealisVSR. Second, they can regularize an upstream autoencoder whose latent space is later modeled by diffusion, as in FrePolad. Third, they can serve as trajectory-level optimization criteria for training-free editing, as in DriveFlow. This suggests that HR-Loss is less a single formula than a design principle: define a frequency decomposition appropriate to the data and enforce asymmetric penalties on the high-frequency part where standard objectives underperform.

A final conceptual boundary is established by Rectified Diffusion. That work shows that rectification alone consists of matched-pair retraining toward first-order path consistency, with no explicit frequency term (Wang et al., 2024). Therefore, any use of the label HR-Loss should be reserved for objectives that add genuine high-frequency selectivity, rather than for rectified diffusion losses in general.