PHFP: A Decoder Module for ICA Segmentation

• The paper introduces PHFP, a frequency-domain refinement module that restores high-frequency details such as vessel edges and bifurcations in ICA segmentation.
• PHFP employs a recursive Haar wavelet decomposition and depthwise convolution to progressively merge low-frequency global structures with enhanced high-frequency features.
• Empirical results demonstrate significant improvements in segmentation metrics (e.g., Dice, HD95) when PHFP is integrated into the U-Net-based architecture.

Searching arXiv for PHFP and closely related work to ground the article in current papers. Progressive High-Frequency Perception (PHFP) is a decoder-side module introduced in SFD-Mamba2Net for invasive coronary angiography (ICA) segmentation and stenosis detection. In that work, PHFP is defined as a mechanism that “employs multi-level wavelet decomposition and deep convolution to progressively refine high-frequency details while integrating low-frequency global structures,” with the explicit aim of recovering vessel edges, thin branches, bifurcations, stenotic contour changes, boundary detail, and overall vascular connectivity that are prone to attenuation during encoder–decoder processing (Mu et al., 10 Sep 2025).

1. Definition, scope, and problem setting

PHFP was proposed for a specific failure mode of encoder–decoder segmentation in ICA. The motivating setting is technically narrow but clinically demanding: ICA images are described as low-contrast, noisy, and populated by extremely fine, tortuous, multi-scale vessels, including micro-vessels with diameters smaller than two pixels. Under repeated downsampling and semantic abstraction, high-frequency information such as vessel edges, thin branches, bifurcations, and stenotic contour changes can weaken or disappear. PHFP is introduced precisely to counter that loss by restoring frequency content rather than relying only on spatial-domain upsampling and skip fusion (Mu et al., 10 Sep 2025).

Within SFD-Mamba2Net, PHFP is not a standalone network and not a general-purpose frequency block attached arbitrarily to the architecture. It is the decoder’s frequency-domain refinement component. The broader framework couples three components with distinct roles: CASE provides early structural priors, AA-DS Mamba2 provides bottleneck long-range context, and PHFP reconstructs fine detail during upsampling. This division of labor is important because PHFP is formulated not as a replacement for semantic modeling, but as a targeted mechanism for detail-preserving reconstruction (Mu et al., 10 Sep 2025).

The term also has a narrow lexical scope. In current arXiv usage, PHFP refers to Progressive High-Frequency Perception in SFD-Mamba2Net, whereas PHCP refers to “Progressive Heterogeneous Collaborative Perception,” an inference-time adaptation framework for heterogeneous collaborative perception in autonomous driving; the two should not be conflated (Si et al., 11 Sep 2025). A different arXiv submission on AdvSwap is sometimes associated with high-frequency information, but the available submission text contains no recoverable scientific content relevant to PHFP (Huang et al., 12 Feb 2025).

2. Architectural placement in the decoder

PHFP is repeated across the decoder rather than applied once. The paper states that “each decoder stage begins with the incorporation of a PHFP module,” and the architecture shows four PHFP blocks, one for each upsampling stage corresponding to decoder levels $D_4, D_3, D_2, D_1$ (Mu et al., 10 Sep 2025).

At decoder layer $i$, the input to PHFP is the fused decoder–encoder feature map

$Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$

where $E_i$ is the encoder feature at the corresponding level and

$$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$

Thus PHFP operates on the already skip-fused representation $Y_i$, not on a purely decoder-local tensor. This placement gives it direct access to both the semantic content propagated from the deeper decoder and the higher-resolution information arriving through long skip connections (Mu et al., 10 Sep 2025).

This placement also clarifies what PHFP is not. It is not the entire decoder stage, and it does not directly subsume the subsequent convolutional refinement block. Instead, it is a front-end unit at the beginning of each upsampling block. A plausible implication is that PHFP is intended to reshape the spectral composition of the fused feature before ordinary semantic refinement resumes.

3. Wavelet hierarchy and the meaning of “progressive”

The internal mechanism of PHFP is a cascaded Wavelet Transform (WT) / Inverse Wavelet Transform (IWT) architecture based on the Haar wavelet. The manuscript explicitly distinguishes one low-frequency subband $LL$ and three high-frequency subbands $LH$, $HL$, and $HH$, corresponding to horizontal, vertical, and diagonal texture responses. For each decoder input feature $i$0, $i$1, PHFP first performs first-level wavelet decomposition: $i$2 It then recursively decomposes only the low-frequency component from the previous level: $i$3

$i$4

The manuscript notes that the equation formatting is imperfect, but the intended operation is explicit: the decomposition proceeds by recursively applying $i$5 to the previous level’s $i$6 branch (Mu et al., 10 Sep 2025).

This recursion is what gives PHFP its “progressive” character. The term does not denote a generic iterative enhancement loop. It refers specifically to hierarchical, multi-level decomposition of low-frequency structure, followed by level-by-level reconstruction. Decomposition deepens the wavelet pyramid through the $i$7 branch, while reconstruction synthesizes the representation back from deep to shallow levels. In that sense, PHFP is a coarse-to-fine wavelet refinement strategy rather than a single-shot frequency split (Mu et al., 10 Sep 2025).

The underlying design principle is asymmetrical. The low-frequency pathway preserves global vessel topology, while the high-frequency pathways preserve and enhance local detail. The paper repeatedly frames this duality as necessary for simultaneously maintaining “overall vascular topology” and perceiving “vascular edges, delicate branches, and stenotic lesions” (Mu et al., 10 Sep 2025).

4. High-frequency enhancement, reconstruction, and output formulation

At each wavelet level, the high-frequency subbands are processed by a $i$8 depthwise convolution $i$9, which the paper describes as serving “feature enhancement and noise suppression.” The high-frequency bands are therefore not passed through transparently; they are selectively denoised and strengthened. Reconstruction is then performed progressively via IWT: $Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$0 where the printed base-case symbol in the manuscript is unclear, but the stated interpretation is that the deepest level initializes reconstruction and upper levels recursively add enhanced high-frequency components and lower-level reconstruction before IWT (Mu et al., 10 Sep 2025).

The final PHFP output is

$Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$1

This is a residual-style fusion between a depthwise-convolved version of the original decoder input and the top-level progressive wavelet reconstruction (Mu et al., 10 Sep 2025).

Several negative specifications are equally important. The paper does not state the decomposition depth $Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$2; it does not provide explicit tensor-shape formulas for subbands; and it does not describe channel attention, batch normalization, activation functions, learned gating, or explicit fusion weights inside PHFP. The module, as described, consists of Haar WT, recursive decomposition of the $Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$3 branch, $Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$4 depthwise convolution on high-frequency bands, IWT-based progressive reconstruction, and final residual fusion. A plausible implication is that the authors intended PHFP to remain structurally simple and frequency-explicit rather than densely parameterized.

5. Empirical behavior in ICA segmentation

The empirical evidence for PHFP comes primarily from ablation and qualitative analysis. In the ablation table, adding PHFP alone to U-Net (“U+P”) improves several segmentation metrics over the plain U-Net baseline: Dice rises from $Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$5 to $Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$6, Accuracy from $Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$7 to $Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$8, F1 from $Y_i = \mathrm{Concat}(U_{i+1}, E_i), \tag{3}$9 to $E_i$0, IoU from $E_i$1 to $E_i$2, Sensitivity from $E_i$3 to $E_i$4, and ASSD improves from $E_i$5 to $E_i$6 pixels. The paper emphasizes that PHFP alone reduces HD95 from $E_i$7 to $E_i$8 pixels, which it interprets as improved boundary localization (Mu et al., 10 Sep 2025).

When combined with other modules, the gains become larger. The variant U+M+P reaches Dice $E_i$9 and HD95 $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$0, while U+A+P reaches Dice $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$1, F1 $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$2, IoU $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$3, Sensitivity $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$4, HD95 $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$5, and ASSD $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$6. The full model reaches Dice $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$7, F1 $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$8, IoU $$U_{i+1} = \mathrm{Conv}_{1\times1}(\mathrm{Interpolate}(D_{i+1})). \tag{4}$$9, HD95 $Y_i$0, and ASSD $Y_i$1. The paper treats PHFP as a useful contributor on its own, but especially effective when combined with structural priors and long-range dependency modeling (Mu et al., 10 Sep 2025).

Variant	Dice	HD95
U-Net	$Y_i$2	$Y_i$3
U+P	$Y_i$4	$Y_i$5
U+A+P	$Y_i$6	$Y_i$7
Full model	$Y_i$8	$Y_i$9

The qualitative interpretation is consistent with these boundary-sensitive improvements. The paper states that PHFP produces activation maps with “greater stability and concentration along the fine vascular branches” and stronger activations in “the main arterial trunks.” It also attributes improved sensitivity to fine vessel edges and pathological regions to the progressive extraction and enhancement of orientation-specific high-frequency subbands, while progressive low-frequency decomposition is said to help represent principal vascular structures and global connectivity. In the comparison with standard convolution, PHFP is further said to reduce “redundant information and artifact accumulation” and to play “a crucial role in maintaining continuity along both the main vessels and minor branches,” especially in curved or small branches (Mu et al., 10 Sep 2025).

6. Relation to adjacent frequency-aware and progressive methods

PHFP belongs to a broader design space in which models defer, structure, or progressively recover fine detail rather than treating all frequencies uniformly from the outset. This broader pattern appears across several unrelated domains, although the implementations differ materially.

In single-image super-resolution, PPON stages reconstruction as content, structure, and perception, so that realistic fine texture is added only after a structure-preserving intermediate has been established. That method does not perform explicit wavelet or Fourier decomposition, but it does embody progressive residual enhancement from low-level fidelity to perceptual detail (Hui et al., 2019). In ultra-high-definition image restoration, ERR goes further and explicitly decomposes restoration into zero-frequency enhancement, low-frequency restoration, and high-frequency refinement, with stage-specific spectral regularizers and a dedicated high-frequency refiner (Zhao et al., 17 Mar 2025). In 3D Gaussian splatting, FreGS introduces progressive frequency regularization and frequency annealing so that Gaussian densification is guided from low to high spectral discrepancies rather than by static all-frequency supervision (Zhang et al., 2024).

Comparable ideas also appear outside vision. FSC-Net for speech bandwidth extension uses “frequency-progressive learning” in which progressively sharper residual targets guide recovery from coarse spectral envelope to fine harmonic detail, again postponing difficult high-frequency generation until a broader spectral scaffold is available (Chen et al., 5 Jun 2026). In CT sinogram inpainting, HiSin performs low-resolution global inference first and defers high-resolution local detail recovery to selective patches, with frequency-aware patch skipping and structure-adaptive step allocation (E et al., 10 Jun 2025).

These comparisons do not make PHFP a generic label for all progressive spectral methods. The specific PHFP module in SFD-Mamba2Net remains distinguished by three properties: decoder-stage repetition, recursive decomposition of the $LL$0 branch in a Haar wavelet hierarchy, and progressive IWT reconstruction with depthwise-convolved high-frequency bands (Mu et al., 10 Sep 2025). Still, the wider literature suggests a recurring principle: when global structure and local detail are optimized in different ways or at different stages, explicit progressive treatment of high-frequency information can improve stability, detail recovery, or computational allocation.