Adaptive Frequency Domain Perceptron (AFDP)

• Adaptive Frequency Domain Perceptron (AFDP) is a frequency-aware feature enhancement block that amplifies high-frequency crack details while suppressing low-frequency noise.
• It operates in the Fourier domain using directional convolutions and learnable soft frequency masking to improve multimodal fusion in lightweight LIDAR networks.
• AFDP enhances crack segmentation performance with minimal computational cost, integrating efficiently into dual-domain fusion pipelines.

Searching arXiv for recent and directly relevant papers on "Adaptive Frequency Domain Perceptron" and related AFDP usages. Adaptive Frequency Domain Perceptron (AFDP) denotes, in the multimodal crack-segmentation literature, a frequency-aware feature enhancement block that operates in the Fourier domain to amplify crack-relevant high-frequency structure and suppress low-frequency background content before multimodal fusion (Liu et al., 30 Jul 2025). The term is most explicitly defined in the LIDAR architecture, where AFDP forms the first stage of the Lightweight Dual Domain Dynamic Collaborative Fusion (LD3CF) module. In adjacent literatures, the same acronym is not uniform: a generalized quantum perceptron with a tunable oscillatory activation has been described as very close in spirit to an AFDP because its effective nonlinearity is frequency-controlled (Benatti et al., 2 Nov 2025), while an unrelated bioinformatics paper uses AFDP to mean “Automated Function Description Prediction” (Jozashoori et al., 2019).

1. Definition and terminological scope

In the LIDAR model, AFDP is a frequency-domain enhancement component designed to enhance high-frequency crack details, suppress low-frequency background noise, and do so in a direction-aware way, especially along horizontal and vertical frequency axes (Liu et al., 30 Jul 2025). Its stated motivation is that crack regions tend to have strong high-frequency responses, whereas background regions are mostly low-frequency. AFDP is therefore introduced not as a generic perceptron in the classical single-neuron sense, but as a feature-processing block that performs frequency discrimination before multimodal fusion.

The architectural setting is important. LIDAR is a “Lightweight Adaptive Cue-Aware Vision Mamba” network for multimodal segmentation of structural cracks. It is composed of a Lightweight Adaptive Cue-Aware Visual State Space module (LacaVSS), a Lightweight Dual Domain Dynamic Collaborative Fusion module (LD3CF), and a Lightweight Dynamically Modulated Multi-Kernel convolution (LDMK) (Liu et al., 30 Jul 2025). AFDP belongs to LD3CF rather than to the backbone as a standalone encoder.

The acronym also requires disambiguation. In “AFDP: An Automated Function Description Prediction Approach to Improve Accuracy of Protein Function Predictions,” AFDP denotes an integrated protein annotation pipeline that combines eggNOG, Swiss-Prot/UniProt retrieval, and the suffix-tree-based stCFExt algorithm to generate representative human-readable protein function descriptions (Jozashoori et al., 2019). That usage is separate from the frequency-domain perceptron sense.

A further terminological complication arises from “Pseudo quantum advantages in perceptron storage capacity,” whose generalized quantum perceptron is described as very close in spirit to an AFDP because its nonlinearity is a sinusoidal activation with adjustable frequency $\lambda$ (Benatti et al., 2 Nov 2025). This does not define AFDP as a canonical name in that paper, but it does establish a closely related frequency-adaptive perceptron formulation.

2. Placement within LIDAR and LD3CF

AFDP is placed at the beginning of LD3CF. The paper states that “the process begins with the AFDP, which receives the multimodal features extracted from LacaVSS” (Liu et al., 30 Jul 2025). The high-level order is

$$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$

Its input is a feature map

$$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$

and, more generally in the appendix,

$x \in \mathbb{R}^{B \times C \times H \times W}.$

Its output is a frequency-enhanced representation in which high-frequency components are enhanced, low-frequency background is suppressed, and the result is returned to the spatial domain for subsequent LD3CF processing (Liu et al., 30 Jul 2025).

This placement clarifies a common misconception. AFDP is not described as an independent segmentation network; it is a preparatory enhancement stage inside a larger multimodal fusion pipeline. Its function is to provide downstream processing with a clearer feature map of the texture cues, after which LD3CF performs dual-pooling fusion and cross-scale dynamic interaction.

The role of AFDP is therefore upstream of multimodal aggregation. The paper’s rationale is that standard multimodal fusion can still leave background noise and dilute critical crack boundaries. By enhancing crack texture and boundary information before fusion, AFDP is intended to improve the quality of the inputs seen by the downstream spatial fusion operators (Liu et al., 30 Jul 2025).

3. Internal mechanism

AFDP combines four elements: a 2D real-valued Fast Fourier Transform (rFFT), direction-aware frequency convolutions, learnable soft frequency masking, and spatial-domain fusion with channel-wise gating (Liu et al., 30 Jul 2025).

The frequency transform is given in the appendix in standard form:

$$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$

Here $f(x,y)$ is the spatial-domain input and $F(u,v)$ is the frequency representation. The stated purpose is to expose directional periodic patterns that are harder to distinguish in the spatial domain.

After the transform, AFDP applies direction-specific convolutions:

$Freq_h = ReLU(BN(Conv_h(FFT(x))))$

and

$Freq_v = ReLU(BN(Conv_v(FFT(x)))).$

The kernels are explicitly specified as $1 \times 3$ for $$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$0 and $$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$1 for $$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$2 (Liu et al., 30 Jul 2025). These operators independently model horizontal and vertical directional frequency components and are intended to capture anisotropic crack structures.

Frequency separation is performed with learnable soft masks rather than hard thresholding. The paper defines a learnable frequency separation radius $$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$3, a temperature scaling factor $$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$4, and directional distances $$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$5 and $$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$6 from the spectral center. The masks are

$$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$7

$$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$8

and

$$\text{LacaVSS} \rightarrow \text{AFDP} \rightarrow \text{dual-pooling fusion} \rightarrow \text{cross-level gate} \rightarrow \text{output}.$$9

The underlying idea is that bins farther from the center are treated as high frequency, central bins as low frequency, and the boundary is adaptive rather than fixed. The paper explicitly connects this to variability of crack texture across datasets and modalities (Liu et al., 30 Jul 2025).

After masking, AFDP reconstructs the components with inverse rFFT, refines them, and fuses them in the spatial domain through a channel-wise gating strategy, producing frequency-enhanced features (Liu et al., 30 Jul 2025). The appendix further states that each refined component is aligned via lightweight convolution, direction-specific attention weights are produced, these are fused into a unified channel-wise attention map, and the output is a residual response that keeps semantics while emphasizing crack textures and boundaries.

Algorithmically, the paper summarizes the sequence as: input feature map, rFFT, horizontal and vertical convolutions, direction-aware soft masks using $$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$0 and $$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$1, separation into high-frequency and low-frequency components, lightweight refinement, inverse rFFT, channel-wise gating or attention, and emission of a frequency-enhanced feature map (Liu et al., 30 Jul 2025).

4. Interaction with multimodal fusion, ablation behavior, and efficiency

AFDP is only the first stage of LD3CF. After AFDP, LD3CF performs a dual-pooling multimodal fusion strategy. For multimodal features, with $$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$2 the RGB modality feature and $$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$3 the auxiliary modality features, the RGB branch is first enhanced as

$$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$4

Each auxiliary modality then interacts with the RGB feature through

$$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$5

followed by

$$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$6

The manuscript notes a notation inconsistency: the text says $$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$7 and $$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$8 are learnable weights, while the equation shows $$\zeta \in \mathbb{R}^{B \times C \times H \times W},$$9 and $x \in \mathbb{R}^{B \times C \times H \times W}.$0 (Liu et al., 30 Jul 2025).

Cross-scale dynamic interaction follows dual pooling:

$x \in \mathbb{R}^{B \times C \times H \times W}.$1

with

$x \in \mathbb{R}^{B \times C \times H \times W}.$2

In this decomposition, AFDP handles frequency-domain enhancement, dual pooling handles multimodal spatial fusion, and the cross-scale gate balances semantic reinforcement against structural preservation (Liu et al., 30 Jul 2025).

The paper’s ablation results quantify AFDP’s contribution within the full LD3CF configuration.

Metric	Full configuration	Change when AFDP is removed
ODS	0.8213	-1.16%
OIS	0.8237	-1.13%
F1	0.8204	-1.26%
mIoU	0.8465	-0.74%

The full configuration corresponds to AFDP on, DualPool on, and CrossGate on (Liu et al., 30 Jul 2025). The paper interprets these results as evidence that AFDP enhances crack-region representation, reinforces high-frequency features, suppresses low-frequency background noise, and produces clearer textures for subsequent processing. It also notes that removing both dual pooling and cross-level gating causes larger degradation, indicating that AFDP is important but operates best as part of the full LD3CF design.

On computational cost, the paper does not isolate AFDP-only parameters or FLOPs, but states that LD3CF as a whole adds only 0.25 GFLOPs and 0.02M parameters (Liu et al., 30 Jul 2025). At the full-model level, LIDAR is reported as 33.33G FLOPs, 5.35M parameters, and 78MB in the dual-modal setting, and 16.66G FLOPs, 2.68M parameters, and 39MB in the RGB unimodal setting. Experiments on three datasets are said to show that LIDAR outperforms other SOTA methods, and on the light-field depth dataset the method achieves 0.8204 in F1 and 0.8465 in mIoU with only 5.35M parameters (Liu et al., 30 Jul 2025).

5. AFDP-like oscillatory perceptrons and storage capacity

A distinct but closely related research direction studies a generalized quantum perceptron architecture with a tunable oscillatory activation. The paper explicitly states that this architecture is very close in spirit to an AFDP because the perceptron’s nonlinearity is not a fixed step or sign rule, but a sinusoidal activation whose frequency is controlled by a tunable parameter $x \in \mathbb{R}^{B \times C \times H \times W}.$3 (Benatti et al., 2 Nov 2025).

The unitary analyzed is

$x \in \mathbb{R}^{B \times C \times H \times W}.$4

and for binary inputs $x \in \mathbb{R}^{B \times C \times H \times W}.$5, measuring $x \in \mathbb{R}^{B \times C \times H \times W}.$6 on the output qubit gives

$x \in \mathbb{R}^{B \times C \times H \times W}.$7

This is the AFDP-like element identified in the paper: a sinusoidal activation function with adjustable frequency $x \in \mathbb{R}^{B \times C \times H \times W}.$8.

The low-frequency limit is classical. As $x \in \mathbb{R}^{B \times C \times H \times W}.$9,

$$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$0

so the model becomes effectively linearized and the classification rule reduces to classical perceptron behavior (Benatti et al., 2 Nov 2025). The storage-capacity analysis is formulated in Gardner’s framework. For the oscillatory perceptron, the relevant version space is

$$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$1

The critical load is $$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$2, and the critical capacity is obtained from the limit $$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$3 in the replica calculation (Benatti et al., 2 Nov 2025).

The paper derives the explicit frequency-dependent storage capacity

$$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$4

It also gives the classical baseline

$$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$5

The main analytical claims are that $$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$6 for all $$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$7, that $$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$8 as $$F(u, v) = \sum_{x=0}^{H-1} \sum_{y=0}^{W-1} f(x, y)\cdot e^{-j 2\pi \left( \frac{ux}{H} + \frac{vy}{W} \right)}.$$9, and that $f(x,y)$0 is smooth at $f(x,y)$1 but nonanalytic there, with all derivatives at $f(x,y)$2 vanishing even though the function is not constant (Benatti et al., 2 Nov 2025).

The paper is equally explicit that this is only a “pseudo quantum advantage.” The enhancement is attributed to the chosen activation function, the resulting measurement-induced nonlinearity, and the altered geometry of the solution space, and is argued to be classically emulable in principle (Benatti et al., 2 Nov 2025). A genuine quantum advantage, in the paper’s framing, would require dependence on resources unavailable to classical systems, such as entanglement, nonclassical interference, or exponentially hard classical simulation. This distinction is significant for AFDP-style interpretation: the central mechanism is adaptive frequency-controlled nonlinearity, not irreducible quantum computation.

6. Related frequency-domain adaptation control and distinct acronym usage

Another nearby line of work appears in frequency-domain adaptive system identification. “End-To-End Deep Learning-Based Adaptation Control for Frequency-Domain Adaptive System Identification” presents DNN-FDAF, which is described as a learned, end-to-end analogue of the AFDP idea: the adaptation rate is controlled in the frequency domain, bin by bin, but the mapping from observed signal features to step-sizes is learned directly from data rather than imposed by heuristic or model-based rules (Haubner et al., 2021).

The underlying identification engine is a frequency-domain adaptive filter with update

$f(x,y)$3

$f(x,y)$4

The DNN controller maps normalized logarithmic power spectra of the current input block and prior error block to two diagonal masking matrices, $f(x,y)$5 and $f(x,y)$6, which modulate the step-size formula (Haubner et al., 2021). The architecture consists of a feedforward layer with $f(x,y)$7 activation, two stacked GRU layers, and two separate feedforward heads with sigmoid activations. The training objective is the average logarithmic normalized Euclidean system distance (NESD), optimized end to end through the adaptive filter update chain.

This work is not a paper on AFDP in the naming sense, but it is AFDP-like in the structural sense used by the paper itself: frequency-selective, feature-driven adaptation control rather than a single global update coefficient (Haubner et al., 2021). A plausible implication is that “AFDP” can also function as a broader design motif for frequency-domain adaptive control, even when the exact acronym is not formalized in the title of the method.

The distinct acronym usage in bioinformatics remains separate. In (Jozashoori et al., 2019), AFDP is an integrated approach for protein function prediction that combines eggNOG, curated Swiss-Prot or UniProt descriptions, and the suffix-tree-based stCFExt algorithm. Its benchmark reports evaluation scores of 0.4323 for stCFExt, 0.2061 for eggNOG, 0.2458 for AHRD, and 0.4254 for the best BLAST hit against Swiss-Prot (Jozashoori et al., 2019). That literature has no connection to frequency-domain perceptrons, Fourier masking, or adaptive frequency-selective neural processing.

Taken together, the cited work presents AFDP in two conceptually different senses: a concrete frequency-domain enhancement module in multimodal vision (Liu et al., 30 Jul 2025), and a broader family resemblance to frequency-controlled or frequency-selective perceptron mechanisms in statistical mechanics and adaptive filtering (Benatti et al., 2 Nov 2025, Haubner et al., 2021). The most precise current use of “Adaptive Frequency Domain Perceptron” as a named architectural block is the one in LIDAR, where it serves as the frequency-domain enhancement core of LD3CF (Liu et al., 30 Jul 2025).