Adaptive Gradient & Phase Edge Operator

• Adaptive Gradient and Phase Edge Operator (AGPEO) is an edge extraction technique that combines multi-orientation gradient analysis with phase congruency to detect weak, diffuse boundaries.
• It integrates multi-scale filter responses via a learnable fusion weight, effectively balancing gradient magnitude and phase coherence.
• AGPEO improves deep network performance by enhancing the extraction of semi-transparent plume edges, leading to notable gains in gas leak detection metrics.

The adaptive gradient and phase edge operator (AGPEO) is an edge extraction mechanism designed to produce robust, content-adaptive edge priors within deep neural network backbones, particularly for visual tasks involving low-contrast and diffuse boundaries. AGPEO integrates multi-directional gradient analysis and multi-scale phase-consistent responses, fusing them via a learnable parameter. Its principal application has been in physics-inspired gas leak detection networks targeting the identification of small, semi-transparent plumes with weak, blurry boundaries in infrared imagery (Li et al., 29 Dec 2025).

1. Mathematical Definition and Core Components

Let $X \in \mathbb{R}^{C \times H \times W}$ denote a feature map extracted from a neural backbone, with $C$ channels and spatial dimensions $H \times W$. AGPEO comprises the following key calculations:

• Multi-directional gradient magnitude: For each pixel $(x, y)$, compute responses to Sobel-style filters $K_\theta$ at orientations $$\Theta = \{0^\circ, 45^\circ, 90^\circ, 135^\circ\}$$, then take the maximal absolute value:

$$G(x, y) = \max_{\theta \in \Theta} |(K_\theta * X)(x, y)|$$

• Phase congruency: For $S$ scales indexed by $s$, with quadrature filter pairs $f_{even,s}, f_{odd,s}$ (e.g., Gabor or Log-Gabor), calculate per-pixel real and imaginary components:

$$R_s^{even}(x, y) = (f_{even,s} * X)(x, y), \quad R_s^{odd}(x, y) = (f_{odd,s} * X)(x, y)$$

The local amplitude and phase are:

$$A_s(x, y) = \sqrt{ [R_s^{even}(x, y)]^2 + [R_s^{odd}(x, y)]^2 }$$

$$\varphi_s(x, y) = \mathrm{atan2}(R_s^{odd}(x, y), R_s^{even}(x, y))$$

Compute the amplitude-weighted mean phase:

$$\bar{\varphi}(x, y) = \mathrm{atan2} \left( \sum_{s} A_s(x, y) \sin \varphi_s(x, y), \sum_{s} A_s(x, y) \cos \varphi_s(x, y) \right)$$

And the phase congruency response:

$$P(x, y) = \frac{ \sum_s A_s(x, y) \cos [\varphi_s(x, y) - \bar{\varphi}(x, y)] }{ \sum_s A_s(x, y) + \epsilon }$$

• Adaptive fusion: Fuse the gradient and phase responses using a learnable scalar $\alpha \in [0, 1]$:

$$E_0(x, y) = \alpha \cdot G(x, y) + (1 - \alpha) \cdot P(x, y)$$

The final edge prior $E_0$ is thus content-adaptive, balancing local derivative magnitude with phase-coherent edge cues.

2. Multi-Directional Gradient Analysis

AGPEO employs depth-wise convolutions with fixed-oriented Sobel or Prewitt kernels to detect edge structures along principal axes (horizontal, vertical, and diagonals). Kernels are typically $3 \times 3$:

• $0^\circ$ (horizontal): $$\begin{bmatrix} -1 & 0 & +1 \ -2 & 0 & +2 \ -1 & 0 & +1 \end{bmatrix}$$
• $90^\circ$ (vertical): Transpose of $0^\circ$
• $45^\circ$ and $135^\circ$: Rotations of $0^\circ$

For each orientation, apply depth-wise convolution and take the pixel-wise maximum across directions. Optional normalization scales $G$ into $[0, 1]$ via data normalization or batch normalization. This multi-orientation strategy is effective at detecting directional edges even when contrast is low, as in faint plumes.

3. Multi-Scale Phase Congruency Computation

Edge detection via phase congruency exploits the property that sharp transitions in natural scenes yield locally coherent phase across scales when decomposed with a quadrature filter bank. For each scale:

• Convolve $X$ with even and odd filters to obtain analytic signal components.
• Compute amplitude and phase per pixel.
• Aggregate amplitudes and weighted phases to estimate the mean local orientation.
• Compute the congruency as the degree of phase alignment; edge locations correspond to high congruency across scales.

This approach is robust to intensity variation and excels in capturing weak, semi-transparent contours that may not appear prominent in gradient magnitude.

4. Fusion Mechanism and Learnable Weighting

The adaptive fusion mechanism interpolates between gradient and phase-based cues. The learnable weight $\alpha$ is initialized (e.g., 0.7) and optimized via back-propagation against the detection loss. Ablation studies—by fixing $\alpha = 0$ (phase-only) or $1$ (gradient-only)—demonstrate that adaptive fusion outperforms either pure approach for edge detection in complex, low-contrast environments.

5. Integration with Multi-Scale Edge Perception Module (MSEPM)

Once $E_0$ is computed, it is processed by the multi-scale edge perception module (MSEPM) to create a hierarchical edge feature representation:

• $E_0$ is passed through a $1 \times 1$ convolution to match the backbone’s shallow feature map dimensions.
• Successive downsampling with $2 \times 2$ max pooling and $1 \times 1$ convolutions produces a sequence $\{\tilde{E}_0, \tilde{E}_1, \ldots, \tilde{E}_N\}$, each aligned with the respective backbone resolution and channel count.
• These features are injected into the corresponding backbone stages via concatenation or addition, enabling the network to leverage edge priors consistently from shallow to deep levels, thereby reinforcing boundary information throughout the hierarchy.

6. Computational Characteristics and Implementation Considerations

The computational overhead of AGPEO is modest compared to modern detection backbones. For typical parameterization ($H = W \approx 640$, $C = 64$, $k = 3$, $S = 3$–$5$ scales), the FLOP count for AGPEO is in the low gigaflop regime, dwarfed by the cost of full convolutional passes through the backbone. Key points:

• All gradient kernels $K_\theta$ are fixed.
• Quadrature filters $f_{even,s}, f_{odd,s}$ are applied over several scales, typically $S \approx 3$.
• The fusion weight $\alpha$ is globally learned, not spatially varying.
• $\epsilon$ is set (e.g., $1 \times 10^{-4}$) for numerical stability.

Final edge priors are typically channel-collapsed (by mean or $1 \times 1$ conv) before injection into the backbone.

7. Applications and Performance Impact

AGPEO is central to the performance of the physics-edge hybrid gas dynamic routing network (PEG-DRNet) for infrared gas leak detection (Li et al., 29 Dec 2025). By producing robust priors aligned with faint, semi-transparent gas plumes’ contours, AGPEO aids the detection pipeline in both low-contrast and cluttered background scenarios. Experimental results demonstrate significant gains in AP, AP$_{50}$, and small-object AP on benchmark datasets, notably IIG and LangGas, as compared to modern CNN and Transformer-based detectors, while incurring minor computational overhead.

A plausible implication is that phase-informed edge priors, when adaptively fused with traditional gradients and injected in multi-scale form, directly improve the extraction and discrimination of weak, diffuse objects in other vision tasks characterized by poor edge salience or partial transparency.