Multi-Directional Perception Convolution (MDPConv)

• MDPConv is a convolutional method that integrates directional gradient cues via multiple branches to capture anisotropic spatial features.
• It employs a re-parameterization strategy that fuses multiple directional kernels into a single efficient kernel for low-latency inference.
• MDPConv has achieved state-of-the-art results in applications such as image deraining, medical segmentation, and geometric deep learning.

Multi-Directional Perception Convolution (MDPConv) comprises a class of convolutional operations explicitly designed to enhance feature extraction by integrating gradient or orientation-based cues along multiple spatial directions. This approach has broad technical motivations, ranging from geometric deep learning on manifolds to image restoration in the presence of directional artifacts. MDPConv modules have been recently formalized in standard Euclidean image processing contexts (notably in DeRainMamba (Zhu et al., 8 Oct 2025)), but their conceptual antecedents include directional convolutions on curved surfaces (Poulenard et al., 2018) and multi-directional integration methods for semantic segmentation (You et al., 2020).

1. Theoretical Foundations

The central premise underlying MDPConv is that spatial structures (edges, textures, contours) contain information that is not isotropic—i.e., the statistical properties of features depend on their direction of propagation. To exploit this, convolutional operators compute differences (gradients) or responses along multiple canonical orientations.

A canonical implementation is found in DeRainMamba (Zhu et al., 8 Oct 2025), where MDPConv explicitly uses five distinct convolutional branches: horizontal, vertical, angular, central, and standard (vanilla). Each branch employs a different convolutional kernel $K_i$, designed to produce sensitivity to a particular class of gradient or orientation:

$$F_{out} = \text{MDPConv}(F_{in}) = \sum_{i=1}^{5} (F_{in} * K_i) = F_{in} * \left(\sum_{i=1}^{5} K_i \right) = F_{in} * K_{eq}$$

where $F_{in}$ and $F_{out}$ denote input and output feature maps, $*$ is the convolution operator, $K_i$ are the branch kernels, and $K_{eq}$ is the unified, re-parameterized kernel for efficient inference.

In geometric contexts (Poulenard et al., 2018), the notion is formalized via directional functions $\varphi(x, v)$ on a surface $X$, for $x \in X$ and $v$ in the unit tangent circle $S^1$. Directional convolution is performed as:

$$(\varphi \star k)(x, v) = \langle (\overline{\exp}_{x \circ x})^*\varphi, \tau_{x,v}^*k \rangle_{L^2}$$

where the kernel is aligned to the reference direction $v$ inside the tangent plane at $x$.

2. Multi-Directional Gradient Extraction

MDPConv modules typically instantiate several differential operators targeting signal change along multiple axes. In DeRainMamba (Zhu et al., 8 Oct 2025):

• Horizontal Differential Convolution (HDC): Captures horizontal gradients.
• Vertical Differential Convolution (VDC): Captures vertical gradients.
• Angular Difference Convolution (ADC): Captures oblique, non-orthogonal gradients.
• Central Difference Convolution (CDC): Captures pixel-level local variation.
• Vanilla Convolution (VC): Retains traditional isotropic convolution.

The use of multiple branches targets anisotropy in the image data, which is especially pertinent for tasks such as deraining or segmentation. For example, rain streaks are directional, typically vertical or slanted, and fine object contours may align with various axes.

In multi-scale variants (You et al., 2020), feature maps are partitioned and flipped across major axes (0°, 90°, 180°, 360°), with each orientation processed separately. This strategy amplifies sensitivity to feature orientations and allows the network to learn direction-specific semantic responses.

3. Efficient Fusion: Re-Parameterization and Inference

To balance expressivity against computational cost, modern designs apply re-parameterization at inference time, merging multiple directional kernels into a single aggregated kernel $K_{eq}$. During training, each branch operates individually, allowing gradient propagation along separately tuned directions. During deployment, the branches are mathematically summed to eliminate redundant computation, yielding an optimized single-kernel convolution.

This approach preserves the diverse gradient cues harvested in training while allowing for efficient, low-latency inference. The effectiveness of this design is empirically substantiated in DeRainMamba (Zhu et al., 8 Oct 2025), where models with re-parameterized MDPConv achieve improvements in PSNR and SSIM with fewer parameters and reduced FLOPs compared to multi-branch architectures operated naively.

4. Feature Mining, Semantic Enrichment, and Noise Suppression

MDPConv encapsulates principles of feature enrichment by integrating directional cues and multi-scale receptive fields. In medical image segmentation networks such as DT-Net (You et al., 2020), the multi-directional integrated convolution (MDIC) divides feature maps into subregions, applies distinct spatial flipping operations, and processes each with convolutional kernels of varying sizes. This results in feature sets that preserve detailed orientation-specific information, minimize redundancy, and maximize semantic coverage.

In addition, associated modules such as threshold convolution eliminate superfluous activations below a fixed threshold, reducing noise and focusing learning on salient spatial regions. This synergy has been shown to produce state-of-the-art results on segmentation benchmarks—such as a liver segmentation accuracy of 93.55% on CHAOS, with higher Dice and specificity than previous baselines (You et al., 2020).

5. Geometric Deep Learning and Manifold Extensions

MDPConv’s theoretical lineage is found in geometric deep learning via directional convolution operators defined on manifolds (Poulenard et al., 2018). Directional functions $\varphi(x, v)$ keep track of angular dependencies across layers, and directional convolution exploits completed exponential maps and parallel transport to propagate orientation information intrinsically across the mesh.

A salient property, proven in (Poulenard et al., 2018), is rotation equivariance: changing the reference direction induces a deterministic shift in the output’s angular coordinate. This property contrasts with traditional methods that align templates arbitrarily or discard directional cues (e.g., angular max-pooling), leading to richer, more robust multimodal feature representations.

Stacking directional convolutional layers allows propagation of orientation-sensitive information over complex surfaces, benefitting tasks such as shape segmentation (where smooth and accurate boundaries are needed) and shape matching (where correspondences must remain robust across resolution changes).

6. Representative Applications and Performance Assessments

MDPConv and its derivatives have been validated in multiple domains:

• Image Deraining: In DeRainMamba (Zhu et al., 8 Oct 2025), incorporation of MDPConv improved PSNR (from 41.13 dB to 41.49 dB) when compared with Mamba-only baselines, and further to 41.71 dB with joint frequency and gradient modules, demonstrating concurrent improvements in SSIM (up to 0.9899).
• Medical Image Segmentation: In DT-Net (You et al., 2020), MDIC and threshold convolution achieved state-of-the-art Dice and specificity scores on CHAOS and BraTS datasets, outperforming U-Net, CE-Net, and others.
• Geometric Classification, Segmentation, and Matching: Directional geodesic convolution (Poulenard et al., 2018) delivered superior accuracy and stability over standard GCNNs on tasks such as CIFAR-10 classification mapped to surfaces, human shape segmentation, and non-rigid shape matching.

The table below summarizes core variants and their application foci:

Module/Variant	Application Domain	Effect
MDPConv (Zhu et al., 8 Oct 2025)	Image Deraining	Spatial detail restoration, anisotropic gradient cue extraction
MDIC (You et al., 2020)	Medical Image Segmentation	Enhanced multi-scale, multi-orientation semantic maps
Directional Convolution (Poulenard et al., 2018)	Geometric Deep Learning	Propagation of orientation-encoded signals on manifolds

7. Future Research Directions

The literature identifies several promising extensions:

• Multi-scale modules: Incorporating variable receptive window sizes, akin to Inception or spatial pyramid pooling, is anticipated to further bolster scale-robustness (Poulenard et al., 2018).
• Alternative orientation representations: Use of Fourier bases for angular functions could provide exact invariance under rotation, although non-linear activations and pooling become operationally complex.
• Fusion of intrinsic and extrinsic information: Combining coordinate-free manifold representations with extrinsic geometric cues may enhance model transferability to domains with strong orientation priors.
• Broadening application scope: MDPConv principles may benefit domains where feature extraction across multiple canonical directions is essential, such as remote sensing, industrial inspection, and natural scene understanding.

A plausible implication is that as architectural integration of directional cues becomes standard, improvements in robustness to rotation, deformation, and noise will propagate across diverse subfields of deep vision and geometric learning.