# Padding-free Convolution based on Preservation of Differential Characteristics of Kernels

arXiv: https://arxiv.org/abs/2309.06370 · HTML: https://mlsanity.com/paper/2309.06370/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 12 Sep 2023.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Cube Padding for Weakly-Supervised Saliency Prediction in 360deg Videos](https://mlsanity.com/paper/1806.01320/) (2018)
- [Mind the Pad -- CNNs can Develop Blind Spots](https://mlsanity.com/paper/2010.02178/) (2020)
- [Partial Convolution based Padding](https://mlsanity.com/paper/1811.11718/) (2018)
- [Position, Padding and Predictions: A Deeper Look at Position Information in CNNs](https://mlsanity.com/paper/2101.12322/) (2021)
- [Convolution with even-sized kernels and symmetric padding](https://mlsanity.com/paper/1903.08385/) (2019)
- [Effects of boundary conditions in fully convolutional networks for learning spatio-temporal dynamics](https://mlsanity.com/paper/2106.11160/) (2021)
- [Learning on the Edge: Explicit Boundary Handling in CNNs](https://mlsanity.com/paper/1805.03106/) (2018)
- [Tracking translation invariance in CNNs](https://mlsanity.com/paper/2104.05997/) (2021)
- [Context-aware Padding for Semantic Segmentation](https://mlsanity.com/paper/2109.07854/) (2021)
- [Unveiling The Mask of Position-Information Pattern Through the Mist of Image Features](https://mlsanity.com/paper/2206.01202/) (2022)
- [Improved Padding in CNNs for Quantitative Susceptibility Mapping](https://mlsanity.com/paper/2106.15331/) (2021)
- [Random Padding Data Augmentation](https://mlsanity.com/paper/2302.08682/) (2023)

## Built on by (most cited first, 1 of 1)

- [Per-channel autoregressive linear prediction padding in tiled CNN processing of 2D spatial data](https://mlsanity.com/paper/2502.12300/) (2025)

## Abstract (arXiv)

Convolution is a fundamental operation in image processing and machine learning. Aimed primarily at maintaining image size, padding is a key ingredient of convolution, which, however, can introduce undesirable boundary effects. We present a non-padding-based method for size-keeping convolution based on the preservation of differential characteristics of kernels. The main idea is to make convolution over an incomplete sliding window "collapse" to a linear differential operator evaluated locally at its central pixel, which no longer requires information from the neighbouring missing pixels. While the underlying theory is rigorous, our final formula turns out to be simple: the convolution over an incomplete window is achieved by convolving its nearest complete window with a transformed kernel. This formula is computationally lightweight, involving neither interpolation or extrapolation nor restrictions on image and kernel sizes. Our method favours data with smooth boundaries, such as high-resolution images and fields from physics. Our experiments include: i) filtering analytical and non-analytical fields from computational physics and, ii) training convolutional neural networks (CNNs) for the tasks of image classification, semantic segmentation and super-resolution reconstruction. In all these experiments, our method has exhibited visible superiority over the compared ones.