Padding-free Convolution based on Preservation of Differential
  Characteristics of Kernels
Kuangdai Leng Jeyan Thiyagalingam
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.