  Convolutional neural networks use regular quadrilateral convolution kernels
to extract features. Since the number of parameters increases quadratically
with the size of the convolution kernel, many popular models use small
convolution kernels, resulting in small local receptive fields in lower layers.
This paper proposes a novel log-polar space convolution (LPSC) method, where
the convolution kernel is elliptical and adaptively divides its local receptive
field into different regions according to the relative directions and
logarithmic distances. The local receptive field grows exponentially with the
number of distance levels. Therefore, the proposed LPSC not only naturally
encodes local spatial structures, but also greatly increases the single-layer
receptive field while maintaining the number of parameters. We show that LPSC
can be implemented with conventional convolution via log-polar space pooling
and can be applied in any network architecture to replace conventional
convolutions. Experiments on different tasks and datasets demonstrate the
effectiveness of the proposed LPSC. Code is available at
https://github.com/BingSu12/Log-Polar-Space-Convolution.
