  Despite the remarkable success of deep learning, an optimal convolution
operation on point clouds remains elusive owing to their irregular data
structure. Existing methods mainly focus on designing an effective continuous
kernel function that can handle an arbitrary point in continuous space. Various
approaches exhibiting high performance have been proposed, but we observe that
the standard pointwise feature is represented by 1D channels and can become
more informative when its representation involves additional spatial feature
dimensions. In this paper, we present Multidimensional Kernel Convolution
(MKConv), a novel convolution operator that learns to transform the point
feature representation from a vector to a multidimensional matrix. Unlike
standard point convolution, MKConv proceeds via two steps. (i) It first
activates the spatial dimensions of local feature representation by exploiting
multidimensional kernel weights. These spatially expanded features can
represent their embedded information through spatial correlation as well as
channel correlation in feature space, carrying more detailed local structure
information. (ii) Then, discrete convolutions are applied to the
multidimensional features which can be regarded as a grid-structured matrix. In
this way, we can utilize the discrete convolutions for point cloud data without
voxelization that suffers from information loss. Furthermore, we propose a
spatial attention module, Multidimensional Local Attention (MLA), to provide
comprehensive structure awareness within the local point set by reweighting the
spatial feature dimensions. We demonstrate that MKConv has excellent
applicability to point cloud processing tasks including object classification,
object part segmentation, and scene semantic segmentation with superior
results.
