Efficient Spatially Adaptive Convolution and Correlation

arXiv preprint, June 2020

Thomas W. Mitchel, Benedict Brown, David Koller,
Tim Weyrich, Szymon Rusinkiewicz, Michael Kazhdan

Applications of extended convolution. Left: Rotation-independent pattern matching was used to locate the pattern in the image at left. The three correct matches correspond to the three peaks in the match-quality image. Center: A rotation-dependent filter applied to a photograph with added noise produces an artistic effect. Right: Scale-dependent smoothing is used to remove compression artifacts from an image while preserving edges.

Fast methods for convolution and correlation underlie a variety of applications in computer vision and graphics, including efficient filtering, analysis, and simulation. However, standard convolution and correlation are inherently limited to fixed filters: spatial adaptation is impossible without sacrificing efficient computation. In early work, Freeman and Adelson have shown how steerable filters can address this limitation, providing a way for rotating the filter as it is passed over the signal. In this work, we provide a general, representation-theoretic, framework that allows for spatially varying linear transformations to be applied to the filter. This framework allows for efficient implementation of extended convolution and correlation for transformation groups such as rotation (in 2D and 3D) and scale, and provides a new interpretation for previous methods including steerable filters and the generalized Hough transform. We present applications to pattern matching, image feature description, vector field visualization, and adaptive image filtering.

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This paper on arXiv

Thomas W. Mitchel, Benedict Brown, David Koller, Tim Weyrich, Szymon Rusinkiewicz, and Michael Kazhdan.
"Efficient Spatially Adaptive Convolution and Correlation."
arXiv:2006.13188, June 2020.

@techreport{Mitchel:2020:ESA,
   author = "Thomas W. Mitchel and Benedict Brown and David Koller and Tim Weyrich
      and Szymon Rusinkiewicz and Michael Kazhdan",
   title = "Efficient Spatially Adaptive Convolution and Correlation",
   institution = "arXiv preprint",
   year = "2020",
   month = jun,
   number = "2006.13188"
}