Efficient computation of channel-coded feature maps through piecewise polynomials

Channel-coded feature maps (CCFMs) represent arbitrary image features using multi-dimensional histograms with soft and overlapping bins. This representation can be seen as a generalization of the SIFT descriptor, where one advantage is that it is better suited for computing derivatives with respect to image transformations. Using these derivatives, a local optimization of image scale, rotation and position relative to a reference view can be computed. If piecewise polynomial bin functions are used, e.g. B-splines, these histograms can be computed by first encoding the data set into a histogram-like representation with non-overlapping multi-dimensional monomials as bin functions. This representation can then be processed using multi-dimensional convolutions to obtain the desired representation. This allows to reuse much of the computations for the derivatives. By comparing the complexity of this method to direct encoding, it is found that the piecewise method is preferable for large images and smaller patches with few channels, which makes it useful, e.g. in early steps of coarse-to-fine approaches.