Linking image compression and image analysis through Taylor–Riesz pyramids

Compressed signal representations find applications in image analysis through the reductions in numerical overheads made possible by downsizing the physical dimensions of an image signal. The pyramid algorithm of Burt and Adelson (1983 IEEE Transactions on Communications 31 532–540) is a popular choice that splits the image signal into two parts. One part is lowpass and recursively downsized, while the other is highpass but retains the original signal dimensions, which is inefficient. As a general rule for image compression without loss of information, a downsizing factor should equal the number of spatial filters. Here we show that a Taylor–Riesz expansion of the image signal allows one to downsize an image signal by an order equal to the n + 1th Taylor–Riesz derivatives of the image signal. Application of the unique relationship allows us to both orthogonalize and decimate image signals by large factors (~16×) that approach those used in current image compression coding schemes. Low-level image properties like contrast, orientation, and energy may be estimated as a part of the downsizing process. The Riesz transform also allows one to steer continuously across 2‑D phase and spatial orientation while allowing access to a local definition of energy, which is itself a lowpass signal, and may itself be downsized. In downsizing both luminance and energy signals, we demonstrate how both coarse-to-fine and global-to-local image information may accessed by combining luminance and energy-based computations. We further demonstrate that the Riesz transform and its generalizations may be regarded as eigen-functions of a polar separable sampling mosaic. Our results suggest that polar separable signal representations based upon 2‑D phase and the Riesz transform lead to a unification of ideas capable of merging techniques for image compression and image analysis.