Simple Multiresolution Filtering and the Spectra of Radio Galaxies and Supernova Remnants

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Abstract
A simple method is explored for examining the structures and spectral index distributions on different scales in Galactic and extragalactic radio sources, based on a multiresolution filtering technique suggested by Starck, Murtagh, & Bijaoui. Multiresolution filtering, whether in this simple algorithm or more sophisticated ones, is also useful for a wide variety of image analysis purposes. For spectral index studies, the filtering is complementary to other methods and is especially useful in separating large‐scale and small‐scale features with different spectral distributions. In this paper, filtered spectral distributions are produced for several sources. In the extragalactic sources, different spectral gradients are found for narrow and broad features as a function of distance from the terminal hot spots. This complicates the standard analyses used to understand relativistic particle acceleration and losses. We show that the high‐field regions can yield either flatter or steeper spectra than low‐field regions and be either brighter or fainter, depending on the particle history. In our supernova remnant example, Cas A, the spectra of the fine‐scale features are found, in the mean, to be flatter than those on larger scales. In addition, a clear asymmetry is found in the spectral indices of the largest scale emission and is related to other physical properties of the remnant.