Analysis of directional regularity by transforms with parabolic scaling

Abstract

In an effort to characterize uniform and pointwise regularities, we obtain necessary decay rates and sufficient decay rates of continuous shearlet transform across scales. They are the same rates as those of the Hart Smith and continuous curvelet transforms. We then consider the situation where regularity on a line in a non-parallel direction is much lower than directional regularity along the line in a neighborhood. Similar to that of the Hart Smith and continuous curvelet transforms, a set of necessary conditions for this direction of singularity is that the continuous shearlet transform decays half an order faster in directions “away” from the direction of the line and that the decay rate in directions “near” the line depends also on the horizontal distance from the line to the parallel line containing the center of the shearlet function. Moreover, we define a new directional regularity based on parabolic scaling. Then we obtain necessary condition and sufficient condition for analysis the directional regularity by discrete wavelet transform