Neighborhood Shape for Rotational Invariance

Hypercube-shaped neighborhoods $\mathcal{N}_t$ produce results with undesirable artifacts exhibiting preferences for grid-aligned features. A solution is to weight the intensities, making neighborhoods more isotropic. We incorporate such fuzzy weights by using an anisotropic feature-space distance metric, $\parallel {\bf z} \parallel_M = \sqrt { {\bf z}' M {\bf z} }$, where ${\bf z}'$ is the transpose of ${\bf z}$, and $M$ is a diagonal $d \times d$ matrix with the elements being the appropriate weights on the influence of the neighbors on the center pixel. Figure 3.3(a)-(b) shows the disk-shaped mask that we use in this dissertation. The intensities near the center are unchanged ($M (i,i) = 1$) while the intensities near the corners are weighted down $M (i,i) < 1$ gradually, via cubic-spline interpolation, to zero. The proposed isotropic mask is a grayscale version of the DUDE [175] strategy of using a binary disc-shaped mask for discrete (half-toned) images.