Neighborhood Shape for Handling Image Boundaries

Typical image boundary conditions, e.g., replicating pixels or toroidal topologies, can produce neighborhoods that distort the feature-space statistics. We handle boundary neighborhoods by collapsing the feature space along the dimensions corresponding to the neighbors falling outside the image. We crop the square regions crossing image boundaries and process them in the lower-dimensional subspace, as in Figure 3.3(c).

Figure 3.3: Neighborhood shapes. (a) Preserving rotational invariance via a neighborhood mask consisting of a flat central circular plateau with cubic splines on the sides. (b) The discrete sampling of the mask (black $\equiv$ 1, white $\equiv$ 0) for a 9 $\times$ 9 pixels neighborhood. (c) Anisotropic neighborhoods at boundaries.

This strategy results in important modifications in the image-processing algorithms. First, the cropped intensity vectors are processed based on the Markov PDFs only in the particular subspace where they reside. Second, we choose the optimal Parzen-window kernel parameter $\sigma$ based only on the observations ${\bf z}_t$ at indices where the neighborhoods are not cropped.