Parzen-Window Sampling Schemes

This section discusses practical, effective strategies for choosing the sample $\mathcal{A}$ during the Parzen-window density estimation.

For images that conform very well to the stationarity assumption, we proposed the following strategy. To estimate the probability $P ({\bf z})$, we construct $\mathcal{A}$ as a random sample uniformly distributed over $\mathcal {R}$. We call this the global-sampling strategy. The random selection results in a stochastic approximation for the PDFs that alleviates the effects of spurious local maxima introduced in the finite-sample Parzen-window density estimate [170]. The uniform sampling works well for certain applications, e.g., while dealing with textured images which, by definition, are derived from stationary MRFs.

We have found that most image statistics are not stationary and, in practice, are more consistent in proximate regions in the image than between distant regions. In other words, images are better approximated as realizations of piecewise stationary-ergodic MRFs [175]. To account for this, we use a local-sampling strategy. In this local-sampling framework, for each voxel $t$, we draw a unique random sample $\mathcal{A} = \mathcal{A}_t$ from an isotropic Gaussian PDF, defined on the image-coordinate space, with mean at the voxel $t$ and variance $\sigma_{\mathrm{spatial}}^2$. Thus, the sample $\mathcal{A}_t$ is biased and contains more voxels near the voxel $t$ being processed. Experiments show that the method performs well for any choice of $\sigma_{\mathrm{spatial}}$ that encompasses more than several hundred voxels. Figure 3.2(a) shows a local random sample for a particular pixel of the Lena image.