Parzen Window Density Estimation

The prior joint intensity distribution can also be modeled using Parzen window density estimation. A mixture of Gaussians model follows from the idea that the different classes should roughly correspond to different anatomical structures and thus provides an approximate segmentation into tissue classes. However, the EM algorithm for estimating the parameters of a mixture of Gaussians is sensitive to the initialization of the parameters and in some cases can result in an inaccurate prior model of the joint intensities.

We therefore also consider modeling the joint intensity distribution based on the Parzen window density estimation using Gaussians as the windowing function. In practice, this model defined by directly sampling the training data provides a better explanation of the intensity relationship than the Gaussian mixtures that require the estimation of various parameters.

Consider our registered training image pair $\langle I_1, I_2 \rangle$. We estimate the joint intensity distribution of an intensity pair i = [i₁, i₂]^T given the prior model, M:

$$P(i \,\vert\, M) = \frac{1}{N} \sum_{\mu \in \langle I_1, I_... ...\sigma^2} e^{-\frac{1}{2\sigma^2}(i-\mu)^T (i-\mu)} \right)\;,$$ (6)

where the $\mu$'s are N samples of corresponding intensity pairs from the training images. Figure 4 illustrates this estimated joint intensity distribution.

  

Figure: Starting position, a middle position, and the final alignment computed by the registration gradient ascent algorithm. Each image shows the SPGR and PD overlayed in block format in the three orthogonal slices. The images in the upper right depict the histogram of the intensity pairs at that alignment. When the images are aligned, the histogram should resemble the distribution in Figure 4.