Results and Validation

Table 6.1: The proposed method is fairly robust to changes in the values of the local-sampling Gaussian variance parameter and the Parzen-window $\sigma$ multiplicative factor. This table gives the Dice metrics for the BrainWeb T1 data with $5 \%$ noise and a $40 \%$ bias field.

Local-sampling Gaussian variance	Gray matter	White matter
100	0.9033	0.9386
225	0.9079	0.9427
400	0.9082	0.9422
625	0.9043	0.9368
Parzen-window $\sigma$ multiplicative factor	Gray matter	White matter
1.0	0.7634	0.9105
2.5	0.8988	0.9502
5.0	0.9106	0.9487
7.5	0.9095	0.9451
10.0	0.9079	0.9427
12.5	0.9066	0.9411
15.0	0.9058	0.9402

This section gives validation results on real and synthetic brain MR images along with the analysis of the method's behavior. It also provides quantitative comparisons with a current state-of-the-art classification method [93,94]. For all the results in this paper, we use a first-order neighborhood system for the MRF model. Thus, each pixel has $6$ neighbors--$2$ neighbors along each of the $3$ coordinate axes for the volumetric MR data. For all of the results in this chapter, we use $\sigma_{\mathrm{spatial}} = 15$ voxels along each cardinal direction. The empirical results in Table 6.1 confirm that the performance of the proposed method degrades gracefully for suboptimal values of this parameter. This local-sampling strategy also plays an important role in implicit inhomogeneity handling by enabling the method to subsume the bias field in the estimated Markov statistics that determine the segmentation. For all voxels $t$, the proposed method sets $\vert\mathcal{A}_t\vert = 500$, based on the method explained in Section 3.5.2. The computation for each iteration is $O (K \vert\mathcal{A}_t\vert \vert\mathcal{T}\vert \vert\mathcal{N}_t\vert)$. The algorithm typically takes about $4$ to $7$ iterations to converge depending on the noise/bias level. The implementation takes about 45 minutes to process a 181-voxels $\times$ 217-voxels $\times$ 181-voxels volume on a single Pentium-IV $2.8$GHz workstation. The implementation runs about twice as fast on a dual-processor shared-memory Pentium machine. The implementation in this chapter relies on the Insight Toolkit [2].

Leemput et al. [94] use the Dice metric [44] to evaluate the classification performance of their state-of-the-art approach, which is based on EM and Gibbs/Markov priors on the segmentation labels. For a direct comparison, we use the same metric. Let $\{\tilde \mathcal{T}_k\}_{k=1}^K$ denote the ground-truth classification and $\{\mathcal{T}^*_k\}_{k=1}^K$ denotes the classification obtained from the proposed method. Then, the Dice metric $D_k$ that quantifies the quality of the classification for class $k$ is $2 \vert \mathcal{T}^*_k \cap \tilde \mathcal{T}_k\vert / (\vert \mathcal{T}^*_k\vert + \vert\tilde \mathcal{T}_k\vert)$, where the $\vert\cdot\vert$ operator gives the cardinality of sets.