Results

This section presents results from experiments with real and synthetic data. The number of regions $K$ is a user parameter and should be chosen appropriately. The neighborhood size, in the current implementation, is also a user parameter. This can be improved by using a multiresolution scheme for the image representation. We use 9 $\times$ 9 pixels neighborhoods for all examples, unless we explicitly state otherwise. We choose $\beta = 2$, $\gamma = 3$, and $\vert\mathcal{A}_t\vert = 1000$. 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 less than $10$ iterations to converge. Each iteration of the proposed method takes about $3$ minutes for a 256 $\times$ 256 pixels image on a standard 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].

Figure 7.2(a) shows a level-set initialization $\{R_k^0\}_{k=1}^K$ as a randomly generated image with $K = 2$ regions. The level-set scheme using threshold dynamics, coupled with the global-sampling strategy as explained in Section 3.5.1, makes the level sets evolve very fast towards the optimal segmentation. We have found that, starting from the random initialization, just a few iterations (less than $10$) are sufficient to reach a virtually-optimal segmentation. However, this sampling strategy sometimes falls short of giving very accurate boundaries. This is because, in practice, the texture boundaries present neighborhoods overlapping both textures and exhibiting subtleties that may not be captured by the global sampling. Figure 7.2(b) depicts this behavior.

We can handle texture boundaries better by selecting a larger portion of the samples in $\mathcal{A}_t$ from a region close to $t$ might help. Hence, we propose a second stage of level-set evolution for a few iterations that incorporates local sampling, in addition to global sampling, and is initialized with the segmentation resulting from the first stage. We found that such a scheme produces consistently better segmentations.

Figure 7.2(c) shows the final segmentation. For each pixel $t$, we have used a random sample of size $\vert\mathcal{A}_t\vert = 250$ taken from a Gaussian distribution, with a standard-deviation $\sigma_{\mathrm{spatial}} = 30$ and mean at the pixel $t$. Furthermore, we have found that the method performs well for any choice of the variance such that the Gaussian distribution encompasses more than several hundred pixels. Note that given this variance, both $\vert\mathcal{A}_t\vert$ and the Parzen-window $\sigma_{\mathrm{spatial}}$ are computed automatically in a data-driven manner, as explained before in Section 3.5.1 and Section 3.5.2.

Figure 7.2: Two-texture segmentation. (a) Random initial segmentation for an image having two Brodatz textures for grass and straw. The black and white intensities denote the two regions. (b) Segmentation after stage 1; global samples only (see text). (c) Segmentation after stage 2; local and global samples (see text).

Figure 7.3 gives examples dealing with multiple-texture segmentation. Figure 7.3(a) shows a randomly generated initialization with three regions that leads to the final segmentation in Figure 7.3(b). In this case the proposed algorithm uses a multiphase extension of the fast threshold-dynamics based scheme [54,53]. Figure 7.3(c) shows another multiple-texture segmentation with four textures.

Figure 7.3: Multiple-texture segmentation. (a) Random initial segmentation containing three regions for the image in (b). (b) Final segmentation for an image with three Brodatz textures, including both irregular and regular textures. (c) Final segmentation for an image with four Brodatz textures.

Figure 7.4 shows electron-microscopy images of cellular structures. Because the original images severely lacked contrast, we preprocessed them using adaptive histogram equalization before applying the proposed texture-segmentation method. Figure 7.4 shows the enhanced images. These images are challenging to segment using edge or intensity information because of reduced textural homogeneity in the regions. The discriminating feature for these cell types is their subtle textures formed by the arrangements of sub-cellular structures. To capture the large-scale structures in the images we used larger neighborhood sizes of 13 $\times$ 13 pixels. We combine this with a higher $\gamma$ for increased boundary regularization. Figure 7.4(a) demonstrates a successful segmentation. In Figure 7.4(b) the two cell types are segmented to a good degree of accuracy; however, notice that the membranes between the cells are grouped together with the middle cell. A third texture region could be used for the membrane, but this is not a trivial extension due to the thin, elongated geometric structure of the membrane and the associated difficulties in the Parzen-window sampling. The hole in the region on the top left forms precisely because the region contains a large elliptical patch that is identical to such patches in the other cell. Figure 7.4(c) shows a successful three-texture segmentation for another image.

Figure 7.4: Final segmentations for electron-microscopy images of rabbit retinal cells for (a),(b) the two-texture case, and (c) the three-texture case.

Figure 7.5(a) shows a zebra example that occurs quite often in the texture-segmentation literature, e.g., [148,144]. Figures 7.5(b) and 7.5(c) show other zebras. Here, the proposed method performs well to differentiate the striped patterns, with varying orientations and scales, from the irregular grass texture. The grass texture depicts homogeneous statistics. The striped patterns on the zebras' bodies, although incorporating many variations, change gradually from one part of the body to another. Hence, neighborhoods from these patterns form one continuous manifold in the associated high-dimensional feature space, which is captured by the method as a single texture class.

Figure 7.5: Final segmentations for real images of Zebras.

Figure 7.6(a) shows the successful segmentation of the Leopard with the random sand texture in the background. Figure 7.6(b) shows an image that actually contains three different kinds of textures, where the background is split into two textures. Because we constrained the number of regions to be two, the method grouped two of the background textures into the same region.

Figure 7.6: Final segmentations for real images of Leopards. Note: The segmentation outline for image (b) is shown in gray.

We can alleviate the sensitivity of the model to the neighborhood size by considering a multiscale adaptive-MRF model, which forms an important future engineering extension to the proposed algorithm. Such a model relies on the assumption of MRFs at each level or scale of a specific multiscale image pyramid [122]. This would significantly enhance the utility of the algorithm to images of varied resolutions comprising fractal-like textures with regularities at all scales.