Discussion

As we will show in subsequent sections, the key ideas in this section apply well in practice. Nevertheless, we can further improve the proposed method via some engineering advances. For instance, the method of nonparametric density estimation with single-scale isotropic Parzen-window kernels is, perhaps, one of the simplest such schemes. Parzen-window density estimation can improve by choosing kernels adaptively to accommodate the signal or noise. This, however, introduces a risk of overtraining. If we try to learn the subtle features in the data when the amount of data is insufficient, then we could end up learning the local noise patterns in feature space. The PDFs learned will not generalize well to predict the structure underlying the image data.

An intrinsic limitation of the model is that its performance degrades for image regions not having stationary statistics, because that is the assumption underpinning the adaptive-MRF model. Nevertheless, one of the interesting empirical outcomes of this dissertation is that the model, and the performance of the algorithms based on the model, performs well even as these conditions are relaxed.

All algorithms in this dissertation entail computation of the Markov probabilities an $O (1)$ times for processing each pixel. This makes the algorithmic complexity of methods based on this adaptive-MRF image model as $O (\vert\mathcal{T}\vert \vert\mathcal{A}_t\vert E^D)$ where $D$ is the image dimension and $E$ is the extent of the neighborhood along a dimension. This grows exponentially with increasing $D$ and, for many applications in this dissertation, the long computation times limit our experiments to $2$D images. The literature suggests some improvements for faster density estimation, e.g., reduction in the computational complexity via the improved fast-gauss transform [185]. Such an approach entails approximating the PDFs in the feature-space by grouping or clustering important chunks of feature-space vectors offline. Thus, although this preprocessing phase is computationally extensive, subsequent density estimates can be computed very fast. Most of our applications, however, take few iterations of processing--around five on average--and, hence, the performance gains by a direct application of the improved fast-Gauss transform are significantly offset by the increase in preprocessing time. Another alternative for speedup is to exploit parallelism. All algorithms proposed in this dissertation are relatively straightforward to parallelize on shared-memory-multiprocessor machines (e.g., dual-processor Pentium workstations; not distributed-shared-memory supercomputers) and shared-memory-multicore machines (e.g., those using dual-core Intel/AMD processors). In general, speedup from the parallelization will depend significantly on the locality of the data references and the cache management. For shared-memory machines with two processors, we obtain a speedup close to two.

The implications of the results in this dissertation are significant. They show that it is possible to construct nonparametric density estimates in the very high-dimensional spaces of image neighborhoods. These results also suggest that the statistical structure in these spaces captures important geometric properties of images. The adaptive-MRF formulation also generalizes in several different ways. All of the mathematics, statistics, and engineering in the proposed adaptive-MRF modeling scheme are appropriate for any kind of densely-sampled data including data on higher-dimensional image domains and vector-valued data. Furthermore, the same scheme could easily apply to other image representations, such as image pyramids, wavelets, or local geometric features.