Pixel Classification

Recall that the key step of thresholding techniques described in section 3.4.1 is the choice of thresholds that is determined either manually or in a semiautomatic manner based on the local statistics such as mean, maximum, or minimum of the given image (or subimages). The basic concept of threshold selection can be generalized, leading to a data-driven paradigm, which determines the threshold automatically based on clustering techniques or artificial neural networks.

Pixel classification methods that use histogram statistics to define single or multiple thresholds to classify an image can be regarded as a generalization of thresholding techniques. It is particularly useful when the pixels have multiple features, which can be expressed in terms of a vector in multidimensional feature space. For instance, the feature vector may consist of gray level, local texture, and color components for each pixel in the image. In the case of single-channel (or single-frame) image, pixel classification is typically based on gray level and image segmentation can be performed in a one-dimensional feature space. Segmentation can be performed in multidimensional feature space through clustering of all features of interest for multichannel (multiple-frame) images or multispectral (multimodality) images.

Clustering, or cluster analysis, has been widely applied in anthropology, archaeology, psychiatry, and zoology, etc, for many years. An example of clustering is the taxonomy of animals and plants whose names have to be the same between different people for effective communication, although it is not necessary that the naming scheme is the best [39]. Clustering is the process of grouping of similar objects into a single cluster, while objects with dissimilar features are grouped into different clusters based on some similarity criteria. The similarity is quantified in terms of an appropriate distance measure. An obvious measure of the similarity is the distance between two vectors in the feature space which can be expressed in terms of Lp norm as d{xi, x i }=(£|| x - x i yp) (3.13)

where x e R™ and xj e R™ are the two vectors in the feature space. It can be seen that the above measure corresponds to Euclidean distance when p = 2 and Mahalanobis distance when p = 1. Another commonly used distance measure is the normalized inner product between two vectors given by

xi ' x j where T denotes the transpose operation. The above distance measure is simply the angle between vectors xi and xj in the feature space.

Each cluster is represented by its centroid (or mean) and variance, which indicates the compactness of the objects within the cluster, and the formation of clusters is optimized according to a cost function that typically takes the similarity within individual cluster and dissimilarity between clusters into account. There are many clustering techniques proposed in the literature (see Ref. [39]). The most famous clustering techniques are ^-means [40], fuzzy c-means [41], ISODATA [42], hierarchical clustering with average linkage method [43], and Gaussian mixture approach [44].

As we will see later in this chapter, the idea of pixel classification in two-dimensional image segmentation using clustering techniques can be extended to multidimensional domain where the images convey not only spatial information of the imaged structures but also their temporal variations, for which clustering plays a pivotal role in identification of different temporal kinetics present in the data, extraction of blood and tissue TACs, ROI delineation, localization of abnormality, kinetic modeling, characterization of tissue kinetics, smoothing, and fast generation of parametric images.

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