II.      Appearance-based Recognition for CAD in CT Colonography

The development of conventional CAD schemes is usually based on model-based recognition, where the design of the detection methods is based on the use of explicit models of the target lesions. For example, the fully automated CAD scheme that was developed in Ref. [5] represents model-based recognition. The CAD scheme extracts the colonic wall based on an anatomical model of the colon, and compares the local shape characteristics of the extracted region with that of a polypoid shape model. The polypoid shape model was based upon two volumetric rotation-invariant shape features, the shape index (SI) and the curvedness (CV). The SI feature characterizes the topological 3-D shape of a local iso-intensity surface in the vicinity of a voxel, and the CV feature characterizes the flatness of that shape. They can be calculated from the principal curvatures   and  of the local 3-D iso-surface at voxel p as follows:



Because of the high detection sensitivity of the CAD scheme, it also includes a Bayesian neural network model to reduce false-positive (FP) detections, where the input features have been modeled to identify typical false positives. For example, a 3-D gradient concentration feature is used to identify folds based on the concentration of the gradient vectors in the vicinity of an operating point.

Although model-based CAD schemes can provide excellent detection performance, such schemes have several fundamental limitations. The model-based paradigm requires explicit mathematical modeling of the problem; however, it is often not obvious how to develop such a model with sufficient detail for practical applications in a clinical setting. This makes the development of model-based systems a demanding and time-consuming task that requires a considerable amount of human labor.  For example, Fig. 1 shows images of 48 polyps in our database of CT colonographic images (see Section V.A). These images demonstrate that the shape of polyps varies substantially across different polyps; thus the characterization and detection of polyps based only on the shape information is limited. Furthermore, a specific model that is developed for a specific application tends to generalize poorly to other problems.  Finally, despite the progress made in CAD methodology during the past years, the CAD schemes that are currently used in a clinical setting tend to generate a large number of FP detections that irritate experienced readers.

Appearance-based recognition is another widely used computer vision methodology, although, currently, it is rarely applied for medical imaging applications.  We are developing an appearance-based method that complements the performance of a model-based CAD scheme in the detection of polyps. In this paper, we propose using an appearance-based method for texture-based feature analysis of the polyp candidates, obtained by a shape-based method, for discrimination of false positives from true polyps. Unlike the model-based approaches, the texture-based feature analysis does not specify the solid target model; instead, the texture features are simply handled as instances in the analysis process.

For efficient feature analysis, extraction of the salient features of polyps is essential because of the size and the 3-D nature of the polyp datasets.  Moreover, the distribution of the image features of polyps is expected to be non-linear. Therefore, we employ Kernel Principal Component Analysis (Kernel PCA) [15, 16, 19, 20, 21], which is well known as a superior data compression method, and its extension to a non-linear feature space, kernel feature space, in this paper.

Fig. 2 illustrates the concept of feature dissimilarity and of classifier selection by use of the kernel feature space.  Here, Fig. 2(a) shows a set of linearly inseparable two-class features in the input space, in which features belonging to a class is labeled by green and those belonging to the other class is labeled by red, and Fig. 2(b) shows a set of linearly separable features using a hyper plane. The features in Fig. 2(a) are transferable to the feature set in Fig. 2(b) by converting the input space into an appropriate higher-dimensional feature space using an operator, so that the features in the two classes can be linearly separable.

The problem is how to select such an ideal operator.  A nonlinear, positive-definite kernel  of an integral operator, e.g., , computes the inner product of the transformed vectors , where  denotes a nonlinear embedding (induced by k) into a possibly infinite dimensional Hilbert space H. Given n sample points in the domain , the image  of  spans a linear subspace of at most  dimensions. Heuristically, the dominant linear correlations in the distribution of the image  may elucidate important nonlinear dependencies in the original data sample . This is advantageous because it permits making PCA non-linear without complicating the original PCA algorithm.  The kernel function k is traditionally chosen in the form of a Gaussian function such as Radial Basis Functions (RBF): .  As visually demonstrated in Fig. 3, when feature points in the input space are mapped, via the RBF kernel function, to higher-dimensional space corresponding to an inner product in an expanded feature space, features belonging to different classes can be well clustered, and thus the kernel space can be efficient in discriminating one class, e.g., false positives, from the other class, e.g., true polyps.