The Surface Classifier approach for the fusion of sensor data has been shown to produce improved classification performance over traditional methods that characterize object classes as a single mean feature vector and associated covariance. The key aspect of this approach is the notion of characterizing object classes as parametric representations of curves, or surfaces, in feature space that capture the underlying correlations between features. By performing calculations in this representation of feature space, the fusion of feature data from the two sensors was seen to be straightforward. In this paper, the Surface Classifier approach is extended to combine multiple observations of these objects into a 'manifold fragment' that is fitted to the surface representing an object's parametric representation in feature space. Additionally, by using a 'Torn-Surface' representation of the object classes, the approach is able to address discontinuities in object class representations and give estimates of non-observed, derived object features (e.g., physical dimensions). As will be shown, with added white noise, classification errors and the errors in estimating the derived features increase but remain very well behaved.
In classification problems where multiple features are extracted from the observations of one or more sensors, the features often exhibit some degree of correlation, or a functional relationship. Frequently, this is expected and arises because of the mapping between the parameters that define the object's equation of state and the sensor observables. Therefore, it is of interest to develop representations of the objects and classification algorithms that exploit the correlations between the features. An approach for developing these types of representations makes use of Differential Geometry. In this approach, the objects are represented as a mean surface in feature space. When the functional relationship between features can be expressed analytically, Differential Geometry is used to develop analytical expressions for class surfaces and classification algorithms. More complex problems require the use of numerical techniques. In this paper, some of the mathematical foundations of this approach are reviewed. In an example, tensor product non-uniform rational b-splines are employed to develop the description of class surfaces along with the associated metric tensor and geodesic equations, leading to classification algorithms. The resulting Surface Classifier performance is compared with that of a traditional Quadratic Classifier.
In this paper, an approach for representing target classes in feature space using Riemannian manifolds is explored. In a formal application of the approach, it is required that several basic assumptions used in the development of differential and Riemannian geometry are satisfied. These assumptions relate to the concepts of allowable parametric representations and allowable coordinate transformations. Developing target class representations which satisfy these assumptions has a direct consequence on the selection of a suitable feature set. Having found a suitable feature set, the approach results in a natural coordinate system in which to calculate distance metrics used in classification algorithms. In this paper, the approach is applied to a situation where an active sensor and a passive sensor are spatially separated and are simultaneously collecting data on a set of targets. It is shown that the use of the natural coordinate system offered by this approach leads to a straightforward and mathematically rigorous method for fusing the sensor data at the feature level.
The use of Riemann surfaces offers an alternative approach to the characterization of object classes. In a situation where data from multiple sensors is available, the sensor observables can be used as the coordinates which define the space occupied by the class surfaces. The curvature of the surfaces will be governed by the underlying correlations between the phenomenologies of interest and the viewing conditions. The result is a natural coordinate system in which to implement classification algorithms. In this paper, a simple two-dimensional example is presented which introduces the underlying mathematics of the approach. A traditional statistical classifier is then used to examine classification performance. Extending the approach to include non-collocated sensors, sensor measurement error, and noise sources is also briefly discussed.
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