Generalized Feature Detection Using the Karhunen-Loeve Transform and Expansion Matching

This paper presents a novel generalized feature extraction method based on the Expansion Matching EXM method and the Karhunen Loeve KL transform This yields an e cient method to locate a large variety of features with reduced number of ltering operations The EXM method is used to design optimal detectors for di erent features The KL representation is used to de ne an optimal basis for representing these EXM feature detectors with minimum truncation error Input images are then analyzed with the the resulting KL bases The KL coe cients obtained from the analysis are used to e ciently reconstruct the response due to any combination of feature detectors The method is applied to real images and successfully extracts a variety of arc and edge features as well as complex junction features formed by combining two or more arc or line features INTRODUCTION Feature extraction forms an important preprocessing step in many computer vision and image understanding tasks The input to the feature extraction system is an image and the output is the symbolic representations of various features of the shapes in the image such as the curvature length position and orientation of boundary segments Generally such pre processing schemes consist of a series of steps including edge detection edge thinning edge linking or grouping feature detection etc Evidently if the speci c feature searched for is a priori unde ned the computational complexity involved is signi cantly large since each feature requires a separate search In this paper we present a novel generalized feature extraction scheme based on the Karhunen Loeve KL transform and the Expansion Matching EXM method developed by Ben Arie and Rao It is evident that most shape features in an image can be accurately described by combinations of line and arc segments with di erent orientations and curvatures In an exhaustive approach feature detectors for each combination of line and arc segments need to be designed Since the exact combinations that need to be detected are a priori unknown the number of such feature detectors involved is very large Consequently such an exhaustive implementation is computationally impractical Instead it is possible to design optimal feature detectors for line and arc segments for various curvatures orientations and lengths where straight line segments are zero curvature arcs These arc segments then constitute a set of model features The complete set of feature detectors for all of these features may then be applied to an image At every location in the image the detected feature is the one with the maximum output above a certain threshold from among the outputs of the di erent feature detectors However this implementation also involves considerable computational e ort due to the large number of ltering operations In the generalized feature detection scheme presented here we propose an alternative implementation wherein This work was supported by the Advanced Research Projects Agency under ARPA ONR Grant No N the EXM method is rst used to design an optimal set of feature detectors over a range of curvatures and orientations EXM is an optimal matching method based on non orthogonal expansion of the image signal onto template similar basis functions Using the EXM method one can design an optimal linear lter for any given feature pattern Such a lter responds with a sharp impulse like matching response at the center of the feature with minimal spurious responses elsewhere EXM is e ciently implemented in the frequency domain as shown in Section It has a de nite advantage over correlation matched ltering in matching Another advantage is that EXM based lters can recognize even severely occluded feature patterns in cluttered background Since the detectors are based on the optimal EXM method they react only to the corresponding features and yield minimal output at other locations As is well known the KL transform is based on de ning an optimal basis for orthogonal expansion of a set of signals so as to minimize the mean squared error MSE in reconstruction from a truncated basis The KL transform may be applied to the ensemble of optimal EXM feature detectors The eigenvectors of the KL expansion form a set of eigen feature detectors A subset of these eigen feature detectors which correspond to the subset of largest eigen values may now be used for feature detection The computational complexity of the task is reduced to correlating a smaller number of eigen feature detectors as compared to the complete set of EXM feature detectors Using the eigenvectors corresponding to the largest eigen values ensures that the MSE in reconstruction of the result of complete set of feature detectors is minimized A data ow diagram of the scheme is shown in Fig The approach is developed in Section and Section The resulting error from using a truncated KL basis is discussed in Section In Section we show the results of applying our approach to detect complex junction features composed of various combinations of arc and line features Hummel used an approach based on the KL transform for detecting arbitrarily oriented edges However he used linear combinations of basis vectors to detect various orientations of a standard edge model This is not the generalized method proposed here using conjunctions of smaller features to recognize complex features