Improved Local Discriminant Bases Using Empirical Probability Density Estimation

Recently, the authors introduced the concept of the socalled Local Discriminant Basis (LDB) for signal and image classification problems [6], [17, Chap. 4], [19], [20]. This method first decomposes available training signals in a time-frequency dictionary (also known as a dictionary of orthonormal bases ) which is a large collection of the basis functions (such as wavelet packets and local trigonometric functions) localized both in time and in frequency. Then, signal energies at the basis coordinates are accumulated for each signal class separately to form a time-frequency energy distribution per class. Based on the differences among these energy distributions (measured by a certain “distance” functional), a complete orthonormal basis called LDB, which “can see” the distinguishing signal features among signal classes, is selected from the dictionary. After the basis is determined, expansion coefficients in the most important several coordinates (features) are fed into a traditional classifier such as linear discriminant analysis (LDA) or classification tree (CT). Finally, the corresponding coefficients of test signals are computed and fed to the classifier to predict their classes. This LDB concept has been increasingly popular and applied to a variety of classification problems including geophysical acoustic waveform classification [18], radar signal classification [11], and classification of neuron firing patterns of monkeys to different stimuli [22]. Through these studies, we have found that the criterion used in the original LDB algorithm—the one based on the differences of the time-frequency energy distributions among signal classes—is not always the best one to use. Consider an artificial problem as follows. Suppose one class of signals consists of a single basis function in a time-frequency dictionary with its amplitude 10 and they are embedded in white Gaussian noise (WGN) with zero mean and unit variance. The other class of signals consists of the same basis function but with its amplitude 10 and again they are embedded in the same WGN process. Then their time-frequency energy distributions are identical. Therefore, we cannot select the right basis function as a discriminator. This simple counterexample suggests that we should also consider the differences of the distributions of the expansion coefficients in each basis coordinate. In this example, all coordinates except the one corresponding to the single basis function have the same Gaussian distribution. The probability density function (pdf) of the projection of input signals onto this one basis function should reveal twin peaks around 10. In this paper we propose a new LDB algorithm based on the differences among coordinate-wise pdfs as a basis selection criterion and we explain similarities and differences among the original LDB algorithm and the new LDB algorithm.