This paper considers the extraction of information from a locally
stationary process modelled by wavelet packets. A method is presented to select subprocesses that characterize the key aspects of the nonstationary process for pattern analysis. The estimated parameters of the selected subprocesses are used to infer the process' time varying behavior. The estimated parameters can be used as features in the attempt to distinguish changing states within a process or differentiate two different locally stationary processes.
For decades the process of aluminum electrolysis has facilitated the production of aluminum. The process occurs within aluminum electrolytic cells, where alumina (Al<SUB>2</SUB>O<SUB>3</SUB>) is dissolved in liquid cryolite (Na<SUB>3</SUB>AlF<SUB>6</SUB>). The dissolved alumina is reduced by the carbon anode and forms carbon dioxide. Complexes containing aluminum ions migrate to the cathode surface (bath-metal interface) where aluminum metal is produced. The monitoring of the electrolysis process is done through the use of the cell resistance. Using resistance set point values that are indirectly related to the desired alumina concentration in the bath (cryolite), the computed resistance can indicate if the cell is operating within acceptable production conditions. The resistance time series is a nonstationary random process. We have applied the principal component method to shortsegments of each time series to identify key components. However the principal components are data dependent. In order to study the time series' localized structure we use a wavelet packet based approach to analyze this nonstationary process. We use Daubechies 3 orthonormal wavelet and scaling function as our basis functions and model each short segment of the resistance time series as a locally stationary wavelet process. The use of wavelet packets increases the separability of the innovations into individual packets. Hence each wavelet packet time series represents a single subprocess. The analysis of individual subprocesses yields information for making inference of how the process evolves during unstable operating conditions.
The application of wavelet transforms to edge detection has improved edge localization. The image produced by the local maxima of the wavelet modulus needs to be thresholded to extract out the relevant edge pixels. This is currently done manually. In this paper, we apply a fuzzy thresholding approach for automatic determination of the threshold level for wavelet maxima. A membership function is used to determine the characterization of the candidate edges based on a particular threshold. The threshold which yields the best characteristic or lowest uncertainty is selected. Non-crisp thresholding is achieved by re-evaluating edge pixel membership values to identify those pixels that may have been improperly classified. This results in the closure of small gaps between edge segments and a reduction in the size and number of larger gaps. For disjoint edge segments with a separation of less than six pixels, their endpoints can be linked by fuzzy reasoning based on membership values, distance, and their wavelet angles. Experimental results on test images have demonstrated the effectiveness of this method.