Multiscale filtering

Input mapping methods can be divided into univariate, multivariate, and probabalistic methods. Univariate methods analyze the inputs by extracting the relationship between the measurements. These methods include various types of single-scale and multiscale filtering such as exponential smoothing, wavelet thresholding, and median filtering. Multivariate methods analyze... 

More recently, the development of wavelets has allowed the development of fast nonlinear or multiscale filtering methods that can adapt to the nature of the measured data. Multiscale methods are an active area of research and have provided a formal mathematical framework for interpreting existing methods. Additional details about wavelet methods can be found in Strang (1989) and Daubechies (1988). 

The methodology for multiscale filtering consists of the following three steps ... 

Nonlinear filtering techniques have been developed to overcome the inability of linear filters to capture features at different scales. Nonlinear filtering methods include FIR-Median Hybrid filtering and multiscale filtering, and... 

Multiscale filtering and compression. Multiscale filtering and compression using wavelets are based on the observation that random errors in a signal are present over all the coefficients while deterministic changes get captured in a small number of relatively large coefficients. Thus, stationary Gaussian noise may be removed by a three-step method [4] ... 

Fig. 5 (a) Bumps signal with while noise, (h) multiscale filtered signal using D2 boundary corrected filter (MSE = 0.1045, CR = 7.7), (c) reconstructed. signal using boxcar compression (MSE = 0.5721, CR = 7.7). 

On-line multiscale filtering of data with Gaussian errors... 

On-line multiscale filtering is based on multiscale filtering of data in a moving window of dyadic length as shown in Fig. 11. 

Hints for tuning the filter parameters in multiscale filtering and compression... 

As in the case of any filtering or compression method, multiscale filtering relies on some information about the data and the nature of the errors to tune its filter parameters, which include the threshold, decomposition depth, the wavelet filter, and the size of the median filter for the robust techniques. Hints for selecting these tuning parameters in off-line and on-line modes are discussed below. 

M.N. Nounou and B.R. Bakshi, Online Multiscale Filtering of Random and Gross Errors without Proeess Models, AICHE Journal 45 (5), 1041-1058, 1999. 

Existing methods for data rectification with process models including, maximum likelihood and Bayesian methods, are inherently single-scale in nature, since they represent the data at the same resolution everywhere in time and frequency. The multiscale Bayesian data rectification method developed in this section combines the benefits of Bayesian rectification and multiscale filtering using orthonormal wavelets. 

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