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In this study, a novel direction selection method using the two-dimensional complex discrete wavelet transform (2D-CDWT) is proposed. In order to achieve arbitrary direction selection, the directional filters are first designed. Calculation procedure of directional selection can be shown as follows: (1) The 16 sub-images are generally generated from the original image by the 2D-CDWT without a down-sampling process and the 12 sub-images that correspond to the high-frequency components are selected. (2) The 12 sub-images are filtered by using the designed directional filter. (3) The down-sampling process is carried out and the resulting images are obtained. Furthermore, this method is applied to the surface analysis of a wafer, and it is confirmed that our method is effective in detecting irregular direction components.

Independent component analysis (ICA) is a useful method for blind source separation of two or more signals. We have previously proposed a new method combining ICA with the complex discrete wavelet transform (CDWT), in which voice and noise signals were separated using a new method. At that time, we used a simulated signal. In this study, we analyze measured biological signals by using a new method, and discuss its effectiveness. As an experiment, we try to separate an electromyogram (EMG) signal from an electrocardiogram (ECG) signal.

It is well-known that a mother wavelet for the discrete wavelet transform (DWT) has the band-pass filter characteristic with octave width in the frequency domain and can be used for octave analysis. However, it is possible that the octave analysis is not necessarily the most suitable to match the analysis signal. In this study, in order to construct the most suitable basis to match the analysis signal, a novel variable-filter band discrete wavelet transform (VFB-DWT) is proposed. It is achieved by using variable-band filters instead of conventional decomposition and reconstruction sequences, which are designed in consideration of the real signal characteristics. Additionally, it is proven that perfect reconstruction of the analysis signal by VFB-DWT is guaranteed using the perfect shift invariant theorem that underlies the theory of the PTI-CDWT having base DWT.

In this study, in order to verify the effectiveness of the variable filter band discrete wavelet transform (VFB-DWT) and construction method of the variable-band filter (VBF), a fetal ECG extraction has been carried out and the main results obtained are as follows. The approach to configuration VBF by selecting the frequency band only where the fetal ECG component is present was effective to configure the optimal base sensible signal. The extraction of the fetal ECG was successful by applying the wavelet shrinkage to VFB-DWT, which used the constructed VBF. The information entropy was selected as an evaluation index, and two kinds of ECG signals are used to evaluate the wavelet transform basis between the wavelet packet transform (WPT) and the VFB-DWT. One is a synthesized signal composed of white noise, the maternal ECG and the fetal ECG. The other signal is the real target signal separated by independent component analysis (ICA) and has the mother's body noise, the maternal ECG and the fetal ECG. The result shows that the basis by VBF of the VFB-DWT is better than the basis of the WPT that was chosen by the best basis algorithm (BBA).

In this paper, we propose a design method for directional selection in the two-dimensional complex wavelet packet transform (2D-CWPT). Current two-dimensional complex discrete wavelet transforms (2D-CDWT) can extract directional components from images, but the number of directions is small, and the directions and resolutions are fixed. Thus the current 2D-CDWTs are not flexible enough. In this study, we propose a new design method of the directional filters that can detect desirable direction components. Additionally flexible directional selection is achieved because the directional filters are added to the 2D-CWPT. Finally, the proposed method is applied to defect detection in semiconductor wafer circuits and an encouraging result is obtained.