We show that characteristic functions of elements of self-similar tilings can be used as scaling functions of multiresolution analysis of L2(Rn). This multiresolution analysis is a generalization of a self-affine tiling multiresolution analysis using a characteristic function of element of self-affine tiling as a scaling function. We give a method of constructing a wavelet basis which realizes such an MRA.

This paper proves a general sampling theorem, which is an extension of Shannon's classical theorem. Let o be a closed subspace of square integrable functions and call o a signal space. The main aim of this paper is giving a necessary and sufficient condition for unique existence of the sampling basis {Sn}o without band-limited assumption. Using the general sampling theorem we rigorously discuss a frequency domain treatment and a general signal space spanned by translations of a single function. Many known sampling theorems in signal spaces, which have applications for multiresolution analysis in wavelets theory are corollaries of the general sampling theorem.

We propose concepts of Paley-Wiener multiresolution analysis and Paley-Wiener wavelet frame based on general, not limited to dyadic, dilations of functions. Such a wavelet frame is an extension both of the Shannon wavelet basis and the Journe-Meyer wavelet basis. A concept of "natural" Paley-Wiener wavelet frame is also proposed to clarify whether a Paley-Wiener wavelet frame can naturally express functions from the point of view of the multiresolution analysis. A method of constructing a natural Paley-Wiener wavelet frame is given. By using this method, illustrative examples of Paley-Wiener wavelet frames with general scales are provided. Finally, we show that functions can be more efficiently expressed by using a Paley-Wiener wavelet frame with general scales.

Scattering data from radar targets are analyzed in the time-frequency domain by using wavelet transform, and the scattering mechanisms are investigated. The wavelet transform used here is a powerful tool for the analysis of scattering data, because it can provide better insights into scattering mechanisms that are not immediately apparent in either the time or frequency domain. First, two types of wavelet transforms that are applied to the time domain data and to the frequency domain data are defined, and the multi-resolution characteristics of them are discussed. Next, the scattering data from a conducting cylinder, two parallel conducting cylinders, a parallel-plate waveguide cavity, and a rectangular cavity in the underground are analyzed by using these wavelet transforms to reveal the scattering mechanisms. In the resulting time-frequency displays, the scattering mechanisms including specular reflection, creeping wave, resonance, and dispersion are clearly observed and identified.

High-speed display of 3-D objects in virtual reality environments is one of the currently important subjects. Shape simplification is considered an efficient method. This paper presents a method of hierarchical cube-based segmentation for shape simplification and multiresolution model construction. The relations among shape simplification, resolution and visual distance are derived firstly. The first level model is generated from scattered range data by cube-base segmentation with the first level cube size. Multiresolution models are then generated by re-sampling polygonal patch vertices of each former level model with hierarchical cube-based segmentation structure. The results show that the algorithm is efficient for constructing multiresolution models of free-form shape 3-D objects from scattered range data and high compression ratio can be obtained with little noticeable difference during the visualization.

We propose a theory of general frame multiresolution analysis (GFMRA) which generalizes both the theory of multiresolution analysis based on an affine orthonormal basis and the theory of frame multiresolution analysis based on an affine frame to a general frame. We also discuss the problem of perfectly representing a function by using a wavelet frame which is not limited to being of affine type. We call it a "generalized affine wavelet frame." We then characterize the GFMRA and provide the necessary and sufficient conditions for the existence of a generalized affine wavelet frame.

MultiWave data compression algorithm is based on the multiresolution wavelet techniqu for decomposing Electrocardiogram (ECG) signals into their coarse and successively more detailed components. At each successive resolution, or scale, the data are convolved with appropriate filters and then the alternate samples are discarded. This procedure results in a data compression rate that increased on a dyadic scale with successive wavelet resolutions. ECG signals recorded from patients with normal sinus rhythm, supraventricular tachycardia, and ventriular tachycardia are analyzed. The data compression rates and the percentage distortion levels at each resolution are obtained. The performance of the MultiWave data compression algorithm is shown to be superior to another algorithm (the Turning Point algorithm) that also carries out data reduction on a dyadic scale.

The properties of the Haar Transform (HT) are discussed based on the Wavelet Transform theory. A system with two channels at resolution 2-1 and 2-2 for detecting paroxysm-spike in human's EEG is presented according to the multiresolution properties of the HT. The system adopts a coarse-to-fine strategy. First, it performs the coarse recognition on the 2-2 channel for selecting the candidate of spike in terms of rather relaxed criterion. Then, if the candidate appears, the fine recognition on the 2-1 channel is carried out for detecting spike in terms of stricter criterion. Three features of spike are extracted by investigating its intrinsic properties based on the HT. In the case of having no knowledge of prior probability of the presence of spike, the Neyman-Pearson criteria is applied to determining thresholds on the basis of the probability distribution of background and spike obtained by the results of statistical analysis to minimize error probability. The HT coefficients at resolution 2-2 and 2-1 can be computed individually and the data are compressed with 4:1 and 2:1 respectively. A half wave is regarded as the basic recognition unit so as to be capable of detecting negative and positive spikes simultaneously. The system provides a means of pattern recognition for non-stationary signal including sharp variation points in the transform domain. It is specially suitable and efficient to recognize the transient wave with small probability of occurrence in non-stationary signal. The practical examples show the performance of the system.

In this paper, subband image coding with symmetric biorthogonal wavelet filters is studied. In order to implement the symmetric biorthogonal wavelet basis, we use the Laplacian Pyramid Model (LPM) and the trigonometric polynomial solution method. These symmetric biorthogonal wavelet basis are used to form filters in each subband. Also coefficients of the filter are optimized with respect to the coding efficiency. From this optimization, we show that the values of a in the LPM generating kernel have the best coding efficiency in the range of 0.7 to 0.75. We also present an optimal bit allocation method based on considerations of the reconstruction filter characteristics. The step size of each subband uniform quantizer is determined by using this bit allocation method. The coding efficiency of the symmetric biorthogonal wavelet filter is compared with those of other filters: QMF, SSKF and Orthonormal wavelet filter. Simulation results demonstrate that the symmetric biorthogonal wavelet filter is useful as a basic means for image analysis/synthesis filters and can give better coding efficiency than other filters.