Three-dimensional wavelet transform

M.G. Wolkenstein and M. Mutter, De-noising Secondary Ion Mass Spectrometry Image Sets Using A Three-Dimensional Wavelet Transformation, submitted to Analytical Chemistry (May 1998). 

J. Wang and M.K. Muang, Medical Image Compression by Using Three-Dimensional Wavelet Transformation, IEEE Trans. Med. Imag., 15 (4) (1996). 

M.G. Wolkenstein and H. Hutter, Compression of Secondary Ion Microscopy Image Sets Using a Three-Dimensional Wavelet Transformation, submitted to Microscopy and Microanalysis (March 1999). 

We can define the wavelet transform W a, b) as the transformation of a signal /in Cartesian space by the three-dimensional integral... 

Having the three-dimensional coordinates of atoms in the molecules, we can convert these into Cartesian RDF descriptors of 128 components (B = 100 A ). To simplify the descriptor we can exclude hydrogen atoms, which do not essentially contribute to the skeleton structure. Finally, a wavelet transform can be applied using a Daubechies wavelet with 20 filter coefficients (D20) to compress the descriptor. A low-pass filter on resolution level 1 results in vectors containing 64 components. These descriptors can be encoded in binary format to allow fast comparison during descriptor search. 

Wavelets and the wavelet transformation refer the representation of a spectral data set in terms of a finite spectral range or a rapidly decaying oscillating waveform. This waveform is scaled and translated to match the original spectmm. Wavelet transformation may be considered to calculate the time-frequency representation, related to the subject of harmonic analysis. The projection of a spectmm on a single wavelet or a series of wavelets reduces the dimensionality of the data set. Wavelet transforms are broadly divided into three classes, the continuous wavelet transform, the discrete wavelet transform and multiresolution-based wavelet transforms. Each class has advantages and disadvantages in terms of the wanted information. 

An easy way to construct a multi-dimensional (e.g. 2-D) wavelet transform is, for example, to implement the tensor products of the 1-D counterparts. That is, we apply the 1-D wavelet transform separately along one dimension at a time. This, as we shall see shortly, results in one scaling function and three different mother wavelet functions. 

The one-dimensional (1-D) discrete wavelet transform (DWT) defined in the first part of the book can be generalised to higher dimensions. The most general case has been studied by Lawton and Resnikoff [1]. An N-dimen-sional (N-D) DWT is described also in [2]. The separable extension of the wavelet transform (WT) to three dimensions, for example, is explained in [2, 3,4]. In this chapter, for simplicity and because of the problems studied, only the theory of the 2-D and 3-D DWT will be outlined, and only separable 2-D and 3-D wavelets will be considered. These wavelets are constructed from one-dimensional wavelets. Separable wavelets are most frequently used in practice, since they lead to significant reduction in the computational complexity. 

The TDAS is a methodology which reduces the dimension of the wavelet matrix due to applying the geometric mean to output from the wavelet transform. It is expected that the TDAS application will allow fault patterns to be identified better as shown in HALIM et al. (2008). However, the complexity involved in the vibration signals may impede this identification. This is one limitation of the TDAS method that needs to be pointed out. Additionally, this method requires a visual interpretation of the three-dimensional graph of the U(s,p), a hard task to be computationally automated. Besides, in some cases, vibration signals can be acquired from more than one source simultaneously. Therefore, the dimension of the vibration data can make use of the TDAS method difficult. 

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