Wavelet matrix

Slide the critical area Peru b, a) (for every scale the corresponding dilated version) over the wavelet matrix. Every point inside a patch is defined as areawise significant, if the critical area containing this point totally lays within the patch. 

We now develop the matrix representation for the wavelet transform that allows us to represent the pyramidal synthesis and analysis lattice equations for finite length signals in a convenient matrix computational framework. We first introduce the concept of a wavelet matrix in the context of infinite signals. 

The wavelet matrix A is defined as an infinite row of 2 x 2 matrix blocks ... 

The first row in the wavelet matrix A, simply eontains the low-pass filter coefficients. The second row in the wavelet matrix A, contains the high-pass filter coefficients. As shown above, it is sometimes more convenient to store the filter eoefficients in the matrix A as a sequence of sub-blocks Aj j —. ..,-2,-1,0,1,2,...). The sub-blocks are simply the filter coefficients found in the lattice decomposition and reconstruction equations ar-... 

We now consider in more detail the factorized form of a wavelet matrix, and show that A can be constructed from some set of normalized vectors, denoted by ui,...,Uq, and v. 

Recall from Section 4, that the wavelet matrix A can be partitioned into m X m submatrices as follows A = (AqAi. .. Aq). Provided that the orthogonality condition ) satisfied, the wavelet matrix can... 

Now for the projection matrices. A symmetric projection matrix of rank p can be written R = UU where Umxp is a matrix with orthonormal columns. For the wavelet matrix to be non-redundant we require rank(R ) < rank(R2) < < rank(Rq). That is the individual ranks of the projection matrices form a monotonically increasing sequence [1]. For simplicity we set rank(R ) = rank(R2) = = rank(Rq) = 1 and... 

Now consider forming the wavelet matrix A. Using the factorized form of the wavelet matrix one has... 

Constrained optimization versus unconstrained optimization. In the adaptive wavelet algorithm, it was possible to avoid using constraints which ensured orthogonality. This is due to some clever algebraic factorizations of the wavelet matrix for which much credit is due to [6]. However, one constraint which we have not discussed in very much... 

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. 

S is a wavelets spectrum, W is an analyzing wavelets matrix, and is the transpose matrix of W. This study uses Coifman function of order 30 as the analyzing wavelets as shown in Fig. 2. This basie wavelet is expanded in the multi-scale to analyze the image D [9]. Because the wavelets transform is an orthonormal transformation, the inverse wavelets transform and its multiresolution is expressed by... 

Muralidharan, K. et al. (2008) Dynamic compound wavelet matrix method for multiphysics and multiscale problems. Pkys. Rev. E, T1 (2), 026714-1 026714-14. 

Mishra, S.K. et al. (2008) Spatiotempo-ral compound wavelet matrix framework for multiscale/multiphysics reactor simulation case study of a heterogeneous reaction/diffusion system. Int.J. Chem. Reactor Eng., 6 (A28), A28-1-A28-42. 

Wavelet transformation (analysis) is considered as another and maybe even more powerful tool than FFT for data transformation in chemoinetrics, as well as in other fields. The core idea is to use a basis function ("mother wavelet") and investigate the time-scale properties of the incoming signal [8], As in the case of FFT, the Wavelet transformation coefficients can be used in subsequent modeling instead of the original data matrix (Figure 4-7). 

Multiplication of this 4x8 transformation matrix with the 8x1 column vector of the signal results in 4 wavelet transform coefficients or N/2 coefficients for a data vector of length N. For c, = C2 = Cj = C4 = 1, these wavelet transform coefficients are equivalent to the moving average of the signal over 4 data points. Consequently,... 

The discrete wavelet transform can be represented in a vector-matrix notation... 

The factor Vl/ 2 is introduced to keep the intensity of the signal unchanged. The 8 first wavelet transform coefficients are the a or smooth components. The last eight coefficients are the d or detail components. In the next step, the level 2 components are calculated by applying the transformation matrix, corresponding to the level on the original signal. This transformation matrix contains 4 wavelet filter... 

Since in many applications minor absorption changes have to be detected against strong, interfering background absorptions of the matrix, advanced chemometric data treatment, involving techniques such as wavelet analysis, principle component analysis (PCA), partial least square (PLS) methods and artificial neural networks (ANN), is a prerequisite. 

Like the FFT, the fast wavelet transform (FWT) is a fast, linear operation that operates on a data vector in which the length is an integer power of two (i.e., a dyadic vector), transforming it into a numerically different vector of the same length. Like the FFT, the FWT is invertible and in fact orthogonal that is, the inverse transform, when viewed as a matrix, is simply the transpose of the transform. Both the FFT and the discrete wavelet transform (DWT) can be regarded as a rotation in function... 

J. Kautsky, A Matrix Approach to Discrete Wavelets, in Wavelets Theory, Algorithms and Applications (C. Chui, L. Montefusco and L. Puccio Eds) (1994), pp. 117-335. 

We now focus on the wavelet transform for discrete signals and, more significantly, for finite length (non-infinite) discrete signals since, in practice, most signals are finite. We develop the matrix representation for the discrete wavelet transform. 

As stated previously, with most applications in analytical chemistry and chemometrics, the data we wish to transform are not continuous and infinite in size but discrete and finite. We cannot simply discretise the continuous wavelet transform equations to provide us with the lattice decomposition and reconstruction equations. Furthermore it is not possible to define a MRA for discrete data. One approach taken is similar to that of the continuous Fourier transform and its associated discrete Fourier series and discrete Fourier transform. That is, we can define a discrete wavelet series by using the fact that discrete data can be viewed as a sequence of weights of a set of continuous scaling functions. This can then be extended to defining a discrete wavelet transform (over a finite interval) by equating it to one period of the data length and generating a discrete wavelet series by its infinite periodic extension. This can be conveniently done in a matrix framework. 

In general, we choose compact wavelets (i.e. only a finite number of coefficients for the dilation and wavelet equations are non-zero) and therefore only a finite number of Aj blocks are non-zero. In this case the matrix W is sparse and banded (see Section 4.2.2). Compactly supported wavelets have good localisation properties but may not always have a high degree of smoothness (e.g. the Haar wavelet). 

Since we obtain an interlaced vector of scaling and wavelet coefficients at each stage of applying the decomposition equation, the vector has to be unshuffled by pre-multiplying with a permutation matrix P. The permutation matrix selects every second element (due to the dyadic nature of the decomposition scheme) and reorders them in sequence. P is given by... 

Let us consider the Discrete Wavelet Transform (DWT), applied to the set of m signals (e.g. spectra) of length n each, presented in the form of matrix X. If all signals are decomposed by DWT with the same filter and to the same decomposition level, they can be presented as m vectors of length n each in the time-frequency domain, forming matrix Z (see Fig. 2). The information content of... 

Now we would like to compress the data in the time-frequency domain, without loss of an essential information about the data variability. This can be done, based on the variance of the wavelet coefficients (matrix Z). For each column of matrix Z, a variance can be calculated and in this way vector V (1 X n) is obtained, which contains n elements, each of them describing the variance of one column of matrix Z (see Fig. 3). The jth element of vector v is defined as ... 

It is also possible to apply a filter F to decompose all m signals, using the Wavelet Packet Transform (WPT). For each signal, a matrix is obtained that contains the wavelets coefficients (see Fig. 6). Element denotes the ith wavelet coefficient at the jth level in the r band of the kth signal decomposition. 

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