Wavelet functions, construction

By iterative application of the FWT to the high-pass filter coefficients, a shape emerges that is an approximation of the wavelet function. The same applies to the iterative convolution of the low-pass filter that produces a shape approximating the scaling function. Figure 4.6 and Figure 4.7 display the construction of the scaling and wavelet functions, respectively ... 

FIGURE 4.7 Construction of wavelet function / with high-pass D4 filter coefficients (above). Below the functional representation of the high-pass filter coefficients (left) and their refinement by iterative calculation (increasing resolution level j) leading to an approximation of the wavelet function / (right). 

We have demonstrated that it is possible to obtain the discrete wavelet transform of both continuous functions and discrete data points without having to construct the scaling or wavelet functions. We only need to work with the filter coefficients. One may begin to wonder where the filter coefficients actually come from. Basically, wavelets with special characteristics such as orthogonality, can be determined by placing restrictions on the filter coefficients. 

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 continuous wavelet transform (WT) is a space-scale analysis that consists in expanding signals in terms of wavelets that are constructed from a single function, the analyzing wavelet /, by means of dilations and translations [13, 27-29]. When using the successive derivatives of the Gaussian function as analyzing wavelets, namely... 

Multiresolution analysis (MRA) [7,8,9] provides a concise framework for explaining many aspects of wavelet theory such as how wavelets can be constructed [1,10]. MRA provides greater insight into the representation of functions using wavelets and helps establish a link between the discrete wavelet transform of continuous functions and discrete signals. MRA also allows for an efficient algorithm for implementing the discrete wavelet transform. This is called the fast wavelet transform and follows a pyramidal... 

Provided we know the scaling coefficients at some resolution level j, the remaining scaling coefficients and wavelet coefficients can be found by the pyramidal filtering algorithm without even having to construct a wavelet or scaling function. We need only work with the filter coefficients Ik and hk. 

As stated in the previous section, optimal wavelet image compression can be achieved by integrating the process of wavelet construction with best-basis search. The best-basis search using a standard off-the-shelf Coifman wavelet (Nf = 2) for four levels is shown in Fig. II. In this case the threshold cost function used was simply the constant value 0.2. The resulting compression ratio obtained was 9.50. 

Fig. 12 shows the result when the task-specific wavelet construction is integrated with the best-basis search (with the same threshold cost function). Here, the compression ratio is 9.71, an improvement of 2.2% compared with... 

The extension of the above algorithm to two-dimensions is straightforward. Let 0(x,y) be a separable spline scaling function, which plays the role of a smoothing filter. We can construct two oriented wavelets by taking the partial derivatives... 

This construction ensures the wavelets and their associated scaling functions to be orthogonal. The scaling and wavelet equation provide a simple tool to derive the fast wavelet transform. If / Z f )) and if we denote scaling coefficients of function/ by yjji, and... 

In this paper we use biorthogonal wavelets on the interval constructed by (Cema Finek 2008 Cerna Finek 2009) which outperforms similar construction in the sense of better conditioning of base functions as well as in better conditioning of wavelet transform. The condition number seems to be nearly optimal most especially in the case of cubic spline wavelets. From the viewpoint of numerical stabihty, ideal wavelet bases are orthogonal wavelet bases. However, they are... 

Wavelets form a tool for constructing stable bases of W = L (M). The basis Wj of the space Vj is constructed by means of dyadic dilations and translations of a single scaling function ... 

For many problems the solution does not extend over the whole space M. Hence one is interested in a basis for bounded intervals, e. g. for the space jL ([0, 1]). There are several papers considering this problem, e.,g. [1, 4]. However, for the simplest choice of (, namely (p = X[o,i]j where X[o,i] denotes the characteristic function for the interval [0,1], the construction of an ONE of L ([0,1]) is very easy. The corresponding orthonormal wavelets are known as the Haar wavelets ip and will be considered in the following. 

A second approach to data compression is to compress infrared spectra with a construct called a wavelet neural network (WNN). The WNN approach stores large amounts of infrared data for fast archiving of spectral data. It is achieved by modifying the machine learning technique of artificial neural networks (ANNs) to capture the shape of infrared spectra using wavelet basis functions. The WNN approach is similar to another approach... 

This construction can be geometrically viewed through Figure 4.6, the function x can be approximated by the basis and 4 and the scaling basis could be further approximated by finer wavelet and scaling basis. 

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