Best basis algorithm

When both the scaling and wavelet coefficients are filtered there is a surplus of information stored in the wavelet packet tree. An advantage of this redundant information is that it provides greater freedom in choosing an orthogonal basis. The best basis algorithm is a routine which endeavours to find a basis in the WPT which optimizes some criterion. 

The best basis algorithm seeks a basis in the WPT which optimizes some criterion function. Thus, the best basis algorithm is a task-specific algorithm in that the particular basis is dependent upon the role for which it will be used. For example, a basis chosen for compressing data may be quite different from a basis that might be used for classifying or calibrating data, since different criterion functions would be optimized. The wavelet packet coefficients which are resultant of the best basis, may then be used for some specific task such as compression or classification for instance. 

The first step in obtaining the wavelet packet coefficients from the best basis is to liroduce the wavelet packet decomposition tree to some level jo- A criterion measure for each of the wavelet packet coefficients in each node (or band) in the wavelet packet decomposition is calculated and is denoted by 

Here descendant nodes are used to categorize any nodes which lie beneath a node at a higher level in the tree. The node which the descendant nodes lie under is called a parent node. If the criterion measure of the parent node is superior to that of the descendant nodes, then the descendant nodes are deleted. If the descendant nodes produce a superior criterion measure, then the descendant nodes are kept and the parent node is deleted. This procedure eontinues all the way to the top of the tree and the coefficients in the best basis will lie in the bands which were not deleted in the elimination process. 

Consider finding the best basis for some signal x = (xq.xi, X7). Once 

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Wavelet Packet Transforms and Best Basis Algorithms... 

Fig. 4 describes the best basis algorithm, or more specifically how to find the wavelet packet coefficients from the best basis algorithm. Step 1 performs the WPT to some prespecified level jo as described previously. Step 2 then initializes a current best basis or best set of bands. Initially, the best set of bands (BB) is simply all the bands at level Jq in the WPT. Steps 3-9 begin to compare the cost measure of the parent nodes against the current best of bands which are descendants of the parent nodes being examined. Here, the aim is to minimize the cost measure. 

Obtaining the wavelet packet coefficients From the best basis algorithm... 

Coifman, R. R., and Wickerhauser, M. V., Entropy-based algorithms for best basis selection. 

R. Coifman and V. Wickerhauser, Entropy-based Algorithms for Best-basis Selection, IEEE Transactions on Information Theory, 38 (1992), 496-518. 

If the best-basis selection algorithm with the entropy criterion is applied to individual signals decomposed by WPT, the following situations can occur ... 

In the second case, the uniform representation of signals in the wavelet domain is not possible and instead, the joint best-basis, in which a small number of wavelets coefficients can describe the majority of data variance, must be selected. This can be done, by applying Coifman and Wickerhausers algorithm of best-basis selection to the so-called variance tree", the elements of which represent the variance of wavelet coefficients with the same addresses (indices) (see Fig. 8) [5]. 

Once the variance tree is constructed, it can be searched for the joint best-basis, using the Coifman-Wickerhauser best-basis selection algorithm with, e.g., the entropy criterion. The entropy cost function (see Chapter 6) for the variance tree coefficients, which occur at the jth level in the i band of the signal decomposition is defined as ... 

In Section 2 we introduce the fundamentals of image compression and overview the various compression algorithms. We review the transformation techniques used in image compression in Section 3. Section 4 describes image compression using optimal task-based and best-basis image compression algorithms. 

Entropy-Based Algorithms for Best Basis Selection. 

This algorithm alternates between the electronic structure problem and the nuclear motion It turns out that to generate an accurate nuclear trajectory using this decoupled algoritlun th electrons must be fuUy relaxed to the ground state at each iteration, in contrast to Ihe Car-Pairinello approach, where some error is tolerated. This need for very accurate basis se coefficients means that the minimum in the space of the coefficients must be located ver accurately, which can be computationally very expensive. However, conjugate gradient rninimisation is found to be an effective way to find this minimum, especially if informatioi from previous steps is incorporated [Payne et cd. 1992]. This reduces the number of minimi sation steps required to locate accurately the best set of basis set coefficients. 

Although a number of studies were made and approximate methods developed for predicting the effect of liquid holdup in the period of the 1950s and 1960s, as summarized in the 6th edition of Peny .s Chemical Engineers Handbook, the complexity of the effect of liqmd holdup is such that it is now best to use computer-based batch-distillation algorithms to determine the effect of holdup on a case-bycase basis. 

Linear PCR can be modified for nonlinear modeling by using nonlinear basis functions 0m that can be polynomials or the supersmoother (Frank, 1990). The projection directions for both linear and nonlinear PCR are identical, since the choice of basis functions does not affect the projection directions indicated by the bracketed term in Eq. (22). Consequently, the nonlinear PCR algorithm is identical to that for the linear PCR algorithm, except for an additional step used to compute the nonlinear basis functions. Using adaptive-shape basis functions provides the flexibility to find the smoothed function that best captures the structure of the unknown function being approximated. 

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