Haar wavelet

The accuracy of the error equations (Eqs. (22) and (23)] also depends on the selected wavelet. A short and compactly supported wavelet such as the Haar wavelet provides the most accurate satisfaction of the error estimate. For longer wavelets, numerical inaccuracies are introduced in the error equations due to end effects. For wavelets that are not compactly supported, such as the Battle-Lemarie family of wavelets, the truncation of the filters contributes to the error of approximation in the reconstructed signal, resulting in a lower compression ratio for the same approximation error. 

Fig. 40.43. Waveforms for the discrete wavelet transform using the Haar wavelet for an 8-points long signal with the scheme of Mallat s pyramid algorithm for calculating the wavelet transform coefficients.

We elaborate on these in the following sections. The pipelines explored here have been used by Konig et al. (23), but these investigators used another feature extraction technique, the Haar wavelet transform. We explored a novel feature extraction technique based on a combinatorial approach. We confirm further the results obtained via the pipelines above using the Haar wavelet transform. This transform was done using the clusters due to the consecutive-ones clustering technique (23). 

According to a proposal of Ingrid Daubechies, the notation DK will be used for a Daubechies Wavelet transform with K coefficients. Actually, D2 is identical to the simplest Wavelet of all, the so-called Haar Wavelet, and, thus, is not originally a member of the Daubechies family. 

More complex wavelets have in common with the Haar wavelet that they correspond with a complementary pair of LP and HP filters that cut the... 

For an impulse response that differs from zero on, let us say, four points, several aspects of the pyramid algorithm are less obvious. We need to be able to drop half the points and still represent the signal using the output of the LP and HP filters. In other words, we need to step the linear convolution of signal and impulse response by two points. The Haar wavelet basis is also special in the sense that, as the impulse responses are only two points wide, we do not lose points at the extremes when performing a linear convolution. For wider impulse responses, something has to be done about those extremes, e.g. a circular convolution, which puts an additional constraint on the shapes of those impulse responses. 

This chapter has briefly eluded to two wavelet families, the Haar and Daubechies wavelets. In fact when Nf = 2 the Daubechies wavelet is identical to the Haar wavelet. In this section we would like to discuss in greater detail more about these wavelet families and other wavelet families. We will also provide a brief comparison between the different properties possessed by these wavelets and other wavelet families. This is important because depending on your application, you may need to choose a wavelet that satisfies special properties. We first review the terms orthogonal and compact support. Following this, we will introduce some more properties, namely smoothness and symmetry of wavelets and also discuss the term vanishing moments. 

Symmetry is a useful property in image processing [14]. Unfortunately, with the exception of the Haar wavelet it is not possible to have orthogonal compactly supported wavelets which are symmetric. Owing to this fact, the... 

Fig. 12 An example of a Haar wavelet (a), and wavelets from the Daubechies (b), symmlet (c) and coiflet (d) families.

So that a wavelet function can efficiently represent characteristics of an underlying function, it is necessary that the wavelet be sufficiently smooth [14], If one considers the Haar wavelet depicted in Fig. 11, one can clearly see that this wavelet suffers from a lack of smoothness. Smoothness is closely related to how many times a wavelet can be differentiated and to the number of vanishing moments possessed by the wavelet vj/(t). One measure of smoothness is the Holder exponent [14], which is defined to equal = q + Y where and Y are the largest values such that... 

Briefly, when using orthogonal compactly supported wavelets it is not straightforward to obtain a wavelet which has symmetrical properties [7,12] and allows for an exact reconstruction. That is of course with the exception of the trivial Haar wavelet. Biorthogonal wavelets relax the assumptions of orthogonality, and allow for a perfect reconstruction with symmetrical wavelets. 

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). 

Fig. 15 (a) A bumps signal contaminated with white noise of variance 0.5 and outlier patches of length 3. (b) robust OLMS filtering, median filter length = 9, Haar wavelet, scale depth = 2 (MSE = 0.8366). 

Perform the wavelet packet transform using filter coefficients associated with the Haar wavelet and scaling functions, then, compute the wavelet packet coefficients associated with the best basis using the entropy cost function for the signal x = (0.0000,0.0491.0.1951,0.4276,0.7071,0.9415,0.9808,0.6716). 

There exists many different kinds or families of wavelets. These wavelet families are defined by their respective filter coefficients which are readily available for the situation when m = 2, and include for example the Daubechies wavelets, Coiflets, Symlets and the Meyer and Haar wavelets. One basic issue to overcome is deciding which set (or family) of filter coefficients will produce the best results for a particular application. It is possible to trial different sets of filter coefficients and proceed with the family of filter coefficients which produces the most desirable results. It can be advantageous however, to design your own task specific filter coefficients rather than using a predefined set. 

Let us consider the discrete wavelet transform with the Haar wavelet. If the DWT with the Haar wavelet is applied to a time series the detail coefficients supply information about the temporal change of the time series. The detail coefficients of different levels correspond to changes on different time scales. Hence, these coefficients may serve as a measure of change of a time series. If a transition occurs in a time series the detail coefficients take values that are larger than the values they take during stable states. Below it is described how this property is used for detection of transitions. 

In the wavelet transform approach, the choice of wavelet has a direct impact on the decomposed image, which indicates that the selection of the wavelet is closely related to the detection and classification performance. And standard wavelets, e. g. Haar wavelet, Daubechies wavelets, Coiflets etc., may not guarantee efficient discrimination of fabric defects. 

Gearhart 2001) applied a httle bit different approach. He used a simple piecewise linear crash pulse as an input to an occupant injury model. Then in sequence he added variations to this simplified pulse in the form of Haar wavelets with Ig amplitude and for each variation of the occupant injury criteria calculated injury mnnbers. He also studied the sensitivity of the occupant injury criteria with regard to variations in the simpUfied pulse. 

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. 

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