Piecewise constant approximation and the Haar Wavelet

Transcription

1 Chapter Piecewise constant approximation and the Haar Wavelet (Group - Sandeep Mullur 4339 and Shanmuganathan Raman 433). Introduction Piecewise constant approximation principle forms the basis for the basic understanding of the wavelets. his is done in order to extract information at various resolutions from the function. In this chapter, we shall see the basic principles behind piecewise constant approximation with regard to Haar Wavelets.. Piecewise Constant Approximation A continuous-time signal can be approximated over an interval of the time-axis by a constant equal to its average value over that interval. x dt he above approximation could be applied to smaller and smaller intervals. If the function is smooth and continuous, as the size of the interval decreases, the approximation closely resembles the original signal. Let x x dt x x / dt in the first sub-interval / / / x dt in the second sub-interval / x / / / hat information in the piecewise constant approximation on an interval of length, which is not available at the next higher interval size, is referred to as the information specific to the resolution of interval. For our convenience, let us tae =, i.e. the n n function is piecewise constant over [, ] and the value of m determines the m m resolution of approximation. he relationship of the various subspaces can be seen from the following expressions. We see that we may start at any, say at =, and write... L ( R) We now define the wavelet spanned subspace W such that W

2 which extends to In general this gives. W W L ( R) W W... 3 W W W As we traverse through the subspaces by piecewise constant approximation, the change in resolution can be seen as below: Dilation Contraction {} L ( R) Lesser Resolution Higher Resolution he concept behind the navigation from one subspace to another subspace is very simple. Let us eep subspace as our reference. If we navigate towards positive values of in, where is an integer, we are approximating the function with average taen over lesser time intervals. his means we are contracting our averaging interval and thereby attaining higher resolutions. If we navigate towards negative values of in, where is an integer, we are approximating the function with average taen over greater time intervals. his means that we are dilating our averaging interval and thereby attaining lesser resolutions..3 Orthogonal Condition he requirement for the scaling function and the wavelet function is that they should be orthogonal to each other. here are several advantages to requiring that the scaling functions and wavelets be orthogonal. Orthogonal basis functions allow simple calculation of expansion coefficients and have a parseval's theorem that allows a partitioning of the signal energy in the wavelet transform domain. he orthogonal

3 complement of in orthogonal to all members ofw. We require for all appropriate,, l.4 he Scaling function is defined asw. his means that all members of,,, l,, l dt Z are Consider the space, which is a subspace of L (R). Here, L (R) is a space containing all square integrable functions. Any function f( belonging to L (R), satisfies the condition that its energy is less than. Now, represents a space in which the functions are piecewise constant over the interval [n, n+], where m= and n. A basis for is given by the scaling function as shown below: Using the above scaling function and its translates, we can construct all functions in. Also, L ( R ). he scaling function is plotted below: ] t he scaling function captures information specific to. In general a scaling function, captures details specific to the space. he span over is for all integers Z. span{ } span{, } 3

4 / t his means that if f then it can be expressed as f a ( t ) But as we move from one space to another, for example, from to, we need some function which gives details specific only to. his detail is provided by the wavelet function..5 he Wavelet Function he incremental information at interval of length / is given by x x ( ) ( / t. x ( / x in each interval is a multiple of the function (. ( and its translates capture the details that are peculiar to the resolution at /. In general, in going from interval m to, the function to capture the difference still remains the same, m except for the interval over which it spreads. he dilates of ( and the translates of its dilates, ( at b) allow us to capture the information specific to other resolutions. he Haar wavelet function is plotted as below: 4

5 t - he function mentioned above and its translates form the basis for the space W. Any function in the space W can be represented by and its translates alone. But, for us, serves one more purpose. It helps us to navigate from one subspace to another, for example, from to by giving the details specific to alone. Further L (R) can be represented by the sum of all subspaces W as shown below:, m, n Z captures the information specific to the sub- In general, ( m t n ) intervals of length. m m: measure of resolution n: index of interval of size m L ( R) W W W 3 W W L ( R)... W W W W... hus, if is smooth and continuous over the interval [, ], it can be written (over that interval) as m m n c m m, n ( t n) 5

6 .6 Multiresolution Formulation A multiresolution formulation comprises of two closely related basic functions. In addition to the wavelet ( we need another basic function called the scaling function (. he simplest possible orthogonal wavelet system is generated from the Haar scaling function and wavelet. A large class of signals can be represented by c ( t ) d, ( t ).7 Example of Multiresolution Analysis Problem Statement Define be the space of piecewise constant functions which are piecewise constant on unit intervals of the form ]n, n+[ in the Multiresolution ladder of subspaces, of the space of square integrable functions. Consider the square integrable function y(. y( - y( = +t, -<t< = -t, <t< Obtain the proection of y( on the space. A basis for this space, ( is given by ( 6

7 Proection of y( is given by inner product of y( with a) ( in (,) b) (t+) in (-,) By simple integration, the proection of y( on can be obtained as.5 for -<t<. ` he proection of y( on.5.8 Exercise - Similarly obtain the proection of y( on the following spaces: a) he space m, for positive integers m. b) he space m, for negative integers m. c) he space W d) he space W m, for positive integers m. e) he space W m, for negative integers m. 7

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