The Discrete Wavelet Transform The Story of Wavelets Theory and Engineering Applications The Discrete Wavelet Transform

An example on MRA f(t) fj(t) Let’s suppose we wish to approximate the function f(t) using a simpler function that has constant values over set intervals of 2j, such that the error ||f(t)-fj(t)|| is minimum for any j. The best approx. would be the average of the function over the interval length. In general, then… f(t) j=- j=-2 fj(t) j=-1 j=0 j=1 j=2 j=

An Example (cont.) As we go from a value of j to a higher value, we obtain a coarser approximation of the function, hence some detail is lost. Let’s denote the detail lost at each approximation level with a function g(t): Detail function Approximation function That is, the original function can be reconstructed simply by adding all the details (!)

An example (Cont.) Approximations can be obtained by “averaging” the original function over time  the duration of the averaging window determine the resolution of the approximation. Original function can be reconstructed by adding all the details lost in approximations (to the coarsest approximation) Note that the functions at any level can be obtained from a prototype, simply by dilating or compressing the prototype function.

Scaling Functions The prototype function used in the above example was a piecewise linear function. Since we can approximate and reconstruct any function using dilated and translated replicas of this prototype  Piecewise linear functions constitute a set of basis functions. (t) 1 The piecewise linear functions, however, can all be obtained by dilations and translations of this prototype function. We call this prototype function the scaling function. Among many such prototypes, the one shown here is the simplest, and it is known as the Haar scaling function.

Scaling Functions 1 t 3/2 1 t 2 2 1 t (2-jt-4) 1/2 1 t some j t k t (2t-3) 3/2 1 t 2 Approximation coefficients (t/2) 2 1 t (2-jt-4) (2t) 1/2 1 t some j a(j,4) a(j,1) a(j,0) t 0 1.2j 4.2j 5.2j … … k 0 1 2 3 4 5 6 … …

Scaling Functions We are now ready to officially define the scaling functions: Orthonormal dyadic discrete wavelets are also associated with scaling functions, which are used to smooth or obtain approximations of a signal. Scaling functions (t) have a similar form to those of wavelet functions (t): where is known as the father wavelet. Scaling functions also have the property Also note that, scaling functions are orthogonal to their translations, but not to their dilations (Recall that) Haar scaling function.

Scaling Functions In general, for any given scaling function, the approximation coefficients can be obtained by the inner product of the signal and the scaling function. A continuous approximation of the signal at scale j can then be obtained from the discrete sum of these coefficients Recall and note that as j- xj(t)x(t) Approximation coefficients at scale j Smoothed, scaling-function-dependent approximation of x(t) at scale j

Detail Coefficients So we can reconstruct the signal from its approximation coefficients. But, how about the detail function? Detail functions, too, can be reconstructed by dilating and translating a prototype function, called the wavelet. Just like the approximation functions fj(t)=xj(t) can be obtained from scaling functions, the detail functions gj(t) can be obtained from the wavelet functions. Furthermore, the wavelets corresponding to the scale functions used for approximation, can be obtained from the scaling function:

The Details… (t) (2t) 1/2 1 t (2t-1) 1/2 1 t 1 - = 1/2 t 1 -1

Putting it All Together Approximation coefficients at scale j Smoothed, scaling-function-dependent approximation of x(t) at scale j Wavelet-function-dependent detail of x(t) at scale j Detail coefficients at scale j Furthermore: & Wavelet Reconstruction

Putting It All Together (cont.) Note that we can also start reconstruction from any level (scale) in between: Approximation coefficients at scale j0 Detail coefficients at scale j0 and below Smoothed, scaling-function-dependent approximation of x(t) at scale j0 Wavelet-function-dependent details of x(t) at scales j0 and below