0.7 Generalizations of the basic multiresolution wavelet system  (Page 18/28)

the expansion coefficients are listed next for the same cases described previously.

Again, case 1 is the minimum norm solution; however, it has no zero components this time because there are no expansion vectors orthogonal tothe signal. Since the signal lies between the 90 ^o and 45 ^o expansion vectors, it is case 3 which has the least two-vector energyrepresentation.

There are an infinite variety of ways to construct the overcomplete frame matrix $\mathbf{X}$ . The one in this example is a four-vector tight frame. Each vector is 45 ^o degrees apart from nearby vectors. Thus they are evenly distributed in the 180 ^o upper plane of the two dimensional space. The lower plane is covered by the negative ofthese frame vectors. A three-vector tight frame would have three columns, each 60 ^o from each other in the two-dimension plane. A 36-vector tight frame would have36 columns spaced 5 ^o from each other. In that system, any signal vector would be very close to an expansion vector.

Still another alternative would be to construct a frame (not tight) with nonorthogonal rows. This would result in columns that are not evenlyspaced but might better describe some particular class of signals. Indeed, one can imagine constructing a frame operator with closely spacedexpansion vectors in the regions where signals are most likely to occur or where they have the most energy.

We next consider a particular modified tight frame constructed so as to give a shift-invariant DWT.

Shift-invariant redundant wavelet transforms and nondecimated filter banks

One of the few flaws in the various wavelet basis decompositions and wavelet transforms is the fact the DWT is not translation-invariant. Ifyou shift a signal, you would like the DWT coefficients to simply shift, but it does more than that. It significantly changes character.

Imagine a DWT of a signal that is a wavelet itself. For example, if the signal were

then the DWT would be

In other words, the series expansion in the orthogonal wavelet basis would have only one nonzero coefficient.

If we shifted the signal to the right so that $y\left(n\right)=\phi (2^4(n-1)-10)$ , there would be many nonzero coefficients because at this shift or translation, the signal is no longer orthogonal to most of thebasis functions. The signal energy would be partitioned over many more coefficients and, therefore, because ofParseval's theorem, be smaller. This would degrade any denoising or compressions usingthresholding schemes. The DWT described in Chapter: Calculation of the Discrete Wavelet Transform is periodic in that at each scale $j$ the periodized DWT repeats itself after a shift of $n=2^j$ , but the period depends on the scale. This can also be seen from the filter bank calculation of theDWT where each scale goes through a different number of decimators and therefore has a different aliasing.