first.last: Build a first/last database for wavelet transforms

Description

This function is not intended for user use, but is used by various functions involved in computing and displaying wavelet transforms.

Usage

first.last(LengthH, DataLength, bc = c("periodic","symmetric"))

Arguments

LengthH

length of the filter used to produce a wavelet decomposition.

DataLength

length of the data before transforming; must be a power of 2, say $2^m$.

character string, determining how the boundaries of the the function are to be handled; one of "periodic" (default) or "symmetric".

Value

A first/last database structure, a list with the following components:
first.last.cA (m+1) x 3 matrix. The first column specifies the real index of the first coefficient of the smoothed data at a level, the 2nd column is the real index of the last coefficient, the last column specifies the offset of the first smoothed datum at that level. The offset is used by the C code to work out where the beginning of the sequence is within a packed vector of the pyramid structure. The first and 2nd columns can be used to work out how many numbers there are at a level.
If bc="periodic" then the pyramid is a true power of 2 pyramid, that is it starts with a power of 2, and the next level is half of the previous. If bc="symmetric" then the pyramid is nearly exactly a power of 2, but not quite, see the Details section for why this is so.
ntotalThe total number of smoothed data/original data points.
first.last.dA mx3 matrix. As for first.last.c but for the wavelet coefficients packed as the D component of a wavelet structure.
ntotal.dThe total number of wavelet coefficients.

RELEASE

Release 2.2 Copyright Guy Nason 1993

BUGS

None, I hope. However, with hindsight, I should have implemented the periodic version first. The symmetric boundary stuff confused a lot of people (including me)!

Details

Suppose you begin with $2^m=2048$ coefficients. At the next level you would expect 1024 smoothed data coefficients, and 1024 wavelet coefficients, and if bc="periodic" this is indeed what happens. However, if bc="symmetric" you actually need more than 1024 (as the wavelets extend over the edges). The first/last database keeps track of where all these "extras" appear and also where they are located in the packed vectors C and D of pyramidal coefficients within wavelet structures.

For example, given a first.last.c row of -2 3 20 The actual coefficients would be $c_{-2}, c_{-1}, c_{0}, c_{1}, c_{2}, c_{3}$.

In other words, there are 6 coefficients, starting at -2 and ending at 3, and the first of these ($c_{-2}$) appears at an offset of 20 from the beginning of the * $ C component vector of the wavelet structure.

You can ``do'' first.last in your head for periodic boundary handling, but for more general boundary treatments (e.g. symmetric) first.last is indispensable.

References

The numbers in first/last databases were worked out from inequalities derived from:

Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets; Communications on Pure and Applied Mathematics, 41, 909-996.

Examples

Run this code

## If you're twisted then you may just want to look at one of these.

first.last(length(filter.select(2)), 64)

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