multiProj {mwaved} | R Documentation |
Reconstructs a function using wavelet coefficients (waveletCoef
object) as input.
multiProj(beta, j1 = log2(length(beta$coef)) - 1)
beta |
A |
j1 |
The finest resolution to be used in the projection (specifies which resolution that the wavelet expansion is truncated). |
Function that takes an input of wavelet coefficients in the form of a waveletCoef
object (see multiCoef
for details) and optionally a desired maximum resolution level, j1
, to create an inhomogeneous wavelet expansion starting from resolution j0
up to resolution j1
.
Namely, it creates the wavelet expansion,
∑_{k = 0}^{2^{j_0} - 1} β_k φ_{j_0,k} + ∑_{j = j_0}^{j_1} ∑_{k = 0}^{2^j - 1} β_{j,k} ψ_{j,k}.
where (φ,ψ) denote the father and mother periodised Meyer wavelet functions and β_{j,k} denotes the mother wavelet coefficient at resolution j and location k and β_{k} denotes the father wavelet coefficients at resolution j=j0 and location k.
The coefficients beta
need to be ordered so that the first 2^\code{j0} elements correspond to father wavelet coefficients at resolution j=\code{j0} and the remaining elements correspond to the mother wavelet coefficients from resolution j=\code{j0} to j = log_2 n - 1. If the maximum resolution level j1 is not specified, the full wavelet expansion will be given.
A numeric vector of size n giving the wavelet function expansion.
library(mwaved) # Make a noiseless doppler function n <- 2^8 x <- (1:n)/n y <- makeDoppler(n) # Determine the wavelet coefficients beta <- multiCoef(y) # plot three raw wavelet expansions truncating in each case at j1 = 3, 4 and 5 respectively plot(x, y, type = 'l', main = 'Doppler and wavelet projections at three different truncations') j0 <- 3 j1 <- 5 j <- j0:j1 lcols <- c(1, j - j0 + 2) ltys <- c(1, 1:length(j)) matlines(x, sapply(j, function(i) multiProj(beta, j1 = i)), type = 'l', col = lcols[-1]) legend("bottomright", legend = c("Signal", paste('j1 =', j)), col = lcols, lty =ltys)