Wimage.info: Finds key indices related to a 2-d multiresolution

Description

Functions for finding the indices and other information for a 2-d basis in a multiresolution wavelet decomposition.

Usage

Wimage.info(m = 128, n = m, cut.min = 4)
Wimage.i2s(ind, m, n, cut.min)
Wimage.s2i(i, j, level, flavor, m, n, cut.min, mat = TRUE)

Arguments

Value

Wimage.info returns a list where all indices pertain to a location or subsets of the mXn matrix of wavelet coefficients.mNumber of rows in imagenNumber of columns of imagecut.mincut.minSA vector with 4 elements. S[1],S[2] give the min/max row indices for the father wavelets and S[3],S[4] give the min/max column indices for the father wavelets.HA matrix where each row is a resolution level and cloumns give s ranges of row and column indices. e.g. H[k,1:4] gives the ranges for the rows and columns for the horizontal basis functions at level k.VA matrix in teh same format as H that gives subsets for the vertical basis functions.DiA matrix in the same format as H that gives subsets for the diagonal basis functions.LA column matrix where rows index resolutaion and column give the grid size for each level of resolution.LmaxTotal number of resolution levels.offset.mA 3 column matrix that has the row offsets that indicate the begining of a block of coeffiecients. Rows are levels of resolution and columns are H,V,Di.offset.nA 3 column matrix that has the column offsets that indicate the begining of a block of coeffiecients. Rows are levels of resolution and columns are H,V,Di.

Details

The wavelet coefficents are computed efficiently as a single large matrix/image but this format make it difficult to identify the indices ofspecific kinds of basis functions. The wavelet basis functions as found with a tensor product multiresolution. ( e.g. Wtransform.image) are orgainzied in levels of resolution and type. At the coarsest level of resolution are father wavelets (dentoed by S for "smooth") roughly centered on a rectanglar grid of locations (the size of the grid controlled by cut.min). At this resolution are three mother wavelets that capture horizontal (H), vertical (V) and diagonal( Di) structure at this scale. These basis functions are also centered on a lattice of grid locations. Subsequent levels only use the H,V, Di mother wavelet templates. For example given a 32X32 image with cut.min=4 There are three levels of resolution and 1024 basis functions total. The coarsest level is 16 S basis functions on a regular 4X4 grid, 16 H, 16 V and 16 Di basis function also on 4X4 grid. The next level has 64 H, V and Di basis functions on an 8X8 grid and the final level has 256 H, V and Di basis functions on a 16X16 grid, giving a grand total of 1024 basis functions.

Without some additional calculations it is possible to organize the results in different resolutions. This function provides the necessary indexing information to do this. See the function plot.Wimag as an exmaple of how this is used.

Indexing the image can happen in 3 ways. 1)as a structure where one specifies the location (i,j) , level, and flavor. 2) as the row and column indices of the image. 3) as a single index if the image is stacked as a long vector. The functions Wimage.s2i and Wimage.i2s convert indices between these forms.

Examples

Run this code

#Find a basis function.
# For a basis on a 64X64 image with cut.min=8 find a 
# horizontal basis function at second level of resolution in the (4,4)
# position. ( There are 16X16 horizontal basis functions at the 2 nd level
#
Wimage.s2i( i=3,j=2, level=2, flavor=1, m=64, n=64, cut.min=8)-> ind
tmp<- matrix( 0, 64,64)
tmp[ind] <- 1
Wtransform.image( tmp, cut.min=8, inv=TRUE)-> look
image.plot( look)

# A check of Wimage.i2s
Wimage.i2s( ind,m=64, n=64, cut.min=8)
# should get i=3,j=2, level=2, flavor=1

# complete check of functions
 Wimage.i2s( 1:512, cut.min=8, m=16, n=32)-> look
 Wimage.s2i( look, cut.min=8, m=16, n=32, mat=FALSE)-> look2
 sum( look2 - (1:512)) # sum should be zero 


# W transform of John Lennon image

     data(lennon)
     m<- nrow(lennon)
     n<- ncol(lennon)

# add an grey strip down middle columns

     lennon[ , 128:132]<- 120

     look<- Wtransform.image( lennon, cut.min=8)

# get info 
     info<- Wimage.info( n, m, cut.min=8)


# Zero out all V basis functions coefficients, 

tmp<- look

Vind<- info$V
for (lev in 1: info$Lmax){
tmp[ Vind[lev,1]: Vind[lev,2], Vind[lev,3]: Vind[lev,4]]<- 0 
}


look2<- Wtransform.image( tmp, cut.min=8, inv=TRUE)

# take a look at vertically  filtered image
set.panel( 2,1)
par( pty="s")
image( lennon, col=grey(seq(0,1,,256)))
image( look2, col=grey(seq(0,1,,256)))

# NOTE:
# What is vertical and horizontal is confusing here
# due to the convention of the R image plot of running rows 
# of the image as the X coordinate
# The grey column in lennon image is plotted as 
# a horizontal line  
#