(Images included are .png images, so no additional distortion was added in save/upload for viewing)
I have used the D4 transform from page 20 of "Ripples in mathematics", which is basically these 5 steps:
c1 = √3 / 4.0 ; c2 = (√3 - 2) / 4.0 ; s[ IEVEN ] += √3 * s[ IODD ] ; s[ IODD ] -= c1*s[ IEVEN ] + c2*s[ IPREVEVEN ] ; s[ IEVEN ] -= s[ INEXTODD ] ; s[ IEVEN ] *= ( √3 - 1 ) / √2 ; s[ IODD ] *= ( √3 + 1 ) / √2 ;
c1 = √3 / 4.0 ; c2 = (√3 - 2) / 4.0 ; s[ IODD ] *= ( √3 - 1 ) / √2 ; s[ IEVEN ] *= ( √3 + 1 ) / √2 ; s[ IEVEN ] += s[INEXTODD] ; s[ IODD ] += c1*s[ IEVEN ] + c2*s[IPREVEVEN] ; s[ IEVEN ] -= √3 * s[ IODD ] ;
I'm compiling and running this using
double precision values from C++. I run this on the rows of the image, then the columns. I use a crude filtration algorithm to remove the lowest 90% of the difference coefficients in the image.
The filtration algorithm is:
- Run through entire transformed image (as a set of numbers)
- Find the largest difference coefficient (
maxVal) (in entire 2d image)
minValToSurviveas 1% of
- If a difference coefficient has a magnitude less than
minValToSurvive, it is zeroed.
Here's my problem is. When I remove only 83% of the lowest difference coefficients from the image (minValToSurvive=0.01*maxVal), you get this:
If I remove the normalization steps:
s[ IEVEN ] *= ( √3 - 1 ) / √2 ; // REMOVE s[ IODD ] *= ( √3 + 1 ) / √2 ;
(in both the fwd and reverse transforms), the result after removing 90% of the components is much better (much less noise)
So I can think of 1 of 2 problems:
- Normalizing the image by the ( √3 - 1 ) / √2 factors is killing precision
- I'm not filtering correctly
Or am I wrong? If I'm filtering (removing insignificant components) incorrectly, what is a better way to filter? If it's the floating point precision, then should I not normalize the transform at every step?