Lode Nachtergaele (nachterg@imec.be) Guest

Posted: Tue Jun 30, 1998 2:54 pm Subject: Software: Applet demonstrates the wavelet transform of an image




#12 Software: Applet demonstrates the wavelet transform of an image
Dear wavelet researcher,
on the web page http://www.tele.dtu.dk/wg1/jpeg2000Demo/jpeg2Demo.html
an applet is provided that demonstrates a wavelet transform in action
on an image.
The applet implements a forward and backward wavelet transform defined
in the MPEG4 standard Version 1 (ISO/IEC JTC/SC29/WG11 N1902,
Fribourg, Oct. 1997, Paragraph 10 Annex A Coding transforms).
The wavelet transform is implemented by means of the lifting scheme
(See papers of Wim Sweldens [DS96] for an explanation about this
scheme.) The implemented synthesis filters corresponds to a Cohen 
Daubechies  Feauveau biorthogonal 3/9 taps filter with the following
coefficients :
Lowpass = {32,64,32}*2^(13/2)
Highpass = {3,6,16,38,90,38,16,6,3}*2^(13/2)
There are 2 vanishing moments in the primal wavelet and 4 in the dual
wavelet [UW97]. The primal wavelet with 2 vanishing moments is :
t_1(z) = 1/2 * z^1  1/2
The dual wavelet with 4 vanishing moments is :
s_1(z) = 3/64 * z^1  19/64 + 19/64 * z  3/64 * z^2
Remark that the denominators are powers of two. This is of interest when
considering implementation efficiency.
The JAVA implementation uses integers and hence is lossless. This
means that the two pictures on the left are exactly the same when the
applet is finished! Moreover, unlike in many other implementations,
the image width and height must not be a power of two.
This applet concentrates on the wavelet transform. No compression is
being done.
Enjoy,
Lode
References:
[DS96] I. Daubechies, W. Sweldens, "Factoring Wavelet Transform into
Lifting Steps," AT&T internal report BL011217096092720TM, September
1996. Online at http://cm.belllabs.com/who/wim/
[UW97] Geert Uytterhoeven, Filip Van Wulpen, Maarten Jansen, Dirk Roose
and Adhemar Bultheel, "WAILI: Wavelets with Integer Lifting", Report
TW262, July 1997. Online at http://www.cs.kuleuven.ac.be/~wavelets/

Lode Nachtergaele nachterg@imec.be
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