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[WM]: Vanishing Moments, Support, Regularity and Symmetry. & JPEG2000!
Dear watermarkers,
I'm investigating a simple non-blind watermark system
using the wavelet coefficients of an image. The
embedding algorithm used is summarized as:
C' = C (1+alpha*W)
where
C' is the marked wavelet coefficients
C is the original wavelet coefficients
alpha is the watermark strength
W is a Gaussian N(0,1) vector.
Note that a 2-level decomposition is used and only the
detail subbands are modified, i.e. H1, V1, D1, H2, V2,
D2 coefficients (H denotes horizontal, V denotes
vertical, D denotes diagonal). The A2 approximation coefficients are NOT
touched.
As I used DWT in the decomposition process, I have
some questions regarding the properties of wavelets
and how they might affect the performance of the
above-decribed system:
Vanishing moments
-----------------
The number of vanishing moments of a wavelet
determines the order of the polynomial that can be approximated. TO my
understanding, higher moments are better since they analyzes a signal
more precisely.
1) That might be true for 1D signals, but in 2D image,
isn't approximating the image using polynomials
strange? What's the basis that an image can be
described using polynomials?
2)If a wavelet has a more vanishing moments than
another wavelet within the same family (e.g.
Daubechies), how does it affect the PSNR of the marked
image wrt to the original image? Higher or lower? Or
does it depend on how "well" the image can be
approximated by polynomials?
Support Size
------------
Small support size are good because they can pick up
isolated singularities within a 1D signal (Mallat's
book: A Wavelet Tour of Signal Processing)
3) But in an image, we seldom find instances of
"isolated singularities" right? The closest thing I
can think of is a noise pixel. Since we are talking
about an "isolated" singularity , a region of complex
texture doesn't seem to qualify as such. In any case,
if we use a wavelet of compact support, does it mean
that the detail coefficients tend to be of higher
amplitude than using a wavelet of wider support (since
details=singularities?? )?
I'm asking this because if indeed more compact
supports mean higher-amplitude detail coeffs, then
using the above embedding algorithm where only detail
coeffs are modified (assuming we use exactly the same
Gaussian watermark and strength), the overall effect
on the wavelet coeffs will be lesser, i.e. a better
PSNR of the reconstructed marked image.
Regularity
----------
I'm a bit confused as to what regularity means.
Is regularity = smoothness, i.e. more regular wavelet
= more smooth wavelet?
The effect of regularity is usually discussed in the
context of quantizing the wavelet coeffs and how the
coeff quantization error affects the spatial error
introduced in the reconstructed image as a result. In
general, using a regular wavelet means that the
reconstructed image will have smoother errors than an
irregular wavelet. The error in both cases will have
the same energy, but because the error using the
regular wavelet is smoother, it APPPEARS less visibily
to the human eye.
4) In the context of watermarking, we add a Gaussian
watermark to the wavelet coeffs. This is similar in
effect to quantization I suppose, since both alter the
wavelet coeffs in some way. So, does this means that
using a more regular wavelet will have improved PSNR
over a less regular one?
Symmetry
--------
5) I've read about symmetric filters, but what are anti-symmetric
filters? They are often lumped together as one group. The other group
being assymmetric filters. Is a symmetric lowpass filter is equivalent
to a symmetric scaling function?
6) Is it a fact that except for the Haar wavelet, most orthogonal
wavelets cannot lead to perfect reconstruction using
symmetric/antisymmetric filters?
7) If so, is this the reason why the biorthogonal
wavelet basis was created? Why are symmetric filters
so important anyway? Does it offer any advantages over asymmetric ones,
such as effect on amplitude of wavelet coeffs?
Biorthogonal wavelet
--------------------
8)My results showed that for most images, e.g. Lena or
Boat, using the bior 5/3 or bior 9/7 wavelet leads to
worse PSNR of the marked image than other wavelets in
general. I'm at a loss as to explain this. Could it be
due to any of the wavelet properties described above?
9) I've also ran JPEG and JPEG2000 attacks (using
Jasper at www.ece.uvic.ca/~mdadams/jasper/) on the
marked image and attempted detection of the watermark.
The detection algorithm is simple, by correlating the
estimated watermark and the original watermark.
Comparison with a threshold is made and if the
correlation is above the threshold, detection is
successful.
JPEG == I ran the attacks on marked image embedded
using the Daubechies, Coiflet, Symmlet, Discrete
Meyer, bior 5/3 and bior 9/7 wavelets.
For almost all quality settings (compresion ratios),
the bior 5/3 and bior 9/7 wavelets gave much higher
correlation compared to the rest. Can anyone explain
why this is the case? I don't see how a DCT-based
compression attack will have less effect if a bior
wavelet were to be used.
JPEG2000 == JPEG2000 attacks were run for the same
bunch of wavelets. Again, the bior 5/3 and bior 9/7 consistently
produced higher correlation values than the rest for all compression
ratios. This greatly mystifies me because to my knowledge, the JPEG2000
Part-1 still image coding standard employs the bior 5/3 and bior 9/7
wavelet transforms for intra-component compression. My understanding is
that in wavelet-based image compression, the goal is to have many
low-amplitude detail coeffs so that they can be discarded, similar to
the DCT-based JPEG. If so, then the point of using the bior 5/3 or 9/7
wavelet in JPEG2000 is, I suppose, to help towards this. But since we
are also embedding the watermark in the detail coeffs, wouldn't
bior-based JPEG2000 compression REMOVE the watermark better if the bior
wavelets were to be used in the embedding process AS WELL?
(This duality or conflict in goals between a
compression system and a watermarking system has been
pointed out by Peter Meerwald.)
==============================================
I must thank anyone who has managed to trudge through
all my questions. Any help would be greatly
appreciated.
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