Hintermueller Christoph
06-14-2007, 09:24 PM
Hi
I'm sorry for the cross posting but i'm not sure which group is the proper
one if any at all.
Could you please give me some hints on the following, including keywordfor
newsgroup, web search and/or searching library indices for finding the
proper books, or at least the proper newsgroup(s) to send my questions to.
What i already do know and use:
1) In 1D i know and use it frequently that two real valued signals can be
combined to a complex signal c(x) = f(x) + jg(x) so that the fourer
transform
of it is C[X] = F[X] + jG[X]. To this i can apply the even odd decomposition
to compute the spectra F(X) and G(X). after filtering , convolution or
deconvolution of the two signals i can recombine them into the complex
specturm C'[X] = F'[X] + jG'[X]. Applying inverse complex fourier transform
i get c'(x) = f'(x) +jg'(x) from which i then can extract the filtered,
convoulted, deconvoluted signals.
2) Further i know that the fourier transform and back transform for
multidimensional signals , like images, can be computed as a combination of
one dimensional fourier transforms along each dimension:
F2(f(x,y) = Fy(Fx(f(x,y))
What i think or at least be curious about that it might be correct:
3) Combining 1) and 2) than should lead to
F2(f(x,y) + jg(x,y)) = Fy(Fx(f(x,y) + jg(x,y)))
In combination with the linarity of the fouriertransform
F2(f(x,y) + jg(x,y)) = Fy(Fx(f(x,y)) +jFy(Fx(f(x,y))
So my questions are:
a) Does 3) really follow fom 1) and 2) ? does this hold?
b) If 3) along which dimension do i have to aply the even odd decomposition?
Along x, y or both? Or is even odd decomposition only aplicable in 1D)
c) What happens if f(x,y) resembles all odd lined of the image I and g(x,y)
represents all even lines of a single image? do i still get the full complex
spectrum of the image or is the spectrum reduced to the positiv half ofthe
spectrum? Can the negative part be computed exploiting the hemertiantity of
the compex spectra?
d) Even if 3) holds can i recombine two 2D spectra the same way as in 1D in
order to gain the filtered, convoluted, deconvoluted real valued images
after
inverse fourier transform.
e) What characteristics of complex signals, and fourier transform should i
exploit to find an aswwer my self if any at all.
f) Where can i get answers for the above questions? Who shall i ask? Which
newsgroup should i send the questions too? Which one is most likely that
sombody can and will answer my questions not just by "this is off topic" or
other non satisfying default answers.
Thanks in Advance for your help
sincerly
Dr. Christoph Hintermüller
I'm sorry for the cross posting but i'm not sure which group is the proper
one if any at all.
Could you please give me some hints on the following, including keywordfor
newsgroup, web search and/or searching library indices for finding the
proper books, or at least the proper newsgroup(s) to send my questions to.
What i already do know and use:
1) In 1D i know and use it frequently that two real valued signals can be
combined to a complex signal c(x) = f(x) + jg(x) so that the fourer
transform
of it is C[X] = F[X] + jG[X]. To this i can apply the even odd decomposition
to compute the spectra F(X) and G(X). after filtering , convolution or
deconvolution of the two signals i can recombine them into the complex
specturm C'[X] = F'[X] + jG'[X]. Applying inverse complex fourier transform
i get c'(x) = f'(x) +jg'(x) from which i then can extract the filtered,
convoulted, deconvoluted signals.
2) Further i know that the fourier transform and back transform for
multidimensional signals , like images, can be computed as a combination of
one dimensional fourier transforms along each dimension:
F2(f(x,y) = Fy(Fx(f(x,y))
What i think or at least be curious about that it might be correct:
3) Combining 1) and 2) than should lead to
F2(f(x,y) + jg(x,y)) = Fy(Fx(f(x,y) + jg(x,y)))
In combination with the linarity of the fouriertransform
F2(f(x,y) + jg(x,y)) = Fy(Fx(f(x,y)) +jFy(Fx(f(x,y))
So my questions are:
a) Does 3) really follow fom 1) and 2) ? does this hold?
b) If 3) along which dimension do i have to aply the even odd decomposition?
Along x, y or both? Or is even odd decomposition only aplicable in 1D)
c) What happens if f(x,y) resembles all odd lined of the image I and g(x,y)
represents all even lines of a single image? do i still get the full complex
spectrum of the image or is the spectrum reduced to the positiv half ofthe
spectrum? Can the negative part be computed exploiting the hemertiantity of
the compex spectra?
d) Even if 3) holds can i recombine two 2D spectra the same way as in 1D in
order to gain the filtered, convoluted, deconvoluted real valued images
after
inverse fourier transform.
e) What characteristics of complex signals, and fourier transform should i
exploit to find an aswwer my self if any at all.
f) Where can i get answers for the above questions? Who shall i ask? Which
newsgroup should i send the questions too? Which one is most likely that
sombody can and will answer my questions not just by "this is off topic" or
other non satisfying default answers.
Thanks in Advance for your help
sincerly
Dr. Christoph Hintermüller