transforms for complex data

[John Grinstead (jgrinstead@mednet.ucla.edu)]

I am trying to denoise a complex data set, in particular the complex data acquired in Magnetic Resonance experiments. In MR, two quadrature receiver coils, out of phase by 90 degrees, give a complex data set. Typically there are some phase errors between the real and (so-called) imaginary channels, so that to create a good image you display the magnitude of the complex image data, but sometimes you may also display the phase image.

Anyway, I was wondering what are good ways to filter complex data, using either wavelets or wavelet packets, and if models that take the complex nature of the data into account are better suited than ones that just filter the real and imaginary channels separately (which I have tried with moderate success).

Another aspect I have been looking into lately, is how models that exploit the dependency across scales of the wavelet coefficients compare to more simple wavelet shrinkage methods (hard or soft thresholding). Again, I am most interested in applying this to complex image data.

Any opinions appreciated, thanks!

-John

[John Grinstead (jgrinstead@mednet.ucla.edu)]

There real and imaginary channels of data both have normally distributed additive gaussian noise, and each channel has independent noise. The magnitude (modulus) and phase images are calculated from the Re and Im channels as normal. I wish to filter the raw data before the magnitude and phase images are calculated.

Even in a zero noise environment, the phase image will not be uniform, due to magnetic field inhomogeneity, subject motion, or other artifacts. The phase image looks like a slowly varying surface, while some regions (where there is air) looks like pure noise. Filtering the real and imaginary channels separately will clean up both the magnitude and phase images simultaneously.

My question arises form some question on whether one should filter the data as a complex value, or filter each channel separatly. Some argue that phase errors or slightly correlated noise between the two channels could lead to problems if filtering the data as a single complex entity.

[Jun Ge (jge@emba.uvm.edu)]

Hi,

I am a graduate student who is currently working on denoising with redundant representations. I would like to refer you to the paper by Dr. Sylvain Sardy:

2000-04 IEEE Trans. Signal Processing:
``minimax threshold for denoising complex signals with waveshrink''

If you like, we can discuss this topic in details through emails.


--JG