2d wavelet signal processing - need some clarification

[DonRoth]

Hello:

I am using the advanced signal processing toolkit  and doing some extensive wavelet processing in both 1-d and 2-d.

In 1-d, when processing waveforms, using discrete wavelet transform-based multiresolution analysis, 2 "coefficients" are produced depending on the "level" chosen.  These coefficients are known as the "approximation" and the "detail."  The approximation at level 1 is the summation of the approximation and detail at level 2.  The approximation at level 2 is the summation of the approximation and detail at level 3.  And so forth.  See fig. 4-5 on p.4-6 of the Wavelet Analysis Tools user manual.

I want to consider the same analogy in the 2-d case.  In the wavelet transform of 2d images at any particular level, 4 coefficients are produced. (See the manual, pgs. 4-7 and higher.) These are labeled low_low, low_high, high_low, and high_high.  My interpretation of this is that low_low is the equivalent of the "approximation" and the other 3 coefficients comprise the "detail."  I thought then by summing the 4 coefficients at one level (egs. level 2) would produce the approximation at the level just above it (level 1).  When I test this out, it does not give the expected result.

I wonder if someone who helped produce the toolkit can clarify how one gets the "approximation" and "detail" coeffcients from the 4 produced coefficients and how they relate to the approximation at one level higher.

Thanks,

Don

[DonRoth]

ps. For the 2-d effort, I have been working with the example entitled "Undecimated Image Decomposition and Reconstruction (UWT).vi.

Message Edited by DonRoth on 05-21-2007 01:13 PM

[DonRoth]

Let's extend the analogy of the 1-d and 2-d cases further.  First, look at the Advanced Signal Processing Express VI entitled "Multiresolution Analysis."  In that VI, one can see that the time domain waveform (1-d) is transformed to its power spectrum which is broken into subbands based on the number of levels chosen.  The subband closest to the origin of the power spectrum represents the "approximation" and the other subbands are associated with the "details".  One can see that there is one approximation level and multiple detail levels associated with the power spectrum subbands regardless of the number of levels chosen.  This express VI allows one to choose whatever combination of approximation and details to reconstruct the signal, thus providing a potentially multiple bandpass /  bandstop reconstructed signal.

Fast forward to the 2-d case where I would like to do the same thing.  The analog to the 1-d power spectrum would be the 2-d spatial fourier transform (see attached).  I am devising a similar visualization method to draw the "subbands" on top of the fourier transform as shown by the blue lines and filled rectangles demarcating mouse-selected regions D3 and D1.  However, for the 2-d case, the way I see it, there are actually 3 details associated with each level as mentioned in the above post (low_high, high_low, high_high) and 1 approximation (low_low).  The questions then are:

1) how does one synthesize the 3 details (low_high, high_low, high_high) into 1 composite detail for a specific level so that the visualization method I propose for the 2-d case is consistent with that shown for the 1-d case in the Multiresolution Analysis Expresss VI? Do we just add the 3 details? When I do this, I get a great composite detail image that combines the details of rows, columns, and diagonals - but still not sure if it is the proper way to combine the 3 details.
2) how does one perform a reconstruction (inverse wavelet transform) with only the selected subbands for the 2-d case? I have attempted some reconstructions using the WA Inverse Undecimated Wavelet Transform, but the inverse transform does not appear to behave correctly without the use of all coefficients (low_low, low_high, high_low, high_high).  In the 1-d case in the Express VI, one can see that only the selected subbands can be used to reconstruct the waveform.

Any help regarding these issues is greatly appreciated.

Sincerely,

Don

Message Edited by DonRoth on 05-22-2007 09:58 AM

[NIJanell]

Hello Don,

It could be that I am simplifying this too much, but why do you want the combine the details into one composite detail when you need the individual details to reconstruct the waveform?  From what I understand, the three detail coefficients represent the horizontal, vertical, and diagonal detail coefficients - therefore combining them would cause the signal to become less meaningful.

There are multiple shipping examples that reconstruct 2D wavelets, but it seems you have already looked at some of them.  None of them were able to help?

Good luck,

Janell R | applications engineer

[DonRoth]

The problem can be boiled down to this: How do we create the wavelet-based 2-d analog for the 1-d case shown in the Multiresolution Analysis Express VI?

Adding the detail coefficients along rows, columns, and diagonal provides 1 detail coefficient and 1 analog coeffient at a single level - again, I am trying to get to the analog of the 1-d case.

Yes, I have pretty much studied all of the examples. Filter banks - 2D.vi is very similar principal to Undecimated Image Decomposition and Reconstruction (UWT).vi except decimation occurs

Sincerely,

Don

Message Edited by DonRoth on 05-23-2007 01:54 PM

[DonRoth]

ps. I meant to say adding the detail coefficients along rows, columns, and diagonal provides 1 detail coefficient.  Then we have 1 detail coefficient and 1 APPROXIMATION coefficient per level.....Don

Message Edited by DonRoth on 05-23-2007 02:14 PM