Flipped waveform in dddtree2 wavelet

One of our customers is using the dual-tree complex wavelet transform. When he runs the command dddtree2 on a set image (see code file) he sees the first level have a mirror orientation to the rest of the images. 

Attached a code which shows this as an example. 

Is this a bug or deliberate code structure? 

Answer 1

[MathWorks Support Team]

The answer to your customer's question regarding the mirror image as result of "dddtree2" function is that this is the expected result.  Detailed explanations are below:

One of the specialists on wavelets here at MathWorks has investigated into the customer's code provide in the script.  In short, the 6 dual-tree wavelets formed from DUALTREE2 do appear to be approximately analytic and have the same orientation in the real and imaginary parts. This is the expected behavior.   The specialist has provided a sample script that examines the wavelets in terms of their analytic properties and hopefully sheds some light into the customer's perspective on his desired workflow (dualtree_test.m).  In addition, the specialist also provided a PDF document describing in detail how the dual-tree complex wavelet transform works in the context of image processing (CWT_Tutorial.pdf).

In the MATLAB script provided, it shows how to obtain the complex-valued dual-tree wavelet. Specifically: 

 

1.) Both the real and imaginary parts have the same orientation and are shifted versions of each other as would be expected since the imaginary part should be approximately the Hilbert transform of the real part for the dual-tree wavelets.

 

2.) Further, I show that the Fourier transform of the wavelet shows an approximately analytic function in the plane as expected.

 

The customer can also look at a demo of the orientation of the wavelets here:

 

https://www.mathworks.com/help/wavelet/ug/dual-tree-complex-wavelet-transforms.html

 

The customer should go down to the text on this reference page to "The filters applied to the columns and rows may be from the same filter".  There is code below that section that shows how to obtain the real and imaginary parts for all six real and six imaginary parts. Note this figure is very much like the figure on p. 140 (Fig. 16) in the attached CWT_Tutorial.pdf paper. Note that there is no universal convention on the formation of the wavelets so the important thing to note is the real and imaginary parts have the same orientation. They are only shifted version of each other. This is evident in Fig. 16 in the paper and also in the results from DDDTREE2 shown in the demo.