Complex wavelet transform

<h2 id="dual-tree-complex-wavelet-transform">Dual-tree complex wavelet transform</h2>
<p>The dual-tree complex wavelet transform (DTCWT) calculates the complex transform of a signal using two separate DWT decompositions (tree <i>a</i> and tree <i>b</i>).  If the filters used in one are specifically designed different from those in the other it is possible for one DWT to produce the real coefficients and the other the imaginary.
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<p>This redundancy of two provides extra information for analysis but at the expense of extra computational power.  It also provides approximate <a href="/facts/Shift-invariant_system/zaKY72py">shift-invariance</a> (unlike the DWT) yet still allows perfect reconstruction of the signal. 
</p><p>The design of the filters is particularly important for the transform to occur correctly and the necessary characteristics are:
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<ul><li>The <a href="/facts/Low-pass_filter/QFsUjw7h">low-pass filters</a> in the two trees must differ by half a sample period</li>
<li>Reconstruction filters are the reverse of analysis</li>
<li>All filters from the same orthonormal set</li>
<li>Tree <i>a</i> filters are the reverse of tree <i>b</i> filters</li>
<li>Both trees have the same frequency response</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Continuous_wavelet_transform/nGUoUAKX">Continuous wavelet transform</a></li>
<li><a href="/facts/Wavelet_series/4vk3W5be">Wavelet series</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20160303195941/http://www.wavelet.org/phpBB2/viewtopic.php?t=7584">An MPhil thesis: Complex wavelet transforms and their applications</a></li>
<li><a href="http://eprints.soton.ac.uk/11007/">CWT for EMG analysis</a></li>
<li><a href="https://web.archive.org/web/20041111015830/http://www.cvgpr.uni-mannheim.de/Publications/TR_13_03.pdf">A paper on DTCWT</a></li>
<li><a href="https://web.archive.org/web/20050816124215/http://www-iplab.ece.ucsb.edu/publications/99IPTexture.pdf">Another full paper</a></li>
<li><a href="http://www-sigproc.eng.cam.ac.uk/~ngk/publications/ngk_biosig04.pdf">3-D DT MRI data visualization</a></li>
<li><a href="https://ieeexplore.ieee.org/document/1369333/;jsessionid=1966DE8BB08D37C94F50A0AB23ABBDD2?arnumber=1369333">Multidimensional, mapping-based complex wavelet transforms </a></li>
<li><a href="https://web.archive.org/web/20160303183211/http://www-syscom.univ-mlv.fr/~chaux/articles/ieeeIPdouble.pdf">Image Analysis Using a Dual-Tree 
  
    
      
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<li><a href="https://web.archive.org/web/20160303183047/http://www-syscom.univ-mlv.fr/~chaux/articles/chauxpesquetduvalIT.pdf">Noise covariance properties in dual-tree wavelet decompositions (2007), preprint, Caroline Chaux, Laurent Duval, Jean-Christophe Pesquet</a></li>
<li><a href="https://arxiv.org/abs/0712.2317">A nonlinear Stein based estimator for multichannel image denoising (2007), preprint, Caroline Chaux, Laurent Duval, Amel Benazza-Benyahia, Jean-Christophe Pesquet</a></li>
<li><a href="https://web.archive.org/web/20140829073608/http://www-syscom.univ-mlv.fr/~chaux/">Caroline Chaux website (
  
    
      
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<li><a href="http://www.laurent-duval.eu/siva-signal-image-links.html#dual-tree-complex-wavelet">Laurent Duval website (
  
    
      
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<li><a href="http://www.ece.msstate.edu/~fowler/">James E. Fowler (dual-tree wavelets for video and hyperspectral image compression)</a></li>
<li><a href="http://www-sigproc.eng.cam.ac.uk/Main/NGK">Nick Kingsbury website (dual-tree wavelets)</a></li>
<li><a href="http://www-syscom.univ-mlv.fr/~pesquet/">Jean-Christophe Pesquet website (
  
    
      
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<li><a href="https://web.archive.org/web/20150129091055/http://taco.poly.edu/selesi/">Ivan Selesnick (dual-tree wavelets)</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Barri, Adriaan; Dooms, Ann; Schelkens, Peter (2012). "The near shift-invariance of the dual-tree complex wavelet transform revisited". Journal of Mathematical Analysis and Applications. 389 (2): 1303–1314. arXiv:1304.7932. doi:10.1016/j.jmaa.2012.01.010. S2CID 119665123. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Lina, JM; Gagnon, L. (1995). Image enhancement with symmetric Daub echies wavelets (PDF). Wavelet Applications in Signal and Image Processing II. Vol. 2569. pp. 196–207. Archived from the original (PDF) on 2016-03-03. <a href="https://web.archive.org/web/20160303175143/http://www.crim.ca/perso/langis.gagnon/articles/spie95.pdf" target="_blank">https://web.archive.org/web/20160303175143/http://www.crim.ca/perso/langis.gagnon/articles/spie95.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Lina, JM (June 1997). "Image Processing with Complex Daubechies Wavelets". Journal of Mathematical Imaging and Vision. 7 (3): 211–22. doi:10.1023/A:1008274210946. <a href="http://portal.acm.org/citation.cfm?id=258030&dl=GUIDE&coll=GUIDE&CFID=10476702&CFTOKEN=44762573" target="_blank">http://portal.acm.org/citation.cfm?id=258030&dl=GUIDE&coll=GUIDE&CFID=10476702&CFTOKEN=44762573</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>N. G. Kingsbury (September 1999). "Image processing with complex wavelets". Phil. Trans. Royal Society London. London. <a href="http://citeseer.ist.psu.edu/kingsbury97image.html" target="_blank">http://citeseer.ist.psu.edu/kingsbury97image.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Kingsbury, N G (May 2001). "Complex wavelets for shift invariant analysis and filtering of signals" (PDF). Applied and Computational Harmonic Analysis. 10 (3): 234–253. CiteSeerX 10.1.1.588.4232. doi:10.1006/acha.2000.0343. <a href="http://www-sigproc.eng.cam.ac.uk/%7Engk/publications/ngk_ACHApap.pdf" target="_blank">http://www-sigproc.eng.cam.ac.uk/%7Engk/publications/ngk_ACHApap.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Selesnick, Ivan W.; Baraniuk, Richard G.; Kingsbury, Nick G. (November 2005). "The Dual-Tree Complex Wavelet Transform" (PDF). IEEE Signal Processing Magazine. 22 (6): 123–151. Bibcode:2005ISPM...22..123S. doi:10.1109/MSP.2005.1550194. hdl:1911/20355. S2CID 833630. <a href="http://www-sigproc.eng.cam.ac.uk/%7Engk/publications/ngk_SPmag_nov05.pdf" target="_blank">http://www-sigproc.eng.cam.ac.uk/%7Engk/publications/ngk_SPmag_nov05.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
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Complex wavelet transform open-in-new

Complex wavelet transform