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| The '''complex wavelet transform (CWT)''' is a [[Complex_number|complex-valued]] extension to the standard [[discrete wavelet transform]] (DWT). It is a two-dimensional [[wavelet]] transform which provides [[multiresolution analysis|multiresolution]], sparse representation, and useful characterization of the structure of an image. Further, it purveys a high degree of shift-invariance in its magnitude. However, a drawback to this transform is that it is exhibits <math>2^{d}</math> (where <math>d</math> is the dimension of the signal being transformed) redundancy compared to a separable (DWT).
| | Golda is what's written on my birth certification even though it is not the name on my birth certificate. I've always loved living in Mississippi. To play lacross is the factor I adore most of all. Invoicing is my occupation.<br><br>My homepage: [http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ spirit messages] |
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| The use of complex wavelets in image processing was originally set up in 1995 by J.M. Lina and L. Gagnon [http://www.crim.ca/perso/langis.gagnon/articles/spie95.pdf] in the framework of the Daubechies orthogonal filters banks[http://portal.acm.org/citation.cfm?id=258030&dl=GUIDE&coll=GUIDE&CFID=10476702&CFTOKEN=44762573]. It was then generalized in 1997 by [[Nick Kingsbury|Prof. Nick Kingsbury]] <ref>{{cite conference
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| |author = N. G. Kingsbury
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| |title = Image processing with complex wavelets
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| |booktitle = Phil. Trans. Royal Society London
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| |location = London
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| |url = http://citeseer.ist.psu.edu/kingsbury97image.html
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| |date = September 1999}}
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| </ref><ref>{{cite journal
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| |first = N G|last = Kingsbury
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| |date=May 2001
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| |volume = 10
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| |issue = 3
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| |pages = 234–253
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| |journal = Journal of Applied and Computational Harmonic Analysis
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| |title = Complex wavelets for shift invariant analysis and filtering of signals
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| |url = http://www-sigproc.eng.cam.ac.uk/%7Engk/publications/ngk_ACHApap.pdf
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| |format=PDF|doi = 10.1006/acha.2000.0343}}
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| </ref><ref>{{cite journal
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| |first = Ivan W.|last = Selesnick
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| |coauthors = Baraniuk, Richard G. and Kingsbury, Nick G.
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| |title = The Dual-Tree Complex Wavelet Transform
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| |date=November 2005
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| |volume = 22
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| |issue = 6
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| |pages = 123–151
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| |journal = IEEE Signal Processing Magazine
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| |url = http://www-sigproc.eng.cam.ac.uk/%7Engk/publications/ngk_SPmag_nov05.pdf
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| |format=PDF|doi = 10.1109/MSP.2005.1550194}}
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| </ref>
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| of [[University of Cambridge|Cambridge University]].
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| In the area of computer vision, by exploiting the concept of visual contexts, one can quickly focus on candidate regions, where objects of interest may be found, and then compute additional features through the CWT for those regions only. These additional features, while not necessary for global regions, are useful in accurate detection and recognition of smaller objects. Similarly, the CWT may be applied to detect the activated voxels of cortex and additionally the [[temporal independent component analysis]] (tICA) may be utilized to extract the underlying independent sources whose number is determined by Bayesian information criterion [http://www.springerlink.com/(t0ojvoayxrkdyk55vru2g245)/app/home/contribution.asp?referrer=parent&backto=issue,51,56;journal,180,3824;linkingpublicationresults,1:105633,1].
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| == Dual-tree complex wavelet transform ==
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| The '''Dual-tree complex wavelet transform''' (DTCWT) calculates the complex transform of a signal using two separate DWT decompositions (tree ''a'' and tree ''b''). 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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| [[Image:Wavelets - DTCWT.png|frame|none|Block diagram for a 3-level DTCWT]]
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| This redundancy of two provides extra information for analysis but at the expense of extra computational power. It also provides approximate [[shift-invariance]] (unlike the DWT) yet still allows perfect reconstruction of the signal.
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| The design of the filters is particularly important for the transform to occur correctly and the necessary characteristics are:
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| * The [[low-pass filter]]s in the two trees must differ by half a sample period
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| * Reconstruction filters are the reverse of analysis
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| * All filters from the same orthonormal set
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| * Tree ''a'' filters are the reverse of tree ''b'' filters
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| * Both trees have the same frequency response
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| ==See also==
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| * [[Wavelet series]]
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| * [[Continuous wavelet transform]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| * [http://www.wavelet.org/phpBB2/viewtopic.php?t=7584 An MPhil thesis: Complex wavelet transforms and their applications]
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| * [http://eprints.soton.ac.uk/11007/ CWT for EMG analysis]
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| * [http://www.cvgpr.uni-mannheim.de/Publications/TR_13_03.pdf A paper on DTCWT]
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| * [http://www-iplab.ece.ucsb.edu/publications/99IPTexture.pdf Another full paper]
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| * [http://www-sigproc.eng.cam.ac.uk/~ngk/publications/ngk_biosig04.pdf 3-D DT MRI data visualization]
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| * [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1369333 Multidimensional, mapping-based complex wavelet transforms ]
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| * [http://www-syscom.univ-mlv.fr/~chaux/articles/ieeeIPdouble.pdf Image Analysis Using a Dual-Tree <math>M</math>-band Wavelet Transform (2006), preprint, Caroline Chaux, Laurent Duval, Jean-Christophe Pesquet]
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| * [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]
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| * [http://arxiv.org/pdf/0712.2317 A nonlinear Stein based estimator for multichannel image denoising (2007), preprint, Caroline Chaux, Laurent Duval, Amel Benazza-Benyahia, Jean-Christophe Pesquet]
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| * [http://www-syscom.univ-mlv.fr/~chaux/ Caroline Chaux website (<math>M</math>-band dual-tree wavelets)]
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| * [http://www.laurent-duval.eu/siva-signal-image-links.html#dual-tree-complex-wavelet Laurent Duval website (<math>M</math>-band dual-tree wavelets)]
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| * [http://www.ece.msstate.edu/~fowler/ James E. Fowler (dual-tree wavelets for video and hyperspectral image compression)]
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| * [http://www.eng.cam.ac.uk/~ngk/ Nick Kingsbury website (dual-tree wavelets)]
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| * [http://www-syscom.univ-mlv.fr/~pesquet/ Jean-Christophe Pesquet website (<math>M</math>-band dual-tree wavelets)]
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| * [http://taco.poly.edu/selesi/ Ivan Selesnick (dual-tree wavelets)]
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| [[Category:Wavelets]]
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Golda is what's written on my birth certification even though it is not the name on my birth certificate. I've always loved living in Mississippi. To play lacross is the factor I adore most of all. Invoicing is my occupation.
My homepage: spirit messages