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[[File:MorletWaveletMathematica.svg|thumb|Real-valued Morlet wavelet]]
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[[File:Wavelet Cmor.svg|thumb|Complex-valued Morlet wavelet]]
 
In [[mathematics]], the '''Morlet wavelet''' (or '''Gabor wavelet''')<ref name="Bernardino">[http://homepages.inf.ed.ac.uk/rbf/CAVIAR/PAPERS/05-ibpria-alex.pdf A Real-Time Gabor Primal Sketch for Visual Attention] "The Gabor kernel satisfies the admissibility condition for wavelets, thus being suited for multi-resolution analysis. Apart from a scale factor, it is also known as the Morlet Wavelet."</ref> is a [[wavelet]] composed of a [[complex exponential]] ([[carrier wave|carrier]]) multiplied by a [[Gaussian window]] (envelope).  This wavelet is closely related to human perception, both hearing<ref name="Mallat"/> and vision.<ref>[[John Daugman|J. G. Daugman]]. Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. ''Journal of the Optical Society of America A'', 2(7):1160–1169, July 1985.</ref>
 
In 1946, physicist [[Dennis Gabor]], applying ideas from [[quantum physics]], introduced the use of Gaussian-windowed sinusoids for time-frequency decomposition, which he referred to as ''[[Gabor atom|atoms]]'', and which provide the best trade-off between spatial and frequency resolution.<ref name="Bernardino"/>  These are used in the [[Gabor transform]], a type of [[short-time Fourier transform]].<ref name="Mallat">[http://cnx.org/content/m23074/latest/ Time-Frequency Dictionaries], Mallat</ref> In 1984, [[Jean Morlet]] introduced Gabor's work to the seismology community and, with Goupillaud and Grossmann, modified it to keep the same wavelet shape over equal octave intervals, resulting in the first formalization of the [[continuous wavelet transform]].<ref>http://rocksolidimages.com/pdf/gabor.pdf</ref>  (See also [[Wavelet#History|Wavelet history]])
 
The wavelet is defined as a constant <math>\kappa_{\sigma}</math> subtracted from a plane wave and then localised by a [[Gaussian]] [[Window_function#Gaussian_window|window]]:<ref name="Ashmead2012">
{{cite journal
| author = John  Ashmead
| title  = Morlet Wavelets in Quantum Mechanics
| journal = Quanta
| year = 2012
| volume = 1
| issue = 1
| pages = 58–70
| doi = 10.12743/quanta.v1i1.5
| url = http://quanta.ws/ojs/index.php/quanta/article/view/5
}}
</ref>
 
:<math>\Psi_{\sigma}(t)=c_{\sigma}\pi^{-\frac{1}{4}}e^{-\frac{1}{2}t^{2}}(e^{i\sigma t}-\kappa_{\sigma})</math>
 
where <math>\kappa_{\sigma}=e^{-\frac{1}{2}\sigma^{2}}</math> is defined by
the admissibility criterion and the normalisation constant <math>c_{\sigma}</math> is:
 
:<math>c_{\sigma}=\left(1+e^{-\sigma^{2}}-2e^{-\frac{3}{4}\sigma^{2}}\right)^{-\frac{1}{2}}</math>
 
The [[Fourier transform]] of the Morlet wavelet is:
 
:<math>\hat{\Psi}_{\sigma}(\omega) = c_\sigma \pi^{-\frac{1}{4}} \left( e^{-\frac{1}{2}(\sigma-\omega)^2} - \kappa_\sigma e^{-\frac{1}{2}\omega^{2}} \right)</math>
 
The "central frequency" <math>\omega_{\Psi}</math> is the position of the global maximum of <math>\hat{\Psi}_{\sigma}(\omega)</math> which, in this case, is given by the solution of the equation:
 
:<math>(\omega_{\Psi}-\sigma)^{2}-1=(\omega_{\Psi}^{2}-1)e^{-\sigma\omega_{\Psi}}</math>
 
The parameter <math>\sigma</math> in the Morlet wavelet allows trade between
time and frequency resolutions. Conventionally, the restriction <math>\sigma>5</math> is used to avoid problems with the Morlet wavelet at low <math>\sigma</math> (high temporal resolution).
 
For signals containing only slowly varying frequency and amplitude
modulations (audio, for example) it is not necessary to use small
values of <math>\sigma</math>. In this case, <math>\kappa_{\sigma}</math> becomes very small (e.g. <math>\sigma>5 \quad \Rightarrow \quad \kappa_{\sigma}<10^{-5}\,</math>) and is, therefore, often neglected. Under the restriction <math>\sigma>5</math>, the frequency of the Morlet wavelet is conventionally taken to be <math>\omega_{\Psi}\simeq\sigma</math>.
 
The wavelet exists as a complex version or a purely real-valued version.  Some distinguish between the "real Morlet" vs the "complex Morlet".<ref>[http://www.mathworks.com/help/toolbox/wavelet/ug/f8-24282.html#f8-23908 Matlab Wavelet Families] - "Morlet Wavelet: morl" and "Complex Morlet Wavelets: cmor"</ref> Others consider the complex version to be the "Gabor wavelet", while the real-valued version is the "Morlet wavelet".<ref>Mathematica documentation: [http://reference.wolfram.com/mathematica/ref/GaborWavelet.html GaborWavelet]</ref><ref>Mathematica documentation: [http://reference.wolfram.com/mathematica/ref/MorletWavelet.html MorletWavelet]</ref>
 
== See also ==
* [[Constant Q transform]]
 
== References ==
<references/>
* P. Goupillaud, A. Grossman, and J. Morlet. ''Cycle-Octave and Related Transforms in Seismic Signal Analysis''. Geoexploration, 23:85-102, 1984
* N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani. Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies. IEEE Trans. Inf. Th., 38:644-664, 1992
 
[[Category:Continuous wavelets]]

Latest revision as of 03:08, 10 January 2015

Let's look an actual registry scan plus several of what we will see when we do one on your computer. This test was done on a computer that has been not functioning because it should, operating at slow speed plus having some issues with freezing up.

StreamCI.dll errors are caused by a amount of different issues, including which the file itself has been moved on the program, the file is outdated or you have installed several third-party sound motorists that are conflicting with all the file. The good news is the fact that if you need to resolve the error you're seeing, you need to look to initially ensure the file & drivers are working okay on your PC plus then resolving any StreamCI.dll errors that may be inside the registry of your computer.

Over time your disk can moreover get fragmented. Fragmentation causes your computer to slow down considering it takes windows much longer to obtain a files location. Fortunately, a PC has a built in disk defragmenter. You are able to run this system by clicking "Start" - "All Programs" - "Accessories" - "System Tools" - "Disk Defragmenter". We may today have the option to choose that forces or partition you need to defragment. This action might take we some time thus it is actually advised to do this on a regular basis so because to avoid further fragmentation and to speed up the windows XP computer.

If you feel you don't have enough money at the time to upgrade, then the number one way is to free up certain space by deleting a few of the unwanted files plus folders.

Whenever it comes to software, this is the vital piece because it is the one running your program plus additional programs required inside the works. Always maintain the cleanliness of your system from obsolete data by getting a wise windows registry cleaner. Protect it from a virus on the net by providing a workable virus protection system. You should have a monthly clean up by running the defragmenter program. This technique it may enhance the performance of your computer plus for you to avoid any mistakes. If you think something is incorrect with all the software, and we don't recognize how to fix it then refer to a technician.

Turn It Off: Chances are when you are like me; then you spend a great deal of time on your computer on a daily basis. Try giving the computer some time to do absolutely nothing; this could sound funny however, when you have an older computer you are asking it to do too much.

Another issue with the damaged variation is the fact that it takes too much time to scan the system and whilst it is scanning, you cannot employ the computer otherwise. Moreover, there is not any technical help to these cracked versions meaning if you get stuck someplace, you can't ask for help. They even do not have any customer service aid lines wherein we could call or send to solve a issues.

Frequently the number one way is to read critiques on them and when countless consumers remark regarding its efficiency, it is very likely to be function. The best piece is that there are many top registry products that work; we just have to take a pick.

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