22:
2059:
1853:
1558:
754:
1932:, such as percussion sounds in audio, or high-frequency components in two-dimensional images, for example an image of stars on a night sky. This means that the transient elements of a data signal can be represented by a smaller amount of information than would be the case if some other transform, such as the more widespread
4191:
The wavelet transform can provide us with the frequency of the signals and the time associated to those frequencies, making it very convenient for its application in numerous fields. For instance, signal processing of accelerations for gait analysis, for fault detection, for the analysis of seasonal
2034:
For most natural images, the spectrum density of lower frequency is higher. As a result, information of the low frequency signal (reference signal) is generally preserved, while the information in the detail signal is discarded. From the perspective of image compression and reconstruction, a wavelet
5204:
For processing temporal signals in real time, it is essential that the wavelet filters do not access signal values from the future as well as that minimal temporal latencies can be obtained. Time-causal wavelets representations have been developed by Szu et al and
Lindeberg, with the latter method
1372:
The fundamental idea of wavelet transforms is that the transformation should allow only changes in time extension, but not shape, imposing a restriction on choosing suitable basis functions. Changes in the time extension are expected to conform to the corresponding analysis frequency of the basis
5584:
Vorbis I is a forward-adaptive monolithic transform CODEC based on the
Modified Discrete Cosine Transform. The codec is structured to allow addition of a hybrid wavelet filterbank in Vorbis II to offer better transient response and reproduction using a transform better suited to localized time
1703:
4868:
1131:
4125:
4553:
2496:
Another important issue for image compression and reconstruction is the system's oscillatory behavior, which might lead to severe undesired artifacts in the reconstructed image. To achieve this, the wavelet filters should have a large peak to sidelobe ratio.
579:
4331:
1942:
has been successfully applied for the compression of electrocardiograph (ECG) signals In this work, the high correlation between the corresponding wavelet coefficients of signals of successive cardiac cycles is utilized employing linear prediction.
4730:
3692:
2001:
in the image (i.e., there is no compression yet since it is only a transform). These coefficients can then be compressed more easily because the information is statistically concentrated in just a few coefficients. This principle is called
1847:
568:
3887:
1691:, but with additional special properties of the wavelets, which show up at the resolution in time at higher analysis frequencies of the basis function. The difference in time resolution at ascending frequencies for the
386:
4182:
using a
Gaussian window function. The exception is when searching for signals of a known, non-sinusoidal shape (e.g., heartbeats); in that case, using matched wavelets can outperform standard STFT/Morlet analyses.
2500:
So far we have discussed about one-dimension transformation of the image compression system. This issue can be extended to two dimension, while a more general term - shiftable multiscale transforms - is proposed.
943:
1946:
Wavelet compression is not effective for all kinds of data. Wavelet compression handles transient signals well. But smooth, periodic signals are better compressed using other methods, particularly traditional
4744:
3460:
2935:
981:
261:
1420:
1268:
843:
4005:
2406:. Even though wavelet with strict shift variance is not realistic, it is possible to select wavelet with only slight shift variance. For example, we can compare the shift variance of two filters:
4236:
584:
4342:
1695:
and the wavelet transform is shown below. Note however, that the frequency resolution is decreasing for increasing frequencies while the temporal resolution increases. This consequence of the
139:
749:{\displaystyle {\begin{aligned}\langle \psi _{jk},\psi _{lm}\rangle &=\int _{-\infty }^{\infty }\psi _{jk}(x){\overline {\psi _{lm}(x)}}dx\\&=\delta _{jl}\delta _{km}\end{aligned}}}
473:
194:
4174:
Wavelets have some slight benefits over
Fourier transforms in reducing computations when examining specific frequencies. However, they are rarely more sensitive, and indeed, the common
3176:
3002:
2054:
Wavelet image compression system involves filters and decimation, so it can be described as a linear shift-variant system. A typical wavelet transformation diagram is displayed below:
1959:. Compressing data that has both transient and periodic characteristics may be done with hybrid techniques that use wavelets along with traditional harmonic analysis. For example, the
426:
3758:
2567:
4231:
2784:
2320:
5087:
3336:
1498:
788:
3070:
2654:
4564:
2404:
2362:
1354:
5213:
Synchro-squeezed transform can significantly enhance temporal and frequency resolution of time-frequency representation obtained using conventional wavelet transform.
4949:
and a wavelet-function. A convolution can be implemented as a multiplication in the frequency domain. With this the following approach of implementation results into:
3506:
3496:
3248:
3212:
3038:
2726:
2690:
2603:
2493:
By observing the impulse responses of the two filters, we can conclude that the second filter is less sensitive to the input location (i.e. it is less shift variant).
2278:
2242:
2206:
2170:
2134:
2098:
1687:
has been filtered. The transformed signal provides information about the time and the frequency. Therefore, wavelet-transformation contains information similar to the
1622:
1585:
1552:
1307:
4221:
1463:
293:
3952:
1167:
5163:
5134:
5107:
5039:
5010:
4911:
2811:
1656:
4980:
4947:
3277:
1709:
This shows that wavelet transformation is good in time resolution of high frequencies, while for slowly varying functions, the frequency resolution is remarkable.
1685:
3096:
4891:
4166:
4146:
3993:
3973:
3908:
1518:
1443:
2786:
goes through decimation by a factor of two. After L levels of decomposition (and decimation), the analysis response is obtained by retaining one out of every
1715:
481:
6094:
Silva, K. M.; Souza, B. A.; Brito, N. S. D. (October 2006). "Fault detection and classification in transmission lines based on wavelet transform and ANN".
6230:
Daubechies, Ingrid; Lu, Jianfeng; Wu, Hau-Tieng (December 12, 2009). "Synchrosqueezed
Wavelet Transforms: a Tool for Empirical Mode Decomposition".
3789:
5439:
Akansu, Ali N.; Haddad, Richard A. (1992), Multiresolution Signal
Decomposition: Transforms, Subbands, and Wavelets, Boston, MA: Academic Press,
301:
851:
4863:{\displaystyle Y_{W}(c,\tau )={\frac {1}{\sqrt {c}}}\cdot \int _{-\infty }^{\infty }y(t)\cdot \Psi \left({\frac {t-\tau }{c}}\right)\,dt}
5571:
3341:
2816:
1527:
The higher the required resolution in time, the lower the resolution in frequency has to be. The larger the extension of the analysis
1126:{\displaystyle \left(a,b)={\frac {1}{\sqrt {|a|}}}\int _{-\infty }^{\infty }{\overline {\psi \left({\frac {x-b}{a}}\right)}}f(x)dx\,}
206:
5352:
4120:{\displaystyle X(a,b)={\frac {1}{\sqrt {a}}}\int _{-\infty }^{\infty }{\overline {\Psi \left({\frac {t-b}{a}}\right)}}x(t)\,dt}
1383:
5501:, for example, may use a 5/3 wavelet for lossless (reversible) transform and a 9/7 wavelet for lossy (irreversible) transform.
6070:
6029:
5864:
5444:
5428:
1175:
800:
6012:
Sheybani, E.; Javidi, G. (December 2009). "Dimensionality
Reduction and Noise Removal in Wireless Sensor Network Datasets".
5888:
Liu, Jie (2012). "Shannon wavelet spectrum analysis on truncated vibration signals for machine incipient fault detection".
5194:
varies substantially over the domain, or more specifically, estimation of functions that are sparse in the wavelet domain.
4548:{\displaystyle \Psi (k,n,m)={\frac {1}{\sqrt {c_{0}^{n}}}}\cdot \Psi \left={\frac {1}{\sqrt {c_{0}^{n}}}}\cdot \Psi \left}
4192:
displacements of landslides, for design of low power pacemakers and also in ultra-wideband (UWB) wireless communications.
5933:"Using wavelet tools to analyse seasonal variations from InSAR time-series data: a case study of the Huangtupo landslide"
2208:). The compression and reconstruction system generally involves low frequency components, which is the analysis filters
5169:
1967:
1956:
5779:
Bruns, Andreas (2004). "Fourier-, Hilbert- and wavelet-based signal analysis: are they really different approaches?".
98:
5831:
5627:
5412:
5396:
5367:
5232:
3501:
To obtain the overall L level analysis/synthesis system, the analysis and synthesis responses are combined as below:
5512:
438:
159:
2007:
5309:
1982:
3101:
6181:"A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time"
5979:
5328:
2943:
1374:
394:
6426:
1696:
2509:
As mentioned earlier, impulse response can be used to evaluate the image compression/reconstruction system.
1970:
to compress audio (which is generally smooth and periodic), however allows the addition of a hybrid wavelet
6447:
5362:
5357:
5347:
5253:
4179:
3700:
2515:
1688:
4326:{\displaystyle {\begin{aligned}c_{n}&=c_{0}^{n}\\\tau _{m}&=m\cdot T\cdot c_{0}^{n}\end{aligned}}}
3697:
Finally, the peak to first sidelobe ratio and the average second sidelobe of the overall impulse response
3915:
5744:
Simoncelli, E.P.; Freeman, W.T.; Adelson, E.H.; Heeger, D.J. (1992). "Shiftable multiscale transforms".
6457:
6452:
5263:
2731:
1939:
1865:
271:
197:
26:
5847:
Martin, E. (2011). "Novel method for stride length estimation with body area network accelerometers".
2283:
5243:
4725:{\displaystyle Y_{DW}(n,m)={\frac {1}{\sqrt {c_{0}^{n}}}}\cdot \sum _{k=0}^{K-1}y(k)\cdot \Psi \left}
1933:
1918:
5050:
3282:
1471:
762:
6144:
Szu, Harold H.; Telfer, Brian A.; Lohmann, Adolf W. (1992). "Causal analytical wavelet transform".
5319:
3043:
2608:
1985:(2010) for discussion of practical issues of current methods using wavelets for video compression.
1886:
6053:
Sheybani, E. O.; Javidi, G. (May 2012). "Multi-resolution filter banks for enhanced SAR imaging".
5653:"Relations between the statistics of natural images and the response properties of cortical cells"
1852:
949:
21:
3687:{\displaystyle h_{AS}^{(L)}(n,n_{i})=\sum _{k}f_{h0}^{(L)}(k-n_{i}/2^{L})f_{g0}^{(L)}(n/2^{L}-k)}
2021:
A few 1D and 2D applications of wavelet compression use a technique called "wavelet footprints".
5610:
Ho Tatt Wei; Jeoti, V. (2004). "A wavelet footprints-based compression scheme for ECG signals".
2367:
2325:
6306:"Wave packet transform and wavelet convolution product involving the index Whittaker transform"
6253:"Synchro-squeezed adaptive wavelet transform with optimum parameters for arbitrary time series"
5849:
2011 IEEE Topical
Conference on Biomedical Wireless Technologies, Networks, and Sensing Systems
5235:, which shows that wavelet for image compression can also be nearly coiflet (nearly orthogonal)
1922:
1320:
267:
146:
65:
5561:
3465:
3217:
3181:
3007:
2695:
2659:
2572:
2247:
2211:
2175:
2139:
2103:
2067:
1604:
1567:
1534:
1276:
5608:
Ho Tatt Wei and Jeoti, V. "A wavelet footprints-based compression scheme for ECG signals".
4200:
1929:
1448:
277:
264:
5168:
There are many different types of wavelet transforms for specific purposes. See also a full
3922:
1142:
6350:
6338:
6264:
6153:
5944:
5897:
5710:
5667:
5248:
5141:
5112:
5092:
5017:
4988:
4896:
2789:
1641:
72:
4956:
4923:
3253:
1661:
8:
5566:
5173:
3075:
2015:
1928:
Using a wavelet transform, the wavelet compression methods are adequate for representing
1557:
6354:
6268:
6207:
6180:
6157:
5948:
5909:
5901:
5714:
5671:
2045:
Should not lead to artifacts in the image reconstructed from the reference signal alone.
6374:
6282:
6231:
6111:
6076:
6035:
5913:
5870:
5804:
5761:
5633:
5543:
5391:
Meyer, Yves (1992), Wavelets and
Operators, Cambridge, UK: Cambridge University Press,
5258:
5238:
5226:
5181:
4876:
4151:
4131:
3978:
3958:
3893:
1521:
1503:
1428:
972:
5701:
Villasenor, John D. (August 1995). "Wavelet Filter
Evaluation for Image Compression".
4916:
and as already mentioned in this context, the wavelet-transformation corresponds to a
2940:
On the other hand, to reconstruct the signal x(n), we can consider a reference signal
1842:{\displaystyle y(t)\;=\;\sin(2\pi f_{0}t)\;+\;\sin(4\pi f_{0}t)\;+\;\sin(8\pi f_{0}t)}
563:{\displaystyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}dx}
6442:
6378:
6366:
6325:
6286:
6212:
6066:
6025:
5960:
5917:
5860:
5856:
5827:
5796:
5726:
5683:
5623:
5535:
5440:
5424:
5408:
5407:
Chui, Charles K. (1992), An Introduction to Wavelets, San Diego, CA: Academic Press,
5392:
5303:
5279:
3782:
1948:
1898:
1882:
1878:
1692:
1465:
150:
53:
6115:
6080:
6039:
5874:
5808:
5765:
5637:
2058:
6406:
6358:
6317:
6272:
6202:
6192:
6161:
6103:
6058:
6017:
5994:
5952:
5905:
5852:
5792:
5788:
5753:
5718:
5675:
5615:
5547:
5527:
5479:
5469:
2011:
2003:
1952:
1874:
6419:
5998:
5652:
2364:; The optimal wavelet are those who bring minimum shift variance and sidelobe to
1528:
791:
6277:
6252:
5931:
Tomás, R.; Li, Z.; Lopez-Sanchez, J. M.; Liu, P.; Singleton, A. (June 1, 2016).
5619:
5597:
3763:
6321:
6197:
6062:
5314:
4175:
61:
6107:
5956:
2136:), a decimation process, an interpolation process, and two synthesis filters (
6436:
6370:
6329:
6305:
5964:
5289:
4336:
Leading to wavelets of the form, the discrete formula for the basis wavelet:
3882:{\displaystyle {\hat {X}}(f)=\int _{-\infty }^{\infty }x(t)e^{-i2\pi ft}\,dt}
1975:
1914:
1910:
432:
154:
5932:
6394:
6216:
6021:
6014:
2009 Second International Conference on Computer and Electrical Engineering
5800:
5730:
5679:
5327:, the image format developed from original wavelet compression research at
5294:
5274:
5222:
5177:
2064:
The transformation system contains two analysis filters (a low pass filter
1658:
can be regarded as an impulse response of a system with which the function
961:
5687:
5539:
381:{\displaystyle \psi _{jk}(x)=2^{\frac {j}{2}}\psi \left(2^{j}x-k\right)\,}
5484:
5458:"A Survey on Change Detection and Time Series Analysis with Applications"
5343:
5334:
4917:
1994:
1971:
1963:
69:
57:
45:
5474:
5457:
5339:
5283:
5191:
2035:
should meet the following criteria while performing image compression:
6410:
5757:
5722:
5598:"A New and Novel Image Compression Algorithm Using Wavelet Footprints"
5531:
5047:
Inverse transformation of the product into the time domain results in
4738:
As apparent from wavelet-transformation representation (shown below)
3250:. Similarly, the procedure is iterated to obtain the reference signal
2049:
2039:
Being able to transform more original image into the reference signal.
1913:. The goal is to store image data in as little space as possible in a
938:{\displaystyle f(x)=\sum _{j,k=-\infty }^{\infty }c_{jk}\psi _{jk}(x)}
6362:
6337:
Prasad, Akhilesh; Maan, Jeetendrasingh; Verma, Sandeep Kumar (2021).
6165:
5596:
N. Malmurugan, A. Shanmugam, S. Jayaraman and V. V. Dinesh Chander.
5498:
5299:
2280:
for reconstruction. To evaluate such system, we can input an impulse
1890:
1712:
Another example: The analysis of three superposed sinusoidal signals
6055:
2012 International Conference on Systems and Informatics (ICSAI2012)
3338:. After L iterations, the synthesis impulse response is calculated:
3760:
can be used to evaluate the wavelet image compression performance.
1906:
30:
6339:"Wavelet transforms associated with the index Whittaker transform"
6236:
5372:
5205:
also involving a memory-efficient time-recursive implementation.
5044:
Multiplication with the transformed signal YFFT of the first step
1902:
76:
38:
3455:{\displaystyle h_{s}^{(L)}(n,n_{i})=f_{g0}^{(L)}(n/2^{L}-n_{j})}
2930:{\displaystyle h_{A}^{(L)}(n,n_{i})=f_{h0}^{(L)}(n-n_{i}/2^{L})}
79:. This article provides a formal, mathematical definition of an
5138:
Back to the second step, until all discrete scaling values for
1960:
5743:
5271:
format uses wavelet-based IW44 algorithm for image compression
2042:
Highest fidelity reconstruction based on the reference signal.
1702:
5324:
3764:
Comparison with Fourier transform and time-frequency analysis
1998:
5423:
Daubechies, Ingrid. (1992), Ten Lectures on Wavelets, SIAM,
4225:
Applied the following discretization of frequency and time:
1993:
First a wavelet transform is applied. This produces as many
203:
The Hilbert basis is constructed as the family of functions
16:
Mathematical technique used in data compression and analysis
5268:
5190:
Locally adaptive statistical estimation of functions whose
4558:
Such discrete wavelets can be used for the transformation:
2692:
is a low pass filter. Similarly, the next reference signal
1894:
5455:
256:{\displaystyle \{\psi _{jk}:\,j,\,k\,\in \,\mathbb {Z} \}}
6251:
Qu, Hongya; Li, Tiantian; Chen, Genda (January 1, 2019).
5456:
Ghaderpour, E.; Pagiatakis, S. D.; Hassan, Q. K. (2021).
5286:
image format designed for speed and processing efficiency
1415:{\displaystyle \Delta t\Delta \omega \geq {\frac {1}{2}}}
573:
this family is orthonormal, it is an orthonormal system:
5978:
Akansu, A. N.; Serdijn, W. A.; Selesnick, I. W. (2010).
5930:
1263:{\displaystyle c_{jk}=\left\left(2^{-j},k2^{-j}\right)}
838:{\displaystyle f\,\in \,L^{2}\left(\mathbb {R} \right)}
6304:
Maan, Jeetendrasingh; Prasad, Akhilesh (May 1, 2024).
5977:
1983:
Diary Of An x264 Developer: The problems with wavelets
5144:
5115:
5095:
5053:
5020:
5014:
Scaling of the wavelet-basis-function by this factor
4991:
4959:
4926:
4899:
4879:
4747:
4567:
4345:
4234:
4203:
4154:
4134:
4008:
3981:
3961:
3925:
3896:
3792:
3703:
3509:
3468:
3344:
3285:
3256:
3220:
3184:
3104:
3078:
3046:
3010:
2946:
2819:
2792:
2734:
2698:
2662:
2611:
2575:
2518:
2370:
2328:
2286:
2250:
2214:
2178:
2142:
2106:
2070:
1718:
1664:
1644:
1607:
1570:
1537:
1506:
1474:
1451:
1431:
1386:
1323:
1279:
1178:
1145:
984:
854:
803:
765:
582:
484:
441:
397:
304:
280:
209:
162:
101:
4736:
Implementation via the FFT (fast Fourier transform)
2029:
3072:, then the reference signal at the previous stage (
2410:Biorthogonal filters for wavelet image compression
2050:
Requirement for shift variance and ringing behavior
5157:
5128:
5101:
5081:
5033:
5004:
4974:
4941:
4905:
4885:
4862:
4724:
4547:
4325:
4215:
4160:
4140:
4119:
3987:
3967:
3946:
3902:
3881:
3752:
3686:
3490:
3454:
3330:
3271:
3242:
3206:
3170:
3090:
3064:
3032:
2996:
2929:
2805:
2778:
2720:
2684:
2656:goes through decimation by a factor of two, while
2648:
2597:
2561:
2398:
2356:
2314:
2272:
2236:
2200:
2164:
2128:
2092:
1841:
1679:
1650:
1616:
1579:
1546:
1512:
1492:
1457:
1437:
1414:
1348:
1301:
1262:
1161:
1125:
937:
837:
782:
748:
562:
467:
420:
380:
287:
255:
188:
133:
2504:
6434:
6143:
5609:
5510:
2244:for image compression and the synthesis filters
948:with convergence of the series understood to be
134:{\displaystyle \psi \,\in \,L^{2}(\mathbb {R} )}
6336:
6093:
5513:"ECG coding by wavelet-based linear prediction"
4186:
6229:
6052:
6011:
2439:.852699, .377402, -.110624, -.023849, .037828
960:. This implies that an orthonormal wavelet is
468:{\displaystyle L^{2}\left(\mathbb {R} \right)}
189:{\displaystyle L^{2}\left(\mathbb {R} \right)}
5980:"Emerging applications of wavelets: A review"
5208:
2484:.615051, .133389, -.067237, .006989, .018914
6417:
6343:Mathematical Methods in the Applied Sciences
5187:Fault detection in electrical power systems.
797:Completeness is satisfied if every function
619:
587:
497:
485:
250:
210:
6392:
5520:IEEE Transactions on Biomedical Engineering
6399:IEEE Computational Science and Engineering
6303:
5700:
5612:2004 IEEE Region 10 Conference TENCON 2004
3171:{\displaystyle r_{L-1}(n)=g_{0}(n-2n_{j})}
2057:
1807:
1803:
1771:
1767:
1735:
1731:
1699:is not correctly displayed in the Figure.
6276:
6235:
6206:
6196:
6178:
6128:
5483:
5473:
4853:
4110:
3872:
2997:{\displaystyle r_{L}(n)=\delta (n-n_{j})}
1122:
827:
811:
807:
779:
457:
421:{\displaystyle j,\,k\,\in \,\mathbb {Z} }
414:
412:
408:
404:
377:
284:
246:
244:
240:
236:
229:
178:
124:
109:
105:
6257:Mechanical Systems and Signal Processing
6250:
5650:
37:For broader coverage of this topic, see
20:
5826:. Mathematical Association of America.
5746:IEEE Transactions on Information Theory
5199:
4985:Selection of a discrete scaling factor
6435:
5846:
5821:
5353:Set partitioning in hierarchical trees
5346:generated using wavelets instead of a
5172:but the common ones are listed below:
1859:
1849:with STFT and wavelet-transformation.
1531:is chosen, the larger is the value of
5778:
5703:IEEE Transactions on Image Processing
5511:Ramakrishnan, A.G.; Saha, S. (1997).
3753:{\displaystyle h_{AS}^{(L)}(n,n_{i})}
3462:, which relates the reference signal
3178:, which is obtained by interpolating
2562:{\displaystyle x(n)=\delta (n-n_{i})}
2453:.788486, .418092, -.040689, -.064539
2605:after one level of decomposition is
1917:. Wavelet compression can be either
1634:High frequency, small scaling factor
6096:IEEE Transactions on Power Delivery
5887:
5041:and subsequent FFT of this function
2006:. After that, the coefficients are
1638:In other words, the basis function
1597:Low frequency, large scaling factor
13:
6297:
6179:Lindeberg, T. (January 23, 2023).
5890:Measurement Science and Technology
5574:from the original on April 3, 2022
5170:list of wavelet-related transforms
4824:
4804:
4799:
4668:
4491:
4400:
4346:
4063:
4055:
4050:
3830:
3825:
1968:modified discrete cosine transform
1851:
1701:
1608:
1571:
1556:
1538:
1393:
1387:
1061:
1056:
895:
890:
642:
637:
516:
511:
14:
6469:
6418:Robi Polikar (January 12, 2001).
6386:
6016:. Vol. 2. pp. 674–677.
5233:Biorthogonal nearly coiflet basis
5089:for different discrete values of
4953:Fourier-transformation of signal
4178:is mathematically identical to a
2779:{\displaystyle r_{1}(n)*h_{0}(n)}
2030:Requirement for image compression
1689:short-time-Fourier-transformation
6427:Concise Introduction to Wavelets
5857:10.1109/BIOWIRELESS.2011.5724356
2315:{\displaystyle \delta (n-n_{i})}
845:may be expanded in the basis as
6244:
6223:
6172:
6137:
6131:All of Nonparametric Statistics
6122:
6087:
6046:
6005:
5971:
5924:
5881:
5840:
5824:A Panorama of Harmonic Analysis
5815:
5781:Journal of Neuroscience Methods
5772:
5737:
5694:
5644:
5602:
5310:Least-squares spectral analysis
2322:and observe its reconstruction
1889:). Notable implementations are
5793:10.1016/j.jneumeth.2004.03.002
5590:
5554:
5504:
5492:
5449:
5433:
5417:
5401:
5385:
5329:Los Alamos National Laboratory
5082:{\displaystyle Y_{W}(c,\tau )}
5076:
5064:
4969:
4963:
4936:
4930:
4818:
4812:
4770:
4758:
4662:
4656:
4593:
4581:
4367:
4349:
4107:
4101:
4024:
4012:
3941:
3929:
3844:
3838:
3811:
3805:
3799:
3747:
3728:
3723:
3717:
3681:
3654:
3649:
3643:
3627:
3593:
3588:
3582:
3553:
3534:
3529:
3523:
3498:and the reconstructed signal.
3485:
3479:
3449:
3415:
3410:
3404:
3385:
3366:
3361:
3355:
3331:{\displaystyle L-2,L-3,....,1}
3266:
3260:
3237:
3231:
3201:
3195:
3165:
3143:
3127:
3121:
3027:
3021:
2991:
2972:
2963:
2957:
2924:
2890:
2885:
2879:
2860:
2841:
2836:
2830:
2773:
2767:
2751:
2745:
2715:
2709:
2679:
2673:
2643:
2637:
2621:
2615:
2592:
2586:
2556:
2537:
2528:
2522:
2505:Derivation of impulse response
2393:
2374:
2351:
2332:
2309:
2290:
2267:
2261:
2231:
2225:
2195:
2189:
2159:
2153:
2123:
2117:
2087:
2081:
1836:
1814:
1800:
1778:
1764:
1742:
1728:
1722:
1674:
1668:
1493:{\displaystyle \omega =2\pi f}
1113:
1107:
1041:
1033:
1020:
1008:
932:
926:
864:
858:
783:{\displaystyle \delta _{jl}\,}
691:
685:
666:
660:
545:
539:
530:
524:
324:
318:
145:if it can be used to define a
128:
120:
1:
6395:"An Introduction to Wavelets"
5910:10.1088/0957-0233/23/5/055604
5378:
5368:Time–frequency representation
4913:represents time shift factor
3065:{\displaystyle 1\leq i\leq L}
2649:{\displaystyle x(n)*h_{0}(n)}
2024:
2010:and the quantized values are
1697:Fourier uncertainty principle
90:
5999:10.1016/j.phycom.2009.07.001
5614:. Vol. A. p. 283.
5363:Stationary wavelet transform
5358:Short-time Fourier transform
5348:short-time Fourier transform
5254:Continuous wavelet transform
4187:Other practical applications
4180:short-time Fourier transform
4093:
1367:
1099:
695:
549:
7:
6278:10.1016/j.ymssp.2018.05.020
5620:10.1109/TENCON.2004.1414412
5216:
2470:.788486, .047699, -.129078
952:. Such a representation of
151:complete orthonormal system
10:
6476:
6322:10.1007/s11139-023-00793-3
6198:10.1007/s00422-022-00953-6
6063:10.1109/ICSAI.2012.6223611
5822:Krantz, Steven G. (1999).
5264:Discrete wavelet transform
5209:Synchro-squeezed transform
2399:{\displaystyle h(n-n_{i})}
2357:{\displaystyle h(n-n_{i})}
1957:Fourier-related transforms
1940:Discrete wavelet transform
1866:Discrete wavelet transform
1863:
969:integral wavelet transform
85:integral wavelet transform
36:
27:discrete wavelet transform
6393:Amara Graps (June 1995).
6108:10.1109/TPWRD.2006.876659
5957:10.1007/s10346-015-0589-y
5244:Complex wavelet transform
2460:
2429:
1988:
1934:discrete cosine transform
1594:Good frequency resolution
1349:{\displaystyle b=k2^{-j}}
52:is a representation of a
6129:Wasserman, L.A. (2005).
5651:J. Field, David (1987).
5562:"Vorbis I specification"
5320:Multiresolution analysis
5109:and a discrete value of
3491:{\displaystyle r_{L}(n)}
3243:{\displaystyle g_{0}(n)}
3207:{\displaystyle r_{L}(n)}
3033:{\displaystyle d_{i}(n)}
3004:. If the detail signals
2721:{\displaystyle r_{2}(n)}
2685:{\displaystyle h_{0}(n)}
2598:{\displaystyle r_{1}(n)}
2273:{\displaystyle g_{0}(n)}
2237:{\displaystyle h_{0}(n)}
2201:{\displaystyle g_{1}(n)}
2165:{\displaystyle g_{0}(n)}
2129:{\displaystyle h_{1}(n)}
2093:{\displaystyle h_{0}(n)}
1631:Bad frequency resolution
1617:{\displaystyle \Delta t}
1580:{\displaystyle \Delta t}
1547:{\displaystyle \Delta t}
1302:{\displaystyle a=2^{-j}}
4216:{\displaystyle c-\tau }
3916:Time–frequency analysis
2569:, the reference signal
2512:For the input sequence
2100:and a high pass filter
1458:{\displaystyle \omega }
1373:function. Based on the
288:{\displaystyle \psi \,}
6420:"The Wavelet Tutorial"
6185:Biological Cybernetics
6057:. pp. 2702–2706.
6022:10.1109/ICCEE.2009.282
5987:Physical Communication
5680:10.1364/JOSAA.4.002379
5159:
5130:
5103:
5083:
5035:
5006:
4976:
4943:
4907:
4887:
4864:
4726:
4652:
4549:
4327:
4217:
4162:
4142:
4121:
3989:
3969:
3948:
3947:{\displaystyle X(t,f)}
3904:
3883:
3754:
3688:
3492:
3456:
3332:
3273:
3244:
3208:
3172:
3092:
3066:
3040:are equal to zero for
3034:
2998:
2931:
2807:
2780:
2722:
2686:
2650:
2599:
2563:
2400:
2358:
2316:
2274:
2238:
2202:
2166:
2130:
2094:
1856:
1843:
1706:
1681:
1652:
1618:
1581:
1561:
1548:
1514:
1494:
1459:
1439:
1416:
1377:of signal processing,
1350:
1303:
1264:
1163:
1162:{\displaystyle c_{jk}}
1127:
939:
899:
839:
784:
750:
564:
469:
431:If under the standard
422:
382:
289:
257:
190:
135:
34:
6310:The Ramanujan Journal
5184:, triangular wavelet.
5160:
5158:{\displaystyle c_{n}}
5131:
5129:{\displaystyle c_{n}}
5104:
5102:{\displaystyle \tau }
5084:
5036:
5034:{\displaystyle c_{n}}
5007:
5005:{\displaystyle c_{n}}
4977:
4944:
4908:
4906:{\displaystyle \tau }
4888:
4865:
4727:
4626:
4550:
4328:
4218:
4163:
4143:
4122:
3990:
3970:
3949:
3905:
3884:
3755:
3689:
3493:
3457:
3333:
3274:
3245:
3214:and convoluting with
3209:
3173:
3093:
3067:
3035:
2999:
2932:
2808:
2806:{\displaystyle 2^{L}}
2781:
2723:
2687:
2651:
2600:
2564:
2401:
2359:
2317:
2275:
2239:
2203:
2167:
2131:
2095:
1855:
1844:
1705:
1682:
1653:
1651:{\displaystyle \psi }
1619:
1582:
1560:
1549:
1515:
1495:
1460:
1440:
1417:
1375:uncertainty principle
1351:
1304:
1265:
1164:
1128:
940:
870:
840:
785:
751:
565:
470:
423:
383:
290:
258:
191:
136:
25:An example of the 2D
24:
5249:Constant-Q transform
5200:Time-causal wavelets
5142:
5113:
5093:
5051:
5018:
4989:
4975:{\displaystyle y(k)}
4957:
4942:{\displaystyle y(t)}
4924:
4897:
4877:
4745:
4565:
4343:
4232:
4201:
4197:Discretizing of the
4152:
4132:
4006:
3979:
3959:
3923:
3894:
3790:
3701:
3507:
3466:
3342:
3283:
3272:{\displaystyle r(n)}
3254:
3218:
3182:
3102:
3076:
3044:
3008:
2944:
2817:
2790:
2732:
2696:
2660:
2609:
2573:
2516:
2422:Filter coefficients
2368:
2326:
2284:
2248:
2212:
2176:
2140:
2104:
2068:
1716:
1680:{\displaystyle x(t)}
1662:
1642:
1628:Good time resolution
1605:
1568:
1535:
1504:
1472:
1449:
1445:represents time and
1429:
1384:
1321:
1277:
1176:
1143:
1138:wavelet coefficients
982:
852:
801:
763:
580:
482:
439:
395:
302:
278:
207:
160:
99:
6448:Functional analysis
6355:2021MMAS...4410734P
6349:(13): 10734–10752.
6269:2019MSSP..114..366Q
6158:1992OptEn..31.1825S
6146:Optical Engineering
5949:2016Lands..13..437T
5902:2012MeScT..23e5604L
5715:1995ITIP....4.1053V
5672:1987JOSAA...4.2379F
5567:Xiph.Org Foundation
5475:10.3390/app11136141
5174:Mexican hat wavelet
4893:is scaling factor,
4808:
4700:
4619:
4523:
4484:
4450:
4434:
4393:
4318:
4270:
4059:
3834:
3727:
3653:
3592:
3533:
3414:
3365:
3091:{\displaystyle L-1}
2889:
2840:
2411:
1966:primarily uses the
1871:Wavelet compression
1860:Wavelet compression
1591:Bad time resolution
1065:
950:convergence in norm
646:
520:
143:orthonormal wavelet
81:orthonormal wavelet
6429:by René Puschinger
5851:. pp. 79–82.
5660:J. Opt. Soc. Am. A
5302:, a wavelet-based
5282:, a wavelet-based
5259:Daubechies wavelet
5239:Chirplet transform
5227:Daubechies wavelet
5182:Daubechies wavelet
5155:
5126:
5099:
5079:
5031:
5002:
4972:
4939:
4903:
4883:
4860:
4791:
4722:
4686:
4605:
4545:
4509:
4470:
4436:
4420:
4379:
4323:
4321:
4304:
4256:
4213:
4168:time shift factor
4158:
4138:
4117:
4042:
3985:
3965:
3944:
3910: : frequency
3900:
3879:
3817:
3750:
3704:
3684:
3630:
3569:
3568:
3510:
3488:
3452:
3391:
3345:
3328:
3269:
3240:
3204:
3168:
3088:
3062:
3030:
2994:
2927:
2866:
2820:
2803:
2776:
2718:
2682:
2646:
2595:
2559:
2409:
2396:
2354:
2312:
2270:
2234:
2198:
2162:
2126:
2090:
2016:run length encoded
1901:for still images,
1857:
1839:
1707:
1677:
1648:
1614:
1577:
1562:
1544:
1522:ordinary frequency
1510:
1490:
1455:
1435:
1412:
1346:
1299:
1260:
1169:are then given by
1159:
1123:
1048:
973:integral transform
935:
835:
780:
746:
744:
629:
560:
503:
465:
418:
378:
285:
253:
186:
131:
35:
6458:Image compression
6453:Signal processing
6411:10.1109/99.388960
6072:978-1-4673-0199-2
6031:978-1-4244-5365-8
5866:978-1-4244-8316-7
5758:10.1109/18.119725
5723:10.1109/83.403412
5666:(12): 2379–2394.
5532:10.1109/10.649997
5526:(12): 1253–1261.
5445:978-0-12-047141-6
5429:978-0-89871-274-2
5304:image compression
4886:{\displaystyle c}
4847:
4786:
4785:
4701:
4621:
4620:
4524:
4486:
4485:
4451:
4395:
4394:
4172:
4171:
4161:{\displaystyle b}
4141:{\displaystyle a}
4096:
4086:
4040:
4039:
4000:Wavelet transform
3988:{\displaystyle f}
3968:{\displaystyle t}
3903:{\displaystyle f}
3802:
3783:Fourier transform
3559:
2491:
2490:
2461:Wavelet filter 2
2430:Wavelet filter 1
1949:harmonic analysis
1936:, had been used.
1887:audio compression
1883:video compression
1879:image compression
1693:Fourier transform
1513:{\displaystyle f}
1466:angular frequency
1438:{\displaystyle t}
1410:
1102:
1092:
1046:
1045:
698:
552:
342:
198:square integrable
54:square-integrable
6465:
6423:
6414:
6382:
6363:10.1002/mma.7440
6333:
6291:
6290:
6280:
6248:
6242:
6241:
6239:
6227:
6221:
6220:
6210:
6200:
6176:
6170:
6169:
6166:10.1117/12.59911
6141:
6135:
6134:
6126:
6120:
6119:
6102:(4): 2058–2063.
6091:
6085:
6084:
6050:
6044:
6043:
6009:
6003:
6002:
5984:
5975:
5969:
5968:
5928:
5922:
5921:
5885:
5879:
5878:
5844:
5838:
5837:
5819:
5813:
5812:
5776:
5770:
5769:
5741:
5735:
5734:
5698:
5692:
5691:
5657:
5648:
5642:
5641:
5606:
5600:
5594:
5588:
5587:
5581:
5579:
5570:. July 4, 2020.
5558:
5552:
5551:
5517:
5508:
5502:
5496:
5490:
5489:
5487:
5477:
5462:Applied Sciences
5453:
5447:
5437:
5431:
5421:
5415:
5405:
5399:
5389:
5164:
5162:
5161:
5156:
5154:
5153:
5135:
5133:
5132:
5127:
5125:
5124:
5108:
5106:
5105:
5100:
5088:
5086:
5085:
5080:
5063:
5062:
5040:
5038:
5037:
5032:
5030:
5029:
5011:
5009:
5008:
5003:
5001:
5000:
4981:
4979:
4978:
4973:
4948:
4946:
4945:
4940:
4912:
4910:
4909:
4904:
4892:
4890:
4889:
4884:
4869:
4867:
4866:
4861:
4852:
4848:
4843:
4832:
4807:
4802:
4787:
4781:
4777:
4757:
4756:
4731:
4729:
4728:
4723:
4721:
4717:
4713:
4709:
4702:
4699:
4694:
4682:
4651:
4640:
4622:
4618:
4613:
4604:
4600:
4580:
4579:
4554:
4552:
4551:
4546:
4544:
4540:
4536:
4532:
4525:
4522:
4517:
4505:
4487:
4483:
4478:
4469:
4465:
4460:
4456:
4452:
4449:
4444:
4435:
4433:
4428:
4409:
4396:
4392:
4387:
4378:
4374:
4332:
4330:
4329:
4324:
4322:
4317:
4312:
4284:
4283:
4269:
4264:
4248:
4247:
4222:
4220:
4219:
4214:
4167:
4165:
4164:
4159:
4147:
4145:
4144:
4139:
4126:
4124:
4123:
4118:
4097:
4092:
4091:
4087:
4082:
4071:
4061:
4058:
4053:
4041:
4035:
4031:
3994:
3992:
3991:
3986:
3974:
3972:
3971:
3966:
3953:
3951:
3950:
3945:
3909:
3907:
3906:
3901:
3888:
3886:
3885:
3880:
3871:
3870:
3833:
3828:
3804:
3803:
3795:
3768:
3767:
3759:
3757:
3756:
3751:
3746:
3745:
3726:
3715:
3693:
3691:
3690:
3685:
3674:
3673:
3664:
3652:
3641:
3626:
3625:
3616:
3611:
3610:
3591:
3580:
3567:
3552:
3551:
3532:
3521:
3497:
3495:
3494:
3489:
3478:
3477:
3461:
3459:
3458:
3453:
3448:
3447:
3435:
3434:
3425:
3413:
3402:
3384:
3383:
3364:
3353:
3337:
3335:
3334:
3329:
3278:
3276:
3275:
3270:
3249:
3247:
3246:
3241:
3230:
3229:
3213:
3211:
3210:
3205:
3194:
3193:
3177:
3175:
3174:
3169:
3164:
3163:
3142:
3141:
3120:
3119:
3097:
3095:
3094:
3089:
3071:
3069:
3068:
3063:
3039:
3037:
3036:
3031:
3020:
3019:
3003:
3001:
3000:
2995:
2990:
2989:
2956:
2955:
2936:
2934:
2933:
2928:
2923:
2922:
2913:
2908:
2907:
2888:
2877:
2859:
2858:
2839:
2828:
2812:
2810:
2809:
2804:
2802:
2801:
2785:
2783:
2782:
2777:
2766:
2765:
2744:
2743:
2727:
2725:
2724:
2719:
2708:
2707:
2691:
2689:
2688:
2683:
2672:
2671:
2655:
2653:
2652:
2647:
2636:
2635:
2604:
2602:
2601:
2596:
2585:
2584:
2568:
2566:
2565:
2560:
2555:
2554:
2412:
2408:
2405:
2403:
2402:
2397:
2392:
2391:
2363:
2361:
2360:
2355:
2350:
2349:
2321:
2319:
2318:
2313:
2308:
2307:
2279:
2277:
2276:
2271:
2260:
2259:
2243:
2241:
2240:
2235:
2224:
2223:
2207:
2205:
2204:
2199:
2188:
2187:
2171:
2169:
2168:
2163:
2152:
2151:
2135:
2133:
2132:
2127:
2116:
2115:
2099:
2097:
2096:
2091:
2080:
2079:
2061:
2004:transform coding
1953:frequency domain
1909:, and the BBC's
1881:(sometimes also
1877:well suited for
1875:data compression
1848:
1846:
1845:
1840:
1832:
1831:
1796:
1795:
1760:
1759:
1686:
1684:
1683:
1678:
1657:
1655:
1654:
1649:
1623:
1621:
1620:
1615:
1586:
1584:
1583:
1578:
1553:
1551:
1550:
1545:
1519:
1517:
1516:
1511:
1499:
1497:
1496:
1491:
1464:
1462:
1461:
1456:
1444:
1442:
1441:
1436:
1421:
1419:
1418:
1413:
1411:
1403:
1355:
1353:
1352:
1347:
1345:
1344:
1308:
1306:
1305:
1300:
1298:
1297:
1269:
1267:
1266:
1261:
1259:
1255:
1254:
1253:
1235:
1234:
1217:
1213:
1209:
1208:
1191:
1190:
1168:
1166:
1165:
1160:
1158:
1157:
1132:
1130:
1129:
1124:
1103:
1098:
1097:
1093:
1088:
1077:
1067:
1064:
1059:
1047:
1044:
1036:
1031:
1027:
1007:
1003:
999:
998:
944:
942:
941:
936:
925:
924:
912:
911:
898:
893:
844:
842:
841:
836:
834:
830:
821:
820:
789:
787:
786:
781:
778:
777:
755:
753:
752:
747:
745:
741:
740:
728:
727:
709:
699:
694:
684:
683:
670:
659:
658:
645:
640:
618:
617:
602:
601:
569:
567:
566:
561:
553:
548:
534:
519:
514:
474:
472:
471:
466:
464:
460:
451:
450:
427:
425:
424:
419:
417:
387:
385:
384:
379:
376:
372:
362:
361:
344:
343:
335:
317:
316:
294:
292:
291:
286:
262:
260:
259:
254:
249:
225:
224:
195:
193:
192:
187:
185:
181:
172:
171:
140:
138:
137:
132:
127:
119:
118:
29:that is used in
6475:
6474:
6468:
6467:
6466:
6464:
6463:
6462:
6433:
6432:
6389:
6300:
6298:Further reading
6295:
6294:
6249:
6245:
6228:
6224:
6177:
6173:
6142:
6138:
6127:
6123:
6092:
6088:
6073:
6051:
6047:
6032:
6010:
6006:
5982:
5976:
5972:
5929:
5925:
5886:
5882:
5867:
5845:
5841:
5834:
5820:
5816:
5777:
5773:
5742:
5738:
5699:
5695:
5655:
5649:
5645:
5630:
5607:
5603:
5595:
5591:
5577:
5575:
5560:
5559:
5555:
5515:
5509:
5505:
5497:
5493:
5454:
5450:
5438:
5434:
5422:
5418:
5406:
5402:
5390:
5386:
5381:
5225:(also known as
5219:
5211:
5202:
5197:
5149:
5145:
5143:
5140:
5139:
5120:
5116:
5114:
5111:
5110:
5094:
5091:
5090:
5058:
5054:
5052:
5049:
5048:
5025:
5021:
5019:
5016:
5015:
4996:
4992:
4990:
4987:
4986:
4958:
4955:
4954:
4925:
4922:
4921:
4898:
4895:
4894:
4878:
4875:
4874:
4833:
4831:
4827:
4803:
4795:
4776:
4752:
4748:
4746:
4743:
4742:
4695:
4690:
4681:
4680:
4676:
4675:
4671:
4641:
4630:
4614:
4609:
4599:
4572:
4568:
4566:
4563:
4562:
4518:
4513:
4504:
4503:
4499:
4498:
4494:
4479:
4474:
4464:
4445:
4440:
4429:
4424:
4410:
4408:
4407:
4403:
4388:
4383:
4373:
4344:
4341:
4340:
4320:
4319:
4313:
4308:
4285:
4279:
4275:
4272:
4271:
4265:
4260:
4249:
4243:
4239:
4235:
4233:
4230:
4229:
4202:
4199:
4198:
4189:
4153:
4150:
4149:
4148:scaling ;
4133:
4130:
4129:
4072:
4070:
4066:
4062:
4060:
4054:
4046:
4030:
4007:
4004:
4003:
3980:
3977:
3976:
3960:
3957:
3956:
3924:
3921:
3920:
3895:
3892:
3891:
3851:
3847:
3829:
3821:
3794:
3793:
3791:
3788:
3787:
3766:
3741:
3737:
3716:
3708:
3702:
3699:
3698:
3669:
3665:
3660:
3642:
3634:
3621:
3617:
3612:
3606:
3602:
3581:
3573:
3563:
3547:
3543:
3522:
3514:
3508:
3505:
3504:
3473:
3469:
3467:
3464:
3463:
3443:
3439:
3430:
3426:
3421:
3403:
3395:
3379:
3375:
3354:
3349:
3343:
3340:
3339:
3284:
3281:
3280:
3255:
3252:
3251:
3225:
3221:
3219:
3216:
3215:
3189:
3185:
3183:
3180:
3179:
3159:
3155:
3137:
3133:
3109:
3105:
3103:
3100:
3099:
3077:
3074:
3073:
3045:
3042:
3041:
3015:
3011:
3009:
3006:
3005:
2985:
2981:
2951:
2947:
2945:
2942:
2941:
2918:
2914:
2909:
2903:
2899:
2878:
2870:
2854:
2850:
2829:
2824:
2818:
2815:
2814:
2797:
2793:
2791:
2788:
2787:
2761:
2757:
2739:
2735:
2733:
2730:
2729:
2728:is obtained by
2703:
2699:
2697:
2694:
2693:
2667:
2663:
2661:
2658:
2657:
2631:
2627:
2610:
2607:
2606:
2580:
2576:
2574:
2571:
2570:
2550:
2546:
2517:
2514:
2513:
2507:
2387:
2383:
2369:
2366:
2365:
2345:
2341:
2327:
2324:
2323:
2303:
2299:
2285:
2282:
2281:
2255:
2251:
2249:
2246:
2245:
2219:
2215:
2213:
2210:
2209:
2183:
2179:
2177:
2174:
2173:
2147:
2143:
2141:
2138:
2137:
2111:
2107:
2105:
2102:
2101:
2075:
2071:
2069:
2066:
2065:
2052:
2032:
2027:
2012:entropy encoded
1991:
1978:of transients.
1868:
1862:
1827:
1823:
1791:
1787:
1755:
1751:
1717:
1714:
1713:
1663:
1660:
1659:
1643:
1640:
1639:
1606:
1603:
1602:
1569:
1566:
1565:
1536:
1533:
1532:
1505:
1502:
1501:
1473:
1470:
1469:
1450:
1447:
1446:
1430:
1427:
1426:
1402:
1385:
1382:
1381:
1370:
1362:dyadic position
1337:
1333:
1322:
1319:
1318:
1315:dyadic dilation
1311:binary dilation
1290:
1286:
1278:
1275:
1274:
1246:
1242:
1227:
1223:
1222:
1218:
1204:
1200:
1199:
1195:
1183:
1179:
1177:
1174:
1173:
1150:
1146:
1144:
1141:
1140:
1078:
1076:
1072:
1068:
1066:
1060:
1052:
1040:
1032:
1026:
994:
990:
989:
985:
983:
980:
979:
917:
913:
904:
900:
894:
874:
853:
850:
849:
826:
822:
816:
812:
802:
799:
798:
792:Kronecker delta
770:
766:
764:
761:
760:
743:
742:
733:
729:
720:
716:
707:
706:
676:
672:
671:
669:
651:
647:
641:
633:
622:
610:
606:
594:
590:
583:
581:
578:
577:
535:
533:
515:
507:
483:
480:
479:
456:
452:
446:
442:
440:
437:
436:
413:
396:
393:
392:
357:
353:
352:
348:
334:
330:
309:
305:
303:
300:
299:
279:
276:
275:
245:
217:
213:
208:
205:
204:
177:
173:
167:
163:
161:
158:
157:
123:
114:
110:
100:
97:
96:
93:
75:generated by a
42:
17:
12:
11:
5:
6473:
6472:
6461:
6460:
6455:
6450:
6445:
6431:
6430:
6424:
6415:
6388:
6387:External links
6385:
6384:
6383:
6334:
6299:
6296:
6293:
6292:
6243:
6222:
6191:(1–2): 21–59.
6171:
6136:
6121:
6086:
6071:
6045:
6030:
6004:
5970:
5943:(3): 437–450.
5923:
5880:
5865:
5839:
5832:
5814:
5787:(2): 321–332.
5771:
5752:(2): 587–607.
5736:
5709:(8): 1053–60.
5693:
5643:
5628:
5601:
5589:
5553:
5503:
5491:
5448:
5432:
5416:
5400:
5383:
5382:
5380:
5377:
5376:
5375:
5370:
5365:
5360:
5355:
5350:
5337:
5332:
5322:
5317:
5315:Morlet wavelet
5312:
5307:
5297:
5292:
5287:
5277:
5272:
5266:
5261:
5256:
5251:
5246:
5241:
5236:
5230:
5218:
5215:
5210:
5207:
5201:
5198:
5196:
5195:
5188:
5185:
5167:
5166:
5152:
5148:
5136:
5123:
5119:
5098:
5078:
5075:
5072:
5069:
5066:
5061:
5057:
5045:
5042:
5028:
5024:
5012:
4999:
4995:
4983:
4971:
4968:
4965:
4962:
4938:
4935:
4932:
4929:
4920:of a function
4902:
4882:
4871:
4870:
4859:
4856:
4851:
4846:
4842:
4839:
4836:
4830:
4826:
4823:
4820:
4817:
4814:
4811:
4806:
4801:
4798:
4794:
4790:
4784:
4780:
4775:
4772:
4769:
4766:
4763:
4760:
4755:
4751:
4734:
4733:
4732:
4720:
4716:
4712:
4708:
4705:
4698:
4693:
4689:
4685:
4679:
4674:
4670:
4667:
4664:
4661:
4658:
4655:
4650:
4647:
4644:
4639:
4636:
4633:
4629:
4625:
4617:
4612:
4608:
4603:
4598:
4595:
4592:
4589:
4586:
4583:
4578:
4575:
4571:
4556:
4555:
4543:
4539:
4535:
4531:
4528:
4521:
4516:
4512:
4508:
4502:
4497:
4493:
4490:
4482:
4477:
4473:
4468:
4463:
4459:
4455:
4448:
4443:
4439:
4432:
4427:
4423:
4419:
4416:
4413:
4406:
4402:
4399:
4391:
4386:
4382:
4377:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4334:
4333:
4316:
4311:
4307:
4303:
4300:
4297:
4294:
4291:
4288:
4286:
4282:
4278:
4274:
4273:
4268:
4263:
4259:
4255:
4252:
4250:
4246:
4242:
4238:
4237:
4212:
4209:
4206:
4194:
4188:
4185:
4176:Morlet wavelet
4170:
4169:
4157:
4137:
4127:
4116:
4113:
4109:
4106:
4103:
4100:
4095:
4090:
4085:
4081:
4078:
4075:
4069:
4065:
4057:
4052:
4049:
4045:
4038:
4034:
4029:
4026:
4023:
4020:
4017:
4014:
4011:
4001:
3997:
3996:
3984:
3964:
3954:
3943:
3940:
3937:
3934:
3931:
3928:
3918:
3912:
3911:
3899:
3889:
3878:
3875:
3869:
3866:
3863:
3860:
3857:
3854:
3850:
3846:
3843:
3840:
3837:
3832:
3827:
3824:
3820:
3816:
3813:
3810:
3807:
3801:
3798:
3785:
3779:
3778:
3775:
3774:Representation
3772:
3765:
3762:
3749:
3744:
3740:
3736:
3733:
3730:
3725:
3722:
3719:
3714:
3711:
3707:
3683:
3680:
3677:
3672:
3668:
3663:
3659:
3656:
3651:
3648:
3645:
3640:
3637:
3633:
3629:
3624:
3620:
3615:
3609:
3605:
3601:
3598:
3595:
3590:
3587:
3584:
3579:
3576:
3572:
3566:
3562:
3558:
3555:
3550:
3546:
3542:
3539:
3536:
3531:
3528:
3525:
3520:
3517:
3513:
3487:
3484:
3481:
3476:
3472:
3451:
3446:
3442:
3438:
3433:
3429:
3424:
3420:
3417:
3412:
3409:
3406:
3401:
3398:
3394:
3390:
3387:
3382:
3378:
3374:
3371:
3368:
3363:
3360:
3357:
3352:
3348:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3268:
3265:
3262:
3259:
3239:
3236:
3233:
3228:
3224:
3203:
3200:
3197:
3192:
3188:
3167:
3162:
3158:
3154:
3151:
3148:
3145:
3140:
3136:
3132:
3129:
3126:
3123:
3118:
3115:
3112:
3108:
3087:
3084:
3081:
3061:
3058:
3055:
3052:
3049:
3029:
3026:
3023:
3018:
3014:
2993:
2988:
2984:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2954:
2950:
2926:
2921:
2917:
2912:
2906:
2902:
2898:
2895:
2892:
2887:
2884:
2881:
2876:
2873:
2869:
2865:
2862:
2857:
2853:
2849:
2846:
2843:
2838:
2835:
2832:
2827:
2823:
2800:
2796:
2775:
2772:
2769:
2764:
2760:
2756:
2753:
2750:
2747:
2742:
2738:
2717:
2714:
2711:
2706:
2702:
2681:
2678:
2675:
2670:
2666:
2645:
2642:
2639:
2634:
2630:
2626:
2623:
2620:
2617:
2614:
2594:
2591:
2588:
2583:
2579:
2558:
2553:
2549:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2506:
2503:
2489:
2488:
2485:
2482:
2479:
2475:
2474:
2471:
2468:
2465:
2462:
2458:
2457:
2454:
2451:
2448:
2444:
2443:
2440:
2437:
2434:
2431:
2427:
2426:
2423:
2420:
2417:
2415:
2395:
2390:
2386:
2382:
2379:
2376:
2373:
2353:
2348:
2344:
2340:
2337:
2334:
2331:
2311:
2306:
2302:
2298:
2295:
2292:
2289:
2269:
2266:
2263:
2258:
2254:
2233:
2230:
2227:
2222:
2218:
2197:
2194:
2191:
2186:
2182:
2161:
2158:
2155:
2150:
2146:
2125:
2122:
2119:
2114:
2110:
2089:
2086:
2083:
2078:
2074:
2051:
2048:
2047:
2046:
2043:
2040:
2031:
2028:
2026:
2023:
1990:
1987:
1861:
1858:
1838:
1835:
1830:
1826:
1822:
1819:
1816:
1813:
1810:
1806:
1802:
1799:
1794:
1790:
1786:
1783:
1780:
1777:
1774:
1770:
1766:
1763:
1758:
1754:
1750:
1747:
1744:
1741:
1738:
1734:
1730:
1727:
1724:
1721:
1676:
1673:
1670:
1667:
1647:
1636:
1635:
1632:
1629:
1613:
1610:
1599:
1598:
1595:
1592:
1576:
1573:
1543:
1540:
1509:
1489:
1486:
1483:
1480:
1477:
1454:
1434:
1423:
1422:
1409:
1406:
1401:
1398:
1395:
1392:
1389:
1369:
1366:
1343:
1340:
1336:
1332:
1329:
1326:
1309:is called the
1296:
1293:
1289:
1285:
1282:
1271:
1270:
1258:
1252:
1249:
1245:
1241:
1238:
1233:
1230:
1226:
1221:
1216:
1212:
1207:
1203:
1198:
1194:
1189:
1186:
1182:
1156:
1153:
1149:
1134:
1133:
1121:
1118:
1115:
1112:
1109:
1106:
1101:
1096:
1091:
1087:
1084:
1081:
1075:
1071:
1063:
1058:
1055:
1051:
1043:
1039:
1035:
1030:
1025:
1022:
1019:
1016:
1013:
1010:
1006:
1002:
997:
993:
988:
958:wavelet series
956:is known as a
946:
945:
934:
931:
928:
923:
920:
916:
910:
907:
903:
897:
892:
889:
886:
883:
880:
877:
873:
869:
866:
863:
860:
857:
833:
829:
825:
819:
815:
810:
806:
776:
773:
769:
757:
756:
739:
736:
732:
726:
723:
719:
715:
712:
710:
708:
705:
702:
697:
693:
690:
687:
682:
679:
675:
668:
665:
662:
657:
654:
650:
644:
639:
636:
632:
628:
625:
623:
621:
616:
613:
609:
605:
600:
597:
593:
589:
586:
585:
571:
570:
559:
556:
551:
547:
544:
541:
538:
532:
529:
526:
523:
518:
513:
510:
506:
502:
499:
496:
493:
490:
487:
463:
459:
455:
449:
445:
416:
411:
407:
403:
400:
389:
388:
375:
371:
368:
365:
360:
356:
351:
347:
341:
338:
333:
329:
326:
323:
320:
315:
312:
308:
283:
252:
248:
243:
239:
235:
232:
228:
223:
220:
216:
212:
184:
180:
176:
170:
166:
130:
126:
122:
117:
113:
108:
104:
92:
89:
50:wavelet series
15:
9:
6:
4:
3:
2:
6471:
6470:
6459:
6456:
6454:
6451:
6449:
6446:
6444:
6441:
6440:
6438:
6428:
6425:
6421:
6416:
6412:
6408:
6404:
6400:
6396:
6391:
6390:
6380:
6376:
6372:
6368:
6364:
6360:
6356:
6352:
6348:
6344:
6340:
6335:
6331:
6327:
6323:
6319:
6315:
6311:
6307:
6302:
6301:
6288:
6284:
6279:
6274:
6270:
6266:
6262:
6258:
6254:
6247:
6238:
6233:
6226:
6218:
6214:
6209:
6204:
6199:
6194:
6190:
6186:
6182:
6175:
6167:
6163:
6159:
6155:
6151:
6147:
6140:
6132:
6125:
6117:
6113:
6109:
6105:
6101:
6097:
6090:
6082:
6078:
6074:
6068:
6064:
6060:
6056:
6049:
6041:
6037:
6033:
6027:
6023:
6019:
6015:
6008:
6000:
5996:
5992:
5988:
5981:
5974:
5966:
5962:
5958:
5954:
5950:
5946:
5942:
5938:
5934:
5927:
5919:
5915:
5911:
5907:
5903:
5899:
5895:
5891:
5884:
5876:
5872:
5868:
5862:
5858:
5854:
5850:
5843:
5835:
5833:0-88385-031-1
5829:
5825:
5818:
5810:
5806:
5802:
5798:
5794:
5790:
5786:
5782:
5775:
5767:
5763:
5759:
5755:
5751:
5747:
5740:
5732:
5728:
5724:
5720:
5716:
5712:
5708:
5704:
5697:
5689:
5685:
5681:
5677:
5673:
5669:
5665:
5661:
5654:
5647:
5639:
5635:
5631:
5629:0-7803-8560-8
5625:
5621:
5617:
5613:
5605:
5599:
5593:
5586:
5573:
5569:
5568:
5563:
5557:
5549:
5545:
5541:
5537:
5533:
5529:
5525:
5521:
5514:
5507:
5500:
5495:
5486:
5485:11573/1655273
5481:
5476:
5471:
5467:
5463:
5459:
5452:
5446:
5442:
5436:
5430:
5426:
5420:
5414:
5413:0-12-174584-8
5410:
5404:
5398:
5397:0-521-42000-8
5394:
5388:
5384:
5374:
5371:
5369:
5366:
5364:
5361:
5359:
5356:
5354:
5351:
5349:
5345:
5341:
5338:
5336:
5333:
5330:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5305:
5301:
5298:
5296:
5293:
5291:
5290:Gabor wavelet
5288:
5285:
5281:
5278:
5276:
5273:
5270:
5267:
5265:
5262:
5260:
5257:
5255:
5252:
5250:
5247:
5245:
5242:
5240:
5237:
5234:
5231:
5228:
5224:
5221:
5220:
5214:
5206:
5193:
5189:
5186:
5183:
5179:
5175:
5171:
5165:are processed
5150:
5146:
5137:
5121:
5117:
5096:
5073:
5070:
5067:
5059:
5055:
5046:
5043:
5026:
5022:
5013:
4997:
4993:
4984:
4966:
4960:
4952:
4951:
4950:
4933:
4927:
4919:
4914:
4900:
4880:
4857:
4854:
4849:
4844:
4840:
4837:
4834:
4828:
4821:
4815:
4809:
4796:
4792:
4788:
4782:
4778:
4773:
4767:
4764:
4761:
4753:
4749:
4741:
4740:
4739:
4735:
4718:
4714:
4710:
4706:
4703:
4696:
4691:
4687:
4683:
4677:
4672:
4665:
4659:
4653:
4648:
4645:
4642:
4637:
4634:
4631:
4627:
4623:
4615:
4610:
4606:
4601:
4596:
4590:
4587:
4584:
4576:
4573:
4569:
4561:
4560:
4559:
4541:
4537:
4533:
4529:
4526:
4519:
4514:
4510:
4506:
4500:
4495:
4488:
4480:
4475:
4471:
4466:
4461:
4457:
4453:
4446:
4441:
4437:
4430:
4425:
4421:
4417:
4414:
4411:
4404:
4397:
4389:
4384:
4380:
4375:
4370:
4364:
4361:
4358:
4355:
4352:
4339:
4338:
4337:
4314:
4309:
4305:
4301:
4298:
4295:
4292:
4289:
4287:
4280:
4276:
4266:
4261:
4257:
4253:
4251:
4244:
4240:
4228:
4227:
4226:
4210:
4207:
4204:
4196:
4195:
4193:
4184:
4181:
4177:
4155:
4135:
4128:
4114:
4111:
4104:
4098:
4088:
4083:
4079:
4076:
4073:
4067:
4047:
4043:
4036:
4032:
4027:
4021:
4018:
4015:
4009:
4002:
3999:
3998:
3982:
3962:
3955:
3938:
3935:
3932:
3926:
3919:
3917:
3914:
3913:
3897:
3890:
3876:
3873:
3867:
3864:
3861:
3858:
3855:
3852:
3848:
3841:
3835:
3822:
3818:
3814:
3808:
3796:
3786:
3784:
3781:
3780:
3776:
3773:
3770:
3769:
3761:
3742:
3738:
3734:
3731:
3720:
3712:
3709:
3705:
3695:
3678:
3675:
3670:
3666:
3661:
3657:
3646:
3638:
3635:
3631:
3622:
3618:
3613:
3607:
3603:
3599:
3596:
3585:
3577:
3574:
3570:
3564:
3560:
3556:
3548:
3544:
3540:
3537:
3526:
3518:
3515:
3511:
3502:
3499:
3482:
3474:
3470:
3444:
3440:
3436:
3431:
3427:
3422:
3418:
3407:
3399:
3396:
3392:
3388:
3380:
3376:
3372:
3369:
3358:
3350:
3346:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3263:
3257:
3234:
3226:
3222:
3198:
3190:
3186:
3160:
3156:
3152:
3149:
3146:
3138:
3134:
3130:
3124:
3116:
3113:
3110:
3106:
3085:
3082:
3079:
3059:
3056:
3053:
3050:
3047:
3024:
3016:
3012:
2986:
2982:
2978:
2975:
2969:
2966:
2960:
2952:
2948:
2938:
2919:
2915:
2910:
2904:
2900:
2896:
2893:
2882:
2874:
2871:
2867:
2863:
2855:
2851:
2847:
2844:
2833:
2825:
2821:
2798:
2794:
2770:
2762:
2758:
2754:
2748:
2740:
2736:
2712:
2704:
2700:
2676:
2668:
2664:
2640:
2632:
2628:
2624:
2618:
2612:
2589:
2581:
2577:
2551:
2547:
2543:
2540:
2534:
2531:
2525:
2519:
2510:
2502:
2498:
2494:
2486:
2483:
2480:
2477:
2476:
2472:
2469:
2466:
2463:
2459:
2455:
2452:
2449:
2446:
2445:
2441:
2438:
2435:
2432:
2428:
2424:
2421:
2418:
2416:
2414:
2413:
2407:
2388:
2384:
2380:
2377:
2371:
2346:
2342:
2338:
2335:
2329:
2304:
2300:
2296:
2293:
2287:
2264:
2256:
2252:
2228:
2220:
2216:
2192:
2184:
2180:
2156:
2148:
2144:
2120:
2112:
2108:
2084:
2076:
2072:
2062:
2060:
2055:
2044:
2041:
2038:
2037:
2036:
2022:
2019:
2017:
2013:
2009:
2005:
2000:
1997:as there are
1996:
1986:
1984:
1979:
1977:
1974:for improved
1973:
1969:
1965:
1962:
1958:
1954:
1950:
1944:
1941:
1937:
1935:
1931:
1926:
1924:
1920:
1916:
1912:
1908:
1904:
1900:
1896:
1892:
1888:
1884:
1880:
1876:
1873:is a form of
1872:
1867:
1854:
1850:
1833:
1828:
1824:
1820:
1817:
1811:
1808:
1804:
1797:
1792:
1788:
1784:
1781:
1775:
1772:
1768:
1761:
1756:
1752:
1748:
1745:
1739:
1736:
1732:
1725:
1719:
1710:
1704:
1700:
1698:
1694:
1690:
1671:
1665:
1645:
1633:
1630:
1627:
1626:
1625:
1611:
1596:
1593:
1590:
1589:
1588:
1574:
1559:
1555:
1541:
1530:
1525:
1523:
1507:
1487:
1484:
1481:
1478:
1475:
1467:
1452:
1432:
1407:
1404:
1399:
1396:
1390:
1380:
1379:
1378:
1376:
1365:
1363:
1359:
1341:
1338:
1334:
1330:
1327:
1324:
1316:
1312:
1294:
1291:
1287:
1283:
1280:
1256:
1250:
1247:
1243:
1239:
1236:
1231:
1228:
1224:
1219:
1214:
1210:
1205:
1201:
1196:
1192:
1187:
1184:
1180:
1172:
1171:
1170:
1154:
1151:
1147:
1139:
1119:
1116:
1110:
1104:
1094:
1089:
1085:
1082:
1079:
1073:
1069:
1053:
1049:
1037:
1028:
1023:
1017:
1014:
1011:
1004:
1000:
995:
991:
986:
978:
977:
976:
974:
970:
965:
963:
959:
955:
951:
929:
921:
918:
914:
908:
905:
901:
887:
884:
881:
878:
875:
871:
867:
861:
855:
848:
847:
846:
831:
823:
817:
813:
808:
804:
795:
793:
774:
771:
767:
737:
734:
730:
724:
721:
717:
713:
711:
703:
700:
688:
680:
677:
673:
663:
655:
652:
648:
634:
630:
626:
624:
614:
611:
607:
603:
598:
595:
591:
576:
575:
574:
557:
554:
542:
536:
527:
521:
508:
504:
500:
494:
491:
488:
478:
477:
476:
461:
453:
447:
443:
434:
433:inner product
429:
409:
405:
401:
398:
391:for integers
373:
369:
366:
363:
358:
354:
349:
345:
339:
336:
331:
327:
321:
313:
310:
306:
298:
297:
296:
281:
273:
269:
266:
241:
237:
233:
230:
226:
221:
218:
214:
201:
199:
182:
174:
168:
164:
156:
155:Hilbert space
152:
148:
147:Hilbert basis
144:
141:is called an
115:
111:
106:
102:
88:
86:
82:
78:
74:
71:
68:by a certain
67:
63:
59:
55:
51:
47:
40:
32:
28:
23:
19:
6405:(2): 50–61.
6402:
6398:
6346:
6342:
6316:(1): 19–36.
6313:
6309:
6260:
6256:
6246:
6225:
6188:
6184:
6174:
6149:
6145:
6139:
6130:
6124:
6099:
6095:
6089:
6054:
6048:
6013:
6007:
5990:
5986:
5973:
5940:
5936:
5926:
5893:
5889:
5883:
5848:
5842:
5823:
5817:
5784:
5780:
5774:
5749:
5745:
5739:
5706:
5702:
5696:
5663:
5659:
5646:
5611:
5604:
5592:
5583:
5576:. Retrieved
5565:
5556:
5523:
5519:
5506:
5494:
5468:(13): 6141.
5465:
5461:
5451:
5435:
5419:
5403:
5387:
5342:, a type of
5295:Haar wavelet
5275:Dual wavelet
5223:Binomial QMF
5212:
5203:
5178:Haar Wavelet
4982:with the FFT
4915:
4872:
4737:
4557:
4335:
4224:
4190:
4173:
3696:
3503:
3500:
2939:
2511:
2508:
2499:
2495:
2492:
2063:
2056:
2053:
2033:
2020:
1995:coefficients
1992:
1980:
1976:reproduction
1945:
1938:
1927:
1870:
1869:
1711:
1708:
1637:
1600:
1563:
1526:
1424:
1371:
1361:
1357:
1314:
1310:
1272:
1137:
1135:
968:
966:
957:
953:
947:
796:
758:
572:
430:
390:
268:translations
263:by means of
202:
149:, that is a
142:
94:
84:
80:
49:
43:
18:
6263:: 366–377.
6152:(9): 1825.
5896:(5): 1–11.
5344:spectrogram
5340:Scaleograms
5335:S transform
4918:convolution
2425:Regularity
1972:filter bank
1964:audio codec
975:defined as
200:functions.
95:A function
83:and of the
70:orthonormal
46:mathematics
6437:Categories
5937:Landslides
5379:References
5284:geospatial
5192:smoothness
3995:frequency
3098:stage) is
2025:Evaluation
1930:transients
1864:See also:
1587:is large,
153:, for the
91:Definition
6379:235556542
6371:1099-1476
6330:1572-9303
6287:126007150
6237:0912.2437
5965:1612-5118
5918:121684952
5578:April 10,
5499:JPEG 2000
5300:JPEG 2000
5097:τ
5074:τ
4901:τ
4841:τ
4838:−
4825:Ψ
4822:⋅
4805:∞
4800:∞
4797:−
4793:∫
4789:⋅
4768:τ
4704:−
4669:Ψ
4666:⋅
4646:−
4628:∑
4624:⋅
4527:−
4492:Ψ
4489:⋅
4415:−
4401:Ψ
4398:⋅
4347:Ψ
4302:⋅
4296:⋅
4277:τ
4211:τ
4208:−
4094:¯
4077:−
4064:Ψ
4056:∞
4051:∞
4048:−
4044:∫
3862:π
3853:−
3831:∞
3826:∞
3823:−
3819:∫
3800:^
3771:Transform
3676:−
3600:−
3561:∑
3437:−
3302:−
3290:−
3279:at stage
3150:−
3114:−
3083:−
3057:≤
3051:≤
2979:−
2970:δ
2897:−
2813:samples:
2755:∗
2625:∗
2544:−
2535:δ
2381:−
2339:−
2297:−
2288:δ
2008:quantized
1891:JPEG 2000
1821:π
1812:
1785:π
1776:
1749:π
1740:
1646:ψ
1624:is small
1609:Δ
1572:Δ
1539:Δ
1485:π
1476:ω
1453:ω
1400:≥
1397:ω
1394:Δ
1388:Δ
1368:Principle
1339:−
1292:−
1248:−
1229:−
1206:ψ
1100:¯
1083:−
1070:ψ
1062:∞
1057:∞
1054:−
1050:∫
996:ψ
962:self-dual
915:ψ
896:∞
891:∞
888:−
872:∑
809:∈
768:δ
731:δ
718:δ
696:¯
674:ψ
649:ψ
643:∞
638:∞
635:−
631:∫
620:⟩
608:ψ
592:ψ
588:⟨
550:¯
517:∞
512:∞
509:−
505:∫
498:⟩
486:⟨
410:∈
367:−
346:ψ
307:ψ
282:ψ
272:dilations
242:∈
215:ψ
107:∈
103:ψ
64:-valued)
6443:Wavelets
6217:36689001
6208:10160219
6116:36881450
6081:16302915
6040:17066179
5993:: 1–18.
5875:37689047
5809:21880274
5801:15262077
5766:43701174
5731:18291999
5638:43806122
5572:Archived
5306:standard
5217:See also
1919:lossless
1907:CineForm
1500:, where
66:function
31:JPEG2000
6351:Bibcode
6265:Bibcode
6154:Bibcode
5945:Bibcode
5898:Bibcode
5711:Bibcode
5688:3430225
5668:Bibcode
5585:events.
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