Seislet transform and seislet frame |

We use a simple synthetic test to verify the compression effectiveness of 1-D seislet frame. A test signal mixing two sinusoids with different frequencies and some random noise is displayed in Figure 13a. We use a prediction-error filter to detect the signal frequencies and to design the corresponding seislet frame. The result is shown in Figure 13b. The 1-D seislet frame algorithm with shaping regularization compresses the sinusoidal signal into two nearly perfect impulses with some dispersive random noise. For comparison, we also apply DFT and DWT to transform the signal (Figures 13d and 13c). In the Fourier transform domain, the signal appears as two impulses corresponding to the chosen frequency components. The resolution is not perfect because of spectral leakage caused by non-periodic input data. In the wavelet domain, the transform coefficients are not compressed well. For further comparison, we plot the coefficients in the three different transform domains, sorted from large to small, on a decibel scale (Figure 14). The significantly faster rate of coefficient decay shows the superiority of the 1-D seislet frame in compressing sinusoidal signals.

tatrace,taft,tdwt,fourier
Mixed
sinusoidal signal (a), 1-D seislet frame (b), 1-D wavelet transform (c),
and 1-D Fourier transform (d).
Figure 13. |
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tlog
Compression comparison between digital Fourier
transform, digital wavelet transform, and 1-D seislet frame. Transform
coefficients are sorted from large to small, normalized, and plotted on a
decibel scale.
Figure 14. |
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Seislet transform and seislet frame |

2013-07-26