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Kolmogorov–Zurbenko filters

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Abstract We reviewed the properties of the Kolmogorov–Zurbenko (KZ) filter and its extensions with applications in high resolution signal and image processing. The KZ filter is defined as an iteration of a moving average (MA) filter. The impulse response function of the KZ filter is a convolution of the rectangular window being used in a MA filter. Zero derivatives at the edges of the impulse response function make it a sharply declining function, providing high frequency resolution. The KZ Fourier transform (KZFT) is derived from the KZ filter by applying it to Fourier transform. Extensions of the KZ filter and the KZFT are demonstrated with examples. Copyright © 2010 John Wiley & Sons, Inc. This article is categorized under: Applications of Computational Statistics > Signal and Image Processing and Coding

Comparisons of the MSE as a function of the underlying smoothness of the spectrum for different spectral windows: o(α) is from the optimal spectral window; k(α) is from the KZ window; p(α) is from Parzen window; u(α) is from rectangular window; t(α) is from Tukey–Hamming window; b(α) is from Bartlett window. For the exact formula of MSE, please see Ref 1.

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Plots of raw third‐order periodogram with L = 500, m = 1000 and (a) k = 1; (b) k = 3. (c) and (d) are the adaptively smoothed version of (a) and (b), respectively.

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KZP (solid line) and true spectral density function (dash line).

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Diagram of the parameters in the definition of KZP and KZTP.

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Contour plot of time–frequency map of two close floating frequency chirp signals by KZFT.

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Tidal wave was reconstructed using the KZFT and shifted forward 5 hours. Wind speed was smoothed using KZ(3 hours, 2) filter. Data are from August 23, 2005 19:00 to August 28, 2005 23:00 local time, when hurricane Katrina was at doorsteps of New Orleans. Horizontal scale is using dates in August 2005 at the points of midnights local time.

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Example using KZFT to reconstruct signal.

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10 log10 of the energy transfer function of KZFT of m = 201, k = 1 (black) and k = 5 (red) at frequency 0.025.

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Example of KZA filter to detect breaks in image data: (a) raw data and (b) filtered data.

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Example of spatial KZ filter: (a) raw data; (b) one iteration of spatial 5 × 5 moving block average; and (c) two iterations of spatial 5 × 5 moving block filter.

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Absolute value of the impulse response function of KZ(21,5) and KZ(21,1).

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Applications of Computational Statistics > Signal and Image Processing and Coding

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