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# 5.3 Linear Filtering

Selected FIR filters.

BoxFilter, DiskFilter, GaussianFilter, ExponentialFilter and SmoothingFilter are all variants of so-called smoothing filters. They produce a response that is the local average of the samples of a signal. Such an operation tends to reduce the local variability of sample values and thus decreases signal noise and smooths the signal. However, averaging also smears image edges resulting in a blurred picture. BoxFilter returns a matrix with constant coefficients equal to the inverse of the filter's dimensions. This FIR filter is commonly known as a moving-average filter.
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The Savitzky-Golay FIR smoothing filters, also known as polynomial smoothing or least-squares smoothing filters, are generalizations of the FIR moving-average filter. These are filters that optimally (in the sense of least squares) fit a set of data points to polynomials of different degrees. These filters preserve edges far better than a moving-average filter at the expense of not removing as much noise. For additional noise reduction techniques, see Section 5.5. The length N of a Savitzky-Golay filter must be odd and at least two greater than the desired polynomial order k (i.e., N>k+1). For any given length N, filters of order k and k+1, where k is even, return the same result. Here we show the coefficients of a Savitzky-Golay filter of order three and dimensions 5 × 5.
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UnsharpFilter option.

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