`skimage.filters._fft_based:butterworth`

source: /dev/scikit-image/src/skimage/filters/_fft_based.py :57

Signature

   def     butterworth (    image    ,    cutoff_frequency_ratio    =  0.005   ,    high_pass    =  True   ,    order    =  2.0   ,    channel_axis    =  None   ,  * ,     squared_butterworth    =  True   ,    npad    =  0     )

Summary

Apply a Butterworth filter to enhance high or low frequency features.

Extended Summary

This filter is defined in the Fourier domain.

Parameters

image : (M[, N[, ..., P]][, C]) ndarray: Input image.
cutoff_frequency_ratio : float, optional: Determines the position of the cut-off relative to the shape of the FFT. Receives a value between [0, 0.5].
high_pass : bool, optional: Whether to perform a high pass filter. If False, a low pass filter is performed.
order : float, optional: Order of the filter which affects the slope near the cut-off. Higher order means steeper slope in frequency space.
channel_axis : int, optional: If there is a channel dimension, provide the index here. If None (default) then all axes are assumed to be spatial dimensions.
squared_butterworth : bool, optional: When True, the square of a Butterworth filter is used. See notes below for more details.
npad : int, optional: Pad each edge of the image by npad pixels using numpy.pad's mode='edge' extension.

Returns

result : ndarray: The Butterworth-filtered image.

Notes

A band-pass filter can be achieved by combining a high-pass and low-pass filter. The user can increase npad if boundary artifacts are apparent.

The "Butterworth filter" used in image processing textbooks (e.g. ^[1], ^[2]) is often the square of the traditional Butterworth filters as described by ^[3], ^[4]. The squared version will be used here if squared_butterworth is set to True. The lowpass, squared Butterworth filter is given by the following expression for the lowpass case:

H_{l o w} (f) = \frac{1}{1 + ( \frac{f}{c f _{s}} ) ^{2 n}}

with the highpass case given by

H_{hi} (f) = 1 - H_{l o w} (f)

where $f = \sum_{d = 0}^{ndim} f_{d}^{2}$ is the absolute value of the spatial frequency, $f_{s}$ is the sampling frequency, $c$ the cutoff_frequency_ratio, and $n$ is the filter order ^[1]. When squared_butterworth=False, the square root of the above expressions are used instead.

Note that cutoff_frequency_ratio is defined in terms of the sampling frequency, $f_{s}$ . The FFT spectrum covers the Nyquist range ( $[- f_{s} /2, f_{s} /2]$ ) so cutoff_frequency_ratio should have a value between 0 and 0.5. The frequency response (gain) at the cutoff is 0.5 when squared_butterworth is true and $1/ 2$ when it is false.

Examples

Apply a high-pass and low-pass Butterworth filter to a grayscale and color image respectively:

from skimage.data import camera, astronaut
from skimage.filters import butterworth
high_pass = butterworth(camera(), 0.07, True, 8)
low_pass = butterworth(astronaut(), 0.01, False, 4, channel_axis=-1)

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Aliases

skimage.filters.butterworth