251 lines
8.6 KiB
Python
251 lines
8.6 KiB
Python
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# Copyright (c) 2006-2012 Filip Wasilewski <http://en.ig.ma/>
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# Copyright (c) 2012-2016 The PyWavelets Developers
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# <https://github.com/PyWavelets/pywt>
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# See COPYING for license details.
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"""
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The thresholding helper module implements the most popular signal thresholding
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functions.
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"""
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from __future__ import division, print_function, absolute_import
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import numpy as np
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__all__ = ['threshold', 'threshold_firm']
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def soft(data, value, substitute=0):
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data = np.asarray(data)
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magnitude = np.absolute(data)
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with np.errstate(divide='ignore'):
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# divide by zero okay as np.inf values get clipped, so ignore warning.
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thresholded = (1 - value/magnitude)
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thresholded.clip(min=0, max=None, out=thresholded)
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thresholded = data * thresholded
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if substitute == 0:
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return thresholded
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else:
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cond = np.less(magnitude, value)
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return np.where(cond, substitute, thresholded)
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def nn_garrote(data, value, substitute=0):
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"""Non-negative Garrote."""
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data = np.asarray(data)
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magnitude = np.absolute(data)
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with np.errstate(divide='ignore'):
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# divide by zero okay as np.inf values get clipped, so ignore warning.
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thresholded = (1 - value**2/magnitude**2)
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thresholded.clip(min=0, max=None, out=thresholded)
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thresholded = data * thresholded
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if substitute == 0:
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return thresholded
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else:
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cond = np.less(magnitude, value)
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return np.where(cond, substitute, thresholded)
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def hard(data, value, substitute=0):
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data = np.asarray(data)
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cond = np.less(np.absolute(data), value)
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return np.where(cond, substitute, data)
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def greater(data, value, substitute=0):
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data = np.asarray(data)
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if np.iscomplexobj(data):
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raise ValueError("greater thresholding only supports real data")
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return np.where(np.less(data, value), substitute, data)
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def less(data, value, substitute=0):
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data = np.asarray(data)
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if np.iscomplexobj(data):
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raise ValueError("less thresholding only supports real data")
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return np.where(np.greater(data, value), substitute, data)
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thresholding_options = {'soft': soft,
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'hard': hard,
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'greater': greater,
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'less': less,
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'garrote': nn_garrote,
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# misspelled garrote for backwards compatibility
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'garotte': nn_garrote,
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}
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def threshold(data, value, mode='soft', substitute=0):
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"""
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Thresholds the input data depending on the mode argument.
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In ``soft`` thresholding [1]_, data values with absolute value less than
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`param` are replaced with `substitute`. Data values with absolute value
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greater or equal to the thresholding value are shrunk toward zero
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by `value`. In other words, the new value is
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``data/np.abs(data) * np.maximum(np.abs(data) - value, 0)``.
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In ``hard`` thresholding, the data values where their absolute value is
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less than the value param are replaced with `substitute`. Data values with
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absolute value greater or equal to the thresholding value stay untouched.
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``garrote`` corresponds to the Non-negative garrote threshold [2]_, [3]_.
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It is intermediate between ``hard`` and ``soft`` thresholding. It behaves
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like soft thresholding for small data values and approaches hard
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thresholding for large data values.
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In ``greater`` thresholding, the data is replaced with `substitute` where
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data is below the thresholding value. Greater data values pass untouched.
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In ``less`` thresholding, the data is replaced with `substitute` where data
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is above the thresholding value. Lesser data values pass untouched.
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Both ``hard`` and ``soft`` thresholding also support complex-valued data.
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Parameters
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----------
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data : array_like
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Numeric data.
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value : scalar
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Thresholding value.
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mode : {'soft', 'hard', 'garrote', 'greater', 'less'}
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Decides the type of thresholding to be applied on input data. Default
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is 'soft'.
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substitute : float, optional
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Substitute value (default: 0).
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Returns
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-------
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output : array
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Thresholded array.
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See Also
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--------
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threshold_firm
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References
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----------
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.. [1] D.L. Donoho and I.M. Johnstone. Ideal Spatial Adaptation via
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Wavelet Shrinkage. Biometrika. Vol. 81, No. 3, pp.425-455, 1994.
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DOI:10.1093/biomet/81.3.425
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.. [2] L. Breiman. Better Subset Regression Using the Nonnegative Garrote.
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Technometrics, Vol. 37, pp. 373-384, 1995.
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DOI:10.2307/1269730
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.. [3] H-Y. Gao. Wavelet Shrinkage Denoising Using the Non-Negative
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Garrote. Journal of Computational and Graphical Statistics Vol. 7,
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No. 4, pp.469-488. 1998.
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DOI:10.1080/10618600.1998.10474789
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Examples
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--------
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>>> import numpy as np
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>>> import pywt
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>>> data = np.linspace(1, 4, 7)
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>>> data
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array([ 1. , 1.5, 2. , 2.5, 3. , 3.5, 4. ])
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>>> pywt.threshold(data, 2, 'soft')
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array([ 0. , 0. , 0. , 0.5, 1. , 1.5, 2. ])
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>>> pywt.threshold(data, 2, 'hard')
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array([ 0. , 0. , 2. , 2.5, 3. , 3.5, 4. ])
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>>> pywt.threshold(data, 2, 'garrote')
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array([ 0. , 0. , 0. , 0.9 , 1.66666667,
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2.35714286, 3. ])
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>>> pywt.threshold(data, 2, 'greater')
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array([ 0. , 0. , 2. , 2.5, 3. , 3.5, 4. ])
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>>> pywt.threshold(data, 2, 'less')
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array([ 1. , 1.5, 2. , 0. , 0. , 0. , 0. ])
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"""
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try:
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return thresholding_options[mode](data, value, substitute)
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except KeyError:
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# Make sure error is always identical by sorting keys
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keys = ("'{0}'".format(key) for key in
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sorted(thresholding_options.keys()))
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raise ValueError("The mode parameter only takes values from: {0}."
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.format(', '.join(keys)))
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def threshold_firm(data, value_low, value_high):
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"""Firm threshold.
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The approach is intermediate between soft and hard thresholding [1]_. It
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behaves the same as soft-thresholding for values below `value_low` and
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the same as hard-thresholding for values above `thresh_high`. For
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intermediate values, the thresholded value is in between that corresponding
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to soft or hard thresholding.
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Parameters
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----------
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data : array-like
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The data to threshold. This can be either real or complex-valued.
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value_low : float
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Any values smaller then `value_low` will be set to zero.
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value_high : float
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Any values larger than `value_high` will not be modified.
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Notes
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-----
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This thresholding technique is also known as semi-soft thresholding [2]_.
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For each value, `x`, in `data`. This function computes::
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if np.abs(x) <= value_low:
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return 0
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elif np.abs(x) > value_high:
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return x
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elif value_low < np.abs(x) and np.abs(x) <= value_high:
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return x * value_high * (1 - value_low/x)/(value_high - value_low)
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``firm`` is a continuous function (like soft thresholding), but is
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unbiased for large values (like hard thresholding).
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If ``value_high == value_low`` this function becomes hard-thresholding.
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If ``value_high`` is infinity, this function becomes soft-thresholding.
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Returns
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-------
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val_new : array-like
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The values after firm thresholding at the specified thresholds.
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See Also
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--------
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threshold
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References
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----------
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.. [1] H.-Y. Gao and A.G. Bruce. Waveshrink with firm shrinkage.
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Statistica Sinica, Vol. 7, pp. 855-874, 1997.
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.. [2] A. Bruce and H-Y. Gao. WaveShrink: Shrinkage Functions and
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Thresholds. Proc. SPIE 2569, Wavelet Applications in Signal and
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Image Processing III, 1995.
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DOI:10.1117/12.217582
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"""
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if value_low < 0:
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raise ValueError("value_low must be non-negative.")
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if value_high < value_low:
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raise ValueError(
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"value_high must be greater than or equal to value_low.")
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data = np.asarray(data)
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magnitude = np.absolute(data)
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with np.errstate(divide='ignore'):
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# divide by zero okay as np.inf values get clipped, so ignore warning.
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vdiff = value_high - value_low
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thresholded = value_high * (1 - value_low/magnitude) / vdiff
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thresholded.clip(min=0, max=None, out=thresholded)
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thresholded = data * thresholded
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# restore hard-thresholding behavior for values > value_high
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large_vals = np.where(magnitude > value_high)
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if np.any(large_vals[0]):
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thresholded[large_vals] = data[large_vals]
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return thresholded
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