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https://github.com/wassname/scikit-image.git
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Add doctest to uft.
This commit is contained in:
François Orieux
+115
-18
@@ -28,11 +28,12 @@ transform. They are especially and useful for convolution [1]: they
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respect the Parseval equality, the value of the null frequency is
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equal to
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.. math:: \frac{1}{\sqrt{n}} \sum_i x_i.
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.. math:: \frac{1}{\sqrt{n}} \sum_i x_i
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The transform is applied from the last axes for performance reason (c
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order array). You may use directly the numpy.fft module for more
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sophisticated purpose.
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or the Fourier tranform have the same energy than the original image
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(see `image_quad_norm` function). The transform is applied from the
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last axes for performance reason (c order array). You may use directly
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the numpy.fft module for more sophisticated purpose.
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References
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----------
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@@ -42,7 +43,7 @@ References
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"""
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from __future__ import division
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from __future__ import division, print_function
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import numpy as np
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@@ -68,6 +69,15 @@ def ufftn(inarray, dim=None):
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-------
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outarray : ndarray (same shape than inarray)
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The unitary N-D Fourier transform of `inarray`.
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Examples
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--------
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>>> input = np.ones((3, 3, 3))
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>>> output = ufftn(input)
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>>> np.allclose(np.sum(input) / np.sqrt(input.size), output[0, 0, 0])
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True
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>>> output.shape
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(3, 3, 3)
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"""
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if dim is None:
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dim = inarray.ndim
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@@ -90,6 +100,15 @@ def uifftn(inarray, dim=None):
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-------
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outarray : ndarray (same shape than inarray)
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The unitary inverse N-D Fourier transform of `inarray`.
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Examples
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--------
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>>> input = np.ones((3, 3, 3))
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>>> output = uifftn(input)
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>>> np.allclose(np.sum(input) / np.sqrt(input.size), output[0, 0, 0])
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True
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>>> output.shape
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(3, 3, 3)
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"""
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if dim is None:
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dim = inarray.ndim
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@@ -118,9 +137,18 @@ def urfftn(inarray, dim=None):
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Notes
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-----
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The `r` function assume an input array of real
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The `urfft` functions assume an input array of real
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values. Consequently, the output have an Hermitian property and
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redondant values are not computed and returned.
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Examples
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--------
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>>> input = np.ones((5, 5, 5))
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>>> output = urfftn(input)
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>>> np.allclose(np.sum(input) / np.sqrt(input.size), output[0, 0, 0])
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True
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>>> output.shape
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(5, 5, 3)
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"""
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if dim is None:
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dim = inarray.ndim
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@@ -128,7 +156,7 @@ def urfftn(inarray, dim=None):
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return outarray / np.sqrt(np.prod(inarray.shape[-dim:]))
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def uirfftn(inarray, dim=None):
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def uirfftn(inarray, dim=None, shape=None):
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"""N-dim real unitary Fourier transform
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This transform consider the Hermitian property of the transform
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@@ -141,23 +169,35 @@ def uirfftn(inarray, dim=None):
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dim : int, optional
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The `dim` last axis along wich to compute the transform. All
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axes by default.
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shape : tuple of int
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The shape of the output. The shape of `rfft` is ambiguous in
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case of odd shape. In this case, the parameter must be
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used. see `np.fft.irfftn`.
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Returns
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-------
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outarray : ndarray (the last dim as (N - 1) *2 lenght)
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outarray : ndarray
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The unitary N-D inverse real Fourier transform of `inarray`.
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Notes
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-----
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The `r` function assume an input array of real
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values. Consequently, the output have an Hermitian property and
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redondant values are not computed and returned.
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The `uirfft` function assume that output array is of real
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values. Consequently, the input is assumed of having an Hermitian
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property and redondant values are implicit.
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Examples
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--------
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>>> input = np.ones((5, 5, 5))
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>>> output = uirfftn(urfftn(input), shape=input.shape)
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>>> np.allclose(input, output)
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True
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>>> output.shape
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(5, 5, 5)
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"""
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if dim is None:
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dim = inarray.ndim
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outarray = np.fft.irfftn(inarray, axes=range(-dim, 0))
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return outarray * np.sqrt(np.prod(inarray.shape[-dim:-1]) *
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(inarray.shape[-1] - 1) * 2)
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outarray = np.fft.irfftn(inarray, shape, axes=range(-dim, 0))
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return outarray * np.sqrt(np.prod(outarray.shape[-dim:]))
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def ufft2(inarray):
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@@ -178,6 +218,15 @@ def ufft2(inarray):
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See Also
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--------
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uifft2, ufftn, urfftn
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Examples
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--------
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>>> input = np.ones((10, 128, 128))
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>>> output = ufft2(input)
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>>> np.allclose(np.sum(input[1, ...]) / np.sqrt(input[1, ...].size), output[1, 0, 0])
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True
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>>> output.shape
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(10, 128, 128)
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"""
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return ufftn(inarray, 2)
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@@ -200,6 +249,15 @@ def uifft2(inarray):
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See Also
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--------
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uifft2, uifftn, uirfftn
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Examples
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--------
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>>> input = np.ones((10, 128, 128))
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>>> output = uifft2(input)
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>>> np.allclose(np.sum(input[1, ...]) / np.sqrt(input[1, ...].size), output[0, 0, 0])
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True
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>>> output.shape
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(10, 128, 128)
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"""
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return uifftn(inarray, 2)
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@@ -224,11 +282,20 @@ def urfft2(inarray):
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See Also
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--------
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ufft2, ufftn, urfftn
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Examples
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--------
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>>> input = np.ones((10, 128, 128))
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>>> output = urfft2(input)
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>>> np.allclose(np.sum(input[1,...]) / np.sqrt(input[1,...].size), output[1, 0, 0])
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True
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>>> output.shape
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(10, 128, 65)
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"""
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return urfftn(inarray, 2)
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def uirfft2(inarray):
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def uirfft2(inarray, shape=None):
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"""2-dim real unitary Fourier transform
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Compute the real inverse Fourier transform on the last 2 axes.
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@@ -248,14 +315,24 @@ def uirfft2(inarray):
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See Also
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--------
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urfft2, uifftn, uirfftn
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Examples
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--------
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>>> input = np.ones((10, 128, 128))
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>>> output = uirfftn(urfftn(input), shape=input.shape)
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>>> np.allclose(input, output)
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True
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>>> output.shape
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(10, 128, 128)
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"""
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return uirfftn(inarray, 2)
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return uirfftn(inarray, 2, shape=shape)
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def image_quad_norm(inarray):
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"""Return quadratic norm of images in Fourier space
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This function detect if the image suppose the hermitian property.
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This function detect if the image suppose the hermitian
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property.
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Parameters
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----------
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@@ -266,6 +343,14 @@ def image_quad_norm(inarray):
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-------
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norm : float
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The quadratic norm of `inarray`.
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Examples
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--------
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>>> input = np.ones((5, 5))
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>>> image_quad_norm(ufft2(input)) == np.sum(np.abs(input)**2)
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True
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>>> image_quad_norm(ufft2(input)) == image_quad_norm(urfft2(input))
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True
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"""
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# If there is an hermitian symmetry
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if inarray.shape[-1] != inarray.shape[-2]:
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@@ -306,6 +391,11 @@ def ir2tf(imp_resp, shape, dim=None, real=True):
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--------
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ufftn, uifftn, urfftn, uirfftn
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Examples
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--------
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>>> np.all(np.array([[4, 0], [0, 0]]) == ir2tf(np.ones((2, 2)), (2, 2)))
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True
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Notes
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-----
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The input array can be composed of multiple dimentionnal IR with
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@@ -348,9 +438,16 @@ def laplacian(ndim, shape):
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-------
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tf : array_like, complex
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The transfert function
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impr : array_like, real
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The laplacian
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Examples
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--------
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>>> tf, ir = laplacian(2, (32, 32))
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>>> np.all(ir == np.array([[0, -1, 0], [-1, 4, -1], [0, -1, 0]]))
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True
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>>> np.all(tf == ir2tf(ir, (32, 32)))
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True
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"""
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impr = np.zeros([3] * ndim)
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for dim in range(ndim):
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