450 lines
13 KiB
Python
450 lines
13 KiB
Python
r"""Function of unitary fourier transform (uft) and utilities
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This module implements the unitary fourier transform, also known as
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the ortho-normal transform. It is especially useful for convolution
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[1], as it respects the Parseval equality. The value of the null
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frequency is equal to
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.. math:: \frac{1}{\sqrt{n}} \sum_i x_i
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so the Fourier transform has the same energy as the original image
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(see ``image_quad_norm`` function). The transform is applied from the
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last axis for performance (assuming a C-order array input).
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References
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----------
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.. [1] B. R. Hunt "A matrix theory proof of the discrete convolution
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theorem", IEEE Trans. on Audio and Electroacoustics,
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vol. au-19, no. 4, pp. 285-288, dec. 1971
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"""
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import numpy as np
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from .._shared.fft import fftmodule as fft
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__keywords__ = "fft, Fourier Transform, orthonormal, unitary"
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def ufftn(inarray, dim=None):
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"""N-dimensional unitary Fourier transform.
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Parameters
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----------
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inarray : ndarray
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The array to transform.
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dim : int, optional
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The last axis along which to compute the transform. All
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axes by default.
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Returns
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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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outarray = fft.fftn(inarray, axes=range(-dim, 0), norm='ortho')
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return outarray
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def uifftn(inarray, dim=None):
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"""N-dimensional unitary inverse Fourier transform.
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Parameters
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----------
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inarray : ndarray
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The array to transform.
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dim : int, optional
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The last axis along which to compute the transform. All
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axes by default.
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Returns
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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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outarray = fft.ifftn(inarray, axes=range(-dim, 0), norm='ortho')
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return outarray
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def urfftn(inarray, dim=None):
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"""N-dimensional real unitary Fourier transform.
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This transform considers the Hermitian property of the transform on
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real-valued input.
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Parameters
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----------
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inarray : ndarray, shape (M, N, ..., P)
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The array to transform.
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dim : int, optional
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The last axis along which to compute the transform. All
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axes by default.
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Returns
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-------
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outarray : ndarray, shape (M, N, ..., P / 2 + 1)
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The unitary N-D real Fourier transform of ``inarray``.
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Notes
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-----
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The ``urfft`` functions assume an input array of real
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values. Consequently, the output has a Hermitian property and
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redundant values are not computed or 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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outarray = fft.rfftn(inarray, axes=range(-dim, 0), norm='ortho')
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return outarray
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def uirfftn(inarray, dim=None, shape=None):
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"""N-dimensional inverse real unitary Fourier transform.
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This transform considers the Hermitian property of the transform
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from complex to real input.
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Parameters
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----------
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inarray : ndarray
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The array to transform.
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dim : int, optional
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The last axis along which to compute the transform. All
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axes by default.
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shape : tuple of int, optional
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The shape of the output. The shape of ``rfft`` is ambiguous in
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case of odd-valued input shape. In this case, this parameter
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should be provided. See ``np.fft.irfftn``.
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Returns
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-------
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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 ``uirfft`` function assumes that the output array is
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real-valued. Consequently, the input is assumed to have a Hermitian
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property and redundant 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 = fft.irfftn(inarray, shape, axes=range(-dim, 0), norm='ortho')
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return outarray
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def ufft2(inarray):
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"""2-dimensional unitary Fourier transform.
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Compute the Fourier transform on the last 2 axes.
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Parameters
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----------
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inarray : ndarray
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The array to transform.
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Returns
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-------
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outarray : ndarray (same shape as inarray)
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The unitary 2-D Fourier transform of ``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),
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... 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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def uifft2(inarray):
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"""2-dimensional inverse unitary Fourier transform.
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Compute the inverse Fourier transform on the last 2 axes.
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Parameters
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----------
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inarray : ndarray
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The array to transform.
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Returns
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-------
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outarray : ndarray (same shape as inarray)
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The unitary 2-D inverse Fourier transform of ``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),
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... 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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def urfft2(inarray):
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"""2-dimensional real unitary Fourier transform
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Compute the real Fourier transform on the last 2 axes. This
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transform considers the Hermitian property of the transform from
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complex to real-valued input.
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Parameters
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----------
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inarray : ndarray, shape (M, N, ..., P)
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The array to transform.
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Returns
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-------
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outarray : ndarray, shape (M, N, ..., 2 * (P - 1))
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The unitary 2-D real Fourier transform of ``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),
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... 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, shape=None):
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"""2-dimensional inverse real unitary Fourier transform.
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Compute the real inverse Fourier transform on the last 2 axes.
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This transform considers the Hermitian property of the transform
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from complex to real-valued input.
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Parameters
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----------
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inarray : ndarray, shape (M, N, ..., P)
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The array to transform.
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shape : tuple of int, optional
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The shape of the output. The shape of ``rfft`` is ambiguous in
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case of odd-valued input shape. In this case, this parameter
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should be provided. See ``np.fft.irfftn``.
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Returns
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-------
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outarray : ndarray, shape (M, N, ..., 2 * (P - 1))
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The unitary 2-D inverse real Fourier transform of ``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, shape=shape)
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def image_quad_norm(inarray):
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"""Return the quadratic norm of images in Fourier space.
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This function detects whether the input image satisfies the
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Hermitian property.
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Parameters
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----------
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inarray : ndarray
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Input image. The image data should reside in the final two
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axes.
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Returns
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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 a Hermitian symmetry
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if inarray.shape[-1] != inarray.shape[-2]:
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return (2 * np.sum(np.sum(np.abs(inarray) ** 2, axis=-1), axis=-1) -
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np.sum(np.abs(inarray[..., 0]) ** 2, axis=-1))
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else:
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return np.sum(np.sum(np.abs(inarray) ** 2, axis=-1), axis=-1)
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def ir2tf(imp_resp, shape, dim=None, is_real=True):
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"""Compute the transfer function of an impulse response (IR).
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This function makes the necessary correct zero-padding, zero
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convention, correct fft2, etc... to compute the transfer function
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of IR. To use with unitary Fourier transform for the signal (ufftn
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or equivalent).
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Parameters
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----------
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imp_resp : ndarray
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The impulse responses.
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shape : tuple of int
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A tuple of integer corresponding to the target shape of the
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transfer function.
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dim : int, optional
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The last axis along which to compute the transform. All
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axes by default.
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is_real : boolean, optional
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If True (default), imp_resp is supposed real and the Hermitian property
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is used with rfftn Fourier transform.
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Returns
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-------
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y : complex ndarray
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The transfer function of shape ``shape``.
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See Also
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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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>>> ir2tf(np.ones((2, 2)), (512, 512)).shape == (512, 257)
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True
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>>> ir2tf(np.ones((2, 2)), (512, 512), is_real=False).shape == (512, 512)
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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-dimensional IR with
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an arbitrary number of IR. The individual IR must be accessed
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through the first axes. The last ``dim`` axes contain the space
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definition.
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"""
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if not dim:
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dim = imp_resp.ndim
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# Zero padding and fill
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irpadded = np.zeros(shape)
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irpadded[tuple([slice(0, s) for s in imp_resp.shape])] = imp_resp
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# Roll for zero convention of the fft to avoid the phase
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# problem. Work with odd and even size.
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for axis, axis_size in enumerate(imp_resp.shape):
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if axis >= imp_resp.ndim - dim:
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irpadded = np.roll(irpadded,
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shift=-int(np.floor(axis_size / 2)),
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axis=axis)
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if is_real:
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return fft.rfftn(irpadded, axes=range(-dim, 0))
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else:
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return fft.fftn(irpadded, axes=range(-dim, 0))
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def laplacian(ndim, shape, is_real=True):
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"""Return the transfer function of the Laplacian.
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Laplacian is the second order difference, on row and column.
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Parameters
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----------
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ndim : int
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The dimension of the Laplacian.
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shape : tuple
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The support on which to compute the transfer function.
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is_real : boolean, optional
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If True (default), imp_resp is assumed to be real-valued and
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the Hermitian property is used with rfftn Fourier transform
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to return the transfer function.
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Returns
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-------
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tf : array_like, complex
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The transfer 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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idx = tuple([slice(1, 2)] * dim +
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[slice(None)] +
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[slice(1, 2)] * (ndim - dim - 1))
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impr[idx] = np.array([-1.0,
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0.0,
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-1.0]).reshape([-1 if i == dim else 1
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for i in range(ndim)])
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impr[(slice(1, 2), ) * ndim] = 2.0 * ndim
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return ir2tf(impr, shape, is_real=is_real), impr
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