o &?e1@sdZddlZddlmZddlmZdddZdddZdd d Z dd d Z d dZ ddZ ddZ dddZddZdddZdddZdS) aFunction of unitary fourier transform (uft) and utilities This module implements the unitary fourier transform, also known as the ortho-normal transform. It is especially useful for convolution [1], as it respects the Parseval equality. The value of the null frequency is equal to .. math:: \frac{1}{\sqrt{n}} \sum_i x_i so the Fourier transform has the same energy as the original image (see ``image_quad_norm`` function). The transform is applied from the last axis for performance (assuming a C-order array input). References ---------- .. [1] B. R. Hunt "A matrix theory proof of the discrete convolution theorem", IEEE Trans. on Audio and Electroacoustics, vol. au-19, no. 4, pp. 285-288, dec. 1971 N)_supported_float_typecC*|dur|j}tj|t| ddd}|S)aIN-dimensional unitary Fourier transform. Parameters ---------- inarray : ndarray The array to transform. dim : int, optional The last axis along which to compute the transform. All axes by default. Returns ------- outarray : ndarray (same shape than inarray) The unitary N-D Fourier transform of ``inarray``. Examples -------- >>> input = np.ones((3, 3, 3)) >>> output = ufftn(input) >>> np.allclose(np.sum(input) / np.sqrt(input.size), output[0, 0, 0]) True >>> output.shape (3, 3, 3) NrorthoaxesZnorm)ndimfftfftnrangeinarraydimoutarrayrX/home/www/facesmatcher.com/pyenv/lib/python3.10/site-packages/skimage/restoration/uft.pyufftnrcCr)aZN-dimensional unitary inverse Fourier transform. Parameters ---------- inarray : ndarray The array to transform. dim : int, optional The last axis along which to compute the transform. All axes by default. Returns ------- outarray : ndarray (same shape than inarray) The unitary inverse N-D Fourier transform of ``inarray``. Examples -------- >>> input = np.ones((3, 3, 3)) >>> output = uifftn(input) >>> np.allclose(np.sum(input) / np.sqrt(input.size), output[0, 0, 0]) True >>> output.shape (3, 3, 3) Nrrr)rr Zifftnr r rrruifftn=rrcCr)aN-dimensional real unitary Fourier transform. This transform considers the Hermitian property of the transform on real-valued input. Parameters ---------- inarray : ndarray, shape (M, N, ..., P) The array to transform. dim : int, optional The last axis along which to compute the transform. All axes by default. Returns ------- outarray : ndarray, shape (M, N, ..., P / 2 + 1) The unitary N-D real Fourier transform of ``inarray``. Notes ----- The ``urfft`` functions assume an input array of real values. Consequently, the output has a Hermitian property and redundant values are not computed or returned. Examples -------- >>> input = np.ones((5, 5, 5)) >>> output = urfftn(input) >>> np.allclose(np.sum(input) / np.sqrt(input.size), output[0, 0, 0]) True >>> output.shape (5, 5, 3) Nrrr)rr rfftnr r rrrurfftn\s"rcCs,|dur|j}tj||t| ddd}|S)aQN-dimensional inverse real unitary Fourier transform. This transform considers the Hermitian property of the transform from complex to real input. Parameters ---------- inarray : ndarray The array to transform. dim : int, optional The last axis along which to compute the transform. All axes by default. shape : tuple of int, optional The shape of the output. The shape of ``rfft`` is ambiguous in case of odd-valued input shape. In this case, this parameter should be provided. See ``np.fft.irfftn``. Returns ------- outarray : ndarray The unitary N-D inverse real Fourier transform of ``inarray``. Notes ----- The ``uirfft`` function assumes that the output array is real-valued. Consequently, the input is assumed to have a Hermitian property and redundant values are implicit. Examples -------- >>> input = np.ones((5, 5, 5)) >>> output = uirfftn(urfftn(input), shape=input.shape) >>> np.allclose(input, output) True >>> output.shape (5, 5, 5) Nrrr)rr Zirfftnr )r rshaperrrruirfftns&rcC t|dS)ap2-dimensional unitary Fourier transform. Compute the Fourier transform on the last 2 axes. Parameters ---------- inarray : ndarray The array to transform. Returns ------- outarray : ndarray (same shape as inarray) The unitary 2-D Fourier transform of ``inarray``. See Also -------- uifft2, ufftn, urfftn Examples -------- >>> input = np.ones((10, 128, 128)) >>> output = ufft2(input) >>> np.allclose(np.sum(input[1, ...]) / np.sqrt(input[1, ...].size), ... output[1, 0, 0]) True >>> output.shape (10, 128, 128) r)rr rrrufft2 rcCr)a2-dimensional inverse unitary Fourier transform. Compute the inverse Fourier transform on the last 2 axes. Parameters ---------- inarray : ndarray The array to transform. Returns ------- outarray : ndarray (same shape as inarray) The unitary 2-D inverse Fourier transform of ``inarray``. See Also -------- uifft2, uifftn, uirfftn Examples -------- >>> input = np.ones((10, 128, 128)) >>> output = uifft2(input) >>> np.allclose(np.sum(input[1, ...]) / np.sqrt(input[1, ...].size), ... output[0, 0, 0]) True >>> output.shape (10, 128, 128) r)rrrrruifft2rrcCr)a2-dimensional real unitary Fourier transform Compute the real Fourier transform on the last 2 axes. This transform considers the Hermitian property of the transform from complex to real-valued input. Parameters ---------- inarray : ndarray, shape (M, N, ..., P) The array to transform. Returns ------- outarray : ndarray, shape (M, N, ..., 2 * (P - 1)) The unitary 2-D real Fourier transform of ``inarray``. See Also -------- ufft2, ufftn, urfftn Examples -------- >>> input = np.ones((10, 128, 128)) >>> output = urfft2(input) >>> np.allclose(np.sum(input[1,...]) / np.sqrt(input[1,...].size), ... output[1, 0, 0]) True >>> output.shape (10, 128, 65) r)rrrrrurfft2s rcCst|d|dS)a2-dimensional inverse real unitary Fourier transform. Compute the real inverse Fourier transform on the last 2 axes. This transform considers the Hermitian property of the transform from complex to real-valued input. Parameters ---------- inarray : ndarray, shape (M, N, ..., P) The array to transform. shape : tuple of int, optional The shape of the output. The shape of ``rfft`` is ambiguous in case of odd-valued input shape. In this case, this parameter should be provided. See ``np.fft.irfftn``. Returns ------- outarray : ndarray, shape (M, N, ..., 2 * (P - 1)) The unitary 2-D inverse real Fourier transform of ``inarray``. See Also -------- urfft2, uifftn, uirfftn Examples -------- >>> input = np.ones((10, 128, 128)) >>> output = uirfftn(urfftn(input), shape=input.shape) >>> np.allclose(input, output) True >>> output.shape (10, 128, 128) r)r)r)r rrrruirfft2s"rcCsx|jd|jdkr+dtjtjt|dddddtjt|ddddStjtjt|dddddS)aEReturn the quadratic norm of images in Fourier space. This function detects whether the input image satisfies the Hermitian property. Parameters ---------- inarray : ndarray Input image. The image data should reside in the final two axes. Returns ------- norm : float The quadratic norm of ``inarray``. Examples -------- >>> input = np.ones((5, 5)) >>> image_quad_norm(ufft2(input)) == np.sum(np.abs(input)**2) True >>> image_quad_norm(ufft2(input)) == image_quad_norm(urfft2(input)) True r)axis).r)rnpsumabsrrrrimage_quad_norm7s $"r&Tc Cs|s|j}t|j}tj||d}||tdd|jD<t|jD]\}}||j|kr=tj|t t |d |d}q"|rCt j nt j }||t| dd} t|tj} | j| dd S) aCompute the transfer function of an impulse response (IR). This function makes the necessary correct zero-padding, zero convention, correct fft2, etc... to compute the transfer function of IR. To use with unitary Fourier transform for the signal (ufftn or equivalent). Parameters ---------- imp_resp : ndarray The impulse responses. shape : tuple of int A tuple of integer corresponding to the target shape of the transfer function. dim : int, optional The last axis along which to compute the transform. All axes by default. is_real : boolean, optional If True (default), imp_resp is supposed real and the Hermitian property is used with rfftn Fourier transform. Returns ------- y : complex ndarray The transfer function of shape ``shape``. See Also -------- ufftn, uifftn, urfftn, uirfftn Examples -------- >>> np.all(np.array([[4, 0], [0, 0]]) == ir2tf(np.ones((2, 2)), (2, 2))) True >>> ir2tf(np.ones((2, 2)), (512, 512)).shape == (512, 257) True >>> ir2tf(np.ones((2, 2)), (512, 512), is_real=False).shape == (512, 512) True Notes ----- The input array can be composed of multiple-dimensional IR with an arbitrary number of IR. The individual IR must be accessed through the first axes. The last ``dim`` axes contain the space definition. )dtypecSsg|]}td|qS)r)slice).0srrr szir2tf..r)shiftr"r)rF)copy)rrr'r#zerostupler enumerateZrollintfloorr rr r Z promote_typesZ complex64Zastype) Zimp_resprris_realZirpadded_dtypeZirpaddedr"Z axis_sizefuncoutZ cplx_dtyperrrir2tfXs / r6cstdg|}t|D]2ttddgtdgtddg|d}tgdfddt|D||<q d||tddf|<t|||d |fS) a8Return the transfer function of the Laplacian. Laplacian is the second order difference, on row and column. Parameters ---------- ndim : int The dimension of the Laplacian. shape : tuple The support on which to compute the transfer function. is_real : boolean, optional If True (default), imp_resp is assumed to be real-valued and the Hermitian property is used with rfftn Fourier transform to return the transfer function. Returns ------- tf : array_like, complex The transfer function. impr : array_like, real The Laplacian. Examples -------- >>> tf, ir = laplacian(2, (32, 32)) >>> np.all(ir == np.array([[0, -1, 0], [-1, 4, -1], [0, -1, 0]])) True >>> np.all(tf == ir2tf(ir, (32, 32))) True rN)gr9csg|] }|kr dndqS)r r8r)r)irrrr+szlaplacian..g@)r3)r#r.r r/r(arrayZreshaper6)rrr3Zimpridxrr;r laplacians   r>)N)NN)NT)T)__doc__numpyr#Z scipy.fftr Z _shared.utilsrrrrrrrrrr&r6r>rrrrs      (,  "% !E