ó  ‰\c@ s°dZddlmZmZddlZdZdd„Zdd„Z dd„Z ddd„Z d „Z d „Z d „Zdd „Zd „Zded„Zed„ZdS(sÍFunction of unitary fourier transform 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:: rac{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 iÿÿÿÿ(tdivisiontprint_functionNs,fft, Fourier Transform, orthonormal, unitarycC s\|dkr|j}ntjj|dt| dƒƒ}|tjtj|j| ƒƒS(sIN-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) taxesiN( tNonetndimtnptffttfftntrangetsqrttprodtshape(tinarraytdimtoutarray((s6lib/python2.7/site-packages/skimage/restoration/uft.pytufftn s  "cC s\|dkr|j}ntjj|dt| dƒƒ}|tjtj|j| ƒƒS(sZN-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) RiN( RRRRtifftnRR R R (R R R((s6lib/python2.7/site-packages/skimage/restoration/uft.pytuifftn?s  "cC s\|dkr|j}ntjj|dt| dƒƒ}|tjtj|j| ƒƒS(s’N-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) RiN( RRRRtrfftnRR R R (R R R((s6lib/python2.7/site-packages/skimage/restoration/uft.pyturfftn^s"  "cC s_|dkr|j}ntjj||dt| dƒƒ}|tjtj|j| ƒƒS(sQN-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) RiN( RRRRtirfftnRR R R (R R R R((s6lib/python2.7/site-packages/skimage/restoration/uft.pytuirfftn†s&  %cC s t|dƒS(sp2-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) i(R(R ((s6lib/python2.7/site-packages/skimage/restoration/uft.pytufft2²scC s t|dƒS(s‹2-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) i(R(R ((s6lib/python2.7/site-packages/skimage/restoration/uft.pytuifft2ÒscC s t|dƒS(s2-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) i(R(R ((s6lib/python2.7/site-packages/skimage/restoration/uft.pyturfft2òscC st|dd|ƒS(sõ2-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. 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) iR (R(R R ((s6lib/python2.7/site-packages/skimage/restoration/uft.pytuirfft2scC s¤|jd|jdkrqdtjtjtj|ƒdddƒddƒtjtj|dƒdddƒStjtjtj|ƒdddƒddƒSdS( sEReturn 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 iÿÿÿÿiþÿÿÿitaxis.iN(.i(R Rtsumtabs(R ((s6lib/python2.7/site-packages/skimage/restoration/uft.pytimage_quad_norm5s2%cC sÿ|s|j}ntj|ƒ}||tg|jD]}td|ƒ^q4ƒ>> 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. itshiftiRRN(RRtzerosttupleR tslicet enumeratetrolltinttfloorRRRR(timp_respR R tis_realtirpaddedtsRt axis_size((s6lib/python2.7/site-packages/skimage/restoration/uft.pytir2tfVs/ /  cC sùtjdg|ƒ}x©t|ƒD]›}ttddƒg|td ƒgtddƒg||dƒ}tjdddgƒjgt|ƒD]}||kr«dnd^q“ƒ||>> 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 iiigð¿giÿÿÿÿg@R'N( RRRR R!RtarraytreshapeR+(RR R'timprR tidxti((s6lib/python2.7/site-packages/skimage/restoration/uft.pyt laplacian—s " 9(t__doc__t __future__RRtnumpyRt __keywords__RRRRRRRRRRtTrueR+R1(((s6lib/python2.7/site-packages/skimage/restoration/uft.pyts    (, " ! !A