9iE1SrSSKrSSKJr SSKJr SSjrSSjrSSjr SSjr S r S r S r SS jrS rSSjrSSjrg)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_typecfUc URn[R"U[U*S5SS9nU$)aN-dimensional unitary Fourier transform. Parameters ---------- inarray : ndarray The array to transform. dim : int, optional The last axis along which to compute the transform. All axes by default. Returns ------- outarray : ndarray (same shape than inarray) The unitary N-D Fourier transform of ``inarray``. Examples -------- >>> input = np.ones((3, 3, 3)) >>> output = ufftn(input) >>> np.allclose(np.sum(input) / np.sqrt(input.size), output[0, 0, 0]) True >>> output.shape (3, 3, 3) rorthoaxesnorm)ndimfftfftnrangeinarraydimoutarrays W/var/www/html/land-doc-ocr/venv/lib/python3.13/site-packages/skimage/restoration/uft.pyufftnrs12 {llxxeSD!n7CH OcfUc URn[R"U[U*S5SS9nU$)aN-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 The unitary inverse nD Fourier transform of ``inarray``. Has the same shape as ``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) rrr)r r ifftnr rs ruifftnr;s14 {llyyucT1~GDH OrcfUc URn[R"U[U*S5SS9nU$)a$N-dimensional real unitary Fourier transform. This transform considers the Hermitian property of the transform on real-valued input. Parameters ---------- inarray : ndarray, shape (M[, ...], 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[, ...], 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) rrr)r r rfftnr rs rurfftnr[s2D {llyyucT1~GDH OrcfUc URn[R"X[U*S5SS9nU$)aN-dimensional inverse real unitary Fourier transform. This transform considers the Hermitian property of the transform from complex to real input. Parameters ---------- inarray : ndarray The array to transform. dim : int, optional The last axis along which to compute the transform. All axes by default. shape : tuple of int, optional The shape of the output. The shape of ``rfft`` is ambiguous in case of odd-valued input shape. In this case, this parameter should be provided. See ``np.fft.irfftn``. Returns ------- outarray : ndarray The unitary N-D inverse real Fourier transform of ``inarray``. Notes ----- The ``uirfft`` function assumes that the output array is real-valued. Consequently, the input is assumed to have a Hermitian property and redundant values are implicit. Examples -------- >>> input = np.ones((5, 5, 5)) >>> output = uirfftn(urfftn(input), shape=input.shape) >>> np.allclose(input, output) True >>> output.shape (5, 5, 5) rrr)r r irfftnr )rrshapers ruirfftnrs2L {llzz'ucT1~GLH Orc[US5$)a2-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)rrs rufft2r!s: ! rc[US5$)a32-dimensional inverse unitary Fourier transform. Compute the inverse Fourier transform on the last 2 axes. Parameters ---------- inarray : ndarray The array to transform. Returns ------- outarray : ndarray (same shape as inarray) The unitary 2-D inverse Fourier transform of ``inarray``. See Also -------- uifft2, uifftn, uirfftn Examples -------- >>> input = np.ones((10, 128, 128)) >>> output = uifft2(input) >>> np.allclose(np.sum(input[1, ...]) / np.sqrt(input[1, ...].size), ... output[0, 0, 0]) True >>> output.shape (10, 128, 128) r)rr s ruifft2r#s: '1 rc[US5$)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[, ...], P) The array to transform. Returns ------- outarray : ndarray, shape (M[, ...], 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)rr s rurfft2r%s> '1 rc[USUS9$)ai2-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[, ...], 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[, ...], 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)rrs ruirfft2r'sD 7AU ++rcURSURS:warS[R"[R"[R"U5S-SS9SS9-[R"[R"US5S-SS9- $[R"[R"[R"U5S-SS9SS9$)aReturn 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)rnpsumabsr s rimage_quad_normr/6s4}}RGMM"--266"&&A!5B?bIIBFF FF76? #q (rM   vvbffRVVG_1;"EErc U(d URn[UR5n[R"XS9nX[ UR Vs/sHn[SU5PM sn5'[UR 5HPupxXpRU- :dM[R"U[[R"US- 55*US9nMR U(a[RO[Rn U "U[U*S5S9n [R "U[R"5n U R%U SS9$s snf)aCompute the transfer function of an impulse response (IR). This function makes the necessary correct zero-padding, zero convention, correct fft2, etc... to compute the transfer function of IR. To use with unitary Fourier transform for the signal (ufftn or equivalent). Parameters ---------- imp_resp : ndarray The impulse responses. shape : tuple of int A tuple of integer corresponding to the target shape of the transfer function. dim : int, optional The last axis along which to compute the transform. All axes by default. is_real : boolean, optional If True (default), imp_resp is supposed real and the Hermitian property is used with rfftn Fourier transform. Returns ------- y : complex ndarray The transfer function of shape ``shape``. See Also -------- ufftn, uifftn, urfftn, uirfftn Examples -------- >>> np.all(np.array([[4, 0], [0, 0]]) == ir2tf(np.ones((2, 2)), (2, 2))) True >>> ir2tf(np.ones((2, 2)), (512, 512)).shape == (512, 257) True >>> ir2tf(np.ones((2, 2)), (512, 512), is_real=False).shape == (512, 512) True Notes ----- The input array can be composed of multiple-dimensional IR with an arbitrary number of IR. The individual IR must be accessed through the first axes. The last ``dim`` axes contain the space definition. )dtyperr)shiftr+)rF)copy)r rr1r,zerostuplerslice enumeraterollintfloorr rr r promote_types complex64astype) imp_resprris_realirpadded_dtypeirpaddedsr+ axis_sizefuncout cplx_dtypes rir2tfrGXs^ mm*8>>:Nxx4H=EU8AE!QK8 9:%X^^4 ==3& &wwxBHHY]4K0L/LSWXH5 399SXXD xucT1~ /C!!.",,?J ::ju: --9sE c[R"S/U-5n[U5Hn[[ SS5/U-[ S5/-[ SS5/X- S- --5n[R "/SQ5R [U5Vs/sH ofU:XaSOSPM sn5X5'M SU-U[ SS54U-'[X1US9U4$s snf) aReturn 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)grKr)g@)r?)r,r4r r5r6arrayreshaperG)r rr?imprridxis r laplacianrQs> 88QC$J DT{ 1a[MC 5;- /5A;-4:PQ>2R R HH./77,1$K 8Kq8R "K 8  #&*D%1+$  g . 44 9sC )N)NN)NT)T)__doc__numpyr, scipy.fftr _shared.utilsrrrrrr!r#r%r'r/rGrQrrrWsW*1>@%P)X@@D",JFD@.F(5r