U @f)M @s0ddlmZmZmZmZmZmZmZmZm Z m Z m Z ddl m Z mZmZmZmZmZmZmZmZddlmZmZddlmZddlmZddd d d d d ddg Zd+ddZiZddZ ddZ!dd Z"dd Z#dd Z$ddZ%ddZ&ddZ'ddZ(d d!Z)d"d#Z*d,d%d Z+d-d&d Z,d.d(dZ-d/d)dZ.d*S)0) logical_andasarraypi zeros_like piecewisearrayarctan2tanzerosarangefloor) sqrtexpgreaterlesscosaddsin less_equal greater_equal) cspline2dsepfir2d)comb)float_factorial spline_filterbspline gauss_splinecubic quadratic cspline1d qspline1dcspline1d_evalqspline1d_eval@c Cs|jj}tdddgdd}|dkrr|d}t|j|}t|j|}t|||}t|||}|d||}n2|dkrt||}t|||}||}ntd |S) a3Smoothing spline (cubic) filtering of a rank-2 array. Filter an input data set, `Iin`, using a (cubic) smoothing spline of fall-off `lmbda`. Parameters ---------- Iin : array_like input data set lmbda : float, optional spline smooghing fall-off value, default is `5.0`. Returns ------- res : ndarray filterd input data Examples -------- We can filter an multi dimentional signal (ex: 2D image) using cubic B-spline filter: >>> import numpy as np >>> from scipy.signal import spline_filter >>> import matplotlib.pyplot as plt >>> orig_img = np.eye(20) # create an image >>> orig_img[10, :] = 1.0 >>> sp_filter = spline_filter(orig_img, lmbda=0.1) >>> f, ax = plt.subplots(1, 2, sharex=True) >>> for ind, data in enumerate([[orig_img, "original image"], ... [sp_filter, "spline filter"]]): ... ax[ind].imshow(data[0], cmap='gray_r') ... ax[ind].set_title(data[1]) >>> plt.tight_layout() >>> plt.show() ?g@f@)FDr(y?)r&dzInvalid data type for Iin) dtypecharrastyperrealimagr TypeError) ZIinZlmbdaZintypeZhcolZckrZckiZoutrZoutioutr2:/tmp/pip-unpacked-wheel-w7avvj8p/scipy/signal/_bsplines.pyrs&        csz tWStk rYnXdd}dd}drBd}nd}|dd|g}|td|dD]"}||dddqf||ddd dtfd d fd d t|D}||ft<||fS) aReturns the function defined over the left-side pieces for a bspline of a given order. The 0th piece is the first one less than 0. The last piece is a function identical to 0 (returned as the constant 0). (There are order//2 + 2 total pieces). Also returns the condition functions that when evaluated return boolean arrays for use with `numpy.piecewise`. cs<|dkrfddS|dkr*fddSfddSdS)Nrcstt|t|SN)rrrxval1val2r2r3]s z>_bspline_piecefunctions..condfuncgen..cs t|Sr4)rr5)r9r2r3r:`cstt|t|Sr4)rrrr5r7r2r3r:bs r2)numr8r9r2r7r3 condfuncgen[s  z,_bspline_piecefunctions..condfuncgenr;ggrr@csdd|dkrdSfddtdDfddtdDfdd}|S) Nr;rc s6g|].}dd|dttd|ddqS)rr;)exact)floatr.0k)fvalorderr2r3 }szA_bspline_piecefunctions..piecefuncgen..rcsg|]} |qSr2r2rB)boundr2r3rGscs6d}tdD] }||||7}q|S)Nrrange)r6resrD)MkcoeffsrFshiftsr2r3thefuncsz>_bspline_piecefunctions..piecefuncgen..thefuncrJ)r=rP)rHrErF)rMrNrOr3 piecefuncgenys   z-_bspline_piecefunctions..piecefuncgencsg|] }|qSr2r2rB)rQr2r3rGsz+_bspline_piecefunctions..)_splinefunc_cacheKeyErrorrKappendr)rFr>lastZ startbound condfuncsr=funclistr2)rHrErFrQr3_bspline_piecefunctionsKs(     rXcs8tt| t|\}}fdd|D}t||S)awB-spline basis function of order n. Parameters ---------- x : array_like a knot vector n : int The order of the spline. Must be non-negative, i.e., n >= 0 Returns ------- res : ndarray B-spline basis function values See Also -------- cubic : A cubic B-spline. quadratic : A quadratic B-spline. Notes ----- Uses numpy.piecewise and automatic function-generator. Examples -------- We can calculate B-Spline basis function of several orders: >>> import numpy as np >>> from scipy.signal import bspline, cubic, quadratic >>> bspline(0.0, 1) 1 >>> knots = [-1.0, 0.0, -1.0] >>> bspline(knots, 2) array([0.125, 0.75, 0.125]) >>> np.array_equal(bspline(knots, 2), quadratic(knots)) True >>> np.array_equal(bspline(knots, 3), cubic(knots)) True csg|] }|qSr2r2)rCfuncaxr2r3rGszbspline..)absrrXr)r6nrWrVZcondlistr2rZr3rs, cCs>t|}|dd}dtdt|t|d d|S)aGaussian approximation to B-spline basis function of order n. Parameters ---------- x : array_like a knot vector n : int The order of the spline. Must be non-negative, i.e., n >= 0 Returns ------- res : ndarray B-spline basis function values approximated by a zero-mean Gaussian function. Notes ----- The B-spline basis function can be approximated well by a zero-mean Gaussian function with standard-deviation equal to :math:`\sigma=(n+1)/12` for large `n` : .. math:: \frac{1}{\sqrt {2\pi\sigma^2}}exp(-\frac{x^2}{2\sigma}) References ---------- .. [1] Bouma H., Vilanova A., Bescos J.O., ter Haar Romeny B.M., Gerritsen F.A. (2007) Fast and Accurate Gaussian Derivatives Based on B-Splines. In: Sgallari F., Murli A., Paragios N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer Science, vol 4485. Springer, Berlin, Heidelberg .. [2] http://folk.uio.no/inf3330/scripting/doc/python/SciPy/tutorial/old/node24.html Examples -------- We can calculate B-Spline basis functions approximated by a gaussian distribution: >>> import numpy as np >>> from scipy.signal import gauss_spline, bspline >>> knots = np.array([-1.0, 0.0, -1.0]) >>> gauss_spline(knots, 3) array([0.15418033, 0.6909883, 0.15418033]) # may vary >>> bspline(knots, 3) array([0.16666667, 0.66666667, 0.16666667]) # may vary rg(@r;)rr rr)r6r]Zsignsqr2r2r3rs0 cCstt|}t|}t|d}|rJ||}dd|dd|||<|t|d@}|r~||}dd|d||<|S)a-A cubic B-spline. This is a special case of `bspline`, and equivalent to ``bspline(x, 3)``. Parameters ---------- x : array_like a knot vector Returns ------- res : ndarray Cubic B-spline basis function values See Also -------- bspline : B-spline basis function of order n quadratic : A quadratic B-spline. Examples -------- We can calculate B-Spline basis function of several orders: >>> import numpy as np >>> from scipy.signal import bspline, cubic, quadratic >>> bspline(0.0, 1) 1 >>> knots = [-1.0, 0.0, -1.0] >>> bspline(knots, 2) array([0.125, 0.75, 0.125]) >>> np.array_equal(bspline(knots, 2), quadratic(knots)) True >>> np.array_equal(bspline(knots, 3), cubic(knots)) True rgUUUUUU??r;gUUUUUU?r\rrranyr6r[rLcond1Zax1cond2Zax2r2r2r3rs(  cCsvtt|}t|}t|d}|r>||}d|d||<|t|d@}|rr||}|ddd||<|S)a-A quadratic B-spline. This is a special case of `bspline`, and equivalent to ``bspline(x, 2)``. Parameters ---------- x : array_like a knot vector Returns ------- res : ndarray Quadratic B-spline basis function values See Also -------- bspline : B-spline basis function of order n cubic : A cubic B-spline. Examples -------- We can calculate B-Spline basis function of several orders: >>> import numpy as np >>> from scipy.signal import bspline, cubic, quadratic >>> bspline(0.0, 1) 1 >>> knots = [-1.0, 0.0, -1.0] >>> bspline(knots, 2) array([0.125, 0.75, 0.125]) >>> np.array_equal(bspline(knots, 2), quadratic(knots)) True >>> np.array_equal(bspline(knots, 3), cubic(knots)) True r^g?r;?r?r`rbr2r2r3r,s(  cCsdd|d|tdd|}ttd|dt|}d|dt|d|}|td|d|tdd||}||fS)Nr`r_0)r r)ZlamxiZomegrhor2r2r3 _coeff_smoothas $,rlcCs.|t|||t||dt|dS)Nr)rr)rDcsrkomegar2r2r3_hcis"rpcCs||d||d||dd||td||d}d||d||t|}t|}|||t|||t||S)Nrr;)rr r\r)rDrnrkroZc0gammaZakr2r2r3_hsns " rsc Cst|\}}dd|t|||}t|}t|f|jj}t|}td||||dt t|d|||||d<td||||dtd||||dt t|d|||||d<t d|D]D}|||d|t|||d||||d||<qt|f|jj} t t ||||t |d||||ddd| |d<t t |d|||t |d||||ddd| |d<t |dddD]F}|||d|t|| |d||| |d| |<q| S)Nrr;rrmr_) rlrlenr r+r,r rprreducerKrs) signallambrkrornKZyprDr]yr2r2r3_cubic_smooth_coeffvsB &   & rzcCsdtd}t|}t|f|jj}|t|}|d|t|||d<td|D] }|||||d||<qXt|f|j}||d||d||d<t|dddD] }|||d||||<q|dS)Nr_rrr;rmr' r rtr r+r,r rrurKrvZzirxZyplusZpowersrDoutputr2r2r3 _cubic_coeffs   rcCsddtd}t|}t|f|jj}|t|}|d|t|||d<td|D] }|||||d||<q\t|f|jj}||d||d||d<t|dddD] }|||d||||<q|dS)Nr;r?rrrmg @r|r}r2r2r3_quadratic_coeffs  rrIcCs|dkrt||St|SdS)a Compute cubic spline coefficients for rank-1 array. Find the cubic spline coefficients for a 1-D signal assuming mirror-symmetric boundary conditions. To obtain the signal back from the spline representation mirror-symmetric-convolve these coefficients with a length 3 FIR window [1.0, 4.0, 1.0]/ 6.0 . Parameters ---------- signal : ndarray A rank-1 array representing samples of a signal. lamb : float, optional Smoothing coefficient, default is 0.0. Returns ------- c : ndarray Cubic spline coefficients. See Also -------- cspline1d_eval : Evaluate a cubic spline at the new set of points. Examples -------- We can filter a signal to reduce and smooth out high-frequency noise with a cubic spline: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.signal import cspline1d, cspline1d_eval >>> rng = np.random.default_rng() >>> sig = np.repeat([0., 1., 0.], 100) >>> sig += rng.standard_normal(len(sig))*0.05 # add noise >>> time = np.linspace(0, len(sig)) >>> filtered = cspline1d_eval(cspline1d(sig), time) >>> plt.plot(sig, label="signal") >>> plt.plot(time, filtered, label="filtered") >>> plt.legend() >>> plt.show() rIN)rzrrvrwr2r2r3r s, cCs|dkrtdnt|SdS)aFCompute quadratic spline coefficients for rank-1 array. Parameters ---------- signal : ndarray A rank-1 array representing samples of a signal. lamb : float, optional Smoothing coefficient (must be zero for now). Returns ------- c : ndarray Quadratic spline coefficients. See Also -------- qspline1d_eval : Evaluate a quadratic spline at the new set of points. Notes ----- Find the quadratic spline coefficients for a 1-D signal assuming mirror-symmetric boundary conditions. To obtain the signal back from the spline representation mirror-symmetric-convolve these coefficients with a length 3 FIR window [1.0, 6.0, 1.0]/ 8.0 . Examples -------- We can filter a signal to reduce and smooth out high-frequency noise with a quadratic spline: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.signal import qspline1d, qspline1d_eval >>> rng = np.random.default_rng() >>> sig = np.repeat([0., 1., 0.], 100) >>> sig += rng.standard_normal(len(sig))*0.05 # add noise >>> time = np.linspace(0, len(sig)) >>> filtered = qspline1d_eval(qspline1d(sig), time) >>> plt.plot(sig, label="signal") >>> plt.plot(time, filtered, label="filtered") >>> plt.legend() >>> plt.show() rIz.Smoothing quadratic splines not supported yet.N) ValueErrorrrr2r2r3r!s- r%cCs t||t|}t||jd}|jdkr0|St|}|dk}||dk}||B}t||| ||<t|d|d||||<||}|jdkr|St||jd} t|dt d} t dD]4} | | } | d|d} | || t || 7} q| ||<|S)aEvaluate a cubic spline at the new set of points. `dx` is the old sample-spacing while `x0` was the old origin. In other-words the old-sample points (knot-points) for which the `cj` represent spline coefficients were at equally-spaced points of: oldx = x0 + j*dx j=0...N-1, with N=len(cj) Edges are handled using mirror-symmetric boundary conditions. Parameters ---------- cj : ndarray cublic spline coefficients newx : ndarray New set of points. dx : float, optional Old sample-spacing, the default value is 1.0. x0 : int, optional Old origin, the default value is 0. Returns ------- res : ndarray Evaluated a cubic spline points. See Also -------- cspline1d : Compute cubic spline coefficients for rank-1 array. Examples -------- We can filter a signal to reduce and smooth out high-frequency noise with a cubic spline: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.signal import cspline1d, cspline1d_eval >>> rng = np.random.default_rng() >>> sig = np.repeat([0., 1., 0.], 100) >>> sig += rng.standard_normal(len(sig))*0.05 # add noise >>> time = np.linspace(0, len(sig)) >>> filtered = cspline1d_eval(cspline1d(sig), time) >>> plt.plot(sig, label="signal") >>> plt.plot(time, filtered, label="filtered") >>> plt.legend() >>> plt.show() )r+rrr;rq) rrArr+sizertr"r r-intrKcliprcjZnewxZdxZx0rLNrcrdZcond3resultZjloweriZthisjZindjr2r2r3r"s*2     cCst|||}t|}|jdkr&|St|}|dk}||dk}||B}t||| ||<t|d|d||||<||}|jdkr|St|} t|dtd} tdD]4} | | } | d|d} | || t || 7} q| ||<|S)aEvaluate a quadratic spline at the new set of points. Parameters ---------- cj : ndarray Quadratic spline coefficients newx : ndarray New set of points. dx : float, optional Old sample-spacing, the default value is 1.0. x0 : int, optional Old origin, the default value is 0. Returns ------- res : ndarray Evaluated a quadratic spline points. See Also -------- qspline1d : Compute quadratic spline coefficients for rank-1 array. Notes ----- `dx` is the old sample-spacing while `x0` was the old origin. In other-words the old-sample points (knot-points) for which the `cj` represent spline coefficients were at equally-spaced points of:: oldx = x0 + j*dx j=0...N-1, with N=len(cj) Edges are handled using mirror-symmetric boundary conditions. Examples -------- We can filter a signal to reduce and smooth out high-frequency noise with a quadratic spline: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.signal import qspline1d, qspline1d_eval >>> rng = np.random.default_rng() >>> sig = np.repeat([0., 1., 0.], 100) >>> sig += rng.standard_normal(len(sig))*0.05 # add noise >>> time = np.linspace(0, len(sig)) >>> filtered = qspline1d_eval(qspline1d(sig), time) >>> plt.plot(sig, label="signal") >>> plt.plot(time, filtered, label="filtered") >>> plt.legend() >>> plt.show() rrr;rer_) rrrrtr#r r-rrKrrrr2r2r3r#bs*4     N)r$)rI)rI)r%r)r%r)/Znumpyrrrrrrrr r r r Znumpy.core.umathr rrrrrrrrZ_splinerrZ scipy.specialrZscipy._lib._utilr__all__rrRrXrrrrrlrprsrzrrr r!r"r#r2r2r2r3s64,    8D3555 2 3 J