a jz?@sddlmZmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZddlmZmZddlmZmZmZddlmZgdZd'dd ZiZd d Zd d ZddZ ddZ!ddZ"ddZ#ddZ$ddZ%ddZ&d(ddZ'd)dd Z(d*d"d#Z)d+d$d%Z*d&S),)asarraypi zeros_likearrayarctan2tanonesarangefloorr_ atleast_1dsqrtexpgreatercosaddsin) cspline2dsepfir2d)lfiltersosfiltlfiltic)BSpline) spline_filter gauss_spline cspline1d qspline1dcspline1d_evalqspline1d_eval@c Cs|jj}tgddd}|dvrp|d}t|j|}t|j|}t|||}t|||}|d||}n2|dvrt||}t|||}||}ntd|S) a4Smoothing 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 filtered input data Examples -------- We can filter an multi dimensional 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@r!f@)FDr$y?)r"dzInvalid data type for Iin) dtypecharrastyperrealimagr TypeError) ZIinZlmbdaZintypeZhcolZckrZckiZoutrZoutioutr.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 >>> knots = np.array([-1.0, 0.0, -1.0]) >>> gauss_spline(knots, 3) array([0.15418033, 0.6909883, 0.15418033]) # may vary rg(@)rr rr)xnZsignsqr.r.r/rJs- rcCs>t|td}tjgddd}||}d||dk|dkB<|S)Nr')rrr0FZ extrapolaterr4r0)rfloatr basis_elementr1br-r.r.r/_cubic|s  r;cCsBtt|td}tjgddd}||}d||dk|dkB<|S)Nr3)gg??Fr6rr<r=)absrr7rr8r9r.r.r/ _quadratics r?cCsdd|d|tdd|}ttd|dt|}d|dt|d|}|td|d|tdd||}||fS)Nr`0)r r)ZlamxiZomegrhor.r.r/ _coeff_smooths $,rGcCs.|t|||t||dt|dS)Nrr5)rr)kcsrFomegar.r.r/_hcs"rKcCs||d||d||dd||td||d}d||d||t|}t|}|||t|||t||S)Nrr0)rrr>r)rHrIrFrJZc0gammaZakr.r.r/_hss " rNcCs"t|\}}dd|t|||}t|}t|}td||||dtt|d||||}td||||dtd||||dtt|d||||}t|tdd|t|||ft||f} | dd} t|dddd|t|||f} | dd} t | |dd| d\} } t||| f} tt ||||t |d||||ddd}tt |d|||t |d||||ddd}t|tdd|t|||ft||f} | dd} t | | ddd| d\} } t| ddd||f} | S)Nrr0rr4r5zi) rGrlenr rKrreducerr ZreshaperrN)signallambrFrJrIKrHZzi_2Zzi_1rPsosZyp_yr.r.r/_cubic_smooth_coeffsB . $   . rZc Cs$dtd}t|}|t|}|dkrZ|d|t||}||d|}t|Stdtd| ftt||}td}td| f}t ||||d\}} ||d||d} t| td| ft| }t | g}t |||ddd|d\}} t|ddd| f}|dS)Nr4rBrrrOr5r# r rRr rrSr rr rrr rTrPrVZpowersZyplusoutputstater:arXZout_lastr.r.r/ _cubic_coeffs"  $ r`c Cs(ddtd}t|}|t|}|dkr^|d|t||}||d|}t|Stdtd| ftt||}td}td| f}t ||||d\}} ||d||d} t| td| ft| }t | g}t |||ddd|d\}} t|ddd| f}|d S) NrQr0g@rrrOr4r5g @r[r\r.r.r/_quadratic_coeffs" $ racCs|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() rbN)rZr`rTrUr.r.r/rs, rcCs|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() rbz.Smoothing quadratic splines not supported yet.N) ValueErrorrarcr.r.r/rAs- rr!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() r3rrr0rL) rr7rr'sizerRrr r)intrangeclipr;cjZnewxZdxZx0resNZcond1Zcond2Zcond3resultZjloweriZthisjZindjr.r.r/rts*2     rcCst|||}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() rrr0r=rB) rrrerRrr r)rfrgrhr?rir.r.r/rs*4     rN)r )rb)rb)r!r)r!r)+numpyrrrrrrrr r r r r rrrrrZ_splinerrZ _signaltoolsrrrZscipy.interpolater__all__rZ_splinefunc_cacherr;r?rGrKrNrZr`rarrrrr.r.r.r/s&L  82*"" 2 3 J