h)MddlmZmZmZmZmZmZmZmZm Z m Z m Z ddl m Z mZmZmZmZmZmZmZmZddlmZmZddlmZddlmZgdZdd ZiZd Z d Z!d Z"d Z#dZ$dZ%dZ&dZ'dZ(dZ)dZ*ddZ+ddZ,ddZ-ddZ.dS)) logical_andasarraypi zeros_like piecewisearrayarctan2tanzerosarangefloor) sqrtexpgreaterlesscosaddsin less_equal greater_equal) cspline2dsepfir2d)comb)float_factorial) spline_filterbspline gauss_splinecubic quadratic cspline1d qspline1dcspline1d_evalqspline1d_eval@c|jj}tgdddz }|dvr}|d}t |j|}t |j|}t|||}t|||}|d|zz|}nJ|dvr7t ||}t|||}||}ntd|S) a3Smoothing spline (cubic) filtering of a rank-2 array. Filter an input data set, `Iin`, using a (cubic) smoothing spline of fall-off `lmbda`. Parameters ---------- Iin : array_like input data set lmbda : float, optional spline smooghing fall-off value, default is `5.0`. Returns ------- res : ndarray filterd input data Examples -------- We can filter an multi dimentional signal (ex: 2D image) using cubic B-spline filter: >>> import numpy as np >>> from scipy.signal import spline_filter >>> import matplotlib.pyplot as plt >>> orig_img = np.eye(20) # create an image >>> orig_img[10, :] = 1.0 >>> sp_filter = spline_filter(orig_img, lmbda=0.1) >>> f, ax = plt.subplots(1, 2, sharex=True) >>> for ind, data in enumerate([[orig_img, "original image"], ... [sp_filter, "spline filter"]]): ... ax[ind].imshow(data[0], cmap='gray_r') ... ax[ind].set_title(data[1]) >>> plt.tight_layout() >>> plt.show() )?g@r'f@)FDr*y?)r(dzInvalid data type for Iin) dtypecharrastyperrealimagr TypeError) Iinlmbdaintypehcolckrckioutroutiouts R/var/www/html/Sam_Eipo/venv/lib/python3.11/site-packages/scipy/signal/_bsplines.pyrrsLY^F # & & ,D jjoo%((%((T4((T4((b4i''// :  U##sD$''jj  3444 Jc  tS#t$rYnwxYwd}dzdz}dzrd}nd}|dd|g}|td|dz D]*}||ddz dz +||dddz dz t fd fd t|D}||ft<||fS) aReturns the function defined over the left-side pieces for a bspline of a given order. The 0th piece is the first one less than 0. The last piece is a function identical to 0 (returned as the constant 0). (There are order//2 + 2 total pieces). Also returns the condition functions that when evaluated return boolean arrays for use with `numpy.piecewise`. c@|dkrfdS|dkrfdSfdS)Nrc\tt|t|SN)rrrxval1val2s r<z>_bspline_piecefunctions..condfuncgen..]s,[At)<)<)6q$)?)?AAr=c$t|SrA)r)rCrEs r<rFz>_bspline_piecefunctions..condfuncgen..`sZ400r=c\tt|t|SrA)rrrrBs r<rFz>_bspline_piecefunctions..condfuncgen..bs*[a)6q$)?)?AAr=)numrDrEs ``r< condfuncgenz,_bspline_piecefunctions..condfuncgen[sm !88AAAAA A AXX0000 0AAAAA Ar=rGggrr@cdz|z dkrdSfdtdzDfdtdzDfd}|S)NrGrc rg|]3}dd|dzzz ttdz|dzz 4S)rrG)exact)floatr).0kfvalorders r< zA_bspline_piecefunctions..piecefuncgen..}sZ***qAE{?eDAQ,G,G,G&H&HH4O***r=rcg|]} |z  SrJrJ)rRrSbounds r<rVzA_bspline_piecefunctions..piecefuncgen..s4445&1*444r=cjd}tdzD]}||||zzzz }|S)Nrrange)rCresrSMkcoeffsrUshiftss r<thefuncz>_bspline_piecefunctions..piecefuncgen..thefuncsHC26]] < <vayAq Me#;;;Jr=r[)rKrar^r_r`rXrTrUs @@@r< piecefuncgenz-_bspline_piecefunctions..piecefuncgenys aZ#  FF1***** a==***4444eBFmm444         r=c&g|] }|SrJrJ)rRrSrbs r<rVz+_bspline_piecefunctions..s!555A Q555r=)_splinefunc_cacheKeyErrorr\appendr) rUrLlast startbound condfuncsrKfunclistrXrTrbs ` @@@r<_bspline_piecefunctionsrkKss  ''      AAA A:>D qy  Q:../I EQq!!Quqy99:::  [[A|c'9::;;; 5 ! !D       6555t555H ()4e Y s   ctt| t|\}}fd|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 c&g|] }|SrJrJ)rRfuncaxs r<rVzbspline..s!///TR///r=)absrrkr)rCnrjricondlistros @r<rrsWX gajj// B1!44Hi////Y///H R8 , ,,r=ct|}|dzdz }dtdtz|zz t|dz dz |z zS)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(@rG)rrrr)rCrqsignsqs r<rrsT`  A!et^F tAFVO$$ $sAF7Q;+?'@'@ @@r=cXtt|}t|}t|d}|r||}dd|dzzd|z zz ||<|t|dz}|r||}dd|z dzz||<|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??rGgUUUUUU?rprrranyrCror]cond1ax1cond2ax2s r<rrsP WQZZB R..C QKKE yy{{>iw1QW==E FT"a[[ E yy{{.iCA~-E Jr=cFtt|}t|}t|d}|r||}d|dzz ||<|t|dz}|r||}|dz dzdz ||<|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 rvg?rG?rMrxrzs r<r r ,sP WQZZB R..C SMME yy{{%iC1H_E FT"c]] "E yy{{,iCiA%+E Jr=c Vdd|zz d|ztdd|zzzz}ttd|zdz t|}d|zdz t|z d|zz }|td|zd|ztdd|zzzz|z z}||fS)Nr`rw0)rr )lamxiomegrhos r< _coeff_smoothras R#XS4C#I #6#66 6B 4c A &&R 1 1D 8a<$r(( "rCx 0C b3hcDS3Y,?,?!??2EFF FC 9r=c|t|z ||zzt||dzzzt|dzS)Nr)rr)rScsromegas r<_hcrisB UOsax (3uA+?+? ? ArNN r=c<||zd||zzzd||zz z dd|z|ztd|zzz |dzzz }d||zz d||zzz t|z }t|}|||zzt||z|t||zzzzS)NrrG)rr rpr)rSrrrc0gammaaks r<_hsrns r'Qs] #q39} 5 q3w}s1u9~~- -q 8 :B s]q39} -E :E QB r >S__us52:/FF GGr=c t|\}}dd|zt|zz ||zz}t|}t|f|jj}t |}td||||dztj t|dz||||zz|d<td||||dztd||||dzztj t|dz||||zz|d<td|D]D}|||zd|zt|z||dz zz||z||dz zz ||<Et|f|jj} tj t||||t|dz|||z|dddz| |dz <tj t|dz |||t|dz|||z|dddz| |dz <t|dz ddD]D}|||zd|zt|z| |dzzz||z| |dzzz | |<E| S)NrrGrrrw) rrlenr r-r.r rrreducer\r) signallambrrrKyprSrqys r<_cubic_smooth_coeffrvst$$JC QWs5zz ! !C#I -B F A tV\& ' 'Bq A BU # #fQi / ZAE2sE22V; < <=BqEBU # #fQi / BU # #fQi /0 ZAE2sE22V; < <=BqE1a[[((fQi!c'CJJ"6AE"BBsRAY&'1 qdFL%&&Az3q"c511q1ub#u5569?"FGGAa!eHz3q1ub#u55q1ub#u5569?"FGGAa!eH1q5"b ! !&&RU QWs5zz1Aa!eH<<c Aa!eH$%! Hr=cdtdz}t|}t|f|jj}|t |z}|d|t j||zzz|d<td|D]}|||||dz zz||<t|f|j}||dz z ||dz z||dz <t|dz ddD]}|||dz||z z||<|dzS)NrwrrrGrr) rrr r-r.r rrr\rzirypluspowersrSoutputs r< _cubic_coeffrs d1ggB F A 1$ ) * *E 6!99_Fay2 6F? ; ;;;E!H 1a[[11!9rE!a%L00a A4 & &F"q&ME!a%L0F1q5M 1q5"b ! !44&Q-%(23q C<r=c ddtdzz}t|}t|f|jj}|t |z}|d|t j||zzz|d<td|D]}|||||dz zz||<t|f|jj}||dz z ||dz z||dz <t|dz ddD]}|||dz||z z||<|dzS)NrGrMrrrg @rrs r<_quadratic_coeffrs% a$s))m B F A 1$ ) * *E 6!99_Fay2 6F? ; ;;;E!H 1a[[11!9rE!a%L00a A4* + +F"q&ME!a%L0F1q5M 1q5"b ! !44&Q-%(23q C<r=rZcL|dkrt||St|S)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() rZ)rrrrs r<r!r!s,X s{{"64000F###r=cJ|dkrtdt|S)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() rZz.Smoothing quadratic splines not supported yet.) ValueErrorrrs r<r"r"s+Z s{{IJJJ'''r=r'ct||z t|z }t||j}|jdkr|St |}|dk}||dz k}||z}t ||| ||<t |d|dz z||z ||<||}|jdkr|St||j} t|dz tdz} tdD]>} | | z} | d|dz } | || t|| z zz } ?| ||<|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-rrrGr) rrQrr-sizerr#r r/intr\cliprcjnewxdxx0r]Nr{r}cond3resultjlowerithisjindjs r<r#r#sad DMMB %)) +D T * * *C x1}}  BA 1HE AENEem ET%[L11CJAQK$u+$=>>CJ ;D yA~~ BH - - -F 4!8__ # #C ( (1 ,F 1XX11 zz!QU##"T(U4%<0000CJ Jr=cpt||z |z }t|}|jdkr|St|}|dk}||dz k}||z}t ||| ||<t |d|dz z||z ||<||}|jdkr|St|} t |dz tdz} tdD]>} | | z} | d|dz } | || t|| z zz } ?| ||<|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() rrrGrrw) rrrrr$r r/rr\rr rs r<r$r$bsOh DMMB " $D T  C x1}}  BA 1HE AENEem ET%[L11CJAQK$u+$=>>CJ ;D yA~~   F 4#:   % %c * *Q .F 1XX55 zz!QU##"T(Yte|4444CJ Jr=N)r%)rZ)r'r)/numpyrrrrrrr r r r r numpy.core.umathrrrrrrrrr_splinerr scipy.specialrscipy._lib._utilr__all__rrdrkrrrr rrrrrrr!r"r#r$rJr=r<rsIIIIIIIIIIIIIIIIIIIIIIIIII9999999999999999999999)(((((((,,,,,, I I I5555pAAAH0-0-0-f2A2A2Aj222j222j HHH   >      /$/$/$/$d0(0(0(0(fGGGGTIIIIIIr=