9i~SSKrSSKJr SSKrSSKJr SSKJrJ r \R"S5r Sr Sr "SS 5r"S S \5r"S S \5r"SS\5rSrSSS\R&SSSS4Sjrg)N)warn)inv)optimizespatialchURS:wdURSU:wa[SUS35eg)NrzInput data must have shape (N, z).ndimshape ValueError)datadims S/var/www/html/land-doc-ocr/venv/lib/python3.13/site-packages/skimage/measure/fit.py_check_data_dimr s6 yyA~A#-:3%rBCC.c`URS:dURSS:a [S5eg)Nr rzInput data must be at least 2D.r )rs r_check_data_atleast_2Drs- yy1} 1 ):;;*rc\rSrSrSrSrg) BaseModelcSUlgNparams)selfs r__init__BaseModel.__init__s  rrN)__name__ __module__ __qualname____firstlineno__r__static_attributes__rrrrsrrcF\rSrSrSrSrS SjrS SjrS SjrS Sjr S r g) LineModelNDa Total least squares estimator for N-dimensional lines. In contrast to ordinary least squares line estimation, this estimator minimizes the orthogonal distances of points to the estimated line. Lines are defined by a point (origin) and a unit vector (direction) according to the following vector equation:: X = origin + lambda * direction Attributes ---------- params : tuple Line model parameters in the following order `origin`, `direction`. Examples -------- >>> x = np.linspace(1, 2, 25) >>> y = 1.5 * x + 3 >>> lm = LineModelND() >>> lm.estimate(np.stack([x, y], axis=-1)) True >>> tuple(np.round(lm.params, 5)) (array([1.5 , 5.25]), array([0.5547 , 0.83205])) >>> res = lm.residuals(np.stack([x, y], axis=-1)) >>> np.abs(np.round(res, 9)) array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]) >>> np.round(lm.predict_y(x[:5]), 3) array([4.5 , 4.562, 4.625, 4.688, 4.75 ]) >>> np.round(lm.predict_x(y[:5]), 3) array([1. , 1.042, 1.083, 1.125, 1.167]) cV[U5 URSS9nX- nURSS:Xa5USUS- n[RR U5nUS:waX4-nO;URSS:a'[RR USS9u pVUSnOgX#4Ulg)a5Estimate line model from data. This minimizes the sum of shortest (orthogonal) distances from the given data points to the estimated line. Parameters ---------- data : (N, dim) array N points in a space of dimensionality dim >= 2. Returns ------- success : bool True, if model estimation succeeds. raxisr rF) full_matricesT)rmeanr nplinalgnormsvdr)rrorigin directionr/_vs restimateLineModelND.estimate?s t$"} ::a=A Q$q')I99>>),Dqy! ZZ]Q iimmDm>GAq!I) rNc"[U5 Uc$URc [S5eURn[U5S:wa [S5eUup4X- X- U-S[R 4U-- n[R RUSS9$)aDetermine residuals of data to model. For each point, the shortest (orthogonal) distance to the line is returned. It is obtained by projecting the data onto the line. Parameters ---------- data : (N, dim) array N points in a space of dimension dim. params : (2,) array, optional Optional custom parameter set in the form (`origin`, `direction`). Returns ------- residuals : (N,) array Residual for each data point. Parameters cannot be Noner !Parameters are defined by 2 sets..rr))rrr lenr-newaxisr.r/)rrrr1r2ress r residualsLineModelND.residualsds$ t$ >{{" !<==[[F v;! @A A"}$-9!< O! !yy~~c~**rcUc$URc [S5eURn[U5S:wa [S5eUupEXRS:Xa[SU35eXU- XR- nXFS[R4U--nU$)aPredict intersection of the estimated line model with a hyperplane orthogonal to a given axis. Parameters ---------- x : (n, 1) array Coordinates along an axis. axis : int Axis orthogonal to the hyperplane intersecting the line. params : (2,) array, optional Optional custom parameter set in the form (`origin`, `direction`). Returns ------- data : (n, m) array Predicted coordinates. Raises ------ ValueError If the line is parallel to the given axis. r8r r9rzLine parallel to axis .)rr r:r-r;)rxr*rr1r2lrs rpredictLineModelND.predicts. >{{" !<==[[F v;! @A A" ?a 5dV<= =   0#rzz/*Y66 rc6URUSUS9SS2S4nU$)a;Predict x-coordinates for 2D lines using the estimated model. Alias for:: predict(y, axis=1)[:, 0] Parameters ---------- y : array y-coordinates. params : (2,) array, optional Optional custom parameter set in the form (`origin`, `direction`). Returns ------- x : array Predicted x-coordinates. rr*rNrrB)ryrr@s r predict_xLineModelND.predict_x&( LL6L 21a4 8rc6URUSUS9SS2S4nU$)a;Predict y-coordinates for 2D lines using the estimated model. Alias for:: predict(x, axis=0)[:, 1] Parameters ---------- x : array x-coordinates. params : (2,) array, optional Optional custom parameter set in the form (`origin`, `direction`). Returns ------- y : array Predicted y-coordinates. rrENrrF)rr@rrGs r predict_yLineModelND.predict_yrJrrr)rN) rr r!r"__doc__r5r=rBrHrLr#r$rrr&r&s$!F#J+@&P.rr&c.\rSrSrSrSrSrSSjrSrg) CircleModelaTotal least squares estimator for 2D circles. The functional model of the circle is:: r**2 = (x - xc)**2 + (y - yc)**2 This estimator minimizes the squared distances from all points to the circle:: min{ sum((r - sqrt((x_i - xc)**2 + (y_i - yc)**2))**2) } A minimum number of 3 points is required to solve for the parameters. Attributes ---------- params : tuple Circle model parameters in the following order `xc`, `yc`, `r`. Notes ----- The estimation is carried out using a 2D version of the spherical estimation given in [1]_. References ---------- .. [1] Jekel, Charles F. Obtaining non-linear orthotropic material models for pvc-coated polyester via inverse bubble inflation. Thesis (MEng), Stellenbosch University, 2016. Appendix A, pp. 83-87. https://hdl.handle.net/10019.1/98627 Examples -------- >>> t = np.linspace(0, 2 * np.pi, 25) >>> xy = CircleModel().predict_xy(t, params=(2, 3, 4)) >>> model = CircleModel() >>> model.estimate(xy) True >>> tuple(np.round(model.params, 5)) (2.0, 3.0, 4.0) >>> res = model.residuals(xy) >>> np.abs(np.round(res, 9)) array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]) c([USS9 [R"UR[R5nUR USS9nUR SS9nX- nUR5nU[R"U5R:a[S[SS9 gX-n[R"US-[R"URSS 4US 9S S9n[R"US-S S9n[R R#XVS S 9upxpU S :wa [S5 gUSSn [$R&"X5n [R("[R "U S-55n X-n X-n X- n [+U 5U 4-Ulg)zEstimate circle model from data using total least squares. Parameters ---------- data : (N, 2) array N points with ``(x, y)`` coordinates, respectively. Returns ------- success : bool True, if model estimation succeeds. r rFcopyrr)zUStandard deviation of data is too small to estimate circle with meaningful precision.category stacklevelrdtypeN)rcondz6Input does not contain enough significant data points.T)rr- promote_typesrZfloat32astyper,stdfinfotinyrRuntimeWarningappendonesr sumr.lstsqrminkowski_distancesqrttupler) rr float_typer1scaleAfCr3rankcenter distancesrs rr5CircleModel.estimate sk !$%%djj"**= {{:E{2"}  288J',, , 4'     IIdQhA(:* MTU V FF47 # D9 d 19 I J1Q..v< GGBGGIqL) *  Fmqd* rc[USS9 URup#nUSS2S4nUSS2S4nU[R"XR- S-Xc- S--5- $)aDetermine residuals of data to model. For each point the shortest distance to the circle is returned. Parameters ---------- data : (N, 2) array N points with ``(x, y)`` coordinates, respectively. Returns ------- residuals : (N,) array Residual for each data point. r rSNrr)rrr-ri)rrxcycrsr@rGs rr=CircleModel.residualsCsZ" !$KK  AJ AJ277AFq=AFq=8999rNcUc URnUup4nX5[R"U5--nXE[R"U5--n[R"USUS4UR S9$)aaPredict x- and y-coordinates using the estimated model. Parameters ---------- t : array Angles in circle in radians. Angles start to count from positive x-axis to positive y-axis in a right-handed system. params : (3,) array, optional Optional custom parameter set. Returns ------- xy : (..., 2) array Predicted x- and y-coordinates. .Nr))rr-cossin concatenater )rtrrvrwrsr@rGs r predict_xyCircleModel.predict_xy]se" >[[F  RVVAY  RVVAY ~~q|Qy\:HHrrr rr r!r"rNr5r=rr#r$rrrPrPs+Z8t:4IrrPc.\rSrSrSrSrSrSSjrSrg) EllipseModelixaTotal least squares estimator for 2D ellipses. The functional model of the ellipse is:: xt = xc + a*cos(theta)*cos(t) - b*sin(theta)*sin(t) yt = yc + a*sin(theta)*cos(t) + b*cos(theta)*sin(t) d = sqrt((x - xt)**2 + (y - yt)**2) where ``(xt, yt)`` is the closest point on the ellipse to ``(x, y)``. Thus d is the shortest distance from the point to the ellipse. The estimator is based on a least squares minimization. The optimal solution is computed directly, no iterations are required. This leads to a simple, stable and robust fitting method. The ``params`` attribute contains the parameters in the following order:: xc, yc, a, b, theta Attributes ---------- params : tuple Ellipse model parameters in the following order `xc`, `yc`, `a`, `b`, `theta`. Examples -------- >>> xy = EllipseModel().predict_xy(np.linspace(0, 2 * np.pi, 25), ... params=(10, 15, 8, 4, np.deg2rad(30))) >>> ellipse = EllipseModel() >>> ellipse.estimate(xy) True >>> np.round(ellipse.params, 2) array([10. , 15. , 8. , 4. , 0.52]) >>> np.round(abs(ellipse.residuals(xy)), 5) array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]) c[USS9 [U5S:a[S[SS9 g[R "UR [R5nURUSS9nURSS 9nX- nUR5nU[R"U5R:a[S [SS9 gX-nUS S 2S4nUS S 2S 4n[R"US-XV-US-/5Rn[R"XV[R"U5/5RnURU-n URU-n URU-n [R "/S Q/SQ/SQ/5n [#U 5X[#U 5-U R-- -n [R$R)U 5upS[R*"USS S 24USS S 245-[R,"US S S 24S5- nUS S 2US:4nSUR.;d[UR155S:wagUR15unnn[#U 5*U R-U-nUR15unnnUS-nUS-nUS-nUU-UU-- US-UU-- - nUU-UU-- US-UU-- - nUUS--UUS---UUS---SU-U-U-- UU-U-- n[R2"UU- S-SUS---5nUS-UU-- UUU-- -nUS-UU-- U*UU-- -n[R2"SU-U- 5n[R2"SU-U- 5n S[R4"SU-UU- - 5-n!UU:aU!S[R6-- n!UU :aU Un nU![R6S- - n!U![R6-n![R8"UUUU U!/5R:n"U"S S===U-sss&U"S S===U- sss&[=SU"55Ulg![R$R&a gf=f)aEstimate ellipse model from data using total least squares. Parameters ---------- data : (N, 2) array N points with ``(x, y)`` coordinates, respectively. Returns ------- success : bool True, if model estimation succeeds. References ---------- .. [1] Halir, R.; Flusser, J. "Numerically stable direct least squares fitting of ellipses". In Proc. 6th International Conference in Central Europe on Computer Graphics and Visualization. WSCG (Vol. 98, pp. 125-132). r rSz3Need at least 5 data points to estimate an ellipse.rVFrTrr)zVStandard deviation of data is too small to estimate ellipse with meaningful precision.Nr)r@)rgr)rrrr\rg?c38# UHn[U5v M g7fr)float).0ps r (EllipseModel.estimate..$s5fE!HHfsT) rr:rrcr-r]rZr^r_r,r`rarbvstackT ones_likearrayrr. LinAlgErroreigmultiplypowerr ravelriarctanpi nan_to_numrealrjr)#rrrkr1rlr@rGD1D2S1S2S3C1Meig_valseig_vecsconda1abca2drngx0y0 numeratorterm denominator1 denominator2widthheightphirs# rr5EllipseModel.estimatesf0 !$ t9q= E'  %%djj"**= {{:E{2"}  288J',, , 5'     AJ AJYY1aeQT* + - - YYbll1o. / 1 1TTBY TTBY TTBYXX(8/J K B2SW rtt 334A  YY]]1-2;;x1~x1~>> QTNAB  a$(m $ =C Oq0((*1a"gX_r !((*1a S S S!ea!em3Q /!ea!em3Q /1Hq1a4x'!ad(2QUQY]BQUQYN wwA!|a!Q$h./1q1u Q8 1q1u $!a%9 I 45Y56BIIsQw1q5122 q5 3; C 6>"E6E 25519 C ruu Bvs;<AAr e r f 5f55 {yy$$  s)QQ0/Q0c*^^^^^^[USS9 URummmmn[R"U5m[R"U5mUSS2S4nUSS2S4nUR SnUUUUUU4Sjn[ R"U4[ RS9n[ R"UT- UT- 5U- n[U5HEn X9n XIn [R"XhU X4S9up[ R"U"XU 55Xy'MG U$) aDetermine residuals of data to model. For each point the shortest distance to the ellipse is returned. Parameters ---------- data : (N, 2) array N points with ``(x, y)`` coordinates, respectively. Returns ------- residuals : (N,) array Residual for each data point. r rSNrrc>[R"[R"U55n[R"[R"U55nT TT -U--TT -U-- nT TT -U--TT -U--nX- S-X&- S--$)Nr )mathr{r-squeezer|) r~xiyictstxtytrrcthetasthetarvrws rfun#EllipseModel.residuals..funEs"**Q-(B"**Q-(Ba&j2o%F R7Ba&j2o%F R7BG>RWN2 2rrY)args)rrrr{r|r r-emptyfloat64arctan2rangerleastsqri)rrthetar@rGNrr=t0irrr~r3rrrrrvrws @@@@@@rr=EllipseModel.residuals(s" !$"kkB1e%% AJ AJ JJqM 3 3$HHaT4 ZZBB '% /qABB##CAbX>DA773qb>2IL rNcrUc URnUup4pVn[R"U5n[R"U5n [R"U5n [R"U5n X5U -U--Xk-U -- n XEU -U--Xj-U --n [R "U SU S4UR S9$)aaPredict x- and y-coordinates using the estimated model. Parameters ---------- t : array Angles in circle in radians. Angles start to count from positive x-axis to positive y-axis in a right-handed system. params : (5,) array, optional Optional custom parameter set. Returns ------- xy : (..., 2) array Predicted x- and y-coordinates. rzr))rr-r{r|rr}r )rr~rrvrwrrrrrrrr@rGs rrEllipseModel.predict_xyfs$ >[[F$e VVAY VVAY%% Vb 1:? 2 Vb 1:? 2~~q|Qy\:HHrrrrr$rrrrxs&PEN<|IrrcbUS:XagUS:Xa[R$X- nSU- nSXB-- n[R"U[S[- S9n[R"U[S[- S9n[R"[R "U5[R "U5- 5$)aDetermine number trials such that at least one outlier-free subset is sampled for the given inlier/outlier ratio. Parameters ---------- n_inliers : int Number of inliers in the data. n_samples : int Total number of samples in the data. min_samples : int Minimum number of samples chosen randomly from original data. probability : float Probability (confidence) that one outlier-free sample is generated. Returns ------- trials : int Number of trials. rr)a_mina_max)r-infclip_EPSILONceillog) n_inliers n_samples min_samples probability inlier_rationomdenoms r_dynamic_max_trialsrs(aA~vv (L k/C ) )E ''#XQ\ :C GGEX >E 77266#;. //rdc Sn [Rn /nUSLnUSLn[RRU 5n [ U[ [ 45(dU4n[US5nSUs=:aU::dO [SUS35eUS:a [S5eUS:a [S5eSU s=::aS::d O [S5eU b*[U 5U:wa[S [U 5S US 35eU bU OU RUUS S 9nU"5nSnUU:GaUS- nUVs/sHnUUPM nnU RUUS S 9nU(a U"U6(dMEUR"U6nUb U(dM`U(aU"U/UQ76(dMw[R"UR"U65nUU:nURU5n[R"U5nUU :d UU :Xa.UU :a(Un Un Un[U[!U UX)55nX:dX::aO UU:aGM[#U5(aHUVs/sHnUUPM nnUR"U6 U(aU"U/UQ76(d [%S5 UU4$SnSn[%S5 UU4$s snfs snf)aFit a model to data with the RANSAC (random sample consensus) algorithm. RANSAC is an iterative algorithm for the robust estimation of parameters from a subset of inliers from the complete data set. Each iteration performs the following tasks: 1. Select `min_samples` random samples from the original data and check whether the set of data is valid (see `is_data_valid`). 2. Estimate a model to the random subset (`model_cls.estimate(*data[random_subset]`) and check whether the estimated model is valid (see `is_model_valid`). 3. Classify all data as inliers or outliers by calculating the residuals to the estimated model (`model_cls.residuals(*data)`) - all data samples with residuals smaller than the `residual_threshold` are considered as inliers. 4. Save estimated model as best model if number of inlier samples is maximal. In case the current estimated model has the same number of inliers, it is only considered as the best model if it has less sum of residuals. These steps are performed either a maximum number of times or until one of the special stop criteria are met. The final model is estimated using all inlier samples of the previously determined best model. Parameters ---------- data : [list, tuple of] (N, ...) array Data set to which the model is fitted, where N is the number of data points and the remaining dimension are depending on model requirements. If the model class requires multiple input data arrays (e.g. source and destination coordinates of ``skimage.transform.AffineTransform``), they can be optionally passed as tuple or list. Note, that in this case the functions ``estimate(*data)``, ``residuals(*data)``, ``is_model_valid(model, *random_data)`` and ``is_data_valid(*random_data)`` must all take each data array as separate arguments. model_class : object Object with the following object methods: * ``success = estimate(*data)`` * ``residuals(*data)`` where `success` indicates whether the model estimation succeeded (`True` or `None` for success, `False` for failure). min_samples : int in range (0, N) The minimum number of data points to fit a model to. residual_threshold : float larger than 0 Maximum distance for a data point to be classified as an inlier. is_data_valid : function, optional This function is called with the randomly selected data before the model is fitted to it: `is_data_valid(*random_data)`. is_model_valid : function, optional This function is called with the estimated model and the randomly selected data: `is_model_valid(model, *random_data)`, . max_trials : int, optional Maximum number of iterations for random sample selection. stop_sample_num : int, optional Stop iteration if at least this number of inliers are found. stop_residuals_sum : float, optional Stop iteration if sum of residuals is less than or equal to this threshold. stop_probability : float in range [0, 1], optional RANSAC iteration stops if at least one outlier-free set of the training data is sampled with ``probability >= stop_probability``, depending on the current best model's inlier ratio and the number of trials. This requires to generate at least N samples (trials): N >= log(1 - probability) / log(1 - e**m) where the probability (confidence) is typically set to a high value such as 0.99, e is the current fraction of inliers w.r.t. the total number of samples, and m is the min_samples value. rng : {`numpy.random.Generator`, int}, optional Pseudo-random number generator. By default, a PCG64 generator is used (see :func:`numpy.random.default_rng`). If `rng` is an int, it is used to seed the generator. initial_inliers : array-like of bool, shape (N,), optional Initial samples selection for model estimation Returns ------- model : object Best model with largest consensus set. inliers : (N,) array Boolean mask of inliers classified as ``True``. References ---------- .. [1] "RANSAC", Wikipedia, https://en.wikipedia.org/wiki/RANSAC Examples -------- Generate ellipse data without tilt and add noise: >>> t = np.linspace(0, 2 * np.pi, 50) >>> xc, yc = 20, 30 >>> a, b = 5, 10 >>> x = xc + a * np.cos(t) >>> y = yc + b * np.sin(t) >>> data = np.column_stack([x, y]) >>> rng = np.random.default_rng(203560) # do not copy this value >>> data += rng.normal(size=data.shape) Add some faulty data: >>> data[0] = (100, 100) >>> data[1] = (110, 120) >>> data[2] = (120, 130) >>> data[3] = (140, 130) Estimate ellipse model using all available data: >>> model = EllipseModel() >>> model.estimate(data) True >>> np.round(model.params) # doctest: +SKIP array([ 72., 75., 77., 14., 1.]) Estimate ellipse model using RANSAC: >>> ransac_model, inliers = ransac(data, EllipseModel, 20, 3, max_trials=50) >>> abs(np.round(ransac_model.params)) # doctest: +SKIP array([20., 30., 10., 6., 2.]) >>> inliers # doctest: +SKIP array([False, False, False, False, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True], dtype=bool) >>> sum(inliers) > 40 True RANSAC can be used to robustly estimate a geometric transformation. In this section, we also show how to use a proportion of the total samples, rather than an absolute number. >>> from skimage.transform import SimilarityTransform >>> rng = np.random.default_rng() >>> src = 100 * rng.random((50, 2)) >>> model0 = SimilarityTransform(scale=0.5, rotation=1, ... translation=(10, 20)) >>> dst = model0(src) >>> dst[0] = (10000, 10000) >>> dst[1] = (-100, 100) >>> dst[2] = (50, 50) >>> ratio = 0.5 # use half of the samples >>> min_samples = int(ratio * len(src)) >>> model, inliers = ransac( ... (src, dst), ... SimilarityTransform, ... min_samples, ... 10, ... initial_inliers=np.ones(len(src), dtype=bool), ... ) # doctest: +SKIP >>> inliers # doctest: +SKIP array([False, False, False, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]) rNz#`min_samples` must be in range (0, ]z.`residual_threshold` must be greater than zeroz&`max_trials` must be greater than zerorz*`stop_probability` must be in range [0, 1]z4RANSAC received a vector of initial inliers (length z+) that didn't match the number of samples (z). The vector of initial inliers should have the same length as the number of samples and contain only True (this sample is an initial inlier) and False (this one isn't) values.F)replacez8Estimated model is not valid. Try increasing max_trials.z"No inliers found. Model not fitted)r-rrandom default_rng isinstancerjlistr:r choicer5absr=dot count_nonzerominranyr)r model_classrresidual_threshold is_data_validis_model_valid max_trialsstop_sample_numstop_residuals_sumstop_probabilityrnginitial_inliersbest_inlier_numbest_inlier_residuals_sum best_inliersvalidate_model validate_data num_samplesspl_idxsmodel num_trialsrsamplessuccessr=inliers residuals_sum inliers_count data_inlierss rransacr sjO "L#4/N!-M ))   $C dUDM * *wd1g,K *{ *>{m1MNNAIJJA~ABB ! &Q &EFF"s?';{'JB?#$%#}%     &  ZZ [%Z @  MEJ z !a )--11X;-::k;:F !8 ..'*  w  ."A"A"A FF5??D12 00! i0 ((1  O +0!$==,O(5 %"L##[+J 2,Bi z !n <156A, 6  % ."F"F"F K L ,    12 , }.j7s J+J)rwarningsrnumpyr- numpy.linalgrscipyrrspacingrrrrr&rPrrrr r$rrrs # ::a=D <  })}@ZI)ZIzMI9MI` 0PFF fr