9iT TSSKrSSKrSSKJrJr SSKrSSKJr SSK J r SSK J r Sr SrS r"S S \5r"S S \5r"SS\5r"SS\5r"SS\5r"SS\5rS Sjr"SS\5r"SS\5r"SS\5r\\\\\\\\S.rSrSrg)!N)ABCabstractmethod)spatial) safe_as_int)NP_COPY_IF_NEEDEDcFURnS[R"SSU--5-S- S- n[[R"U55nX#:wa[ SU35e[R "US-5n[R"XUS-45USS2SS24'U$)z8Affine matrix from linearized (d, d + 1) matrix entries.rz2Invalid number of elements for linearized matrix: N)sizenpsqrtintround ValueErroreyereshape)vnparamddimensionalitymatrixs \/var/www/html/land-doc-ocr/venv/lib/python3.13/site-packages/skimage/transform/_geometric.py_affine_matrix_from_vectorr s VVF RWWQV^ $ $)A-A!%N CF8 L  VVNQ& 'FZZNQ4F#GHF3B36N McURup[R"USS9nX- n[R"[R"US-5U- 5nUS:XaR[R "US-US-4[R 5[R"U[R 54$[R"U5U- nU[R"[R"U5USS2[R4*4SS9-n[R"US/U-S/-/4SS9n[R"UR[R"U5/5n X-Rn U SS2SU24n XSS2US24-n X4$)aCenter and normalize image points. The points are transformed in a two-step procedure that is expressed as a transformation matrix. The matrix of the resulting points is usually better conditioned than the matrix of the original points. Center the image points, such that the new coordinate system has its origin at the centroid of the image points. Normalize the image points, such that the mean distance from the points to the origin of the coordinate system is sqrt(D). If the points are all identical, the returned values will contain nan. Parameters ---------- points : (N, D) array The coordinates of the image points. Returns ------- matrix : (D+1, D+1) array_like The transformation matrix to obtain the new points. new_points : (N, D) array The transformed image points. References ---------- .. [1] Hartley, Richard I. "In defense of the eight-point algorithm." Pattern Analysis and Machine Intelligence, IEEE Transactions on 19.6 (1997): 580-593. raxisrr N)shapermeanrsumfullnan full_like concatenaternewaxisvstackTones) pointsnrcentroidcenteredrms norm_factor part_matrixrpoints_h new_points_h new_pointss r_center_and_normalize_pointsr5snD <>rrVN) __name__ __module__ __qualname____firstlineno____doc__rrZpropertyr`rd__static_attributes__rVrrrSrSs:<    BB?rrScR\rSrSrSrS SS.SjjrSr\S5rS r S r S r S r g)FundamentalMatrixTransforma7 Fundamental matrix transformation. The fundamental matrix relates corresponding points between a pair of uncalibrated images. The matrix transforms homogeneous image points in one image to epipolar lines in the other image. The fundamental matrix is only defined for a pair of moving images. In the case of pure rotation or planar scenes, the homography describes the geometric relation between two images (`ProjectiveTransform`). If the intrinsic calibration of the images is known, the essential matrix describes the metric relation between the two images (`EssentialMatrixTransform`). References ---------- .. [1] Hartley, Richard, and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge university press, 2003. Parameters ---------- matrix : (3, 3) array_like, optional Fundamental matrix. Attributes ---------- params : (3, 3) array Fundamental matrix. Examples -------- >>> import numpy as np >>> import skimage as ski >>> tform_matrix = ski.transform.FundamentalMatrixTransform() Define source and destination points: >>> src = np.array([1.839035, 1.924743, ... 0.543582, 0.375221, ... 0.473240, 0.142522, ... 0.964910, 0.598376, ... 0.102388, 0.140092, ... 15.994343, 9.622164, ... 0.285901, 0.430055, ... 0.091150, 0.254594]).reshape(-1, 2) >>> dst = np.array([1.002114, 1.129644, ... 1.521742, 1.846002, ... 1.084332, 0.275134, ... 0.293328, 0.588992, ... 0.839509, 0.087290, ... 1.779735, 1.116857, ... 0.878616, 0.602447, ... 0.642616, 1.028681]).reshape(-1, 2) Estimate the transformation matrix: >>> tform_matrix.estimate(src, dst) True >>> tform_matrix.params array([[-0.21785884, 0.41928191, -0.03430748], [-0.07179414, 0.04516432, 0.02160726], [ 0.24806211, -0.42947814, 0.02210191]]) Compute the Sampson distance: >>> tform_matrix.residuals(src, dst) array([0.0053886 , 0.00526101, 0.08689701, 0.01850534, 0.09418259, 0.00185967, 0.06160489, 0.02655136]) Apply inverse transformation: >>> tform_matrix.inverse(dst) array([[-0.0513591 , 0.04170974, 0.01213043], [-0.21599496, 0.29193419, 0.00978184], [-0.0079222 , 0.03758889, -0.00915389], [ 0.14187184, -0.27988959, 0.02476507], [ 0.05890075, -0.07354481, -0.00481342], [-0.21985267, 0.36717464, -0.01482408], [ 0.01339569, -0.03388123, 0.00497605], [ 0.03420927, -0.1135812 , 0.02228236]]) NrrcUc[R"US-5nOK[R"U5nURSS- nURUS-US-4:wa [ S5eXlUS:wa[ URS35eg)Nr r&Invalid shape of transformation matrixrzJ is only implemented for 2D coordinates (i.e. 3D transformation matrices).)rrr9r rparamsNotImplementedError __class__rXrrs r__init__#FundamentalMatrixTransform.__init__6s >VVNQ./FZZ'F#\\!_q0N|| 2NQ4FGG !IJJ Q %>>"#55  rc[R"U5n[R"U[R"URS5/5nX R R -$)zApply forward transformation. Parameters ---------- coords : (N, 2) array_like Source coordinates. Returns ------- coords : (N, 3) array Epipolar lines in the destination image. r)rr9 column_stackr*r rsr))rXrYcoords_homogeneouss rrZ#FundamentalMatrixTransform.__call__FsHF#__fbggfll1o6N-OP!KKMM11rcH[U5"URRS9$)zReturn a transform object representing the inverse. See Hartley & Zisserman, Ch. 8: Epipolar Geometry and the Fundamental Matrix, for an explanation of why F.T gives the inverse. r)typersr)r_s rr`"FundamentalMatrixTransform.inverseXsDz//rc[R"U5n[R"U5nURUR:wa [S5eURSS:a [S5e[ U5up1[ U5upB[R"URSS45nXSS2SS 24'USS2SS24==USS2S[R4-ss'XSS2SS 24'USS2SS 24==USS2S [R4-ss'XSS2S S24'[RRU5u pgUS SS24RSS5nXU4$![ aV [R "S[R5UlS[R "S[R5/-s$f=f)aTSetup and solve the homogeneous epipolar constraint matrix:: dst' * F * src = 0. Parameters ---------- src : (N, 2) array_like Source coordinates. dst : (N, 2) array_like Destination coordinates. Returns ------- F_normalized : (3, 3) array The normalized solution to the homogeneous system. If the system is not well-conditioned, this matrix contains NaNs. src_matrix : (3, 3) array The transformation matrix to obtain the normalized source coordinates. dst_matrix : (3, 3) array The transformation matrix to obtain the normalized destination coordinates. z%src and dst shapes must be identical.rz,src.shape[0] must be equal or larger than 8.rr Nrr r )rr9r rr5ZeroDivisionErrorr#r$rsr*r'r;r=r) rXrArB src_matrix dst_matrixrJ_rM F_normalizeds r_setup_constraint_matrix3FundamentalMatrixTransform._setup_constraint_matrixas2jjojjo 99 !DE E 99QF?cURX5up4n[RRU5upgnSUS'U[R"U5-U-n UR U -U-Ulg)aEstimate fundamental matrix using 8-point algorithm. The 8-point algorithm requires at least 8 corresponding point pairs for a well-conditioned solution, otherwise the over-determined solution is estimated. Parameters ---------- src : (N, 2) array_like Source coordinates. dst : (N, 2) array_like Destination coordinates. Returns ------- success : bool True, if model estimation succeeds. rrTrrr;r=r?r)rs) rXrArBrrrrKrLrMFs restimate#FundamentalMatrixTransform.estimatesh*04/L/LS/V, *))-- -a!  NQ  llQ&3 rc<[R"U[R"URS5/5n[R"U[R"URS5/5nURUR -nURR UR -n[R "XER -SS9n[R"U5[R"USS-USS--USS--USS--5- $)aCompute the Sampson distance. The Sampson distance is the first approximation to the geometric error. Parameters ---------- src : (N, 2) array Source coordinates. dst : (N, 2) array Destination coordinates. Returns ------- residuals : (N,) array Sampson distance. rr rr) rrzr*r rsr)r"absr)rXrArBsrc_homogeneousdst_homogeneousF_srcFt_dst dst_F_srcs rrd$FundamentalMatrixTransform.residualss$//3 ! 0E*FG//3 ! 0E*FG o///!2!22FF?WW41= vvi 277 !HME!HM )F1IN :VAY!^ K$   rrsN) rfrgrhrirjrwrZrkr`rrrdrlrVrrrnrns?Oba 2$0044lB rrnc@^\rSrSrSrSSS.U4SjjjrSrSrU=r$) EssentialMatrixTransformia Essential matrix transformation. The essential matrix relates corresponding points between a pair of calibrated images. The matrix transforms normalized, homogeneous image points in one image to epipolar lines in the other image. The essential matrix is only defined for a pair of moving images capturing a non-planar scene. In the case of pure rotation or planar scenes, the homography describes the geometric relation between two images (`ProjectiveTransform`). If the intrinsic calibration of the images is unknown, the fundamental matrix describes the projective relation between the two images (`FundamentalMatrixTransform`). References ---------- .. [1] Hartley, Richard, and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge university press, 2003. Parameters ---------- rotation : (3, 3) array_like, optional Rotation matrix of the relative camera motion. translation : (3, 1) array_like, optional Translation vector of the relative camera motion. The vector must have unit length. matrix : (3, 3) array_like, optional Essential matrix. Attributes ---------- params : (3, 3) array Essential matrix. Examples -------- >>> import numpy as np >>> import skimage as ski >>> >>> tform_matrix = ski.transform.EssentialMatrixTransform( ... rotation=np.eye(3), translation=np.array([0, 0, 1]) ... ) >>> tform_matrix.params array([[ 0., -1., 0.], [ 1., 0., 0.], [ 0., 0., 0.]]) >>> src = np.array([[ 1.839035, 1.924743], ... [ 0.543582, 0.375221], ... [ 0.47324 , 0.142522], ... [ 0.96491 , 0.598376], ... [ 0.102388, 0.140092], ... [15.994343, 9.622164], ... [ 0.285901, 0.430055], ... [ 0.09115 , 0.254594]]) >>> dst = np.array([[1.002114, 1.129644], ... [1.521742, 1.846002], ... [1.084332, 0.275134], ... [0.293328, 0.588992], ... [0.839509, 0.08729 ], ... [1.779735, 1.116857], ... [0.878616, 0.602447], ... [0.642616, 1.028681]]) >>> tform_matrix.estimate(src, dst) True >>> tform_matrix.residuals(src, dst) array([0.42455187, 0.01460448, 0.13847034, 0.12140951, 0.27759346, 0.32453118, 0.00210776, 0.26512283]) rrpc 8>[TU]X4S9 UGb2[R"U5nUc [ S5e[R"U5nUR S:wa [ S5e[ [RRU5S- 5S:a [ S5eURS:wa [ S 5e[ [RRU5S- 5S:a [ S 5e[R"S US *USUS S US *US*US S / 5RSS5nXQ-Ul gUb8[R"U5nUR S:wa [ S 5eX0l g[R"S5Ul g)N)rrz&Both rotation and translation requiredrz Invalid shape of rotation matrixr gư>z*Rotation matrix must have unit determinantrz#Invalid shape of translation vectorz(Translation vector must have unit lengthrrrr)superrwrr9rr rr;r<r normarrayrrsr)rXrotation translationrrt_xrus rrw!EssentialMatrixTransform.__init__sz F  zz(+H" !IJJ**[1K~~' !CDD299==*Q./$6 !MNN1$ !FGG299>>+.23d: !KLL(( ^ONN ^O ^ON  gam .DK  ZZ'F||v% !IJJ K&&)DKrcURX5up4n[RRU5upgnUSUS-S- US'USUS'SUS'U[R"U5-U-n UR U -U-Ulg)aEstimate essential matrix using 8-point algorithm. The 8-point algorithm requires at least 8 corresponding point pairs for a well-conditioned solution, otherwise the over-determined solution is estimated. Parameters ---------- src : (N, 2) array_like Source coordinates. dst : (N, 2) array_like Destination coordinates. Returns ------- success : bool True, if model estimation succeeds. rr g@rTr) rXrArB E_normalizedrrrKrLrMEs rr!EssentialMatrixTransform.estimateFs*04/L/LS/V, *))-- -a!qt s"!t!!  NQ  llQ&3 rrNNN) rfrgrhrirjrwrrl __classcell__)rus@rrrs,CL7;'$NO'$'$R!!rrc\rSrSrSrSSS.Sjjr\S5rSrSS jr S r \S 5r SS jr S r SrSrSr\S5rSrg)ProjectiveTransformijaiProjective transformation. Apply a projective transformation (homography) on coordinates. For each homogeneous coordinate :math:`\mathbf{x} = [x, y, 1]^T`, its target position is calculated by multiplying with the given matrix, :math:`H`, to give :math:`H \mathbf{x}`:: [[a0 a1 a2] [b0 b1 b2] [c0 c1 1 ]]. E.g., to rotate by theta degrees clockwise, the matrix should be:: [[cos(theta) -sin(theta) 0] [sin(theta) cos(theta) 0] [0 0 1]] or, to translate x by 10 and y by 20:: [[1 0 10] [0 1 20] [0 0 1 ]]. Parameters ---------- matrix : (D+1, D+1) array_like, optional Homogeneous transformation matrix. dimensionality : int, optional The number of dimensions of the transform. This is ignored if ``matrix`` is not None. Attributes ---------- params : (D+1, D+1) array Homogeneous transformation matrix. NrrpcUc[R"US-5nOK[R"U5nURSS- nURUS-US-4:wa [ S5eXl[ URS- 5Ulg)Nr rz&invalid shape of transformation matrix) rrr9r rrsranger _coeffsrvs rrwProjectiveTransform.__init__sz >VVNQ./FZZ'F#\\!_q0N|| 2NQ4FGG !IJJ V[[1_- rcT[RRUR5$r)rr;invrsr_s r _inv_matrixProjectiveTransform._inv_matrixsyy}}T[[))rcURSS- n[R"U[SS9n[R"U[R "URSS45/SS9nXBR -n[R"[5RXUSS2U4S:HU4'USS2SU24==USS2X3S-24-ss'USS2SU24$)Nrr r)copyndminr) r rrrr&r*r)finfofloateps)rXrYrndimrArBs r _apply_matProjectiveTransform._apply_mats||A"&'8Bnnfbggv||A.B&CD1MHHn (*xx':':4LA t #$ AuuH Qax/00 1ete8}rcVUc UR$URRU5$r)rsastype)rXr8rs r __array__ProjectiveTransform.__array__s& =;; ;;%%e, ,rc8URXR5$)zApply forward transformation. Parameters ---------- coords : (N, D) array_like Source coordinates. Returns ------- coords_out : (N, D) array Destination coordinates. )rrsrWs rrZProjectiveTransform.__call__sv{{33rc4[U5"URS9$)r^r~)rrr_s rr`ProjectiveTransform.inversesDz!1!122rc P[R"U5n[R"U5nURupE[U5upa[U5upr[R"[R "Xg-55(d3[R "US-US-4[R5Ulg[R"XE-US-S-45n[U5Hn XX-U S-U-2XS--XS--U-24'SXU-U S-U-2XS--U-4'XX-U S-U-2U*S- S24'SXU-U S-U-2S4'XU-U S-U-2U*S- S24==USS2XS-24*-ss'M USS2[UR5S/-4nUc#[RRU5u pO[R"U5n[R"[R "[R""U[R$"U5- 5U55n [RRX-5u p[R&"U SS5(a3[R "US-US-4[R5Ulg[R"US-US-45n U SSS24*U S- U R([UR5S/-'SXU4'[RR+U5U -U-n XS-n Xlg) a}Estimate the transformation from a set of corresponding points. You can determine the over-, well- and under-determined parameters with the total least-squares method. Number of source and destination coordinates must match. The transformation is defined as:: X = (a0*x + a1*y + a2) / (c0*x + c1*y + 1) Y = (b0*x + b1*y + b2) / (c0*x + c1*y + 1) These equations can be transformed to the following form:: 0 = a0*x + a1*y + a2 - c0*x*X - c1*y*X - X 0 = b0*x + b1*y + b2 - c0*x*Y - c1*y*Y - Y which exist for each set of corresponding points, so we have a set of N * 2 equations. The coefficients appear linearly so we can write A x = 0, where:: A = [[x y 1 0 0 0 -x*X -y*X -X] [0 0 0 x y 1 -x*Y -y*Y -Y] ... ... ] x.T = [a0 a1 a2 b0 b1 b2 c0 c1 c3] In case of total least-squares the solution of this homogeneous system of equations is the right singular vector of A which corresponds to the smallest singular value normed by the coefficient c3. Weights can be applied to each pair of corresponding points to indicate, particularly in an overdetermined system, if point pairs have higher or lower confidence or uncertainties associated with them. From the matrix treatment of least squares problems, these weight values are normalised, square-rooted, then built into a diagonal matrix, by which A is multiplied. In case of the affine transformation the coefficients c0 and c1 are 0. Thus the system of equations is:: A = [[x y 1 0 0 0 -X] [0 0 0 x y 1 -Y] ... ... ] x.T = [a0 a1 a2 b0 b1 b2 c3] Parameters ---------- src : (N, 2) array_like Source coordinates. dst : (N, 2) array_like Destination coordinates. weights : (N,) array_like, optional Relative weight values for each pair of points. Returns ------- success : bool True, if model estimation succeeds. r Frr Nr r rT)rr9r r5allisfiniter#r$rszerosrlistrr;r=r?tilermaxiscloseflatr)rXrArBweightsr,rrrrJddimrrMWHs rrProjectiveTransform.estimatesBjjojjoyy6s; 6s; vvbkk*"9:;;''1q5!a%."&&9DK HHaea!e\* +!HDPSdh$(a'Q$a%.1:L)LL M?@AQh$(a'Q!);; <8;dh$(a'!a"4 5/1AQh$(a'+ , Qh$(a'!a1 2s1dQh>O;O7P6P P 2  adll#rd** + ?iimmA&GAq!jj)G"&&/(A BAFGAiimmAE*GAq ::ai # #''1q5!a%."&&9DK HHa!eQU^ $./r3B3wZK!F),CtDLL!RD()Q$ IIMM* % )J 6 vY rc[U[5(aJ[U5[U5:Xa URnO[nU"URUR-5$[ S5e)z)Combine this transformation with another.z2Cannot combine transformations of differing types.) isinstancerrrurs TypeError)rXothertforms r__add__ProjectiveTransform.__add__IsT e0 1 1DzT%[(+ 34 4ST Trcx[R"URSS9nS[R"US5-nU$)z.common 'paramstr' used by __str__ and __repr__z, ) separatorzmatrix= z )r array2stringrstextwrapindent)rXnpstringparamstrs r__nice__ProjectiveTransform.__nice__Vs0??4;;$?6!BBrc UR5nURRnUnSUSUS[[ U55S3$)z5Add standard repr formatting around a __nice__ string<(z) at >)rrurfhexidrXr classnameclassstrs r__repr__ProjectiveTransform.__repr__\sD==?NN++ 8*AhZuSD]O1==rcdUR5nURRnUnSUSUS3$)z4Add standard str formatting around a __nice__ stringrrz)>)rrurfrs r__str__ProjectiveTransform.__str__cs6==?NN++ 8*AhZr**rc:URRSS- $)z)The dimensionality of the transformation.rr )rsr r_s rr"ProjectiveTransform.dimensionalityjs{{  #a''rrrsr)NN)rfrgrhrirjrwrkrrrrZr`rrrrrrrlrVrrrrjs{%N .a .** - 4 33zx U >+((rrct\rSrSrSrS SS.Sjjr\S5r\S5r\S 5r \S 5r S r g) AffineTransformipa Affine transformation. Has the following form:: X = a0 * x + a1 * y + a2 = sx * x * [cos(rotation) + tan(shear_y) * sin(rotation)] - sy * y * [tan(shear_x) * cos(rotation) + sin(rotation)] + translation_x Y = b0 * x + b1 * y + b2 = sx * x * [sin(rotation) - tan(shear_y) * cos(rotation)] - sy * y * [tan(shear_x) * sin(rotation) - cos(rotation)] + translation_y where ``sx`` and ``sy`` are scale factors in the x and y directions. This is equivalent to applying the operations in the following order: 1. Scale 2. Shear 3. Rotate 4. Translate The homogeneous transformation matrix is:: [[a0 a1 a2] [b0 b1 b2] [0 0 1]] In 2D, the transformation parameters can be given as the homogeneous transformation matrix, above, or as the implicit parameters, scale, rotation, shear, and translation in x (a2) and y (b2). For 3D and higher, only the matrix form is allowed. In narrower transforms, such as the Euclidean (only rotation and translation) or Similarity (rotation, translation, and a global scale factor) transforms, it is possible to specify 3D transforms using implicit parameters also. Parameters ---------- matrix : (D+1, D+1) array_like, optional Homogeneous transformation matrix. If this matrix is provided, it is an error to provide any of scale, rotation, shear, or translation. scale : {s as float or (sx, sy) as array, list or tuple}, optional Scale factor(s). If a single value, it will be assigned to both sx and sy. Only available for 2D. .. versionadded:: 0.17 Added support for supplying a single scalar value. rotation : float, optional Rotation angle, clockwise, as radians. Only available for 2D. shear : float or 2-tuple of float, optional The x and y shear angles, clockwise, by which these axes are rotated around the origin [2]. If a single value is given, take that to be the x shear angle, with the y angle remaining 0. Only available in 2D. translation : (tx, ty) as array, list or tuple, optional Translation parameters. Only available for 2D. dimensionality : int, optional The dimensionality of the transform. This is not used if any other parameters are provided. Attributes ---------- params : (D+1, D+1) array Homogeneous transformation matrix. Raises ------ ValueError If both ``matrix`` and any of the other parameters are provided. Examples -------- >>> import numpy as np >>> import skimage as ski >>> img = ski.data.astronaut() Define source and destination points: >>> src = np.array([[150, 150], ... [250, 100], ... [150, 200]]) >>> dst = np.array([[200, 200], ... [300, 150], ... [150, 400]]) Estimate the transformation matrix: >>> tform = ski.transform.AffineTransform() >>> tform.estimate(src, dst) True Apply the transformation: >>> warped = ski.transform.warp(img, inverse_map=tform.inverse) References ---------- .. [1] Wikipedia, "Affine transformation", https://en.wikipedia.org/wiki/Affine_transformation#Image_transformation .. [2] Wikipedia, "Shear mapping", https://en.wikipedia.org/wiki/Shear_mapping NrrpcT[SX#XE455n[XfS--5UlU(aUb [S5eU(aUS:a [S5eUb[R "U5nUR S:wd URSURS:wa [S5eURSS- nXfS--n[U5UlXlgU(GaUcSnUcSnUcSnUcS n[R"U5(aU=pOUup[R"U5(aUSpOUupU [R"U5[R"U 5[R"U5---n U *[R"U 5[R"U5-[R"U5--nUSnU [R"U5[R"U 5[R"U5-- -nU *[R"U 5[R"U5-[R"U5- -nUSn[R"XU/UUU//S Q/5Ulg[R"US-5Ulg) Nc3(# UHoSLv M g7frrV.0params r +AffineTransform.__init__..s +P% +Pr ZYou cannot specify the transformation matrix and the implicit parameters at the same time.rz(Parameter input is only supported in 2D.r'Invalid shape of transformation matrix.r r rrrrr )anyrrrrr9rr rsisscalarmathcostansinrr)rXrrPrshearrrrsrsxsyshear_xshear_ya0a1a2b0b1b2s rrwAffineTransform.__init__sN ,1U+P   ^/ABC f(=  nq(GH H  ZZ'F{{a6<<?fll1o#E !JKK!'a1!4'A+=> =DL K }}"$ {{5!!R{{5!!$)1#( txx)DHHW,=@R,RRSB)DHHX,>>(ASSTBQBtxx)DHHW,=@R,RRSB)DHHX,>>(ASSTBQB((RRL2r2, #JKDK&&!!34DKrcURS:waC[R"[R"URS-SS95SUR$[R"URS-SS9nUS[ R "UR5S-S-- US'[R"U5SUR$)Nrrrr )rrrr"rsr r r)rXsss rrPAffineTransform.scales   ! #77266$++q.q9:;PT=P=PQ Q QQ/BqETXXdjj1Q6:;BqE772;4!4!45 5rcURS:wa [S5e[R"URSURS5$)Nrz??rcURS:wa [S5e[R"URS*URS5nXR - $)Nrz9The shear property is only implemented for 2D transforms.rr r)rrtr rrsr)rXbetas rrAffineTransform.shear-sT   ! #%K zz4;;t,,dkk$.?@mm##rcPURSUR2UR4$Nrrsrr_s rrAffineTransform.translation6'{{1t222D4G4GGHHrr)NNNNN) rfrgrhrirjrwrkrPrrrrlrVrrrrps~hX ?5?5B66@@$$IIrrc:\rSrSrSrSrSrSr\S5r Sr g) PiecewiseAffineTransformi;aPiecewise affine transformation. Control points are used to define the mapping. The transform is based on a Delaunay triangulation of the points to form a mesh. Each triangle is used to find a local affine transform. Attributes ---------- affines : list of AffineTransform objects Affine transformations for each triangle in the mesh. inverse_affines : list of AffineTransform objects Inverse affine transformations for each triangle in the mesh. c<SUlSUlSUlSUlgr) _tesselation_inverse_tesselationaffinesinverse_affinesr_s rrw!PiecewiseAffineTransform.__init__Ks! $(! #rc [R"U5n[R"U5nURSn[R"U5UlSn/UlUR RHGn[US9nXFRXSS24X%SS245-nUR RU5 MI [R"U5Ul /Ul URRHGn[US9nXFRX%SS24XSS245-nURRU5 MI U$)aXEstimate the transformation from a set of corresponding points. Number of source and destination coordinates must match. Parameters ---------- src : (N, D) array_like Source coordinates. dst : (N, D) array_like Destination coordinates. Returns ------- success : bool True, if all pieces of the model are successfully estimated. r TrpN) rr9r rDelaunayr-r/ simplicesrrappendr.r0)rXrArBrsuccesstriaffines rr!PiecewiseAffineTransform.estimateQs$jjojjoyy|$,,S1 $$..C$D9F s6{CQK@ @G LL   '/%,$4$4S$9!!,,66C$D9F s6{CQK@ @G  ' ' /7 rct[R"U5n[R"U[R5nURR U5nSX#S:HSS24'[ [URR55H*nURUnX4:HnU"XSS245X&SS24'M, U$)zApply forward transformation. Coordinates outside of the mesh will be set to `- 1`. Parameters ---------- coords : (N, D) array_like Source coordinates. Returns ------- coords : (N, 2) array Transformed coordinates. r N) rr9 empty_liker:r- find_simplexrlenr4r/)rXrYoutsimplexindexr8 index_masks rrZ!PiecewiseAffineTransform.__call__s F#mmFBJJ/##008!#rM1 3t00::;CA  = rc[U5"5nURUlURUlURUlURUlU$)r^)rr.r-r0r/)rXrs rr` PiecewiseAffineTransform.inversesIT !66%)%6%6",, $  r)r.r-r/r0N rfrgrhrirjrwrrZrkr`rlrVrrr+r+;s, $ -^!Frr+cn[RRRSXS9R 5$)aProduce an Euler rotation matrix from the given intrinsic rotation angles for the axes x, y and z. Parameters ---------- angles : array of float, shape (3,) The transformation angles in radians. degrees : bool, optional If True, then the given angles are assumed to be in degrees. Default is False. Returns ------- R : array of float, shape (3, 3) The Euler rotation matrix. XYZanglesdegrees)r transformRotation from_euler as_matrixrHs r_euler_rotation_matrixrOs4    % % 0 0 f 1 ikrcR\rSrSrSrS SS.SjjrSr\S5r\S 5r S r g) EuclideanTransformiaEuclidean transformation, also known as a rigid transform. Has the following form:: X = a0 * x - b0 * y + a1 = = x * cos(rotation) - y * sin(rotation) + a1 Y = b0 * x + a0 * y + b1 = = x * sin(rotation) + y * cos(rotation) + b1 where the homogeneous transformation matrix is:: [[a0 -b0 a1] [b0 a0 b1] [0 0 1 ]] The Euclidean transformation is a rigid transformation with rotation and translation parameters. The similarity transformation extends the Euclidean transformation with a single scaling factor. In 2D and 3D, the transformation parameters may be provided either via `matrix`, the homogeneous transformation matrix, above, or via the implicit parameters `rotation` and/or `translation` (where `a1` is the translation along `x`, `b1` along `y`, etc.). Beyond 3D, if the transformation is only a translation, you may use the implicit parameter `translation`; otherwise, you must use `matrix`. Parameters ---------- matrix : (D+1, D+1) array_like, optional Homogeneous transformation matrix. rotation : float or sequence of float, optional Rotation angle, clockwise, as radians. If given as a vector, it is interpreted as Euler rotation angles [1]_. Only 2D (single rotation) and 3D (Euler rotations) values are supported. For higher dimensions, you must provide or estimate the transformation matrix. translation : (x, y[, z, ...]) sequence of float, length D, optional Translation parameters for each axis. dimensionality : int, optional The dimensionality of the transform. Attributes ---------- params : (D+1, D+1) array Homogeneous transformation matrix. References ---------- .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions NrrpcUSL=(d USLnU(aUb [S5eUbH[R"U5nURSURS:wa [S5eXlgU(GaWUc@[ U5nUS:XaSnOeUS:Xa[R "S5nOH[SUS35e[R"U5(d[ U5S:wa[SUS35eUcS U-nUS:Xax[R"[R"U5[R"U5*S/[R"U5[R"U5S//S Q/5UlOBUS:Xa<[R"US-5Ul[U5URSU2SU24'X0RSU2U4'g[R"US-5Ulg) Nrrr rrrz-Parameters cannot be specified for dimension z transformsrr)rrr9r rsr=rr rr r r rrO)rXrrrr params_givens rrwEuclideanTransform.__init__s t+F{$/F F.=  ZZ'F||A&,,q/1 !JKK K !$[!1!Q& H#q(!xx{H$G)*+7 {{8,,X!1C$G)*+7""^3 " hh(+dhhx.@-@!D(+TXXh-?C!  1$ ff^a%78 @VA O^O_n_<==HKK.(.8 9&&!!34DKrc[XS5Ul[R"[R"UR55(+$)Estimate the transformation from a set of corresponding points. You can determine the over-, well- and under-determined parameters with the total least-squares method. Number of source and destination coordinates must match. Parameters ---------- src : (N, 2) array_like Source coordinates. dst : (N, 2) array_like Destination coordinates. Returns ------- success : bool True, if model estimation succeeds. FrQrsrrisnanrcs rrEuclideanTransform.estimate.s3*s/ 66"((4;;/000rcURS:Xa1[R"URSURS5$URS:XaURSS2SS24$[ S5e)Nrrrrz3Rotation only implemented for 2D and 3D transforms.)rr rrsrtr_s rrEuclideanTransform.rotationHsl   ! #::dkk$/T1BC C  A %;;rr2A2v& &%E rcPURSUR2UR4$r&r'r_s rrEuclideanTransform.translationTr)rrr) rfrgrhrirjrwrrkrrrlrVrrrQrQsJ2j7;45NO45l14  IIrrQcH\rSrSrSrS SS.SjjrSr\S5rS r g) SimilarityTransformiYa+Similarity transformation. Has the following form in 2D:: X = a0 * x - b0 * y + a1 = = s * x * cos(rotation) - s * y * sin(rotation) + a1 Y = b0 * x + a0 * y + b1 = = s * x * sin(rotation) + s * y * cos(rotation) + b1 where ``s`` is a scale factor and the homogeneous transformation matrix is:: [[a0 -b0 a1] [b0 a0 b1] [0 0 1 ]] The similarity transformation extends the Euclidean transformation with a single scaling factor in addition to the rotation and translation parameters. Parameters ---------- matrix : (dim+1, dim+1) array_like, optional Homogeneous transformation matrix. scale : float, optional Scale factor. Implemented only for 2D and 3D. rotation : float, optional Rotation angle, clockwise, as radians. Implemented only for 2D and 3D. For 3D, this is given in ZYX Euler angles. translation : (dim,) array_like, optional x, y[, z] translation parameters. Implemented only for 2D and 3D. Attributes ---------- params : (dim+1, dim+1) array Homogeneous transformation matrix. Nrrpc,SUl[SX#U455nU(aUb [S5eUbi[R"U5nUR S:wd UR SUR S:wa [S5eXlUR SS- nU(aUS;a [S5e[R"US-[S 9nUcSnUc US:XaSOS nUcS U-nUS:XaBS n[R"U5[R"U5pXXw4'U *U 4XUSSS 24'O[U5USS2SS24'USU2SU24==U-ss'XASU2U4'XlgURc[R"US-5Ulgg)Nc3(# UHoSLv M g7frrVrs rr/SimilarityTransform.__init__..sS4R5$&4Rrrrrr r)rrz(Parameters only supported for 2D and 3D.r7)rrrrSr"r r) rsrrrr9rr rrr r rO) rXrrPrrrrsaxcrOs rrwSimilarityTransform.__init__s SUk4RSS f(=  ZZ'F{{a6<<?fll1o#E !JKK$ !'a1!4 V+ !KLLVVNQ.eXXy)45FZZ'F <<?a IJ J rc [R"U5n[R"U5nUSS2S4nUSS2S4nUSS2S4nUSS2S4nURSn [U5nUS-US--n [R"U S-U S-45n Sn [ US-5HGn [ U S-5H2nX]U- -Xn--U SU 2U 4'X]U- -Xn--XS2XS--4'U S- n M4 MI X{SU 2S4'XU S2S4'Uc$[R RU 5u nnO[R"U5n[R"[R"[R"U[R"U5- 5S55n[R RUU -5u nnUSSS24*US- nURSU S-45Ul g)aEstimate the transformation from a set of corresponding points. You can determine the over-, well- and under-determined parameters with the total least-squares method. Number of source and destination coordinates must match. The transformation is defined as:: X = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i )) Y = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i )) These equations can be transformed to the following form:: 0 = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i )) - X 0 = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i )) - Y which exist for each set of corresponding points, so we have a set of N * 2 equations. The coefficients appear linearly so we can write A x = 0, where:: A = [[1 x y x**2 x*y y**2 ... 0 ... 0 -X] [0 ... 0 1 x y x**2 x*y y**2 -Y] ... ... ] x.T = [a00 a10 a11 a20 a21 a22 ... ann b00 b10 b11 b20 b21 b22 ... bnn c3] In case of total least-squares the solution of this homogeneous system of equations is the right singular vector of A which corresponds to the smallest singular value normed by the coefficient c3. Weights can be applied to each pair of corresponding points to indicate, particularly in an overdetermined system, if point pairs have higher or lower confidence or uncertainties associated with them. From the matrix treatment of least squares problems, these weight values are normalised, square-rooted, then built into a diagonal matrix, by which A is multiplied. Parameters ---------- src : (N, 2) array_like Source coordinates. dst : (N, 2) array_like Destination coordinates. order : int, optional Polynomial order (number of coefficients is order + 1). weights : (N,) array_like, optional Relative weight values for each pair of points. Returns ------- success : bool True, if model estimation succeeds. Nrr rr rT)rr9r rrrr;r=r?rrrrrs)rXrArBorderrxsysxdydrowsurJpidxjirrMrrss rrPolynomialTransform.estimatestjjojjo AY AY AY AYyy|E" QY519 % HHdQhA& 'uqy!A1q5\!#A!6%4%+*,Q-"%*?%Q&' "" %4%) $%)  ?iimmA&GAq!jj)G"&&/(A BAFGAiimmAE*GAq!BG*qy(nnaa[1 rc N[R"U5nUSS2S4nUSS2S4n[URR 55n[ S[ R"SSSU- -- 5-S- 5n[R"UR5nSn[US-5Hwn[US-5Hbn USS2S4==URSU4X(U - --X9--- ss'USS2S4==URSU4X(U - --X9--- ss'US- nMd My U$)zApply forward transformation. Parameters ---------- coords : (N, 2) array_like source coordinates Returns ------- coords : (N, 2) array Transformed coordinates. Nrr rr r) rr9r=rsravelrr rrr r) rXrYxyrwrqrBrxryrzs rrZPolynomialTransform.__call___sF# 1a4L 1a4L  !!# $R$))AQU O449:hhv||$uqy!A1q5\AqD T[[D1Aa%L@14GG AqD T[[D1Aa%L@14GG  ""  rc[S5e)NzThere is no explicit way to do the inverse polynomial transformation. Instead, estimate the inverse transformation parameters by exchanging source and destination coordinates,then apply the forward transformation.)rtr_s rr`PolynomialTransform.inverse~s! 5  rrr)rNrErVrrrmrms3* a `D>  rrm) euclidean similarityr8zpiecewise-affine projective fundamental essential polynomialcUR5nU[;a[SUS35e[U"URSS9nUR"X/UQ70UD6 U$)a4Estimate 2D geometric transformation parameters. You can determine the over-, well- and under-determined parameters with the total least-squares method. Number of source and destination coordinates must match. Parameters ---------- ttype : {'euclidean', similarity', 'affine', 'piecewise-affine', 'projective', 'polynomial'} Type of transform. kwargs : array_like or int Function parameters (src, dst, n, angle):: NAME / TTYPE FUNCTION PARAMETERS 'euclidean' `src, `dst` 'similarity' `src, `dst` 'affine' `src, `dst` 'piecewise-affine' `src, `dst` 'projective' `src, `dst` 'polynomial' `src, `dst`, `order` (polynomial order, default order is 2) Also see examples below. Returns ------- tform : :class:`_GeometricTransform` Transform object containing the transformation parameters and providing access to forward and inverse transformation functions. Examples -------- >>> import numpy as np >>> import skimage as ski >>> # estimate transformation parameters >>> src = np.array([0, 0, 10, 10]).reshape((2, 2)) >>> dst = np.array([12, 14, 1, -20]).reshape((2, 2)) >>> tform = ski.transform.estimate_transform('similarity', src, dst) >>> np.allclose(tform.inverse(tform(src)), src) True >>> # warp image using the estimated transformation >>> image = ski.data.camera() >>> ski.transform.warp(image, inverse_map=tform.inverse) # doctest: +SKIP >>> # create transformation with explicit parameters >>> tform2 = ski.transform.SimilarityTransform(scale=1.1, rotation=1, ... translation=(10, 20)) >>> # unite transformations, applied in order from left to right >>> tform3 = tform + tform2 >>> np.allclose(tform3(src), tform2(tform(src))) True zthe transformation type 'z' is not implementedr rp)lower TRANSFORMSrr r)ttyperArBargskwargsrs restimate_transformrs`| KKME J5eWrs #'. K\J Z+?#+?\p !4p fP9PfC(-C(LHI)HIVp2pf*SI,SIl}R,}R@l -l `$%0%-)%  EP/r