1-PhT ddlZddlZddlmZmZddlZddlmZddl m Z ddl m Z dZ dZd ZGd d eZGd d eZGddeZGddeZGddeZGddeZd!dZGddeZGddeZGddeZeeeeeeeedZdZd ZdS)"N)ABCabstractmethod)spatial) safe_as_int)NP_COPY_IF_NEEDEDc@|j}dtjdd|zzzdz dz }ttj|}||krt d|tj|dz}tj|||dzf|ddddf<|S)z8Affine matrix from linearized (d, d + 1) matrix entries.rz2Invalid number of elements for linearized matrix: N)sizenpsqrtintround ValueErroreyereshape)vnparamddimensionalitymatrixs \/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/skimage/transform/_geometric.py_affine_matrix_from_vectorr s VF RWQV^ $ $ $)A-A!%%NN LF L L   VNQ& ' 'FZNNQ4F#GHHF3B36N Mc|j\}}tj|d}||z }tjtj|dz|z }|dkrFtj|dz|dzftjtj|tjfStj||z }|tjtj ||ddtj f fdz}tj|dg|zdgzgfd}tj |j tj |g} || zj } | ddd|f} | | dd|dfz} || fS)a@Center 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_pointsr5sD >>rN) __name__ __module__ __qualname____doc__rrYpropertyr^rarUrrrSrSsj<<   ^  BB^XB?????rrScNeZdZdZd dddZdZedZdZd Z d Z dS) 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]]) Nrrc|tj|dz}nFtj|}|jddz }|j|dz|dzfkrt d||_|dkrt |jddS)Nr r&Invalid shape of transformation matrixrzJ is only implemented for 2D coordinates (i.e. 3D transformation matrices).)rrr9r rparamsNotImplementedError __class__rWrrs r__init__z#FundamentalMatrixTransform.__init__6s >VNQ.//FFZ''F#\!_q0N| 2NQ4FGGG !IJJJ Q  %>555   rctj|}tj|tj|jdg}||jjzS)aApply 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 rlr))rWrXcoords_homogeneouss rrYz#FundamentalMatrixTransform.__call__FsGF##_fbgfl1o6N6N-OPP!DKM11rcHt||jjS)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)typerlr)r]s rr^z"FundamentalMatrixTransform.inverseXs!tDzz////rctj|}tj|}|j|jkrtd|jddkrtd t |\}}t |\}}nW#t $rJtjdtj|_dtjdtjgzcYSwxYwtj |jddf}||dddd f<|ddddfxx|dddtj fzcc<||dddd f<|dddd fxx|ddd tj fzcc<||ddd df<tj |\}}}|d ddf dd}|||fS)aSetup and solve the homogeneous epipolar constraint matrix:: dst' * F * src = 0. Parameters ---------- src : (N, 2) array_like Source coordinates. dst : (N, 2) array_like Destination coordinates. Returns ------- F_normalized : (3, 3) array The normalized solution to the homogeneous system. If the system is not well-conditioned, this matrix contains NaNs. src_matrix : (3, 3) array The transformation matrix to obtain the normalized source coordinates. dst_matrix : (3, 3) array The transformation matrix to obtain the normalized destination coordinates. z%src and dst shapes must be identical.rz,src.shape[0] must be equal or larger than 8.rzrz Nrr r )rr9r rr5ZeroDivisionErrorr#r$rlr*r'r;r=r) rWrArB src_matrix dst_matrixrJ_rM F_normalizeds r_setup_constraint_matrixz3FundamentalMatrixTransform._setup_constraint_matrixas2joojoo 9 ! !DEE E 9Q>> 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]) Nrric t|||Utj|}|t dtj|}|jdkrt dt tj|dz dkrt d|j dkrt d t tj |dz dkrt d tj d |d  |d|d d |d  |d |d d g  dd}||z|_ dS|7tj|}|jdkrt d ||_ dStjd|_ dS)N)rrz&Both rotation and translation requiredryz Invalid shape of rotation matrixr gư>z*Rotation matrix must have unit determinantrzz#Invalid shape of translation vectorz(Translation vector must have unit lengthrrrk)superrprr9rr rr;r<r normarrayrrlr)rWrotation translationrrt_xrns rrpz!EssentialMatrixTransform.__init__s ~FFF  z(++H" !IJJJ*[11K~'' !CDDD29==**Q.//$66 !MNNN1$$ !FGGG29>>+..233d:: !KLLL( ^ONN ^O ^ON   gamm .DKKK  Z''F|v%% !IJJJ DKKK&))DKKKrc&|||\}}}tj|\}}}|d|dzdz |d<|d|d<d|d<|tj|z|z} |j| z|z|_dS)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) rWrArB E_normalizedrrrKrLrMEs rrz!EssentialMatrixTransform.estimateFs*04/L/LSRU/V/V, j*)-- --1a!qt s"!t!!  NQ  lQ&3 trNNN)rbrcrdrerpr __classcell__)rns@rrrslCCL7;'$NO'$'$'$'$'$'$'$R!!!!!!!rrceZdZdZddddZedZdZddZd Z ed Z dd Z d Z d Z dZdZedZdS)ProjectiveTransformaProjective 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. Nrric |tj|dz}nFtj|}|jddz }|j|dz|dzfkrt d||_t |jdz |_dS)Nr rz&invalid shape of transformation matrix) rrr9r rrlranger _coeffsros rrpzProjectiveTransform.__init__s >VNQ.//FFZ''F#\!_q0N| 2NQ4FGGG !IJJJ V[1_-- rcJtj|jSr)rr;invrlr]s r _inv_matrixzProjectiveTransform._inv_matrixsy}}T[)))rc|jddz }tj|td}tj|tj|jddfgd}||jz}tjtj ||dd|fdk|f<|ddd|fxx|dd||dzfzcc<|ddd|fS)Nrr r)copyndminr) r rrrr&r*r)finfofloateps)rWrXrndimrArBs r _apply_matzProjectiveTransform._apply_mats|A"&'8BBBnfbgv|A.B&C&CD1MMMFHn (*x':C4LA t #$ AAAuuH QQQtax/00 111ete8}rcH||jS|j|Sr)rlastype)rWr8rs r __array__zProjectiveTransform.__array__s% =; ;%%e,, ,rc8|||jS)zApply forward transformation. Parameters ---------- coords : (N, D) array_like Source coordinates. Returns ------- coords_out : (N, D) array Destination coordinates. )rrlrVs rrYzProjectiveTransform.__call__svt{333rc>t||jS)r\ru)rvrr]s rr^zProjectiveTransform.inverses tDzz!12222rc Ftj|}tj|}|j\}}t|\}}t|\}}tjtj||zs.tj|dz|dzftj|_dStj ||z|dzdzf}t|D]} ||| |z| dz|z| |dzz| |dzz|zf<d|| |z| dz|z| |dzz|zf<||| |z| dz|z| dz df<d|| |z| dz|zdf<|| |z| dz|z| dz dfxx|dd| | dzf zcc<|ddt|j dgzf}|$tj |\} } } ntj|}tjtjtj|tj|z |} tj | |z\} } } tj| ddr.tj|dz|dzftj|_dStj |dz|dzf} | dddf | dz | jt|j dgz<d| ||f<tj || z|z} | | dz} | |_dS) 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 = (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$rlzerosrlistrr;r=r?tilermaxiscloseflatr)rWrArBweightsr,rrrrJddimrrMWHs rrzProjectiveTransform.estimatesVBjoojooy16s;; C6s;; Cvbk*z"9::;; '1q5!a%."&99DK5 Ha!ea!e\* + +!HH Q QDPSAdQh$(a'Q$!a%.1:L)LL M?@AdQh$(a'Q!);; <8;AdQh$(a'!a"4 5/1AdQh$(a'+ , dQh$(a'!a1 2 2 2s111ddQh>O;O7P6P P 2 2 2 2 aaadl##rd** + ?immA&&GAq!!j))G"&//(A B BAFFGGAimmAE**GAq! :ai # # '1q5!a%."&99DK5 Ha!eQU^ $ $./r3B3wZK!F),CtDL!!RD()!Q$ IMM* % % )J 6 QvY trct|trGt|t|kr|j}nt}||j|jzSt d)z)Combine this transformation with another.z2Cannot combine transformations of differing types.) isinstancerrvrnrl TypeError)rWothertforms r__add__zProjectiveTransform.__add__Ise e0 1 1 UDzzT%[[((+5 344 4STT Trcltj|jd}dtj|dz}|S)z.common 'paramstr' used by __str__ and __repr__z, ) separatorzmatrix= z )r array2stringrltextwrapindent)rWnpstringparamstrs r__nice__zProjectiveTransform.__nice__Vs3?4;$???6!B!BBrc |}|jj}|}d|d|dtt |dS)z5Add standard repr formatting around a __nice__ string<(z) at >)rrnrbhexidrWr classnameclassstrs r__repr__zProjectiveTransform.__repr__\sK==??N+ =8==h==SD]]====rcX|}|jj}|}d|d|dS)z4Add standard str formatting around a __nice__ stringrrz)>)rrnrbrs r__str__zProjectiveTransform.__str__cs7==??N+ *8**h****rc,|jjddz S)z)The dimensionality of the transformation.rr )rlr r]s rrz"ProjectiveTransform.dimensionalityjs{ #a''rr)NN)rbrcrdrerprfrrrrYr^rrrrrrrUrrrrjs%%N .a . . . . .**X* ---- 444 33X3zzzzx U U U >>>+++((X(((rrceZdZdZ d dddZedZedZedZed Z dS) AffineTransforma 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 Nrrictd||||fD}t||dzz|_|r|td|r|dkrtd|t j|}|jdks|jd|jdkrtd|jddz }||dzz}t||_||_dS|r|d}|d}|d}|d }t j |r|x} } n|\} } t j |r|d} } n|\} } | tj |tj | tj |zzz} | tj | tj |ztj |zz}|d}| tj |tj | tj |zz z}| tj | tj |ztj |z z}|d}t j| ||g|||ggd g|_dSt j|dz|_dS) Nc3K|]}|duV dSrrU.0params r z+AffineTransform.__init__..s8  "'E       rr 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 rlisscalarmathcostansinrr)rWrrPrshearrrrlrsxsyshear_xshear_ya0a1a2b0b1b2s rrpzAffineTransform.__init__s  ,18UK+P      ^~/ABCC  f(=  ) 5nq((GHH H  Z''F{a6<?fl1o#E#E !JKKK!'a1!4'>A+=> ==DL DKKK  5}}"$ {5!! RRB{5!! )$)1#( tx))DHW,=,=@R@R,RRSB))DHX,>,>>(ASASSTBQBtx))DHW,=,=@R@R,RRSB))DHX,>,>>(ASASSTBQB(RRL2r2, #JKKDKKK&!!344DKKKrcf|jdkr=tjtj|jdzdd|jStj|jdzd}|dt j|jdzdzz |d<tj|d|jS)Nrrrr )rrrr"rlrrr)rWsss rrPzAffineTransform.scales  ! # #726$+q.q999::;PT=P;PQ Q QQ///BqETXdj11Q6:;BqE72;;4!445 5rc|jdkrtdtj|jd|jdS)Nrz???rc|jdkrtdtj|jd |jd}||jz S)Nrz9The shear property is only implemented for 2D transforms.rr r)rrmrrrlr)rWbetas rrzAffineTransform.shear-sV  ! # #%K z4;t,,dk$.?@@dm##rc8|jd|j|jfSNrrlrr]s rrzAffineTransform.translation6{1t22D4GGHHr)NNNNN) rbrcrdrerprfrPrrrrUrrrrpshhX ?5?5?5?5?5?5B66X6@@X@$$X$IIXIIIrrc:eZdZdZdZdZdZedZdS)PiecewiseAffineTransformaPiecewise 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>d|_d|_d|_d|_dSr) _tesselation_inverse_tesselationaffinesinverse_affinesr]s rrpz!PiecewiseAffineTransform.__init__Ks' $(! #rc tj|}tj|}|jd}tj||_d}g|_|jjD]Y}t|}|| ||ddf||ddfz}|j |Ztj||_ g|_ |j jD]Y}t|}|| ||ddf||ddfz}|j |Z|S)aEstimate the transformation from a set of corresponding points. Number of source and destination coordinates must match. Parameters ---------- src : (N, D) array_like Source coordinates. dst : (N, D) array_like Destination coordinates. Returns ------- success : bool True, if all pieces of the model are successfully estimated. r TriN) rr9r rDelaunayr r  simplicesrrappendr r )rWrArBrsuccesstriaffines rrz!PiecewiseAffineTransform.estimateQsK$joojooy|$,S11 $. ( (C$D999F vs36{CQQQK@@ @G L   ' ' ' '%,$4S$9$9!!,6 0 0C$D999F vs36{CQQQK@@ @G  ' ' / / / /rcjtj|}tj|tj}|j|}d||dkddf<t t|jjD]1}|j |}||k}|||ddf||ddf<2|S)a4Apply 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_simplexrlenrr )rWrXoutsimplexindexr index_masks rrYz!PiecewiseAffineTransform.__call__s F##mFBJ//#0088!#GrM111 3t0:;;<< ? ?E\%(F E)J!'z111}(=!>!>C AAA   rct|}|j|_|j|_|j|_|j|_|S)r\)rvr r r r )rWrs rr^z PiecewiseAffineTransform.inversesDT !6%)%6", $  rN rbrcrdrerprrYrfr^rUrrrr;sh  $$$ ---^!!!FXrrFcttjjd||S)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_matrixr s r_euler_rotation_matrixr's7   % 0 0 fg 1  ikkrcTeZdZdZ d dddZdZedZedZdS) EuclideanTransformatEuclidean 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 Nrric|dup|du}|r|td|Htj|}|jd|jdkrtd||_dS|rL|Ft |}|dkrd}nh|dkrtjd}nMtd|dtj|s&t |dkrtd|d|d |z}|dkrntjtj |tj | dgtj |tj |dggd g|_n?|dkr9tj |dz|_t||jd|d|f<||jd||f<dStj |dz|_dS) Nrrr rrrzz-Parameters cannot be specified for dimension z transformsrr)rrr9r rlrrrrrrrrr')rWrrrr params_givens rrpzEuclideanTransform.__init__sD t+F{$/F / 5F.=  Z''F|A&,q/11 !JKKK DKKK % 5!$[!1!1!Q&& HH#q((!x{{HH$7)777 {8,,X!1C1C$7)777""^3 "" h(++dhx.@.@-@!D(++TXh-?-?C!   1$$ f^a%788 @VAA O^O_n_<==HDK.(.8 9 9 9&!!344DKKKrct||d|_tjtj|j S)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. FrQrlrrisnanr`s rrzEuclideanTransform.estimate.s8*sC// 6"(4;//0000rc|jdkr+tj|jd|jdS|jdkr|jddddfSt d)Nrrrrzz3Rotation only implemented for 2D and 3D transforms.)rrrrlrmr]s rrzEuclideanTransform.rotationHsl  ! # #:dk$/T1BCC C  A % %;rr2A2v& &%E rc8|jd|j|jfSrrr]s rrzEuclideanTransform.translationTrrr) rbrcrdrerprrfrrrUrrr)r)s22j7;45NO4545454545l1114  X IIXIIIrr)cDeZdZdZ ddddZdZedZdS) SimilarityTransformaSimilarity transformation. Has the following form in 2D:: X = a0 * x - b0 * y + a1 = = s * x * cos(rotation) - s * y * sin(rotation) + a1 Y = b0 * x + a0 * y + b1 = = s * x * sin(rotation) + s * y * cos(rotation) + b1 where ``s`` is a scale factor and the homogeneous transformation matrix is:: [[a0 -b0 a1] [b0 a0 b1] [0 0 1 ]] The similarity transformation extends the Euclidean transformation with a single scaling factor in addition to the rotation and translation parameters. Parameters ---------- matrix : (dim+1, dim+1) array_like, optional Homogeneous transformation matrix. scale : float, optional Scale factor. Implemented only for 2D and 3D. rotation : float, optional Rotation angle, clockwise, as radians. Implemented only for 2D and 3D. For 3D, this is given in ZYX Euler angles. translation : (dim,) array_like, optional x, y[, z] translation parameters. Implemented only for 2D and 3D. Attributes ---------- params : (dim+1, dim+1) array Homogeneous transformation matrix. Nrric,d|_td|||fD}|r|td|atj|}|jdks|jd|jdkrtd||_|jddz }|r|dvrtdtj|dzt }|d}| |dkrdnd }|d |z}|dkrEd }tj |tj |} }||||f<| | f|||ddd f<nt||ddddf<|d|d|fxx|zcc<||d||f<||_dS|jtj|dz|_dSdS)Nc3K|]}|duV dSrrUrs rrz/SimilarityTransform.__init__..s'SS5U$&SSSSSSrrrrr r)rrzz(Parameters only supported for 2D and 3D.r7)rrrr+rr rz) rlrrrr9rr rrrrr') rWrrPrrrrlaxcrOs rrpzSimilarityTransform.__init__s  SSUHk4RSSSSS  5f(=  Z''F{a6<?fl1o#E#E !JKKK$ !'a1!4  5V++ !KLLLVNQ.e<<Xyyy)))455FFZ''F <?a  IJJ J rc tj|}tj|}|dddf}|dddf}|dddf}|dddf}|jd} t|}|dz|dzz} tj| dz| dzf} d} t |dzD]L} t | dzD]7}|| |z z||zz| d| | f<|| |z z||zz| | d| | dzzf<| dz } 8M|| d| df<|| | ddf<|$tj| \}}}ntj|}tjtj tj |tj |z d}tj|| z\}}}|dddf |dz }| d| dzf|_ dS)aEstimate the transformation from a set of corresponding points. You can determine the over-, well- and under-determined parameters with the total least-squares method. Number of source and destination coordinates must match. The transformation is defined as:: X = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i )) Y = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i )) These equations can be transformed to the following form:: 0 = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i )) - X 0 = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i )) - Y which exist for each set of corresponding points, so we have a set of N * 2 equations. The coefficients appear linearly so we can write A x = 0, where:: A = [[1 x y x**2 x*y y**2 ... 0 ... 0 -X] [0 ... 0 1 x y x**2 x*y y**2 -Y] ... ... ] x.T = [a00 a10 a11 a20 a21 a22 ... ann b00 b10 b11 b20 b21 b22 ... bnn c3] In case of total least-squares the solution of this homogeneous system of equations is the right singular vector of A which corresponds to the smallest singular value normed by the coefficient c3. Weights can be applied to each pair of corresponding points to indicate, particularly in an overdetermined system, if point pairs have higher or lower confidence or uncertainties associated with them. From the matrix treatment of least squares problems, these weight values are normalised, square-rooted, then built into a diagonal matrix, by which A is multiplied. Parameters ---------- src : (N, 2) array_like Source coordinates. dst : (N, 2) array_like Destination coordinates. order : int, optional Polynomial order (number of coefficients is order + 1). weights : (N,) array_like, optional Relative weight values for each pair of points. Returns ------- success : bool True, if model estimation succeeds. Nrr rr rT)rr9r rrrr;r=r?rrrrrl)rWrArBorderrxsysxdydrowsurJpidxjirrMrrls rrzPolynomialTransform.estimates$tjoojoo AY AY AY AYy|E"" QY519 % HdQhA& ' 'uqy!!  A1q5\\  !#AQ!6%4%+*,Q-"a%*?$%%Q&'   %4%) $%%)  ?immA&&GAq!!j))G"&//(A B BAFFGGAimmAE**GAq!BG*qy(nnaa[11 trc ^tj|}|dddf}|dddf}t|j}t dt jddd|z zz zdz }tj|j }d}t|dzD]|}t|dzD]g} |dddfxx|jd|f||| z zz|| zzz cc<|dddfxx|jd|f||| z zz|| zzz cc<|dz }h}|S)zApply forward transformation. Parameters ---------- coords : (N, 2) array_like source coordinates Returns ------- coords : (N, 2) array Transformed coordinates. Nrr r{r r) rr9rrlravelrrrrr r) rWrXxyrFr@rBrGrHrIs rrYzPolynomialTransform.__call___s\F## 111a4L 111a4L  !!## $ $R$)AQU O4449::hv|$$uqy!!  A1q5\\  AAAqD T[D1A!a%L@1a4GG AAAqD T[D1A!a%L@1a4GG     rc td)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.)rmr]s rr^zPolynomialTransform.inverse~s! 5   rr)rNrrUrrr=r=s* a     ````D>  X   rr=) euclidean similarityrzpiecewise-affine projective fundamental essential polynomialc|}|tvrtd|dt||jd}|j||g|Ri||S)aEstimate 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 ri)lower TRANSFORMSrr r)ttyperArBargskwargsrs restimate_transformr\s{| KKMME JReRRRSSS u SYq\ : : :EEN3-d---f--- Lrc2t||S)a"Apply 2D matrix transform. Parameters ---------- coords : (N, 2) array_like x, y coordinates to transform matrix : (3, 3) array_like Homogeneous transformation matrix. Returns ------- coords : (N, 2) array Transformed coordinates. )r)rXrs rmatrix_transformr^s ' v & &v . ..r)F)rrabcrrnumpyrscipyr _shared.utilsr_shared.compatrrr5rQrSrhrrrrr'r)r4r=rXr\r^rUrrrds ########''''''......   KKK\J J J Z+?+?+?+?+?#+?+?+?\p p p p p !4p p p fPPPPP9PPPfC(C(C(C(C(-C(C(C(LHIHIHIHIHI)HIHIHIVppppp2pppf*SISISISISI,SISISIl}R}R}R}R}R,}R}R}R@l l l l l -l l l `$%0%-)%   EEEP/////r