+ Ai&H^RIt^RIt^RIHtHt^RIHt^RIH t RRlt RRlt RRR/Rllt RRR/R llt RR ltR tRR/R ltRRR/R lltRRR/RlltR#)N)_supported_float_typecheck_nD) _moments_cy)moments_raw_to_centralc\V^VR7#)aCalculate all raw image moments up to a certain order. The following properties can be calculated from raw image moments: * Area as: ``M[0, 0]``. * Centroid as: {``M[1, 0] / M[0, 0]``, ``M[0, 1] / M[0, 0]``}. Note that raw moments are neither translation, scale, nor rotation invariant. Parameters ---------- coords : (N, D) double or uint8 array Array of N points that describe an image of D dimensionality in Cartesian space. order : int, optional Maximum order of moments. Default is 3. Returns ------- M : (``order + 1``, ``order + 1``, ...) array Raw image moments. (D dimensions) References ---------- .. [1] Johannes Kilian. Simple Image Analysis By Moments. Durham University, version 0.2, Durham, 2001. Examples -------- >>> coords = np.array([[row, col] ... for row in range(13, 17) ... for col in range(14, 18)], dtype=np.float64) >>> M = moments_coords(coords) >>> centroid = (M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]) >>> centroid (14.5, 15.5) )order)moments_coords_central)coordsrs&&V/var/www/html/photoedit/myenv/lib/python3.14/site-packages/skimage/measure/_moments.pymoments_coordsr sL "&!5 99c\V\4'd\P!VRR7p\ V^4VP ^,p\ VP4pVf\P!V^\R7pVPVRR7V, p\P!\V^,4Uu.uF qPV,NK upRR7pVPVP RV^, ,,4p^p\V4F5pVRV3,p\P!V^^V,4pWh,pK7 \P!V^R7p V #uupi)aCalculate all central image moments up to a certain order. The following properties can be calculated from raw image moments: * Area as: ``M[0, 0]``. * Centroid as: {``M[1, 0] / M[0, 0]``, ``M[0, 1] / M[0, 0]``}. Note that raw moments are neither translation, scale nor rotation invariant. Parameters ---------- coords : (N, D) double or uint8 array Array of N points that describe an image of D dimensionality in Cartesian space. A tuple of coordinates as returned by ``np.nonzero`` is also accepted as input. center : tuple of float, optional Coordinates of the image centroid. This will be computed if it is not provided. order : int, optional Maximum order of moments. Default is 3. Returns ------- Mc : (``order + 1``, ``order + 1``, ...) array Central image moments. (D dimensions) References ---------- .. [1] Johannes Kilian. Simple Image Analysis By Moments. Durham University, version 0.2, Durham, 2001. Examples -------- >>> coords = np.array([[row, col] ... for row in range(13, 17) ... for col in range(14, 18)]) >>> moments_coords_central(coords) array([[16., 0., 20., 0.], [ 0., 0., 0., 0.], [20., 0., 25., 0.], [ 0., 0., 0., 0.]]) As seen above, for symmetric objects, odd-order moments (columns 1 and 3, rows 1 and 3) are zero when centered on the centroid, or center of mass, of the object (the default). If we break the symmetry by adding a new point, this no longer holds: >>> coords2 = np.concatenate((coords, [[17, 17]]), axis=0) >>> np.round(moments_coords_central(coords2), ... decimals=2) # doctest: +NORMALIZE_WHITESPACE array([[17. , 0. , 22.12, -2.49], [ 0. , 3.53, 1.73, 7.4 ], [25.88, 6.02, 36.63, 8.83], [ 4.15, 19.17, 14.8 , 39.6 ]]) Image moments and central image moments are equivalent (by definition) when the center is (0, 0): >>> np.allclose(moments_coords(coords), ... moments_coords_central(coords, (0, 0))) True )axis)rdtypeFcopyNNN)) isinstancetuplenpstackrshaperrmeanfloatastyperangereshapemoveaxissum) r centerrndim float_typeccalcr isolated_axisMcs &&& r r r 3s)~&%  &r* VQ <<?D&v||4J ~au5]]:E] 2V ;FXX% *:;*:Qqyy*:;" EF^^FLL44!8+<< =F Dd q$w  M1a$h? # 1 B I)>> image = np.zeros((20, 20), dtype=np.float64) >>> image[13:17, 13:17] = 1 >>> M = moments(image) >>> centroid = (M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]) >>> centroid (14.5, 14.5) rr)r)moments_centralr#)imagerr)s&&$r momentsr/sX 5$"35 RRr c Vf\WVR7p\V4#\VP4pVf#\P !VP VR7pVPVRR7p\\VP 44pVP p V ^,p \P!V^,VR7p \VP4Fwr\P!WR7W<,,W,, pVR\P3,V ,p\P!VVRV V .,W^,R,VW.VRV V .,W^,R,RR7pK V#) uCalculate all central image moments up to a certain order. The center coordinates (cr, cc) can be calculated from the raw moments as: {``M[1, 0] / M[0, 0]``, ``M[0, 1] / M[0, 0]``}. Note that central moments are translation invariant but not scale and rotation invariant. Parameters ---------- image : (N[, ...]) double or uint8 array Rasterized shape as image. center : tuple of float, optional Coordinates of the image centroid. This will be computed if it is not provided. order : int, optional The maximum order of moments computed. spacing : tuple of float, shape (ndim,) The pixel spacing along each axis of the image. Returns ------- mu : (``order + 1``, ``order + 1``) array Central image moments. References ---------- .. [1] Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009. .. [2] B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005. .. [3] T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. .. [4] https://en.wikipedia.org/wiki/Image_moment Examples -------- >>> image = np.zeros((20, 20), dtype=np.float64) >>> image[13:17, 13:17] = 1 >>> M = moments(image) >>> centroid = (M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]) >>> moments_central(image, centroid) array([[16., 0., 20., 0.], [ 0., 0., 0., 0.], [20., 0., 25., 0.], [ 0., 0., 0., 0.]]) Nr+rFrrgreedy)optimize)r/rrrronesr#rlistrarange enumeraternewaxiseinsum)r.r"rr)kwargs moments_raw float_dtyper&L sum_label order_labelordersdim dim_lengthdeltapowers_of_deltas&&&$, r r-r-s=b~e'B %k22' 4K''%**K8 << %< 0D U5:: A Ia-K YYuqy 4F$U[[1 *87<G&+U2:: .&8yy  dsGyk !AAgiL 0   $ dsG{m #aa l 2   2 Kr c\P!\P!VP4V8*4'd \ R4hVf!\P !VP 4p\P!V4pVP4^,p\V4p\P!\V^,4VP R7Fwp\V4^8d\PW6&K'W,V\V4,, V\V4VP , ^,,, W6&Ky V#)uCalculate all normalized central image moments up to a certain order. Note that normalized central moments are translation and scale invariant but not rotation invariant. Parameters ---------- mu : (M[, ...], M) array Central image moments, where M must be greater than or equal to ``order``. order : int, optional Maximum order of moments. Default is 3. spacing : tuple of float, shape (ndim,) The pixel spacing along each axis of the image. Returns ------- nu : (``order + 1``[, ...], ``order + 1``) array Normalized central image moments. References ---------- .. [1] Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009. .. [2] B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005. .. [3] T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. .. [4] https://en.wikipedia.org/wiki/Image_moment Examples -------- >>> image = np.zeros((20, 20), dtype=np.float64) >>> image[13:17, 13:17] = 1 >>> m = moments(image) >>> centroid = (m[0, 1] / m[0, 0], m[1, 0] / m[0, 0]) >>> mu = moments_central(image, centroid) >>> moments_normalized(mu) array([[ nan, nan, 0.078125 , 0. ], [ nan, 0. , 0. , 0. ], [0.078125 , 0. , 0.00610352, 0. ], [0. , 0. , 0. , 0. ]]) z)Shape of image moments must be >= `order`)repeat)ranyarrayr ValueErrorr4r# zeros_likeravelmin itertoolsproductrr!nan)murr)numu0scalepowerss&&& r moments_normalizedrUsZ vvbhhrxx E)**DEE''"''" r B ((*Q-C LE##E%!)$4RWWE v;?BJ*uF ';;F bgg-12BJ F Ir cVPR8Xd\PM\Pp\P !VP VRR74#)uCalculate Hu's set of image moments (2D-only). Note that this set of moments is proved to be translation, scale and rotation invariant. Parameters ---------- nu : (M, M) array Normalized central image moments, where M must be >= 4. Returns ------- nu : (7,) array Hu's set of image moments. References ---------- .. [1] M. K. Hu, "Visual Pattern Recognition by Moment Invariants", IRE Trans. Info. Theory, vol. IT-8, pp. 179-187, 1962 .. [2] Wilhelm Burger, Mark Burge. Principles of Digital Image Processing: Core Algorithms. Springer-Verlag, London, 2009. .. [3] B. Jähne. Digital Image Processing. Springer-Verlag, Berlin-Heidelberg, 6. edition, 2005. .. [4] T. H. Reiss. Recognizing Planar Objects Using Invariant Image Features, from Lecture notes in computer science, p. 676. Springer, Berlin, 1993. .. [5] https://en.wikipedia.org/wiki/Image_moment Examples -------- >>> image = np.zeros((20, 20), dtype=np.float64) >>> image[13:17, 13:17] = 0.5 >>> image[10:12, 10:12] = 1 >>> mu = moments_central(image) >>> nu = moments_normalized(mu) >>> np.round(moments_hu(nu), 4) # doctest: +FLOAT_CMP array([0.7454, 0.3512, 0.104 , 0.0406, 0.0026, 0.0241, 0. ]) float32Fr)rrrWfloat64r moments_hur)rQrs& r rYrYVs=N((i/BJJRZZE  ! !"))E)"> ??r c\VRVP,^VR7pV\\P!VP\ R74,VRVP,,, pV#)aReturn the (weighted) centroid of an image. Parameters ---------- image : array The input image. spacing : tuple of float, shape (ndim,) The pixel spacing along each axis of the image. Returns ------- center : tuple of float, length ``image.ndim`` The centroid of the (nonzero) pixels in ``image``. Examples -------- >>> image = np.zeros((20, 20), dtype=np.float64) >>> image[13:17, 13:17] = 0.5 >>> image[10:12, 10:12] = 1 >>> centroid(image) array([13.16666667, 13.16666667]) )r"rr)r1r,)r-r#rreyeint)r.r)Mr"s&$ r centroidr^s[. dUZZ&7q'RA %uzz- ./ D5::    Mr c>Vf\V^VR7pVRVP,,p\P!VPVP3VPR7p\ ^\P !VP\R7,4p\P!V4pRVPn \P!W,4W,, V, VR&\P!\VP4^4Fzp\P!VP\R7p^V\V4&V\ V4,)V, WG&V\ V4,)V, VP V&K| V#)uCompute the inertia tensor of the input image. Parameters ---------- image : array The input image. mu : array, optional The pre-computed central moments of ``image``. The inertia tensor computation requires the central moments of the image. If an application requires both the central moments and the inertia tensor (for example, `skimage.measure.regionprops`), then it is more efficient to pre-compute them and pass them to the inertia tensor call. spacing : tuple of float, shape (ndim,) The pixel spacing along each axis of the image. Returns ------- T : array, shape ``(image.ndim, image.ndim)`` The inertia tensor of the input image. :math:`T_{i, j}` contains the covariance of image intensity along axes :math:`i` and :math:`j`. References ---------- .. [1] https://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_tensor .. [2] Bernd Jähne. Spatio-Temporal Image Processing: Theory and Scientific Applications. (Chapter 8: Tensor Methods) Springer, 1993. r+r1Trr,)r-r#rzerosrrr[r\diagflags writeabler!rM combinationsrr5T) r.rPr)rRresultcorners2ddimsmu_indexs &&$ r inertia_tensorrks*: z  G  TEJJ  C XXuzz5::.bhh ?FQ #667H AAGG FF2< 2< /3 6AaD&&uUZZ'8!<88EJJc2 d5?++c1 U8_--3 = Mr cVf\WVR7p\PPV4p\P!V^RVR7p\ VRR7#)aCompute the eigenvalues of the inertia tensor of the image. The inertia tensor measures covariance of the image intensity along the image axes. (See `inertia_tensor`.) The relative magnitude of the eigenvalues of the tensor is thus a measure of the elongation of a (bright) object in the image. Parameters ---------- image : array The input image. mu : array, optional The pre-computed central moments of ``image``. T : array, shape ``(image.ndim, image.ndim)`` The pre-computed inertia tensor. If ``T`` is given, ``mu`` and ``image`` are ignored. spacing : tuple of float, shape (ndim,) The pixel spacing along each axis of the image. Returns ------- eigvals : list of float, length ``image.ndim`` The eigenvalues of the inertia tensor of ``image``, in descending order. Notes ----- Computing the eigenvalues requires the inertia tensor of the input image. This is much faster if the central moments (``mu``) are provided, or, alternatively, one can provide the inertia tensor (``T``) directly. N)r))outT)reverse)rkrlinalgeigvalshclipsorted)r.rPrer)eigvalss&&&$ r inertia_tensor_eigvalsrtsM@ y 5g 6ii  #G gggq$G4G '4 ((r ))Nru)ruN)N)NN)rMnumpyr _shared.utilsrrr_moments_analyticalrr r r/r-rUrYr^rkrtr r r{sr;7&:RdN,St,S^LDL^;|(@Vt@5d5p()d()r