9iESSKrSSKrSSKJrJr SSKJr SSKJ r SSjr SSjr SSS .S jjr SSS .S jjr SS jrS rSS .SjrSSS .SjjrSSS .Sjjrg)N)_supported_float_typecheck_nD) _moments_cy)moments_raw_to_centralc[USUS9$)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) r)order)moments_coords_central)coordsr s X/var/www/html/land-doc-ocr/venv/lib/python3.13/site-packages/skimage/measure/_moments.pymoments_coordsr sL "&!5 99cv[U[5(a[R"USS9n[ US5 UR Sn[ UR5nUc[R"US[S9nURUSS 9U- n[R"[US-5Vs/sHoPU-PM snSS9nURUR S US- --5nSn[U5H+nUSS2U4n[R"USSU-5nXh-nM- [R"USS9n U $s snf) 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 )axisrrNr)rdtypeFcopy)r) isinstancetuplenpstackrshaperrmeanfloatastyperangereshapemoveaxissum) r centerr ndim float_typeccalcr isolated_axisMcs r r r 3s'~&%  &r* VQ <<?D&v||4J ~au5]]:E] 2V ;FXX% *:;*:Qqy*:;" EF^^FLL44!8+<< =F Dd q$w  M1a$h? # 1 B I)>> image = np.zeros((20, 20), dtype=np.float64) >>> image[13:17, 13:17] = 1 >>> M = moments(image) >>> centroid = (M[1, 0] / M[0, 0], M[0, 1] / M[0, 0]) >>> centroid (14.5, 14.5) rr r*)moments_centralr#)imager r*s r momentsr0sX 5$"35 RRrc :Uc[XUS9n[U5$[UR5nUc[R "UR US9nURUSS9n[UR5Hup[R"XS9X8-X- n U SS2[R4[R"US-US9-n [R"XxUR 5n[R"X{5n[R"USU5nM U$)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-rFrrr)r0rrrronesr#r enumeraterarangenewaxisrollaxisdot) r/r"r r*kwargs moments_raw float_dtyper&dim dim_lengthdeltapowers_of_deltas r r.r.sb~e'B %k22' 4K''%**K8 << %< 0D$U[[1 *87<G&+U2:: .")) AI[3  {{4ejj1vvd,{{4S)2 KrcT[R"[R"UR5U:*5(a [ S5eUc [R "UR 5n[R"U5nUR5Sn[U5n[R"[US-5UR S9HZn[U5S:a[RX6'M&XU[U5-- U[U5UR - S--- X6'M\ U$)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`rr)repeatr)ranyarrayr ValueErrorr3r# zeros_likeravelmin itertoolsproductrr!nan)mur r*numu0scalepowerss r moments_normalizedrPsZ vvbhhrxx E)**DEE''"''" r B ((*Q-C LE##E%!)$4RWWE v;?BJ*uF ';;F bgg-12BJ F IrcURS:Xa[RO[Rn[R "UR USS95$)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) >>> moments_hu(nu) array([0.74537037, 0.35116598, 0.10404918, 0.04064421, 0.00264312, 0.02408546, 0. ]) float32Fr)rrrRfloat64r moments_hur)rLrs r rTrTLs=P((i/BJJRZZE  ! !"))E)"> ??rc[USUR-SUS9nU[[R"UR[ S95USUR-- nU$)aReturn the (weighted) centroid of an image. Parameters ---------- image : array The input image. spacing: tuple of float, shape (ndim,) The pixel spacing along each axis of the image. Returns ------- center : tuple of float, length ``image.ndim`` The centroid of the (nonzero) pixels in ``image``. Examples -------- >>> image = np.zeros((20, 20), dtype=np.float64) >>> image[13:17, 13:17] = 0.5 >>> image[10:12, 10:12] = 1 >>> centroid(image) array([13.16666667, 13.16666667]) r,r)r"r r*r2)r.r#rreyeint)r/r*Mr"s r centroidrYxs[. dUZZ&7q'RA %uzz- ./ D5::    MrcUc [USUS9nUSUR-n[R"URUR4URS9n[ S[R "UR[S9-5n[R"U5nSURl [R"X5X- U- USS&[R"[UR5S5Hgn[R"UR[S9nSU[U5'U[ U5*U- XG'U[ U5*U- UR U'Mi U$)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. Nrr-r,r2Tr)r.r#rzerosrrrVrWdiagflags writeabler!rH combinationsrlistT) r/rKr*rMresultcorners2ddimsmu_indexs r inertia_tensorrgs*: z  G  TEJJ  C XXuzz5::.bhh ?FQ #667H AAGG FF2< 2< /3 6AaD&&uUZZ'8!<88EJJc2 d5?++c1 U8_--3 = MrcUc [XUS9n[RRU5n[R"USSUS9n[ USS9$)aCompute the eigenvalues of the inertia tensor of the image. The inertia tensor measures covariance of the image intensity along the image axes. (See `inertia_tensor`.) The relative magnitude of the eigenvalues of the tensor is thus a measure of the elongation of a (bright) object in the image. Parameters ---------- image : array The input image. mu : array, optional The pre-computed central moments of ``image``. T : array, shape ``(image.ndim, image.ndim)`` The pre-computed inertia tensor. If ``T`` is given, ``mu`` and ``image`` are ignored. spacing: tuple of float, shape (ndim,) The pixel spacing along each axis of the image. Returns ------- eigvals : list of float, length ``image.ndim`` The eigenvalues of the inertia tensor of ``image``, in descending order. Notes ----- Computing the eigenvalues requires the inertia tensor of the input image. This is much faster if the central moments (``mu``) are provided, or, alternatively, one can provide the inertia tensor (``T``) directly. Nr)r)outT)reverse)rgrlinalgeigvalshclipsorted)r/rKrar*eigvalss r inertia_tensor_eigvalsrpsM@ y 5g 6ii  #G gggq$G4G '4 ((r))Nrq)rqN)N)NN)rHnumpyr _shared.utilsrrr_moments_analyticalrrr r0r.rPrTrYrgrprr rwsp;7&:RdN,St,S^BDBJ;|)@X $@5d5p()d()r