# core¶

Core objects

 test Run tests for module using nose.

## Module: core.geometry¶

Utility functions for algebra etc

 cart2sphere(x, y, z) Return angles for Cartesian 3D coordinates x, y, and z cart_distance(pts1, pts2) Cartesian distance between pts1 and pts2 circumradius(a, b, c) a, b and c are 3-dimensional vectors which are the vertices of a compose_matrix([scale, shear, angles, ...]) Return 4x4 transformation matrix from sequence of transformations. compose_transformations(*mats) Compose multiple 4x4 affine transformations in one 4x4 matrix decompose_matrix(matrix) Return sequence of transformations from transformation matrix. dist_to_corner(affine) Calculate the maximal distance from the center to a corner of a voxel, euler_matrix(ai, aj, ak[, axes]) Return homogeneous rotation matrix from Euler angles and axis sequence. lambert_equal_area_projection_cart(x, y, z) Lambert Equal Area Projection from cartesian vector to plane lambert_equal_area_projection_polar(theta, phi) Lambert Equal Area Projection from polar sphere to plane nearest_pos_semi_def(B) Least squares positive semi-definite tensor estimation normalized_vector(vec[, axis]) Return vector divided by its Euclidean (L2) norm perpendicular_directions(v[, num, half]) Computes n evenly spaced perpendicular directions relative to a given rodrigues_axis_rotation(r, theta) Rodrigues formula setup_test() Set numpy print options to “legacy” for new versions of numpy sph2latlon(theta, phi) Convert spherical coordinates to latitude and longitude. sphere2cart(r, theta, phi) Spherical to Cartesian coordinates sphere_distance(pts1, pts2[, radius, ...]) Distance across sphere surface between pts1 and pts2 vec2vec_rotmat(u, v) rotation matrix from 2 unit vectors vector_cosine(vecs1, vecs2) Cosine of angle between two (sets of) vectors vector_norm(vec[, axis, keepdims]) Return vector Euclidean (L2) norm

## Module: core.gradients¶

 GradientTable(gradients[, big_delta, ...]) Diffusion gradient information HemiSphere([x, y, z, theta, phi, xyz, ...]) Points on the unit sphere. auto_attr(func) Decorator to create OneTimeProperty attributes. check_multi_b(gtab, n_bvals[, non_zero, bmag]) Check if you have enough different b-values in your gradient table disperse_charges(hemi, iters[, const]) Models electrostatic repulsion on the unit sphere generate_bvecs(N[, iters]) Generates N bvectors. gradient_table(bvals[, bvecs, big_delta, ...]) A general function for creating diffusion MR gradients. gradient_table_from_bvals_bvecs(bvals, bvecs) Creates a GradientTable from a bvals array and a bvecs array inv(a[, overwrite_a, check_finite]) Compute the inverse of a matrix. polar(a[, side]) Compute the polar decomposition. reorient_bvecs(gtab, affines) Reorient the directions in a GradientTable. vector_norm(vec[, axis, keepdims]) Return vector Euclidean (L2) norm

## Module: core.graph¶

A simple graph class

 Graph() A simple graph class

## Module: core.histeq¶

 histeq(arr[, num_bins]) Performs an histogram equalization on arr.

## Module: core.ndindex¶

 as_strided(x[, shape, strides, subok, writeable]) Create a view into the array with the given shape and strides. ndindex(shape) An N-dimensional iterator object to index arrays.

## Module: core.onetime¶

Descriptor support for NIPY.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

• Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
• Redistributions in binary form must reproduce the above
copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
• Neither the name of the NIPY Developers nor the names of any
contributors may be used to endorse or promote products derived from this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Utilities to support special Python descriptors [1,2], in particular the use of a useful pattern for properties we call ‘one time properties’. These are object attributes which are declared as properties, but become regular attributes once they’ve been read the first time. They can thus be evaluated later in the object’s life cycle, but once evaluated they become normal, static attributes with no function call overhead on access or any other constraints.

A special ResetMixin class is provided to add a .reset() method to users who may want to have their objects capable of resetting these computed properties to their ‘untriggered’ state.

### References¶

[2] Python data model, http://docs.python.org/reference/datamodel.html

 OneTimeProperty(func) A descriptor to make special properties that become normal attributes. ResetMixin A Mixin class to add a .reset() method to users of OneTimeProperty. auto_attr(func) Decorator to create OneTimeProperty attributes. setattr_on_read(func) Decorator to create OneTimeProperty attributes.

## Module: core.optimize¶

A unified interface for performing and debugging optimization problems.

Only L-BFGS-B and Powell is supported in this class for versions of Scipy < 0.12. All optimizers are available for scipy >= 0.12.

 LooseVersion([vstring]) Version numbering for anarchists and software realists. NonNegativeLeastSquares(*args, **kwargs) A sklearn-like interface to scipy.optimize.nnls Optimizer(fun, x0[, args, method, jac, ...]) Attributes SKLearnLinearSolver(*args, **kwargs) Provide a sklearn-like uniform interface to algorithms that solve problems minimize(fun, x0[, args, method, jac, hess, ...]) Minimization of scalar function of one or more variables. sparse_nnls(y, X[, momentum, step_size, ...]) Solve y=Xh for h, using gradient descent, with X a sparse matrix spdot(A, B) The same as np.dot(A, B), except it works even if A or B or both are sparse matrices. with_metaclass(meta, *bases) Create a base class with a metaclass.

## Module: core.profile¶

Class for profiling cython code

 Profiler([call]) Profile python/cython files or functions optional_package(name[, trip_msg]) Return package-like thing and module setup for package name

## Module: core.rng¶

Random number generation utilities

 LEcuyer() Generate uniformly distributed random numbers using the 32-bit generator from figure 3 of: L’Ecuyer, P. WichmannHill1982() Algorithm AS 183 Appl. WichmannHill2006() B.A. architecture([executable, bits, linkage]) Queries the given executable (defaults to the Python interpreter binary) for various architecture information. floor(x) Return the floor of x as an Integral.

## Module: core.sphere¶

 HemiSphere([x, y, z, theta, phi, xyz, ...]) Points on the unit sphere. Sphere([x, y, z, theta, phi, xyz, faces, edges]) Points on the unit sphere. xrange alias of range auto_attr(func) Decorator to create OneTimeProperty attributes. cart2sphere(x, y, z) Return angles for Cartesian 3D coordinates x, y, and z disperse_charges(hemi, iters[, const]) Models electrostatic repulsion on the unit sphere euler_characteristic_check(sphere[, chi]) Checks the euler characteristic of a sphere faces_from_sphere_vertices(vertices) Triangulate a set of vertices on the sphere. interp_rbf(data, sphere_origin, sphere_target) Interpolate data on the sphere, using radial basis functions. remove_similar_vertices Remove vertices that are less than theta degrees from any other sphere2cart(r, theta, phi) Spherical to Cartesian coordinates unique_edges(faces[, return_mapping]) Extract all unique edges from given triangular faces. unique_sets(sets[, return_inverse]) Remove duplicate sets. vector_norm(vec[, axis, keepdims]) Return vector Euclidean (L2) norm

## Module: core.sphere_stats¶

Statistics on spheres

 permutations permutations(iterable[, r]) –> permutations object angular_similarity(S, T) Computes the cosine distance of the best match between compare_orientation_sets(S, T) Computes the mean cosine distance of the best match between eigenstats(points[, alpha]) Principal direction and confidence ellipse random_uniform_on_sphere([n, coords]) Random unit vectors from a uniform distribution on the sphere.

## Module: core.subdivide_octahedron¶

Create a unit sphere by subdividing all triangles of an octahedron recursively.

The unit sphere has a radius of 1, which also means that all points in this sphere (assumed to have centre at [0, 0, 0]) have an absolute value (modulus) of 1. Another feature of the unit sphere is that the unit normals of this sphere are exactly the same as the vertices.

This recursive method will avoid the common problem of the polar singularity, produced by 2d (lon-lat) parameterization methods.

 HemiSphere([x, y, z, theta, phi, xyz, ...]) Points on the unit sphere. create_unit_hemisphere([recursion_level]) Creates a unit sphere by subdividing a unit octahedron, returns half the sphere. create_unit_sphere([recursion_level]) Creates a unit sphere by subdividing a unit octahedron.

## Module: core.wavelet¶

 afb3D(x, af1[, af2, af3]) 3D Analysis Filter Bank afb3D_A(x, af, d) 3D Analysis Filter Bank cshift3D(x, m, d) 3D Circular Shift dwt3D(x, J, af) 3-D Discrete Wavelet Transform idwt3D(w, J, sf) Inverse 3-D Discrete Wavelet Transform permutationinverse(perm) Function generating inverse of the permutation sfb3D(lo, hi, sf1[, sf2, sf3]) 3D Synthesis Filter Bank sfb3D_A(lo, hi, sf, d) 3D Synthesis Filter Bank

### test¶

dipy.core.test(self, label='fast', verbose=1, extra_argv=None, doctests=False, coverage=False, raise_warnings=None)

Run tests for module using nose.

Parameters: label : {‘fast’, ‘full’, ‘’, attribute identifier}, optional Identifies the tests to run. This can be a string to pass to the nosetests executable with the ‘-A’ option, or one of several special values. Special values are: * ‘fast’ - the default - which corresponds to the nosetests -A option of ‘not slow’. ‘full’ - fast (as above) and slow tests as in the ‘no -A’ option to nosetests - this is the same as ‘’. None or ‘’ - run all tests. attribute_identifier - string passed directly to nosetests as ‘-A’. verbose : int, optional Verbosity value for test outputs, in the range 1-10. Default is 1. extra_argv : list, optional List with any extra arguments to pass to nosetests. doctests : bool, optional If True, run doctests in module. Default is False. coverage : bool, optional If True, report coverage of NumPy code. Default is False. (This requires the coverage module: raise_warnings : None, str or sequence of warnings, optional This specifies which warnings to configure as ‘raise’ instead of being shown once during the test execution. Valid strings are: “develop” : equals (Warning,) “release” : equals (), don’t raise on any warnings. The default is to use the class initialization value. result : object Returns the result of running the tests as a nose.result.TextTestResult object.

Notes

Each NumPy module exposes test in its namespace to run all tests for it. For example, to run all tests for numpy.lib:

>>> np.lib.test()


Examples

>>> result = np.lib.test()
Running unit tests for numpy.lib
...
Ran 976 tests in 3.933s


OK

>>> result.errors
[]
>>> result.knownfail
[]


### cart2sphere¶

dipy.core.geometry.cart2sphere(x, y, z)

Return angles for Cartesian 3D coordinates x, y, and z

See doc for sphere2cart for angle conventions and derivation of the formulae.

$$0\le\theta\mathrm{(theta)}\le\pi$$ and $$-\pi\le\phi\mathrm{(phi)}\le\pi$$

Parameters: x : array_like x coordinate in Cartesian space y : array_like y coordinate in Cartesian space z : array_like z coordinate r : array radius theta : array inclination (polar) angle phi : array azimuth angle

### cart_distance¶

dipy.core.geometry.cart_distance(pts1, pts2)

Cartesian distance between pts1 and pts2

If either of pts1 or pts2 is 2D, then we take the first dimension to index points, and the second indexes coordinate. More generally, we take the last dimension to be the coordinate dimension.

Parameters: pts1 : (N,R) or (R,) array_like where N is the number of points and R is the number of coordinates defining a point (R==3 for 3D) pts2 : (N,R) or (R,) array_like where N is the number of points and R is the number of coordinates defining a point (R==3 for 3D). It should be possible to broadcast pts1 against pts2 d : (N,) or (0,) array Cartesian distances between corresponding points in pts1 and pts2

sphere_distance
distance between points on sphere surface

Examples

>>> cart_distance([0,0,0], [0,0,3])
3.0


dipy.core.geometry.circumradius(a, b, c)

a, b and c are 3-dimensional vectors which are the vertices of a triangle. The function returns the circumradius of the triangle, i.e the radius of the smallest circle that can contain the triangle. In the degenerate case when the 3 points are collinear it returns half the distance between the furthest apart points.

Parameters: a, b, c : (3,) array_like the three vertices of the triangle circumradius : float the desired circumradius

### compose_matrix¶

dipy.core.geometry.compose_matrix(scale=None, shear=None, angles=None, translate=None, perspective=None)

Return 4x4 transformation matrix from sequence of transformations.

Code modified from the work of Christoph Gohlke link provided here http://www.lfd.uci.edu/~gohlke/code/transformations.py.html

This is the inverse of the decompose_matrix function.

Parameters: scale : (3,) array_like Scaling factors. shear : array_like Shear factors for x-y, x-z, y-z axes. angles : array_like Euler angles about static x, y, z axes. translate : array_like Translation vector along x, y, z axes. perspective : array_like Perspective partition of matrix. matrix : 4x4 array

Examples

>>> import math
>>> import numpy as np
>>> import dipy.core.geometry as gm
>>> scale = np.random.random(3) - 0.5
>>> shear = np.random.random(3) - 0.5
>>> angles = (np.random.random(3) - 0.5) * (2*math.pi)
>>> trans = np.random.random(3) - 0.5
>>> persp = np.random.random(4) - 0.5
>>> M0 = gm.compose_matrix(scale, shear, angles, trans, persp)


### compose_transformations¶

dipy.core.geometry.compose_transformations(*mats)

Compose multiple 4x4 affine transformations in one 4x4 matrix

Parameters: mat1 : array, (4, 4) mat2 : array, (4, 4) ... matN : array, (4, 4) matN x ... x mat2 x mat1 : array, (4, 4)

### decompose_matrix¶

dipy.core.geometry.decompose_matrix(matrix)

Return sequence of transformations from transformation matrix.

Code modified from the excellent work of Christoph Gohlke link provided here: http://www.lfd.uci.edu/~gohlke/code/transformations.py.html

Parameters: matrix : array_like Non-degenerative homogeneous transformation matrix scale : (3,) ndarray Three scaling factors. shear : (3,) ndarray Shear factors for x-y, x-z, y-z axes. angles : (3,) ndarray Euler angles about static x, y, z axes. translate : (3,) ndarray Translation vector along x, y, z axes. perspective : ndarray Perspective partition of matrix. ValueError If matrix is of wrong type or degenerative.

Examples

>>> import numpy as np
>>> T0=np.diag([2,1,1,1])
>>> scale, shear, angles, trans, persp = decompose_matrix(T0)


### dist_to_corner¶

dipy.core.geometry.dist_to_corner(affine)

Calculate the maximal distance from the center to a corner of a voxel, given an affine

Parameters: affine : 4 by 4 array. The spatial transformation from the measurement to the scanner space. dist: float The maximal distance to the corner of a voxel, given voxel size encoded in the affine.

### euler_matrix¶

dipy.core.geometry.euler_matrix(ai, aj, ak, axes='sxyz')

Return homogeneous rotation matrix from Euler angles and axis sequence.

Code modified from the work of Christoph Gohlke link provided here http://www.lfd.uci.edu/~gohlke/code/transformations.py.html

Parameters: ai, aj, ak : Euler’s roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple matrix : ndarray (4, 4) Code modified from the work of Christoph Gohlke link provided here http://www.lfd.uci.edu/~gohlke/code/transformations.py.html

Examples

>>> import numpy
>>> R = euler_matrix(1, 2, 3, 'syxz')
>>> numpy.allclose(numpy.sum(R[0]), -1.34786452)
True
>>> R = euler_matrix(1, 2, 3, (0, 1, 0, 1))
>>> numpy.allclose(numpy.sum(R[0]), -0.383436184)
True
>>> ai, aj, ak = (4.0*math.pi) * (numpy.random.random(3) - 0.5)
>>> for axes in _AXES2TUPLE.keys():
...    _ = euler_matrix(ai, aj, ak, axes)
>>> for axes in _TUPLE2AXES.keys():
...    _ = euler_matrix(ai, aj, ak, axes)


### lambert_equal_area_projection_cart¶

dipy.core.geometry.lambert_equal_area_projection_cart(x, y, z)

Lambert Equal Area Projection from cartesian vector to plane

Return positions in $$(y_1,y_2)$$ plane corresponding to the directions of the vectors with cartesian coordinates xyz under the Lambert Equal Area Projection mapping (see Mardia and Jupp (2000), Directional Statistics, p. 161).

The Lambert EAP maps the upper hemisphere to the planar disc of radius 1 and the lower hemisphere to the planar annulus between radii 1 and 2, The Lambert EAP maps the upper hemisphere to the planar disc of radius 1 and the lower hemisphere to the planar annulus between radii 1 and 2. and vice versa.

See doc for sphere2cart for angle conventions

Parameters: x : array_like x coordinate in Cartesion space y : array_like y coordinate in Cartesian space z : array_like z coordinate y : (N,2) array planar coordinates of points following mapping by Lambert’s EAP.

### lambert_equal_area_projection_polar¶

dipy.core.geometry.lambert_equal_area_projection_polar(theta, phi)

Lambert Equal Area Projection from polar sphere to plane

Return positions in (y1,y2) plane corresponding to the points with polar coordinates (theta, phi) on the unit sphere, under the Lambert Equal Area Projection mapping (see Mardia and Jupp (2000), Directional Statistics, p. 161).

See doc for sphere2cart for angle conventions

• $$0 \le \theta \le \pi$$ and $$0 \le \phi \le 2 \pi$$
• $$|(y_1,y_2)| \le 2$$

The Lambert EAP maps the upper hemisphere to the planar disc of radius 1 and the lower hemisphere to the planar annulus between radii 1 and 2, and vice versa.

Parameters: theta : array_like theta spherical coordinates phi : array_like phi spherical coordinates y : (N,2) array planar coordinates of points following mapping by Lambert’s EAP.

### nearest_pos_semi_def¶

dipy.core.geometry.nearest_pos_semi_def(B)

Least squares positive semi-definite tensor estimation

Parameters: B : (3,3) array_like B matrix - symmetric. We do not check the symmetry. npds : (3,3) array Estimated nearest positive semi-definite array to matrix B.

References

 [R14] Niethammer M, San Jose Estepar R, Bouix S, Shenton M, Westin CF. On diffusion tensor estimation. Conf Proc IEEE Eng Med Biol Soc. 2006;1:2622-5. PubMed PMID: 17946125; PubMed Central PMCID: PMC2791793.

Examples

>>> B = np.diag([1, 1, -1])
>>> nearest_pos_semi_def(B)
array([[ 0.75,  0.  ,  0.  ],
[ 0.  ,  0.75,  0.  ],
[ 0.  ,  0.  ,  0.  ]])


### normalized_vector¶

dipy.core.geometry.normalized_vector(vec, axis=-1)

Return vector divided by its Euclidean (L2) norm

Parameters: vec : array_like shape (3,) nvec : array shape (3,) vector divided by L2 norm

Examples

>>> vec = [1, 2, 3]
>>> l2n = np.sqrt(np.dot(vec, vec))
>>> nvec = normalized_vector(vec)
>>> np.allclose(np.array(vec) / l2n, nvec)
True
>>> vec = np.array([[1, 2, 3]])
>>> vec.shape == (1, 3)
True
>>> normalized_vector(vec).shape == (1, 3)
True


### perpendicular_directions¶

dipy.core.geometry.perpendicular_directions(v, num=30, half=False)

Computes n evenly spaced perpendicular directions relative to a given vector v

Parameters: v : array (3,) Array containing the three cartesian coordinates of vector v num : int, optional Number of perpendicular directions to generate half : bool, optional If half is True, perpendicular directions are sampled on half of the unit circumference perpendicular to v, otherwive perpendicular directions are sampled on the full circumference. Default of half is False psamples : array (n, 3) array of vectors perpendicular to v

Notes

Perpendicular directions are estimated using the following two step procedure:

1) the perpendicular directions are first sampled in a unit circumference parallel to the plane normal to the x-axis.

2) Samples are then rotated and aligned to the plane normal to vector v. The rotational matrix for this rotation is constructed as reference frame basis which axis are the following:

• The first axis is vector v
• The second axis is defined as the normalized vector given by the

cross product between vector v and the unit vector aligned to the x-axis - The third axis is defined as the cross product between the previous computed vector and vector v.

Following this two steps, coordinates of the final perpendicular directions are given as:

$\left [ -\sin(a_{i}) \sqrt{{v_{y}}^{2}+{v_{z}}^{2}} \; , \; \frac{v_{x}v_{y}\sin(a_{i})-v_{z}\cos(a_{i})} {\sqrt{{v_{y}}^{2}+{v_{z}}^{2}}} \; , \; \frac{v_{x}v_{z}\sin(a_{i})-v_{y}\cos(a_{i})} {\sqrt{{v_{y}}^{2}+{v_{z}}^{2}}} \right ]$

This procedure has a singularity when vector v is aligned to the x-axis. To solve this singularity, perpendicular directions in procedure’s step 1 are defined in the plane normal to y-axis and the second axis of the rotated frame of reference is computed as the normalized vector given by the cross product between vector v and the unit vector aligned to the y-axis. Following this, the coordinates of the perpendicular directions are given as:

left [ -frac{left (v_{x}v_{y}sin(a_{i})+v_{z}cos(a_{i}) right )} {sqrt{{v_{x}}^{2}+{v_{z}}^{2}}} ; , ; sin(a_{i}) sqrt{{v_{x}}^{2}+{v_{z}}^{2}} ; , ; frac{v_{y}v_{z}sin(a_{i})+v_{x}cos(a_{i})} {sqrt{{v_{x}}^{2}+{v_{z}}^{2}}} right ]

For more details on this calculation, see  here <http://gsoc2015dipydki.blogspot.it/2015/07/rnh-post-8-computing-perpendicular.html>_.

### rodrigues_axis_rotation¶

dipy.core.geometry.rodrigues_axis_rotation(r, theta)

Rodrigues formula

Rotation matrix for rotation around axis r for angle theta.

The rotation matrix is given by the Rodrigues formula:

R = Id + sin(theta)*Sn + (1-cos(theta))*Sn^2

with:

       0  -nz  ny
Sn =   nz   0 -nx
-ny  nx   0


where n = r / ||r||

In case the angle ||r|| is very small, the above formula may lead to numerical instabilities. We instead use a Taylor expansion around theta=0:

R = I + sin(theta)/tetha Sr + (1-cos(theta))/teta2 Sr^2

R = I + (1-theta2/6)*Sr + (1/2-theta2/24)*Sr^2

Parameters: r : array_like shape (3,), axis theta : float, angle in degrees R : array, shape (3,3), rotation matrix

Examples

>>> import numpy as np
>>> from dipy.core.geometry import rodrigues_axis_rotation
>>> v=np.array([0,0,1])
>>> u=np.array([1,0,0])
>>> R=rodrigues_axis_rotation(v,40)
>>> ur=np.dot(R,u)
40.0


### setup_test¶

dipy.core.geometry.setup_test()

Set numpy print options to “legacy” for new versions of numpy

If imported into a file, nosetest will run this before any doctests.

References

### sph2latlon¶

dipy.core.geometry.sph2latlon(theta, phi)

Convert spherical coordinates to latitude and longitude.

Returns: lat, lon : ndarray Latitude and longitude.

### sphere2cart¶

dipy.core.geometry.sphere2cart(r, theta, phi)

Spherical to Cartesian coordinates

This is the standard physics convention where theta is the inclination (polar) angle, and phi is the azimuth angle.

Imagine a sphere with center (0,0,0). Orient it with the z axis running south-north, the y axis running west-east and the x axis from posterior to anterior. theta (the inclination angle) is the angle to rotate from the z-axis (the zenith) around the y-axis, towards the x axis. Thus the rotation is counter-clockwise from the point of view of positive y. phi (azimuth) gives the angle of rotation around the z-axis towards the y axis. The rotation is counter-clockwise from the point of view of positive z.

Equivalently, given a point P on the sphere, with coordinates x, y, z, theta is the angle between P and the z-axis, and phi is the angle between the projection of P onto the XY plane, and the X axis.

Geographical nomenclature designates theta as ‘co-latitude’, and phi as ‘longitude’

Parameters: r : array_like radius theta : array_like inclination or polar angle phi : array_like azimuth angle x : array x coordinate(s) in Cartesion space y : array y coordinate(s) in Cartesian space z : array z coordinate

Notes

See these pages:

for excellent discussion of the many different conventions possible. Here we use the physics conventions, used in the wikipedia page.

Derivations of the formulae are simple. Consider a vector x, y, z of length r (norm of x, y, z). The inclination angle (theta) can be found from: cos(theta) == z / r -> z == r * cos(theta). This gives the hypotenuse of the projection onto the XY plane, which we will call Q. Q == r*sin(theta). Now x / Q == cos(phi) -> x == r * sin(theta) * cos(phi) and so on.

We have deliberately named this function sphere2cart rather than sph2cart to distinguish it from the Matlab function of that name, because the Matlab function uses an unusual convention for the angles that we did not want to replicate. The Matlab function is trivial to implement with the formulae given in the Matlab help.

### sphere_distance¶

dipy.core.geometry.sphere_distance(pts1, pts2, radius=None, check_radius=True)

Distance across sphere surface between pts1 and pts2

Parameters: pts1 : (N,R) or (R,) array_like where N is the number of points and R is the number of coordinates defining a point (R==3 for 3D) pts2 : (N,R) or (R,) array_like where N is the number of points and R is the number of coordinates defining a point (R==3 for 3D). It should be possible to broadcast pts1 against pts2 radius : None or float, optional Radius of sphere. Default is to work out radius from mean of the length of each point vector check_radius : bool, optional If True, check if the points are on the sphere surface - i.e check if the vector lengths in pts1 and pts2 are close to radius. Default is True. d : (N,) or (0,) array Distances between corresponding points in pts1 and pts2 across the spherical surface, i.e. the great circle distance

cart_distance
cartesian distance between points
vector_cosine
cosine of angle between vectors

Examples

>>> print('%.4f' % sphere_distance([0,1],[1,0]))
1.5708
>>> print('%.4f' % sphere_distance([0,3],[3,0]))
4.7124


### vec2vec_rotmat¶

dipy.core.geometry.vec2vec_rotmat(u, v)

rotation matrix from 2 unit vectors

u, v being unit 3d vectors return a 3x3 rotation matrix R than aligns u to v.

In general there are many rotations that will map u to v. If S is any rotation using v as an axis then R.S will also map u to v since (S.R)u = S(Ru) = Sv = v. The rotation R returned by vec2vec_rotmat leaves fixed the perpendicular to the plane spanned by u and v.

The transpose of R will align v to u.

Parameters: u : array, shape(3,) v : array, shape(3,) R : array, shape(3,3)

Examples

>>> import numpy as np
>>> from dipy.core.geometry import vec2vec_rotmat
>>> u=np.array([1,0,0])
>>> v=np.array([0,1,0])
>>> R=vec2vec_rotmat(u,v)
>>> np.dot(R,u)
array([ 0.,  1.,  0.])
>>> np.dot(R.T,v)
array([ 1.,  0.,  0.])


### vector_cosine¶

dipy.core.geometry.vector_cosine(vecs1, vecs2)

Cosine of angle between two (sets of) vectors

The cosine of the angle between two vectors v1 and v2 is given by the inner product of v1 and v2 divided by the product of the vector lengths:

v_cos = np.inner(v1, v2) / (np.sqrt(np.sum(v1**2)) *
np.sqrt(np.sum(v2**2)))

Parameters: vecs1 : (N, R) or (R,) array_like N vectors (as rows) or single vector. Vectors have R elements. vecs1 : (N, R) or (R,) array_like N vectors (as rows) or single vector. Vectors have R elements. It should be possible to broadcast vecs1 against vecs2 vcos : (N,) or (0,) array Vector cosines. To get the angles you will need np.arccos

Notes

The vector cosine will be the same as the correlation only if all the input vectors have zero mean.

### vector_norm¶

dipy.core.geometry.vector_norm(vec, axis=-1, keepdims=False)

Return vector Euclidean (L2) norm

Parameters: vec : array_like Vectors to norm. axis : int Axis over which to norm. By default norm over last axis. If axis is None, vec is flattened then normed. keepdims : bool If True, the output will have the same number of dimensions as vec, with shape 1 on axis. norm : array Euclidean norms of vectors.

Examples

>>> import numpy as np
>>> vec = [[8, 15, 0], [0, 36, 77]]
>>> vector_norm(vec)
array([ 17.,  85.])
>>> vector_norm(vec, keepdims=True)
array([[ 17.],
[ 85.]])
>>> vector_norm(vec, axis=0)
array([  8.,  39.,  77.])


### GradientTable¶

class dipy.core.gradients.GradientTable(gradients, big_delta=None, small_delta=None, b0_threshold=0)

Bases: object

Parameters: gradients : array_like (N, 3) Diffusion gradients. The direction of each of these vectors corresponds to the b-vector, and the length corresponds to the b-value. b0_threshold : float Gradients with b-value less than or equal to b0_threshold are considered as b0s i.e. without diffusion weighting.

Notes

The GradientTable object is immutable. Do NOT assign attributes. If you have your gradient table in a bval & bvec format, we recommend using the factory function gradient_table

Attributes

 gradients ((N,3) ndarray) diffusion gradients bvals ((N,) ndarray) The b-value, or magnitude, of each gradient direction. qvals: (N,) ndarray The q-value for each gradient direction. Needs big and small delta. bvecs ((N,3) ndarray) The direction, represented as a unit vector, of each gradient. b0s_mask ((N,) ndarray) Boolean array indicating which gradients have no diffusion weighting, ie b-value is close to 0. b0_threshold (float) Gradients with b-value less than or equal to b0_threshold are considered to not have diffusion weighting.

Methods

 b0s_mask() bvals() bvecs() qvals()
__init__(gradients, big_delta=None, small_delta=None, b0_threshold=0)

b0s_mask()
bvals()
bvecs()
info
qvals()

### HemiSphere¶

class dipy.core.gradients.HemiSphere(x=None, y=None, z=None, theta=None, phi=None, xyz=None, faces=None, edges=None, tol=1e-05)

Points on the unit sphere.

A HemiSphere is similar to a Sphere but it takes antipodal symmetry into account. Antipodal symmetry means that point v on a HemiSphere is the same as the point -v. Duplicate points are discarded when constructing a HemiSphere (including antipodal duplicates). edges and faces are remapped to the remaining points as closely as possible.

The HemiSphere can be constructed using one of three conventions:

HemiSphere(x, y, z)
HemiSphere(xyz=xyz)
HemiSphere(theta=theta, phi=phi)

Parameters: x, y, z : 1-D array_like Vertices as x-y-z coordinates. theta, phi : 1-D array_like Vertices as spherical coordinates. Theta and phi are the inclination and azimuth angles respectively. xyz : (N, 3) ndarray Vertices as x-y-z coordinates. faces : (N, 3) ndarray Indices into vertices that form triangular faces. If unspecified, the faces are computed using a Delaunay triangulation. edges : (N, 2) ndarray Edges between vertices. If unspecified, the edges are derived from the faces. tol : float Angle in degrees. Vertices that are less than tol degrees apart are treated as duplicates.

Sphere

Attributes

 x y z

Methods

 edges() faces() find_closest(xyz) Find the index of the vertex in the Sphere closest to the input vector, from_sphere(klass, sphere[, tol]) Create instance from a Sphere mirror() Create a full Sphere from a HemiSphere subdivide([n]) Create a more subdivided HemiSphere vertices()
__init__(x=None, y=None, z=None, theta=None, phi=None, xyz=None, faces=None, edges=None, tol=1e-05)

Create a HemiSphere from points

faces()
find_closest(xyz)

Find the index of the vertex in the Sphere closest to the input vector, taking into account antipodal symmetry

Parameters: xyz : array-like, 3 elements A unit vector
classmethod from_sphere(klass, sphere, tol=1e-05)

Create instance from a Sphere

mirror()

Create a full Sphere from a HemiSphere

subdivide(n=1)

Create a more subdivided HemiSphere

See Sphere.subdivide for full documentation.

### auto_attr¶

dipy.core.gradients.auto_attr(func)

Decorator to create OneTimeProperty attributes.

Parameters: func : method The method that will be called the first time to compute a value. Afterwards, the method’s name will be a standard attribute holding the value of this computation.

Examples

>>> class MagicProp(object):
...     @auto_attr
...     def a(self):
...         return 99
...
>>> x = MagicProp()
>>> 'a' in x.__dict__
False
>>> x.a
99
>>> 'a' in x.__dict__
True


### check_multi_b¶

dipy.core.gradients.check_multi_b(gtab, n_bvals, non_zero=True, bmag=None)

Parameters: gtab : GradientTable class instance. n_bvals : int The number of different b-values you are checking for. non_zero : bool Whether to check only non-zero bvalues. In this case, we will require at least n_bvals non-zero b-values (where non-zero is defined depending on the gtab object’s b0_threshold attribute) bmag : int The order of magnitude of the b-values used. The function will normalize the b-values relative $$10^{bmag - 1}$$. Default: derive this value from the maximal b-value provided: $$bmag=log_{10}(max(bvals))$$. bool : Whether there are at least n_bvals different b-values in the gradient table used.

### disperse_charges¶

dipy.core.gradients.disperse_charges(hemi, iters, const=0.2)

Models electrostatic repulsion on the unit sphere

Places charges on a sphere and simulates the repulsive forces felt by each one. Allows the charges to move for some number of iterations and returns their final location as well as the total potential of the system at each step.

Parameters: hemi : HemiSphere Points on a unit sphere. iters : int Number of iterations to run. const : float Using a smaller const could provide a more accurate result, but will need more iterations to converge. hemi : HemiSphere Distributed points on a unit sphere. potential : ndarray The electrostatic potential at each iteration. This can be useful to check if the repulsion converged to a minimum.

### generate_bvecs¶

dipy.core.gradients.generate_bvecs(N, iters=5000)

Generates N bvectors.

Uses dipy.core.sphere.disperse_charges to model electrostatic repulsion on a unit sphere.

Parameters: N : int The number of bvectors to generate. This should be equal to the number of bvals used. iters : int Number of iterations to run. bvecs : (N,3) ndarray The generated directions, represented as a unit vector, of each gradient.

dipy.core.gradients.gradient_table(bvals, bvecs=None, big_delta=None, small_delta=None, b0_threshold=0, atol=0.01)

A general function for creating diffusion MR gradients.

It reads, loads and prepares scanner parameters like the b-values and b-vectors so that they can be useful during the reconstruction process.

Parameters: bvals : can be any of the four options an array of shape (N,) or (1, N) or (N, 1) with the b-values. a path for the file which contains an array like the above (1). an array of shape (N, 4) or (4, N). Then this parameter is considered to be a b-table which contains both bvals and bvecs. In this case the next parameter is skipped. a path for the file which contains an array like the one at (3). bvecs : can be any of two options an array of shape (N, 3) or (3, N) with the b-vectors. a path for the file which contains an array like the previous. big_delta : float acquisition timing duration (default None) small_delta : float acquisition timing duration (default None) b0_threshold : float All b-values with values less than or equal to bo_threshold are considered as b0s i.e. without diffusion weighting. atol : float All b-vectors need to be unit vectors up to a tolerance. gradients : GradientTable A GradientTable with all the gradient information.

Notes

1. Often b0s (b-values which correspond to images without diffusion weighting) have 0 values however in some cases the scanner cannot provide b0s of an exact 0 value and it gives a bit higher values e.g. 6 or 12. This is the purpose of the b0_threshold in the __init__.
2. We assume that the minimum number of b-values is 7.
3. B-vectors should be unit vectors.

Examples

>>> from dipy.core.gradients import gradient_table
>>> bvals=1500*np.ones(7)
>>> bvals[0]=0
>>> sq2=np.sqrt(2)/2
>>> bvecs=np.array([[0, 0, 0],
...                 [1, 0, 0],
...                 [0, 1, 0],
...                 [0, 0, 1],
...                 [sq2, sq2, 0],
...                 [sq2, 0, sq2],
...                 [0, sq2, sq2]])
>>> gt.bvecs.shape == bvecs.shape
True
>>> gt.bvecs.shape == bvecs.T.shape
False


dipy.core.gradients.gradient_table_from_bvals_bvecs(bvals, bvecs, b0_threshold=0, atol=0.01, **kwargs)

Creates a GradientTable from a bvals array and a bvecs array

Parameters: Returns: bvals : array_like (N,) The b-value, or magnitude, of each gradient direction. bvecs : array_like (N, 3) The direction, represented as a unit vector, of each gradient. b0_threshold : float Gradients with b-value less than or equal to bo_threshold are considered to not have diffusion weighting. atol : float Each vector in bvecs must be a unit vectors up to a tolerance of atol. gradients : GradientTable A GradientTable with all the gradient information. **kwargs : dict Other keyword inputs are passed to GradientTable.

### inv¶

dipy.core.gradients.inv(a, overwrite_a=False, check_finite=True)

Compute the inverse of a matrix.

Parameters: a : array_like Square matrix to be inverted. overwrite_a : bool, optional Discard data in a (may improve performance). Default is False. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. ainv : ndarray Inverse of the matrix a. LinAlgError If a is singular. ValueError If a is not square, or not 2-dimensional.

Examples

>>> from scipy import linalg
>>> a = np.array([[1., 2.], [3., 4.]])
>>> linalg.inv(a)
array([[-2. ,  1. ],
[ 1.5, -0.5]])
>>> np.dot(a, linalg.inv(a))
array([[ 1.,  0.],
[ 0.,  1.]])


### polar¶

dipy.core.gradients.polar(a, side='right')

Compute the polar decomposition.

Returns the factors of the polar decomposition [R15] u and p such that a = up (if side is “right”) or a = pu (if side is “left”), where p is positive semidefinite. Depending on the shape of a, either the rows or columns of u are orthonormal. When a is a square array, u is a square unitary array. When a is not square, the “canonical polar decomposition” [R16] is computed.

Parameters: a : (m, n) array_like The array to be factored. side : {‘left’, ‘right’}, optional Determines whether a right or left polar decomposition is computed. If side is “right”, then a = up. If side is “left”, then a = pu. The default is “right”. u : (m, n) ndarray If a is square, then u is unitary. If m > n, then the columns of a are orthonormal, and if m < n, then the rows of u are orthonormal. p : ndarray p is Hermitian positive semidefinite. If a is nonsingular, p is positive definite. The shape of p is (n, n) or (m, m), depending on whether side is “right” or “left”, respectively.

References

 [R15] (1, 2) R. A. Horn and C. R. Johnson, “Matrix Analysis”, Cambridge University Press, 1985.
 [R16] (1, 2) N. J. Higham, “Functions of Matrices: Theory and Computation”, SIAM, 2008.

Examples

>>> from scipy.linalg import polar
>>> a = np.array([[1, -1], [2, 4]])
>>> u, p = polar(a)
>>> u
array([[ 0.85749293, -0.51449576],
[ 0.51449576,  0.85749293]])
>>> p
array([[ 1.88648444,  1.2004901 ],
[ 1.2004901 ,  3.94446746]])


A non-square example, with m < n:

>>> b = np.array([[0.5, 1, 2], [1.5, 3, 4]])
>>> u, p = polar(b)
>>> u
array([[-0.21196618, -0.42393237,  0.88054056],
[ 0.39378971,  0.78757942,  0.4739708 ]])
>>> p
array([[ 0.48470147,  0.96940295,  1.15122648],
[ 0.96940295,  1.9388059 ,  2.30245295],
[ 1.15122648,  2.30245295,  3.65696431]])
>>> u.dot(p)   # Verify the decomposition.
array([[ 0.5,  1. ,  2. ],
[ 1.5,  3. ,  4. ]])
>>> u.dot(u.T)   # The rows of u are orthonormal.
array([[  1.00000000e+00,  -2.07353665e-17],
[ -2.07353665e-17,   1.00000000e+00]])


Another non-square example, with m > n:

>>> c = b.T
>>> u, p = polar(c)
>>> u
array([[-0.21196618,  0.39378971],
[-0.42393237,  0.78757942],
[ 0.88054056,  0.4739708 ]])
>>> p
array([[ 1.23116567,  1.93241587],
[ 1.93241587,  4.84930602]])
>>> u.dot(p)   # Verify the decomposition.
array([[ 0.5,  1.5],
[ 1. ,  3. ],
[ 2. ,  4. ]])
>>> u.T.dot(u)  # The columns of u are orthonormal.
array([[  1.00000000e+00,  -1.26363763e-16],
[ -1.26363763e-16,   1.00000000e+00]])


### reorient_bvecs¶

dipy.core.gradients.reorient_bvecs(gtab, affines)

Reorient the directions in a GradientTable.

When correcting for motion, rotation of the diffusion-weighted volumes might cause systematic bias in rotationally invariant measures, such as FA and MD, and also cause characteristic biases in tractography, unless the gradient directions are appropriately reoriented to compensate for this effect [Leemans2009].

Parameters: gtab : GradientTable The nominal gradient table with which the data were acquired. affines : list or ndarray of shape (n, 4, 4) or (n, 3, 3) Each entry in this list or array contain either an affine transformation (4,4) or a rotation matrix (3, 3). In both cases, the transformations encode the rotation that was applied to the image corresponding to one of the non-zero gradient directions (ordered according to their order in gtab.bvecs[~gtab.b0s_mask]) gtab : a GradientTable class instance with the reoriented directions

References

 [Leemans2009] (1, 2) The B-Matrix Must Be Rotated When Correcting for Subject Motion in DTI Data. Leemans, A. and Jones, D.K. (2009). MRM, 61: 1336-1349

### vector_norm¶

dipy.core.gradients.vector_norm(vec, axis=-1, keepdims=False)

Return vector Euclidean (L2) norm

Parameters: vec : array_like Vectors to norm. axis : int Axis over which to norm. By default norm over last axis. If axis is None, vec is flattened then normed. keepdims : bool If True, the output will have the same number of dimensions as vec, with shape 1 on axis. norm : array Euclidean norms of vectors.

Examples

>>> import numpy as np
>>> vec = [[8, 15, 0], [0, 36, 77]]
>>> vector_norm(vec)
array([ 17.,  85.])
>>> vector_norm(vec, keepdims=True)
array([[ 17.],
[ 85.]])
>>> vector_norm(vec, axis=0)
array([  8.,  39.,  77.])


### Graph¶

class dipy.core.graph.Graph

Bases: object

A simple graph class

Methods

 add_edge(n, m[, ws, wp]) add_node(n[, attr]) all_paths(graph, start[, end, path]) children(n) del_node(n) del_node_and_edges(n) down(n) down_short(n) parents(n) shortest_path(graph, start[, end, path]) up(n) up_short(n)
__init__()

A graph class with nodes and edges :-)

This class allows us to:

1. find the shortest path
2. find all paths
4. get parent & children nodes

Examples

>>> from dipy.core.graph import Graph
>>> g=Graph()
>>> g.up_short('d')
['d', 'b', 'a']

add_edge(n, m, ws=True, wp=True)
add_node(n, attr=None)
all_paths(graph, start, end=None, path=[])
children(n)
del_node(n)
del_node_and_edges(n)
down(n)
down_short(n)
parents(n)
shortest_path(graph, start, end=None, path=[])
up(n)
up_short(n)

### histeq¶

dipy.core.histeq.histeq(arr, num_bins=256)

Performs an histogram equalization on arr. This was taken from: http://www.janeriksolem.net/2009/06/histogram-equalization-with-python-and.html

Parameters: arr : ndarray Image on which to perform histogram equalization. num_bins : int Number of bins used to construct the histogram. result : ndarray Histogram equalized image.

### as_strided¶

dipy.core.ndindex.as_strided(x, shape=None, strides=None, subok=False, writeable=True)

Create a view into the array with the given shape and strides.

Warning

This function has to be used with extreme care, see notes.

Parameters: x : ndarray Array to create a new. shape : sequence of int, optional The shape of the new array. Defaults to x.shape. strides : sequence of int, optional The strides of the new array. Defaults to x.strides. subok : bool, optional New in version 1.10. If True, subclasses are preserved. writeable : bool, optional New in version 1.12. If set to False, the returned array will always be readonly. Otherwise it will be writable if the original array was. It is advisable to set this to False if possible (see Notes). view : ndarray

broadcast_to
broadcast an array to a given shape.
reshape
reshape an array.

Notes

as_strided creates a view into the array given the exact strides and shape. This means it manipulates the internal data structure of ndarray and, if done incorrectly, the array elements can point to invalid memory and can corrupt results or crash your program. It is advisable to always use the original x.strides when calculating new strides to avoid reliance on a contiguous memory layout.

Furthermore, arrays created with this function often contain self overlapping memory, so that two elements are identical. Vectorized write operations on such arrays will typically be unpredictable. They may even give different results for small, large, or transposed arrays. Since writing to these arrays has to be tested and done with great care, you may want to use writeable=False to avoid accidental write operations.

For these reasons it is advisable to avoid as_strided when possible.

### ndindex¶

dipy.core.ndindex.ndindex(shape)

An N-dimensional iterator object to index arrays.

Given the shape of an array, an ndindex instance iterates over the N-dimensional index of the array. At each iteration a tuple of indices is returned; the last dimension is iterated over first.

Parameters: shape : tuple of ints The dimensions of the array.

Examples

>>> from dipy.core.ndindex import ndindex
>>> shape = (3, 2, 1)
>>> for index in ndindex(shape):
...     print(index)
(0, 0, 0)
(0, 1, 0)
(1, 0, 0)
(1, 1, 0)
(2, 0, 0)
(2, 1, 0)


### OneTimeProperty¶

class dipy.core.onetime.OneTimeProperty(func)

Bases: object

A descriptor to make special properties that become normal attributes.

This is meant to be used mostly by the auto_attr decorator in this module.

__init__(func)

Create a OneTimeProperty instance.

Parameters: func : method The method that will be called the first time to compute a value. Afterwards, the method’s name will be a standard attribute holding the value of this computation.

### ResetMixin¶

class dipy.core.onetime.ResetMixin

Bases: object

A Mixin class to add a .reset() method to users of OneTimeProperty.

By default, auto attributes once computed, become static. If they happen to depend on other parts of an object and those parts change, their values may now be invalid.

This class offers a .reset() method that users can call explicitly when they know the state of their objects may have changed and they want to ensure that all their special attributes should be invalidated. Once reset() is called, all their auto attributes are reset to their OneTimeProperty descriptors, and their accessor functions will be triggered again.

Warning

If a class has a set of attributes that are OneTimeProperty, but that can be initialized from any one of them, do NOT use this mixin! For instance, UniformTimeSeries can be initialized with only sampling_rate and t0, sampling_interval and time are auto-computed. But if you were to reset() a UniformTimeSeries, it would lose all 4, and there would be then no way to break the circular dependency chains.

If this becomes a problem in practice (for our analyzer objects it isn’t, as they don’t have the above pattern), we can extend reset() to check for a _no_reset set of names in the instance which are meant to be kept protected. But for now this is NOT done, so caveat emptor.

Examples

>>> class A(ResetMixin):
...     def __init__(self,x=1.0):
...         self.x = x
...
...     @auto_attr
...     def y(self):
...         print('*** y computation executed ***')
...         return self.x / 2.0
...

>>> a = A(10)


About to access y twice, the second time no computation is done: >>> a.y * y computation executed * 5.0 >>> a.y 5.0

Changing x >>> a.x = 20

a.y doesn’t change to 10, since it is a static attribute: >>> a.y 5.0

We now reset a, and this will then force all auto attributes to recompute the next time we access them: >>> a.reset()

About to access y twice again after reset(): >>> a.y * y computation executed * 10.0 >>> a.y 10.0

Methods

 reset() Reset all OneTimeProperty attributes that may have fired already.
__init__()

Initialize self. See help(type(self)) for accurate signature.

reset()

Reset all OneTimeProperty attributes that may have fired already.

### auto_attr¶

dipy.core.onetime.auto_attr(func)

Decorator to create OneTimeProperty attributes.

Parameters: func : method The method that will be called the first time to compute a value. Afterwards, the method’s name will be a standard attribute holding the value of this computation.

Examples

>>> class MagicProp(object):
...     @auto_attr
...     def a(self):
...         return 99
...
>>> x = MagicProp()
>>> 'a' in x.__dict__
False
>>> x.a
99
>>> 'a' in x.__dict__
True


dipy.core.onetime.setattr_on_read(func)

Decorator to create OneTimeProperty attributes.

Parameters: func : method The method that will be called the first time to compute a value. Afterwards, the method’s name will be a standard attribute holding the value of this computation.

Examples

>>> class MagicProp(object):
...     @auto_attr
...     def a(self):
...         return 99
...
>>> x = MagicProp()
>>> 'a' in x.__dict__
False
>>> x.a
99
>>> 'a' in x.__dict__
True


### LooseVersion¶

class dipy.core.optimize.LooseVersion(vstring=None)

Bases: distutils.version.Version

Version numbering for anarchists and software realists. Implements the standard interface for version number classes as described above. A version number consists of a series of numbers, separated by either periods or strings of letters. When comparing version numbers, the numeric components will be compared numerically, and the alphabetic components lexically. The following are all valid version numbers, in no particular order:

1.5.1 1.5.2b2 161 3.10a 8.02 3.4j 1996.07.12 3.2.pl0 3.1.1.6 2g6 11g 0.960923 2.2beta29 1.13++ 5.5.kw 2.0b1pl0

In fact, there is no such thing as an invalid version number under this scheme; the rules for comparison are simple and predictable, but may not always give the results you want (for some definition of “want”).

Methods

 parse(vstring)
__init__(vstring=None)
component_re = re.compile('(\\d+ | [a-z]+ | \\.)', re.VERBOSE)
parse(vstring)

### NonNegativeLeastSquares¶

class dipy.core.optimize.NonNegativeLeastSquares(*args, **kwargs)

A sklearn-like interface to scipy.optimize.nnls

Methods

 fit(X, y) Fit the NonNegativeLeastSquares linear model to data predict(X) Predict using the result of the model
__init__(*args, **kwargs)
fit(X, y)

Fit the NonNegativeLeastSquares linear model to data

### Optimizer¶

class dipy.core.optimize.Optimizer(fun, x0, args=(), method='L-BFGS-B', jac=None, hess=None, hessp=None, bounds=None, constraints=(), tol=None, callback=None, options=None, evolution=False)

Bases: object

Attributes

Methods

__init__(fun, x0, args=(), method='L-BFGS-B', jac=None, hess=None, hessp=None, bounds=None, constraints=(), tol=None, callback=None, options=None, evolution=False)

A class for handling minimization of scalar function of one or more variables.

Parameters: fun : callable Objective function. x0 : ndarray Initial guess. args : tuple, optional Extra arguments passed to the objective function and its derivatives (Jacobian, Hessian). method : str, optional Type of solver. Should be one of ‘Nelder-Mead’ ‘Powell’ ‘CG’ ‘BFGS’ ‘Newton-CG’ ‘Anneal’ ‘L-BFGS-B’ ‘TNC’ ‘COBYLA’ ‘SLSQP’ ‘dogleg’ ‘trust-ncg’ jac : bool or callable, optional Jacobian of objective function. Only for CG, BFGS, Newton-CG, dogleg, trust-ncg. If jac is a Boolean and is True, fun is assumed to return the value of Jacobian along with the objective function. If False, the Jacobian will be estimated numerically. jac can also be a callable returning the Jacobian of the objective. In this case, it must accept the same arguments as fun. hess, hessp : callable, optional Hessian of objective function or Hessian of objective function times an arbitrary vector p. Only for Newton-CG, dogleg, trust-ncg. Only one of hessp or hess needs to be given. If hess is provided, then hessp will be ignored. If neither hess nor hessp is provided, then the hessian product will be approximated using finite differences on jac. hessp must compute the Hessian times an arbitrary vector. bounds : sequence, optional Bounds for variables (only for L-BFGS-B, TNC and SLSQP). (min, max) pairs for each element in x, defining the bounds on that parameter. Use None for one of min or max when there is no bound in that direction. constraints : dict or sequence of dict, optional Constraints definition (only for COBYLA and SLSQP). Each constraint is defined in a dictionary with fields: type : str Constraint type: ‘eq’ for equality, ‘ineq’ for inequality. fun : callable The function defining the constraint. jac : callable, optional The Jacobian of fun (only for SLSQP). args : sequence, optional Extra arguments to be passed to the function and Jacobian. Equality constraint means that the constraint function result is to be zero whereas inequality means that it is to be non-negative. Note that COBYLA only supports inequality constraints. tol : float, optional Tolerance for termination. For detailed control, use solver-specific options. callback : callable, optional Called after each iteration, as callback(xk), where xk is the current parameter vector. Only available using Scipy >= 0.12. options : dict, optional A dictionary of solver options. All methods accept the following generic options: maxiter : int Maximum number of iterations to perform. disp : bool Set to True to print convergence messages. For method-specific options, see show_options(‘minimize’, method). evolution : bool, optional save history of x for each iteration. Only available using Scipy >= 0.12.

scipy.optimize.minimize

evolution
fopt
message
nfev
nit
print_summary()
xopt

### SKLearnLinearSolver¶

class dipy.core.optimize.SKLearnLinearSolver(*args, **kwargs)

Bases: abc.NewBase

Provide a sklearn-like uniform interface to algorithms that solve problems of the form: $$y = Ax$$ for $$x$$

Sub-classes of SKLearnLinearSolver should provide a ‘fit’ method that have the following signature: SKLearnLinearSolver.fit(X, y), which would set an attribute SKLearnLinearSolver.coef_, with the shape (X.shape[1],), such that an estimate of y can be calculated as: y_hat = np.dot(X, SKLearnLinearSolver.coef_.T)

Methods

 fit(X, y) Implement for all derived classes predict(X) Predict using the result of the model
__init__(*args, **kwargs)
fit(X, y)

Implement for all derived classes

predict(X)

Predict using the result of the model

Parameters: X : array-like (n_samples, n_features) Samples. C : array, shape = (n_samples,) Predicted values.

### minimize¶

dipy.core.optimize.minimize(fun, x0, args=(), method=None, jac=None, hess=None, hessp=None, bounds=None, constraints=(), tol=None, callback=None, options=None)

Minimization of scalar function of one or more variables.

In general, the optimization problems are of the form:

minimize f(x) subject to

g_i(x) >= 0,  i = 1,...,m
h_j(x)  = 0,  j = 1,...,p


where x is a vector of one or more variables. g_i(x) are the inequality constraints. h_j(x) are the equality constrains.

Optionally, the lower and upper bounds for each element in x can also be specified using the bounds argument.

Parameters: fun : callable Objective function. x0 : ndarray Initial guess. args : tuple, optional Extra arguments passed to the objective function and its derivatives (Jacobian, Hessian). method : str or callable, optional Type of solver. Should be one of ‘Nelder-Mead’ (see here) ‘Powell’ (see here) ‘CG’ (see here) ‘BFGS’ (see here) ‘Newton-CG’ (see here) ‘L-BFGS-B’ (see here) ‘TNC’ (see here) ‘COBYLA’ (see here) ‘SLSQP’ (see here) ‘dogleg’ (see here) ‘trust-ncg’ (see here) custom - a callable object (added in version 0.14.0), see below for description. If not given, chosen to be one of BFGS, L-BFGS-B, SLSQP, depending if the problem has constraints or bounds. jac : bool or callable, optional Jacobian (gradient) of objective function. Only for CG, BFGS, Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg. If jac is a Boolean and is True, fun is assumed to return the gradient along with the objective function. If False, the gradient will be estimated numerically. jac can also be a callable returning the gradient of the objective. In this case, it must accept the same arguments as fun. hess, hessp : callable, optional Hessian (matrix of second-order derivatives) of objective function or Hessian of objective function times an arbitrary vector p. Only for Newton-CG, dogleg, trust-ncg. Only one of hessp or hess needs to be given. If hess is provided, then hessp will be ignored. If neither hess nor hessp is provided, then the Hessian product will be approximated using finite differences on jac. hessp must compute the Hessian times an arbitrary vector. bounds : sequence, optional Bounds for variables (only for L-BFGS-B, TNC and SLSQP). (min, max) pairs for each element in x, defining the bounds on that parameter. Use None for one of min or max when there is no bound in that direction. constraints : dict or sequence of dict, optional Constraints definition (only for COBYLA and SLSQP). Each constraint is defined in a dictionary with fields: type : str Constraint type: ‘eq’ for equality, ‘ineq’ for inequality. fun : callable The function defining the constraint. jac : callable, optional The Jacobian of fun (only for SLSQP). args : sequence, optional Extra arguments to be passed to the function and Jacobian. Equality constraint means that the constraint function result is to be zero whereas inequality means that it is to be non-negative. Note that COBYLA only supports inequality constraints. tol : float, optional Tolerance for termination. For detailed control, use solver-specific options. options : dict, optional A dictionary of solver options. All methods accept the following generic options: maxiter : int Maximum number of iterations to perform. disp : bool Set to True to print convergence messages. For method-specific options, see show_options(). callback : callable, optional Called after each iteration, as callback(xk), where xk is the current parameter vector. res : OptimizeResult The optimization result represented as a OptimizeResult object. Important attributes are: x the solution array, success a Boolean flag indicating if the optimizer exited successfully and message which describes the cause of the termination. See OptimizeResult for a description of other attributes.

minimize_scalar
Interface to minimization algorithms for scalar univariate functions
show_options
Additional options accepted by the solvers

Notes

This section describes the available solvers that can be selected by the ‘method’ parameter. The default method is BFGS.

Unconstrained minimization

Method Nelder-Mead uses the Simplex algorithm [R17], [R18]. This algorithm is robust in many applications. However, if numerical computation of derivative can be trusted, other algorithms using the first and/or second derivatives information might be preferred for their better performance in general.

Method Powell is a modification of Powell’s method [R19], [R20] which is a conjugate direction method. It performs sequential one-dimensional minimizations along each vector of the directions set (direc field in options and info), which is updated at each iteration of the main minimization loop. The function need not be differentiable, and no derivatives are taken.

Method CG uses a nonlinear conjugate gradient algorithm by Polak and Ribiere, a variant of the Fletcher-Reeves method described in [R21] pp. 120-122. Only the first derivatives are used.

Method BFGS uses the quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [R21] pp. 136. It uses the first derivatives only. BFGS has proven good performance even for non-smooth optimizations. This method also returns an approximation of the Hessian inverse, stored as hess_inv in the OptimizeResult object.

Method Newton-CG uses a Newton-CG algorithm [R21] pp. 168 (also known as the truncated Newton method). It uses a CG method to the compute the search direction. See also TNC method for a box-constrained minimization with a similar algorithm.

Method dogleg uses the dog-leg trust-region algorithm [R21] for unconstrained minimization. This algorithm requires the gradient and Hessian; furthermore the Hessian is required to be positive definite.

Method trust-ncg uses the Newton conjugate gradient trust-region algorithm [R21] for unconstrained minimization. This algorithm requires the gradient and either the Hessian or a function that computes the product of the Hessian with a given vector.

Constrained minimization

Method L-BFGS-B uses the L-BFGS-B algorithm [R22], [R23] for bound constrained minimization.

Method TNC uses a truncated Newton algorithm [R21], [R24] to minimize a function with variables subject to bounds. This algorithm uses gradient information; it is also called Newton Conjugate-Gradient. It differs from the Newton-CG method described above as it wraps a C implementation and allows each variable to be given upper and lower bounds.

Method COBYLA uses the Constrained Optimization BY Linear Approximation (COBYLA) method [R25], [10], [11]. The algorithm is based on linear approximations to the objective function and each constraint. The method wraps a FORTRAN implementation of the algorithm. The constraints functions ‘fun’ may return either a single number or an array or list of numbers.

Method SLSQP uses Sequential Least SQuares Programming to minimize a function of several variables with any combination of bounds, equality and inequality constraints. The method wraps the SLSQP Optimization subroutine originally implemented by Dieter Kraft [12]. Note that the wrapper handles infinite values in bounds by converting them into large floating values.

Custom minimizers

It may be useful to pass a custom minimization method, for example when using a frontend to this method such as scipy.optimize.basinhopping or a different library. You can simply pass a callable as the method parameter.

The callable is called as method(fun, x0, args, **kwargs, **options) where kwargs corresponds to any other parameters passed to minimize (such as callback, hess, etc.), except the options dict, which has its contents also passed as method parameters pair by pair. Also, if jac has been passed as a bool type, jac and fun are mangled so that fun returns just the function values and jac is converted to a function returning the Jacobian. The method shall return an OptimizeResult object.

The provided method callable must be able to accept (and possibly ignore) arbitrary parameters; the set of parameters accepted by minimize may expand in future versions and then these parameters will be passed to the method. You can find an example in the scipy.optimize tutorial.

New in version 0.11.0.

References

 [R17] (1, 2) Nelder, J A, and R Mead. 1965. A Simplex Method for Function Minimization. The Computer Journal 7: 308-13.
 [R18] (1, 2) Wright M H. 1996. Direct search methods: Once scorned, now respectable, in Numerical Analysis 1995: Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis (Eds. D F Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK. 191-208.
 [R19] (1, 2) Powell, M J D. 1964. An efficient method for finding the minimum of a function of several variables without calculating derivatives. The Computer Journal 7: 155-162.
 [R20] (1, 2) Press W, S A Teukolsky, W T Vetterling and B P Flannery. Numerical Recipes (any edition), Cambridge University Press.
 [R21] (1, 2, 3, 4, 5, 6, 7, 8) Nocedal, J, and S J Wright. 2006. Numerical Optimization. Springer New York.
 [R22] (1, 2) Byrd, R H and P Lu and J. Nocedal. 1995. A Limited Memory Algorithm for Bound Constrained Optimization. SIAM Journal on Scientific and Statistical Computing 16 (5): 1190-1208.
 [R23] (1, 2) Zhu, C and R H Byrd and J Nocedal. 1997. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization. ACM Transactions on Mathematical Software 23 (4): 550-560.
 [R24] (1, 2) Nash, S G. Newton-Type Minimization Via the Lanczos Method. 1984. SIAM Journal of Numerical Analysis 21: 770-778.
 [R25] (1, 2) Powell, M J D. A direct search optimization method that models the objective and constraint functions by linear interpolation. 1994. Advances in Optimization and Numerical Analysis, eds. S. Gomez and J-P Hennart, Kluwer Academic (Dordrecht), 51-67.
 [10] (1, 2) Powell M J D. Direct search algorithms for optimization calculations. 1998. Acta Numerica 7: 287-336.
 [11] (1, 2) Powell M J D. A view of algorithms for optimization without derivatives. 2007.Cambridge University Technical Report DAMTP 2007/NA03
 [12] (1, 2) Kraft, D. A software package for sequential quadratic programming. 1988. Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace Center – Institute for Flight Mechanics, Koln, Germany.

Examples

Let us consider the problem of minimizing the Rosenbrock function. This function (and its respective derivatives) is implemented in rosen (resp. rosen_der, rosen_hess) in the scipy.optimize.

>>> from scipy.optimize import minimize, rosen, rosen_der


A simple application of the Nelder-Mead method is:

>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> res = minimize(rosen, x0, method='Nelder-Mead', tol=1e-6)
>>> res.x
array([ 1.,  1.,  1.,  1.,  1.])


Now using the BFGS algorithm, using the first derivative and a few options:

>>> res = minimize(rosen, x0, method='BFGS', jac=rosen_der,
...                options={'gtol': 1e-6, 'disp': True})
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 26
Function evaluations: 31
>>> res.x
array([ 1.,  1.,  1.,  1.,  1.])
>>> print(res.message)
Optimization terminated successfully.
>>> res.hess_inv
array([[ 0.00749589,  0.01255155,  0.02396251,  0.04750988,  0.09495377],  # may vary
[ 0.01255155,  0.02510441,  0.04794055,  0.09502834,  0.18996269],
[ 0.02396251,  0.04794055,  0.09631614,  0.19092151,  0.38165151],
[ 0.04750988,  0.09502834,  0.19092151,  0.38341252,  0.7664427 ],
[ 0.09495377,  0.18996269,  0.38165151,  0.7664427,   1.53713523]])


Next, consider a minimization problem with several constraints (namely Example 16.4 from [R21]). The objective function is:

>>> fun = lambda x: (x[0] - 1)**2 + (x[1] - 2.5)**2


There are three constraints defined as:

>>> cons = ({'type': 'ineq', 'fun': lambda x:  x[0] - 2 * x[1] + 2},
...         {'type': 'ineq', 'fun': lambda x: -x[0] - 2 * x[1] + 6},
...         {'type': 'ineq', 'fun': lambda x: -x[0] + 2 * x[1] + 2})


And variables must be positive, hence the following bounds:

>>> bnds = ((0, None), (0, None))


The optimization problem is solved using the SLSQP method as:

>>> res = minimize(fun, (2, 0), method='SLSQP', bounds=bnds,
...                constraints=cons)


It should converge to the theoretical solution (1.4 ,1.7).

### sparse_nnls¶

dipy.core.optimize.sparse_nnls(y, X, momentum=1, step_size=0.01, non_neg=True, check_error_iter=10, max_error_checks=10, converge_on_sse=0.99)

Solve y=Xh for h, using gradient descent, with X a sparse matrix

Parameters: y : 1-d array of shape (N) The data. Needs to be dense. X : ndarray. May be either sparse or dense. Shape (N, M) The regressors momentum : float, optional (default: 1). The persistence of the gradient. step_size : float, optional (default: 0.01). The increment of parameter update in each iteration non_neg : Boolean, optional (default: True) Whether to enforce non-negativity of the solution. check_error_iter : int (default:10) How many rounds to run between error evaluation for convergence-checking. max_error_checks : int (default: 10) Don’t check errors more than this number of times if no improvement in r-squared is seen. converge_on_sse : float (default: 0.99) a percentage improvement in SSE that is required each time to say that things are still going well. h_best : The best estimate of the parameters.

### spdot¶

dipy.core.optimize.spdot(A, B)

The same as np.dot(A, B), except it works even if A or B or both are sparse matrices.

Parameters: A, B : arrays of shape (m, n), (n, k) The matrix product AB. If both A and B are sparse, the result will be a sparse matrix. Otherwise, a dense result is returned See discussion here: http://mail.scipy.org/pipermail/scipy-user/2010-November/027700.html

### with_metaclass¶

dipy.core.optimize.with_metaclass(meta, *bases)

Create a base class with a metaclass.

### Profiler¶

class dipy.core.profile.Profiler(call=None, *args)

Bases: object

Profile python/cython files or functions

If you are profiling cython code you need to add # cython: profile=True on the top of your .pyx file

and for the functions that you do not want to profile you can use this decorator in your cython files

@cython.profile(False)

Parameters: caller : file or function call args : function arguments

References

Examples

from dipy.core.profile import Profiler import numpy as np p=Profiler(np.sum,np.random.rand(1000000,3)) fname=’test.py’ p=Profiler(fname) p.print_stats(10) p.print_stats(‘det’)

Attributes

 stats (function, stats.print_stats(10) will prin the 10 slower functions)

Methods

 print_stats([N]) Print stats for profiling
__init__(call=None, *args)
print_stats(N=10)

Print stats for profiling

You can use it in all different ways developed in pstats for example print_stats(10) will give you the 10 slowest calls or print_stats(‘function_name’) will give you the stats for all the calls with name ‘function_name’

Parameters: N : stats.print_stats argument

### optional_package¶

dipy.core.profile.optional_package(name, trip_msg=None)

Return package-like thing and module setup for package name

Parameters: name : str package name trip_msg : None or str message to give when someone tries to use the return package, but we could not import it, and have returned a TripWire object instead. Default message if None. pkg_like : module or TripWire instance If we can import the package, return it. Otherwise return an object raising an error when accessed have_pkg : bool True if import for package was successful, false otherwise module_setup : function callable usually set as setup_module in calling namespace, to allow skipping tests.

### LEcuyer¶

dipy.core.rng.LEcuyer()

Generate uniformly distributed random numbers using the 32-bit generator from figure 3 of:

L’Ecuyer, P. Efficient and portable combined random number generators, C.A.C.M., vol. 31, 742-749 & 774-?, June 1988.

The cycle length is claimed to be 2.30584E+18

### WichmannHill1982¶

dipy.core.rng.WichmannHill1982()

Algorithm AS 183 Appl. Statist. (1982) vol.31, no.2

Returns a pseudo-random number rectangularly distributed between 0 and 1. The cycle length is 6.95E+12 (See page 123 of Applied Statistics (1984) vol.33), not as claimed in the original article.

ix, iy and iz should be set to integer values between 1 and 30000 before the first entry.

Integer arithmetic up to 5212632 is required.

### WichmannHill2006¶

dipy.core.rng.WichmannHill2006()

B.A. Wichmann, I.D. Hill, Generating good pseudo-random numbers, Computational Statistics & Data Analysis, Volume 51, Issue 3, 1 December 2006, Pages 1614-1622, ISSN 0167-9473, DOI: 10.1016/j.csda.2006.05.019. (http://www.sciencedirect.com/science/article/B6V8V-4K7F86W-2/2/a3a33291b8264e4c882a8f21b6e43351) for advice on generating many sequences for use together, and on alternative algorithms and codes

Examples

>>> from dipy.core import rng
>>> rng.ix, rng.iy, rng.iz, rng.it = 100001, 200002, 300003, 400004
>>> N = 1000
>>> a = [rng.WichmannHill2006() for i in range(N)]


### architecture¶

dipy.core.rng.architecture(executable='/Users/koudoro/anaconda/envs/dipy_dev_3/bin/python', bits='', linkage='')

Queries the given executable (defaults to the Python interpreter binary) for various architecture information.

Returns a tuple (bits, linkage) which contains information about the bit architecture and the linkage format used for the executable. Both values are returned as strings.

Values that cannot be determined are returned as given by the parameter presets. If bits is given as ‘’, the sizeof(pointer) (or sizeof(long) on Python version < 1.5.2) is used as indicator for the supported pointer size.

The function relies on the system’s “file” command to do the actual work. This is available on most if not all Unix platforms. On some non-Unix platforms where the “file” command does not exist and the executable is set to the Python interpreter binary defaults from _default_architecture are used.

### floor¶

dipy.core.rng.floor(x)

Return the floor of x as an Integral. This is the largest integer <= x.

### HemiSphere¶

class dipy.core.sphere.HemiSphere(x=None, y=None, z=None, theta=None, phi=None, xyz=None, faces=None, edges=None, tol=1e-05)

Points on the unit sphere.

A HemiSphere is similar to a Sphere but it takes antipodal symmetry into account. Antipodal symmetry means that point v on a HemiSphere is the same as the point -v. Duplicate points are discarded when constructing a HemiSphere (including antipodal duplicates). edges and faces are remapped to the remaining points as closely as possible.

The HemiSphere can be constructed using one of three conventions:

HemiSphere(x, y, z)
HemiSphere(xyz=xyz)
HemiSphere(theta=theta, phi=phi)

Parameters: x, y, z : 1-D array_like Vertices as x-y-z coordinates. theta, phi : 1-D array_like Vertices as spherical coordinates. Theta and phi are the inclination and azimuth angles respectively. xyz : (N, 3) ndarray Vertices as x-y-z coordinates. faces : (N, 3) ndarray Indices into vertices that form triangular faces. If unspecified, the faces are computed using a Delaunay triangulation. edges : (N, 2) ndarray Edges between vertices. If unspecified, the edges are derived from the faces. tol : float Angle in degrees. Vertices that are less than tol degrees apart are treated as duplicates.

Attributes

 x y z

Methods

 edges() faces() find_closest(xyz) Find the index of the vertex in the Sphere closest to the input vector, from_sphere(klass, sphere[, tol]) Create instance from a Sphere mirror() Create a full Sphere from a HemiSphere subdivide([n]) Create a more subdivided HemiSphere vertices()
__init__(x=None, y=None, z=None, theta=None, phi=None, xyz=None, faces=None, edges=None, tol=1e-05)

Create a HemiSphere from points

faces()
find_closest(xyz)

Find the index of the vertex in the Sphere closest to the input vector, taking into account antipodal symmetry

Parameters: xyz : array-like, 3 elements A unit vector
classmethod from_sphere(klass, sphere, tol=1e-05)

Create instance from a Sphere

mirror()

Create a full Sphere from a HemiSphere

subdivide(n=1)

Create a more subdivided HemiSphere

See Sphere.subdivide for full documentation.

### Sphere¶

class dipy.core.sphere.Sphere(x=None, y=None, z=None, theta=None, phi=None, xyz=None, faces=None, edges=None)

Bases: object

Points on the unit sphere.

The sphere can be constructed using one of three conventions:

Sphere(x, y, z)
Sphere(xyz=xyz)
Sphere(theta=theta, phi=phi)

Parameters: x, y, z : 1-D array_like Vertices as x-y-z coordinates. theta, phi : 1-D array_like Vertices as spherical coordinates. Theta and phi are the inclination and azimuth angles respectively. xyz : (N, 3) ndarray Vertices as x-y-z coordinates. faces : (N, 3) ndarray Indices into vertices that form triangular faces. If unspecified, the faces are computed using a Delaunay triangulation. edges : (N, 2) ndarray Edges between vertices. If unspecified, the edges are derived from the faces.

Attributes

Methods

 edges() faces() find_closest(xyz) Find the index of the vertex in the Sphere closest to the input vector subdivide([n]) Subdivides each face of the sphere into four new faces. vertices()
__init__(x=None, y=None, z=None, theta=None, phi=None, xyz=None, faces=None, edges=None)
edges()
faces()
find_closest(xyz)

Find the index of the vertex in the Sphere closest to the input vector

Parameters: xyz : array-like, 3 elements A unit vector
subdivide(n=1)

Subdivides each face of the sphere into four new faces.

New vertices are created at a, b, and c. Then each face [x, y, z] is divided into faces [x, a, c], [y, a, b], [z, b, c], and [a, b, c].

   y
/               /               a/____
/\    /            /  \  /             /____\/____          x      c     z

Parameters: n : int, optional The number of subdivisions to preform. new_sphere : Sphere The subdivided sphere.
vertices()
x
y
z

### xrange¶

dipy.core.sphere.xrange

alias of range

### auto_attr¶

dipy.core.sphere.auto_attr(func)

Decorator to create OneTimeProperty attributes.

Parameters: func : method The method that will be called the first time to compute a value. Afterwards, the method’s name will be a standard attribute holding the value of this computation.

Examples

>>> class MagicProp(object):
...     @auto_attr
...     def a(self):
...         return 99
...
>>> x = MagicProp()
>>> 'a' in x.__dict__
False
>>> x.a
99
>>> 'a' in x.__dict__
True


### cart2sphere¶

dipy.core.sphere.cart2sphere(x, y, z)

Return angles for Cartesian 3D coordinates x, y, and z

See doc for sphere2cart for angle conventions and derivation of the formulae.

$$0\le\theta\mathrm{(theta)}\le\pi$$ and $$-\pi\le\phi\mathrm{(phi)}\le\pi$$

Parameters: x : array_like x coordinate in Cartesian space y : array_like y coordinate in Cartesian space z : array_like z coordinate r : array radius theta : array inclination (polar) angle phi : array azimuth angle

### disperse_charges¶

dipy.core.sphere.disperse_charges(hemi, iters, const=0.2)

Models electrostatic repulsion on the unit sphere

Places charges on a sphere and simulates the repulsive forces felt by each one. Allows the charges to move for some number of iterations and returns their final location as well as the total potential of the system at each step.

Parameters: hemi : HemiSphere Points on a unit sphere. iters : int Number of iterations to run. const : float Using a smaller const could provide a more accurate result, but will need more iterations to converge. hemi : HemiSphere Distributed points on a unit sphere. potential : ndarray The electrostatic potential at each iteration. This can be useful to check if the repulsion converged to a minimum.

### euler_characteristic_check¶

dipy.core.sphere.euler_characteristic_check(sphere, chi=2)

Checks the euler characteristic of a sphere

If $$f$$ = number of faces, $$e$$ = number_of_edges and $$v$$ = number of vertices, the Euler formula says $$f-e+v = 2$$ for a mesh on a sphere. More generally, whether $$f -e + v == \chi$$ where $$\chi$$ is the Euler characteristic of the mesh.

• Open chain (track) has $$\chi=1$$
• Closed chain (loop) has $$\chi=0$$
• Disk has $$\chi=1$$
• Sphere has $$\chi=2$$
• HemiSphere has $$\chi=1$$
Parameters: sphere : Sphere A Sphere instance with vertices, edges and faces attributes. chi : int, optional The Euler characteristic of the mesh to be checked check : bool True if the mesh has Euler characteristic $$\chi$$

Examples

>>> euler_characteristic_check(unit_octahedron)
True
>>> hemisphere = HemiSphere.from_sphere(unit_icosahedron)
>>> euler_characteristic_check(hemisphere, chi=1)
True


### faces_from_sphere_vertices¶

dipy.core.sphere.faces_from_sphere_vertices(vertices)

Triangulate a set of vertices on the sphere.

Parameters: vertices : (M, 3) ndarray XYZ coordinates of vertices on the sphere. faces : (N, 3) ndarray Indices into vertices; forms triangular faces.

### interp_rbf¶

dipy.core.sphere.interp_rbf(data, sphere_origin, sphere_target, function='multiquadric', epsilon=None, smooth=0.1, norm='angle')

Interpolate data on the sphere, using radial basis functions.

Parameters: data : (N,) ndarray Function values on the unit sphere. sphere_origin : Sphere Positions of data values. sphere_target : Sphere M target positions for which to interpolate. function : {‘multiquadric’, ‘inverse’, ‘gaussian’} Radial basis function. epsilon : float Radial basis function spread parameter. Defaults to approximate average distance between nodes. a good start smooth : float values greater than zero increase the smoothness of the approximation with 0 as pure interpolation. Default: 0.1 norm : str A string indicating the function that returns the “distance” between two points. ‘angle’ - The angle between two vectors ‘euclidean_norm’ - The Euclidean distance v : (M,) ndarray Interpolated values.

scipy.interpolate.Rbf

### remove_similar_vertices¶

dipy.core.sphere.remove_similar_vertices()

Remove vertices that are less than theta degrees from any other

Returns vertices that are at least theta degrees from any other vertex. Vertex v and -v are considered the same so if v and -v are both in vertices only one is kept. Also if v and w are both in vertices, w must be separated by theta degrees from both v and -v to be unique.

Parameters: vertices : (N, 3) ndarray N unit vectors. theta : float The minimum separation between vertices in degrees. return_mapping : {False, True}, optional If True, return mapping as well as vertices and maybe indices (see below). return_indices : {False, True}, optional If True, return indices as well as vertices and maybe mapping (see below). unique_vertices : (M, 3) ndarray Vertices sufficiently separated from one another. mapping : (N,) ndarray For each element vertices[i] ($$i \in 0..N-1$$), the index $$j$$ to a vertex in unique_vertices that is less than theta degrees from vertices[i]. Only returned if return_mapping is True. indices : (N,) ndarray indices gives the reverse of mapping. For each element unique_vertices[j] ($$j \in 0..M-1$$), the index $$i$$ to a vertex in vertices that is less than theta degrees from unique_vertices[j]. If there is more than one element of vertices that is less than theta degrees from unique_vertices[j], return the first (lowest index) matching value. Only return if return_indices is True.

### sphere2cart¶

dipy.core.sphere.sphere2cart(r, theta, phi)

Spherical to Cartesian coordinates

This is the standard physics convention where theta is the inclination (polar) angle, and phi is the azimuth angle.

Imagine a sphere with center (0,0,0). Orient it with the z axis running south-north, the y axis running west-east and the x axis from posterior to anterior. theta (the inclination angle) is the angle to rotate from the z-axis (the zenith) around the y-axis, towards the x axis. Thus the rotation is counter-clockwise from the point of view of positive y. phi (azimuth) gives the angle of rotation around the z-axis towards the y axis. The rotation is counter-clockwise from the point of view of positive z.

Equivalently, given a point P on the sphere, with coordinates x, y, z, theta is the angle between P and the z-axis, and phi is the angle between the projection of P onto the XY plane, and the X axis.

Geographical nomenclature designates theta as ‘co-latitude’, and phi as ‘longitude’

Parameters: r : array_like radius theta : array_like inclination or polar angle phi : array_like azimuth angle x : array x coordinate(s) in Cartesion space y : array y coordinate(s) in Cartesian space z : array z coordinate

Notes

See these pages:

for excellent discussion of the many different conventions possible. Here we use the physics conventions, used in the wikipedia page.

Derivations of the formulae are simple. Consider a vector x, y, z of length r (norm of x, y, z). The inclination angle (theta) can be found from: cos(theta) == z / r -> z == r * cos(theta). This gives the hypotenuse of the projection onto the XY plane, which we will call Q. Q == r*sin(theta). Now x / Q == cos(phi) -> x == r * sin(theta) * cos(phi) and so on.

We have deliberately named this function sphere2cart rather than sph2cart to distinguish it from the Matlab function of that name, because the Matlab function uses an unusual convention for the angles that we did not want to replicate. The Matlab function is trivial to implement with the formulae given in the Matlab help.

### unique_edges¶

dipy.core.sphere.unique_edges(faces, return_mapping=False)

Extract all unique edges from given triangular faces.

Parameters: faces : (N, 3) ndarray Vertex indices forming triangular faces. return_mapping : bool If true, a mapping to the edges of each face is returned. edges : (N, 2) ndarray Unique edges. mapping : (N, 3) For each face, [x, y, z], a mapping to it’s edges [a, b, c].  y / / a/  / / /__________ x c z 

### unique_sets¶

dipy.core.sphere.unique_sets(sets, return_inverse=False)

Remove duplicate sets.

Parameters: sets : array (N, k) N sets of size k. return_inverse : bool If True, also returns the indices of unique_sets that can be used to reconstruct sets (the original ordering of each set may not be preserved).

### vector_norm¶

dipy.core.sphere.vector_norm(vec, axis=-1, keepdims=False)

Return vector Euclidean (L2) norm

Parameters: vec : array_like Vectors to norm. axis : int Axis over which to norm. By default norm over last axis. If axis is None, vec is flattened then normed. keepdims : bool If True, the output will have the same number of dimensions as vec, with shape 1 on axis. norm : array Euclidean norms of vectors.

Examples

>>> import numpy as np
>>> vec = [[8, 15, 0], [0, 36, 77]]
>>> vector_norm(vec)
array([ 17.,  85.])
>>> vector_norm(vec, keepdims=True)
array([[ 17.],
[ 85.]])
>>> vector_norm(vec, axis=0)
array([  8.,  39.,  77.])


### permutations¶

class dipy.core.sphere_stats.permutations

Bases: object

permutations(iterable[, r]) –> permutations object

Return successive r-length permutations of elements in the iterable.

permutations(range(3), 2) –> (0,1), (0,2), (1,0), (1,2), (2,0), (2,1)

__init__()

Initialize self. See help(type(self)) for accurate signature.

### angular_similarity¶

dipy.core.sphere_stats.angular_similarity(S, T)

Computes the cosine distance of the best match between points of two sets of vectors S and T

Parameters: S : array, shape (m,d) T : array, shape (n,d) max_cosine_distance:float

Examples

>>> import numpy as np
>>> from dipy.core.sphere_stats import angular_similarity
>>> S=np.array([[1,0,0],[0,1,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,0,1]])
>>> angular_similarity(S,T)
2.0
>>> T=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> S=np.array([[1,0,0],[0,0,1]])
>>> angular_similarity(S,T)
2.0
>>> S=np.array([[-1,0,0],[0,1,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,0,-1]])
>>> angular_similarity(S,T)
2.0
>>> T=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> S=np.array([[1,0,0],[0,1,0],[0,0,1]])
>>> angular_similarity(S,T)
3.0
>>> S=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,np.sqrt(2)/2.,np.sqrt(2)/2.],[0,0,1]])
>>> angular_similarity(S,T)
2.7071067811865475
>>> S=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> T=np.array([[1,0,0]])
>>> angular_similarity(S,T)
1.0
>>> S=np.array([[0,1,0],[1,0,0]])
>>> T=np.array([[0,0,1]])
>>> angular_similarity(S,T)
0.0
>>> S=np.array([[0,1,0],[1,0,0]])
>>> T=np.array([[0,np.sqrt(2)/2.,np.sqrt(2)/2.]])


Now we use print to reduce the precision of of the printed output (so the doctests don’t detect unimportant differences)

>>> print('%.12f' % angular_similarity(S,T))
0.707106781187
>>> S=np.array([[0,1,0]])
>>> T=np.array([[0,np.sqrt(2)/2.,np.sqrt(2)/2.]])
>>> print('%.12f' % angular_similarity(S,T))
0.707106781187
>>> S=np.array([[0,1,0],[0,0,1]])
>>> T=np.array([[0,np.sqrt(2)/2.,np.sqrt(2)/2.]])
>>> print('%.12f' % angular_similarity(S,T))
0.707106781187


### compare_orientation_sets¶

dipy.core.sphere_stats.compare_orientation_sets(S, T)

Computes the mean cosine distance of the best match between points of two sets of vectors S and T (angular similarity)

Parameters: S : array, shape (m,d) First set of vectors. T : array, shape (n,d) Second set of vectors. max_mean_cosine : float Maximum mean cosine distance.

Examples

>>> from dipy.core.sphere_stats import compare_orientation_sets
>>> S=np.array([[1,0,0],[0,1,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,0,1]])
>>> compare_orientation_sets(S,T)
1.0
>>> T=np.array([[0,1,0],[1,0,0],[0,0,1]])
>>> S=np.array([[1,0,0],[0,0,1]])
>>> compare_orientation_sets(S,T)
1.0
>>> from dipy.core.sphere_stats import compare_orientation_sets
>>> S=np.array([[-1,0,0],[0,1,0],[0,0,1]])
>>> T=np.array([[1,0,0],[0,0,-1]])
>>> compare_orientation_sets(S,T)
1.0


### eigenstats¶

dipy.core.sphere_stats.eigenstats(points, alpha=0.05)

Principal direction and confidence ellipse

Implements equations in section 6.3.1(ii) of Fisher, Lewis and Embleton, supplemented by equations in section 3.2.5.

Parameters: points : arraey_like (N,3) array of points on the sphere of radius 1 in $$\mathbb{R}^3$$ alpha : real or None 1 minus the coverage for the confidence ellipsoid, e.g. 0.05 for 95% coverage. centre : vector (3,) centre of ellipsoid b1 : vector (2,) lengths of semi-axes of ellipsoid

### random_uniform_on_sphere¶

dipy.core.sphere_stats.random_uniform_on_sphere(n=1, coords='xyz')

Random unit vectors from a uniform distribution on the sphere.

Parameters: n : int Number of random vectors coords : {‘xyz’, ‘radians’, ‘degrees’} ‘xyz’ for cartesian form ‘radians’ for spherical form in rads ‘degrees’ for spherical form in degrees X : array, shape (n,3) if coords=’xyz’ or shape (n,2) otherwise Uniformly distributed vectors on the unit sphere.

Notes

The uniform distribution on the sphere, parameterized by spherical coordinates $$(\theta, \phi)$$, should verify $$\phi\sim U[0,2\pi]$$, while $$z=\cos(\theta)\sim U[-1,1]$$.

References

Examples

>>> from dipy.core.sphere_stats import random_uniform_on_sphere
>>> X.shape == (4, 2)
True
>>> X = random_uniform_on_sphere(4, 'xyz')
>>> X.shape == (4, 3)
True


### HemiSphere¶

class dipy.core.subdivide_octahedron.HemiSphere(x=None, y=None, z=None, theta=None, phi=None, xyz=None, faces=None, edges=None, tol=1e-05)

Points on the unit sphere.

A HemiSphere is similar to a Sphere but it takes antipodal symmetry into account. Antipodal symmetry means that point v on a HemiSphere is the same as the point -v. Duplicate points are discarded when constructing a HemiSphere (including antipodal duplicates). edges and faces are remapped to the remaining points as closely as possible.

The HemiSphere can be constructed using one of three conventions:

HemiSphere(x, y, z)
HemiSphere(xyz=xyz)
HemiSphere(theta=theta, phi=phi)

Parameters: x, y, z : 1-D array_like Vertices as x-y-z coordinates. theta, phi : 1-D array_like Vertices as spherical coordinates. Theta and phi are the inclination and azimuth angles respectively. xyz : (N, 3) ndarray Vertices as x-y-z coordinates. faces : (N, 3) ndarray Indices into vertices that form triangular faces. If unspecified, the faces are computed using a Delaunay triangulation. edges : (N, 2) ndarray Edges between vertices. If unspecified, the edges are derived from the faces. tol : float Angle in degrees. Vertices that are less than tol degrees apart are treated as duplicates.

Sphere

Attributes

 x y z

Methods

 edges() faces() find_closest(xyz) Find the index of the vertex in the Sphere closest to the input vector, from_sphere(klass, sphere[, tol]) Create instance from a Sphere mirror() Create a full Sphere from a HemiSphere subdivide([n]) Create a more subdivided HemiSphere vertices()
__init__(x=None, y=None, z=None, theta=None, phi=None, xyz=None, faces=None, edges=None, tol=1e-05)

Create a HemiSphere from points

faces()
find_closest(xyz)

Find the index of the vertex in the Sphere closest to the input vector, taking into account antipodal symmetry

Parameters: xyz : array-like, 3 elements A unit vector
classmethod from_sphere(klass, sphere, tol=1e-05)

Create instance from a Sphere

mirror()

Create a full Sphere from a HemiSphere

subdivide(n=1)

Create a more subdivided HemiSphere

See Sphere.subdivide for full documentation.

### create_unit_hemisphere¶

dipy.core.subdivide_octahedron.create_unit_hemisphere(recursion_level=2)

Creates a unit sphere by subdividing a unit octahedron, returns half the sphere.

Parameters: recursion_level : int Level of subdivision, recursion_level=1 will return an octahedron, anything bigger will return a more subdivided sphere. The sphere will have $$(4^recursion_level+2)/2$$ vertices. HemiSphere : Half of a unit sphere.

### create_unit_sphere¶

dipy.core.subdivide_octahedron.create_unit_sphere(recursion_level=2)

Creates a unit sphere by subdividing a unit octahedron.

Starts with a unit octahedron and subdivides the faces, projecting the resulting points onto the surface of a unit sphere.

Parameters: recursion_level : int Level of subdivision, recursion_level=1 will return an octahedron, anything bigger will return a more subdivided sphere. The sphere will have $$4^recursion_level+2$$ vertices. Sphere : The unit sphere.

create_unit_hemisphere, Sphere

### afb3D¶

dipy.core.wavelet.afb3D(x, af1, af2=None, af3=None)

3D Analysis Filter Bank

Parameters: x : 3D ndarray N1 by N2 by N3 array matrix, where 1) N1, N2, N3 all even 2) N1 >= 2*len(af1) 3) N2 >= 2*len(af2) 4) N3 >= 2*len(af3) afi : 2D ndarray analysis filters for dimension i afi[:, 1] - lowpass filter afi[:, 2] - highpass filter lo : 1D array lowpass subband hi : 1D array highpass subbands, h[d]- d = 1..7

### afb3D_A¶

dipy.core.wavelet.afb3D_A(x, af, d)
3D Analysis Filter Bank
(along one dimension only)
Parameters: x : 3D ndarray N1xN2xN2 matrix, where min(N1,N2,N3) > 2*length(filter) (Ni are even) af : 2D ndarray analysis filter for the columns af[:, 1] - lowpass filter af[:, 2] - highpass filter d : int dimension of filtering (d = 1, 2 or 3) lo : 1D array lowpass subbands hi : 1D array highpass subbands

### cshift3D¶

dipy.core.wavelet.cshift3D(x, m, d)

3D Circular Shift

Parameters: x : 3D ndarray N1 by N2 by N3 array m : int amount of shift d : int dimension of shift (d = 1,2,3) y : 3D ndarray array x will be shifed by m samples down along dimension d

### dwt3D¶

dipy.core.wavelet.dwt3D(x, J, af)

3-D Discrete Wavelet Transform

Parameters: x : 3D ndarray N1 x N2 x N3 matrix 1) Ni all even 2) min(Ni) >= 2^(J-1)*length(af) J : int number of stages af : 2D ndarray analysis filters w : cell array wavelet coefficients

### idwt3D¶

dipy.core.wavelet.idwt3D(w, J, sf)

Inverse 3-D Discrete Wavelet Transform

Parameters: w : cell array wavelet coefficient J : int number of stages sf : 2D ndarray synthesis filters y : 3D ndarray output array

### permutationinverse¶

dipy.core.wavelet.permutationinverse(perm)

Function generating inverse of the permutation

Parameters: perm : 1D array inverse : 1D array permutation inverse of the input

### sfb3D¶

dipy.core.wavelet.sfb3D(lo, hi, sf1, sf2=None, sf3=None)

3D Synthesis Filter Bank

Parameters: lo : 1D array lowpass subbands hi : 1D array highpass subbands sfi : 2D ndarray synthesis filters for dimension i y : 3D ndarray output array

### sfb3D_A¶

dipy.core.wavelet.sfb3D_A`(lo, hi, sf, d)
3D Synthesis Filter Bank
(along single dimension only)
Parameters: lo : 1D array lowpass subbands hi : 1D array highpass subbands sf : 2D ndarray synthesis filters d : int dimension of filtering y : 3D ndarray the N1xN2xN3 matrix