core

References

[1] How-To Guide for Descriptors, Raymond Hettinger. http://users.rcn.com/python/download/Descriptor.htm

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

_TUPLE2AXES

dipy.core.geometry._TUPLE2AXES()

dict() -> new empty dictionary dict(mapping) -> new dictionary initialized from a mapping object’s

(key, value) pairs

dict(iterable) -> new dictionary initialized as if via:

d = {} for k, v in iterable:

d[k] = v

dict(**kwargs) -> new dictionary initialized with the name=value pairs

in the keyword argument list. For example: dict(one=1, two=2)

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

rarray_like

radius

thetaarray_like

inclination or polar angle

phiarray_like

azimuth angle

Returns

xarray

x coordinate(s) in Cartesion space

yarray

y coordinate(s) in Cartesian space

zarray

z coordinate

Notes

See these pages:

• http://en.wikipedia.org/wiki/Spherical_coordinate_system

• http://mathworld.wolfram.com/SphericalCoordinates.html

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.

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 Returns ——- r : array radius theta : array inclination (polar) angle phi : array azimuth angle

sph2latlon

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

Convert spherical coordinates to latitude and longitude.

Returns

lat, lonndarray

Latitude and longitude.

normalized_vector

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

Return vector divided by its Euclidean (L2) norm

See unit vector and Euclidean norm

Parameters

vec : array_like shape (3,)

Returns

nvecarray 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

vector_norm

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

Return vector Euclidean (L2) norm

See unit vector and Euclidean norm

Parameters

vecarray_like

Vectors to norm.

axisint

Axis over which to norm. By default norm over last axis. If axis is None, vec is flattened then normed.

keepdimsbool

If True, the output will have the same number of dimensions as vec, with shape 1 on axis.

Returns

normarray

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.])

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

leading to:

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

Parameters

r : array_like shape (3,), axis theta : float, angle in degrees

Returns

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)
>>> np.round(np.rad2deg(np.arccos(np.dot(ur,u))))
40.0

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.

Returns

npds(3,3) array

Estimated nearest positive semi-definite array to matrix B.

Examples

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

References

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

radiusNone or float, optional

Radius of sphere. Default is to work out radius from mean of the length of each point vector

check_radiusbool, 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.

Returns

d(N,) or (0,) array

Distances between corresponding points in pts1 and pts2 across the spherical surface, i.e. the great circle distance

See Also

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

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

Returns

d(N,) or (0,) array

Cartesian distances between corresponding points in pts1 and pts2

See Also

sphere_distance : distance between points on sphere surface

Examples

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

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

Returns

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.

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 Returns ——- y : (N,2) array planar coordinates of points following mapping by Lambert’s EAP.

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 Returns ——- y : (N,2) array planar coordinates of points following mapping by Lambert’s EAP.

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

Returns

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)

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.

sheararray_like

Shear factors for x-y, x-z, y-z axes.

anglesarray_like

Euler angles about static x, y, z axes.

translatearray_like

Translation vector along x, y, z axes.

perspectivearray_like

Perspective partition of matrix.

Returns

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)

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

matrixarray_like

Non-degenerate homogeneous transformation matrix

Returns

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.

perspectivendarray

Perspective partition of matrix.

Raises

ValueError

If matrix is of wrong type or degenerate.

Examples

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

circumradius

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

Returns

circumradiusfloat

the desired circumradius

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,)

Returns

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.])

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)

Returns

matN x … x mat2 x mat1 : array, (4, 4)

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

varray (3,)

Array containing the three cartesian coordinates of vector v

numint, optional

Number of perpendicular directions to generate

halfbool, 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

Returns

psamplesarray (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>`_.

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

affine4 by 4 array.

The spatial transformation from the measurement to the scanner space.

Returns

dist: float

The maximal distance to the corner of a voxel, given voxel size encoded in the affine.

is_hemispherical

dipy.core.geometry.is_hemispherical(vecs)

Test whether all points on a unit sphere lie in the same hemisphere.

Parameters

vecsnumpy.ndarray

2D numpy array with shape (N, 3) where N is the number of points. All points must lie on the unit sphere.

Returns

is_hemibool

If True, one can find a hemisphere that contains all the points. If False, then the points do not lie in any hemisphere

polenumpy.ndarray

If is_hemi == True, then pole is the “central” pole of the input vectors. Otherwise, pole is the zero vector.

References

https://rstudio-pubs-static.s3.amazonaws.com/27121_a22e51b47c544980bad594d5e0bb2d04.html # noqa

GradientTable

class dipy.core.gradients.GradientTable(gradients, big_delta=None, small_delta=None, b0_threshold=50, btens=None)

Bases: object

Diffusion gradient information

Parameters

gradientsarray_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_thresholdfloat

Gradients with b-value less than or equal to b0_threshold are considered as b0s i.e. without diffusion weighting.

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_thresholdfloat

Gradients with b-value less than or equal to b0_threshold are considered to not have diffusion weighting.

btens(N,3,3) ndarray

The b-tensor of each gradient direction.

See Also

gradient_table

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

__init__(gradients, big_delta=None, small_delta=None, b0_threshold=50, btens=None)

Constructor for GradientTable class

b0s_mask()

bvals()

bvecs()

gradient_strength()

property info

qvals()

tau()

logger

dipy.core.gradients.logger()

Instances of the Logger class represent a single logging channel. A “logging channel” indicates an area of an application. Exactly how an “area” is defined is up to the application developer. Since an application can have any number of areas, logging channels are identified by a unique string. Application areas can be nested (e.g. an area of “input processing” might include sub-areas “read CSV files”, “read XLS files” and “read Gnumeric files”). To cater for this natural nesting, channel names are organized into a namespace hierarchy where levels are separated by periods, much like the Java or Python package namespace. So in the instance given above, channel names might be “input” for the upper level, and “input.csv”, “input.xls” and “input.gnu” for the sub-levels. There is no arbitrary limit to the depth of nesting.

unique_bvals

dipy.core.gradients.unique_bvals(bvals, bmag=None, rbvals=False)

This function gives the unique rounded b-values of the data dipy.core.gradients.unique_bvals is deprecated, Please use dipy.core.gradients.unique_bvals_magnitude instead * deprecated from version: 1.2 * Raises <class ‘dipy.utils.deprecator.ExpiredDeprecationError’> as of version: 1.4 Parameters ———- bvals : ndarray Array containing the b-values bmag : int The order of magnitude that the bvalues have to differ to be considered an unique b-value. B-values are also rounded up to this order of magnitude. Default: derive this value from the maximal b-value provided: \(bmag=log_{10}(max(bvals)) - 1\). rbvals : bool, optional If True function also returns all individual rounded b-values. Default: False Returns ——- ubvals : ndarray Array containing the rounded unique b-values

gradient_table_from_bvals_bvecs

dipy.core.gradients.gradient_table_from_bvals_bvecs(bvals, bvecs, b0_threshold=50, atol=0.01, btens=None, **kwargs)

Creates a GradientTable from a bvals array and a bvecs array

Parameters

bvalsarray_like (N,)

The b-value, or magnitude, of each gradient direction.

bvecsarray_like (N, 3)

The direction, represented as a unit vector, of each gradient.

b0_thresholdfloat

Gradients with b-value less than or equal to bo_threshold are considered to not have diffusion weighting.

atolfloat

Each vector in bvecs must be a unit vectors up to a tolerance of atol.

btenscan be any of three options

1. a string specifying the shape of the encoding tensor for all volumes in data. Options: ‘LTE’, ‘PTE’, ‘STE’, ‘CTE’ corresponding to linear, planar, spherical, and “cigar-shaped” tensor encoding. Tensors are rotated so that linear and cigar tensors are aligned with the corresponding gradient direction and the planar tensor’s normal is aligned with the corresponding gradient direction. Magnitude is scaled to match the b-value.

2. an array of strings of shape (N,), (N, 1), or (1, N) specifying encoding tensor shape for each volume separately. N corresponds to the number volumes in data. Options for elements in array: ‘LTE’, ‘PTE’, ‘STE’, ‘CTE’ corresponding to linear, planar, spherical, and “cigar-shaped” tensor encoding. Tensors are rotated so that linear and cigar tensors are aligned with the corresponding gradient direction and the planar tensor’s normal is aligned with the corresponding gradient direction. Magnitude is scaled to match the b-value.

3. an array of shape (N,3,3) specifying the b-tensor of each volume exactly. N corresponds to the number volumes in data. No rotation or scaling is performed.

Other Parameters

**kwargsdict

Other keyword inputs are passed to GradientTable.

Returns

gradientsGradientTable

A GradientTable with all the gradient information.

See Also

GradientTable, gradient_table

gradient_table_from_qvals_bvecs

dipy.core.gradients.gradient_table_from_qvals_bvecs(qvals, bvecs, big_delta, small_delta, b0_threshold=50, 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

qvalsan array of shape (N,),

q-value given in 1/mm

bvecs : can be any of two options

1. an array of shape (N, 3) or (3, N) with the b-vectors.

2. a path for the file which contains an array like the previous.

big_deltafloat or array of shape (N,)

acquisition pulse separation time in seconds

small_deltafloat

acquisition pulse duration time in seconds

b0_thresholdfloat

All b-values with values less than or equal to bo_threshold are considered as b0s i.e. without diffusion weighting.

atolfloat

All b-vectors need to be unit vectors up to a tolerance.

Returns

gradientsGradientTable

A GradientTable with all the gradient information.

Examples

>>> from dipy.core.gradients import gradient_table_from_qvals_bvecs
>>> qvals = 30. * np.ones(7)
>>> big_delta = .03  # pulse separation of 30ms
>>> small_delta = 0.01  # pulse duration of 10ms
>>> qvals[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 = gradient_table_from_qvals_bvecs(qvals, bvecs,
...                                      big_delta, small_delta)

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.

gradient_table_from_gradient_strength_bvecs

dipy.core.gradients.gradient_table_from_gradient_strength_bvecs(gradient_strength, bvecs, big_delta, small_delta, b0_threshold=50, 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

gradient_strengthan array of shape (N,),

gradient strength given in T/mm

bvecs : can be any of two options

1. an array of shape (N, 3) or (3, N) with the b-vectors.

2. a path for the file which contains an array like the previous.

big_deltafloat or array of shape (N,)

acquisition pulse separation time in seconds

small_deltafloat

acquisition pulse duration time in seconds

b0_thresholdfloat

All b-values with values less than or equal to bo_threshold are considered as b0s i.e. without diffusion weighting.

atolfloat

All b-vectors need to be unit vectors up to a tolerance.

Returns

gradientsGradientTable

A GradientTable with all the gradient information.

Examples

>>> from dipy.core.gradients import (
...    gradient_table_from_gradient_strength_bvecs)
>>> gradient_strength = .03e-3 * np.ones(7)  # clinical strength at 30 mT/m
>>> big_delta = .03  # pulse separation of 30ms
>>> small_delta = 0.01  # pulse duration of 10ms
>>> gradient_strength[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 = gradient_table_from_gradient_strength_bvecs(
...     gradient_strength, bvecs, big_delta, small_delta)

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.

gradient_table

dipy.core.gradients.gradient_table(bvals, bvecs=None, big_delta=None, small_delta=None, b0_threshold=50, atol=0.01, btens=None)

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

1. an array of shape (N,) or (1, N) or (N, 1) with the b-values.

2. a path for the file which contains an array like the above (1).

3. 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.

4. a path for the file which contains an array like the one at (3).

bvecs : can be any of two options

1. an array of shape (N, 3) or (3, N) with the b-vectors.

2. a path for the file which contains an array like the previous.

big_deltafloat

acquisition pulse separation time in seconds (default None)

small_deltafloat

acquisition pulse duration time in seconds (default None)

b0_thresholdfloat

All b-values with values less than or equal to bo_threshold are considered as b0s i.e. without diffusion weighting.

atolfloat

All b-vectors need to be unit vectors up to a tolerance.

btens : can be any of three options

1. a string specifying the shape of the encoding tensor for all volumes in data. Options: ‘LTE’, ‘PTE’, ‘STE’, ‘CTE’ corresponding to linear, planar, spherical, and “cigar-shaped” tensor encoding. Tensors are rotated so that linear and cigar tensors are aligned with the corresponding gradient direction and the planar tensor’s normal is aligned with the corresponding gradient direction. Magnitude is scaled to match the b-value.

2. an array of strings of shape (N,), (N, 1), or (1, N) specifying encoding tensor shape for each volume separately. N corresponds to the number volumes in data. Options for elements in array: ‘LTE’, ‘PTE’, ‘STE’, ‘CTE’ corresponding to linear, planar, spherical, and “cigar-shaped” tensor encoding. Tensors are rotated so that linear and cigar tensors are aligned with the corresponding gradient direction and the planar tensor’s normal is aligned with the corresponding gradient direction. Magnitude is scaled to match the b-value.

3. an array of shape (N,3,3) specifying the b-tensor of each volume exactly. N corresponds to the number volumes in data. No rotation or scaling is performed.

Returns

gradientsGradientTable

A GradientTable with all the gradient information.

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 = gradient_table(bvals, bvecs)
>>> gt.bvecs.shape == bvecs.shape
True
>>> gt = gradient_table(bvals, bvecs.T)
>>> gt.bvecs.shape == bvecs.T.shape
False

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.

reorient_bvecs

dipy.core.gradients.reorient_bvecs(gtab, affines, atol=0.01)

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

gtabGradientTable

The nominal gradient table with which the data were acquired.

affineslist 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])

atol: see gradient_table()

Returns

gtab : a GradientTable class instance with the reoriented directions

References

[Leemans2009]

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

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

Nint

The number of bvectors to generate. This should be equal to the number of bvals used.

itersint

Number of iterations to run.

Returns

bvecs(N,3) ndarray

The generated directions, represented as a unit vector, of each gradient.

round_bvals

dipy.core.gradients.round_bvals(bvals, bmag=None)

“This function rounds the b-values Parameters ———- bvals : ndarray Array containing the b-values bmag : int The order of magnitude to round the b-values. If not given b-values will be rounded relative to the order of magnitude \(bmag = (bmagmax - 1)\), where bmaxmag is the magnitude order of the larger b-value. Returns ——- rbvals : ndarray Array containing the rounded b-values

unique_bvals_tolerance

dipy.core.gradients.unique_bvals_tolerance(bvals, tol=20)

Gives the unique b-values of the data, within a tolerance gap

The b-values must be regrouped in clusters easily separated by a distance greater than the tolerance gap. If all the b-values of a cluster fit within the tolerance gap, the highest b-value is kept.

Parameters

bvalsndarray

Array containing the b-values

tolint

The tolerated gap between the b-values to extract and the actual b-values.

Returns

ubvalsndarray

Array containing the unique b-values using the median value for each cluster

get_bval_indices

dipy.core.gradients.get_bval_indices(bvals, bval, tol=20)

Get indices where the b-value is bval

Parameters

bvals: ndarray

Array containing the b-values

bval: float or int

b-value to extract indices

tol: int

The tolerated gap between the b-values to extract and the actual b-values.

Returns

Array of indices where the b-value is bval

unique_bvals_magnitude

dipy.core.gradients.unique_bvals_magnitude(bvals, bmag=None, rbvals=False)

This function gives the unique rounded b-values of the data Parameters ———- bvals : ndarray Array containing the b-values bmag : int The order of magnitude that the bvalues have to differ to be considered an unique b-value. B-values are also rounded up to this order of magnitude. Default: derive this value from the maximal b-value provided: \(bmag=log_{10}(max(bvals)) - 1\). rbvals : bool, optional If True function also returns all individual rounded b-values. Default: False Returns ——- ubvals : ndarray Array containing the rounded unique b-values

check_multi_b

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

Check if you have enough different b-values in your gradient table 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 \(10^{bmag}\). Default: derive this value from the maximal b-value provided: \(bmag=log_{10}(max(bvals)) - 1\). Returns ——- bool : Whether there are at least n_bvals different b-values in the gradient table used.

btens_to_params

dipy.core.gradients.btens_to_params(btens, ztol=1e-10)

Compute trace, anisotropy and asymmetry parameters from b-tensors.

Parameters

btens(3, 3) OR (N, 3, 3) numpy.ndarray

input b-tensor, or b-tensors, where N = number of b-tensors

ztolfloat

Any parameters smaller than this value are considered to be 0

Returns

bval: numpy.ndarray

b-value(s) (trace(s))

bdelta: numpy.ndarray

normalized tensor anisotropy(s)

b_eta: numpy.ndarray

tensor asymmetry(s)

Notes

This function can be used to get b-tensor parameters directly from the GradientTable btens attribute.

Examples

>>> lte = np.array([[1, 0, 0], [0, 0, 0], [0, 0, 0]])
>>> bval, bdelta, b_eta = btens_to_params(lte)
>>> print("bval={}; bdelta={}; b_eta={}".format(bdelta, bval, b_eta))
bval=[ 1.]; bdelta=[ 1.]; b_eta=[ 0.]

params_to_btens

dipy.core.gradients.params_to_btens(bval, bdelta, b_eta)

Compute b-tensor from trace, anisotropy and asymmetry parameters.

Parameters

bval: int or float

b-value (>= 0)

bdelta: int or float

normalized tensor anisotropy (>= -0.5 and <= 1)

b_eta: int or float

tensor asymmetry (>= 0 and <= 1)

Returns

(3, 3) numpy.ndarray

output b-tensor

Notes

Implements eq. 7.11. p. 231 in [1].

References

anisotropy, in: R. Valiullin (Ed.), Diffusion NMR of Confined Systems: Fluid Transport in Porous Solids and Heterogeneous Materials, Royal Society of Chemistry, Cambridge, UK, 2016.

ornt_mapping

dipy.core.gradients.ornt_mapping(ornt1, ornt2)

Calculate the mapping needing to get from orn1 to orn2.

reorient_vectors

dipy.core.gradients.reorient_vectors(bvecs, current_ornt, new_ornt, axis=0)

Change the orientation of gradients or other vectors.

Moves vectors, storted along axis, from current_ornt to new_ornt. For example the vector [x, y, z] in “RAS” will be [-x, -y, z] in “LPS”.

R: Right A: Anterior S: Superior L: Left P: Posterior I: Inferior

reorient_on_axis

dipy.core.gradients.reorient_on_axis(bvecs, current_ornt, new_ornt, axis=0)

orientation_from_string

dipy.core.gradients.orientation_from_string(string_ornt)

Return an array representation of an ornt string.

orientation_to_string

dipy.core.gradients.orientation_to_string(ornt)

Return a string representation of a 3d ornt.

Graph

class dipy.core.graph.Graph

Bases: object

A simple graph class

__init__()

A graph class with nodes and edges :-)

This class allows us to:

1. find the shortest path

2. find all paths

3. add/delete nodes and edges

4. get parent & children nodes

Examples

>>> from dipy.core.graph import Graph
>>> g=Graph()
>>> g.add_node('a',5)
>>> g.add_node('b',6)
>>> g.add_node('c',10)
>>> g.add_node('d',11)
>>> g.add_edge('a','b')
>>> g.add_edge('b','c')
>>> g.add_edge('c','d')
>>> g.add_edge('b','d')
>>> 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=None)

children(n)

del_node(n)

del_node_and_edges(n)

down(n)

down_short(n)

parents(n)

shortest_path(graph, start, end=None, path=None)

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

arrndarray

Image on which to perform histogram equalization.

num_binsint

Number of bins used to construct the histogram.

Returns

resultndarray

Histogram equalized image.

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

shapetuple 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)

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

__init__()

reset()

Reset all OneTimeProperty attributes that may have fired already.

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.

auto_attr

dipy.core.onetime.auto_attr(func)

Decorator to create OneTimeProperty attributes.

Parameters

funcmethod

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

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

__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

funcallable

Objective function.

x0ndarray

Initial guess.

argstuple, optional

Extra arguments passed to the objective function and its derivatives (Jacobian, Hessian).

methodstr, 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’

jacbool 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, hesspcallable, 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.

boundssequence, 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.

constraintsdict or sequence of dict, optional

Constraints definition (only for COBYLA and SLSQP). Each constraint is defined in a dictionary with fields:

typestr

Constraint type: ‘eq’ for equality, ‘ineq’ for inequality.

funcallable

The function defining the constraint.

jaccallable, optional

The Jacobian of fun (only for SLSQP).

argssequence, 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.

tolfloat, optional

Tolerance for termination. For detailed control, use solver-specific options.

callbackcallable, optional

Called after each iteration, as callback(xk), where xk is the current parameter vector. Only available using Scipy >= 0.12.

optionsdict, optional

A dictionary of solver options. All methods accept the following generic options:

maxiterint

Maximum number of iterations to perform.

dispbool

Set to True to print convergence messages.

For method-specific options, see show_options(‘minimize’, method).

evolutionbool, optional

save history of x for each iteration. Only available using Scipy >= 0.12.

See Also

scipy.optimize.minimize

property evolution

property fopt

property message

property nfev

property nit

print_summary()

property xopt

SKLearnLinearSolver

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

Bases: object

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)

__init__(*args, **kwargs)

abstract fit(X, y)

Implement for all derived classes

predict(X)

Predict using the result of the model

Parameters

Xarray-like (n_samples, n_features)

Samples.

Returns

Carray, shape = (n_samples,)

Predicted values.

NonNegativeLeastSquares

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

Bases: SKLearnLinearSolver

A sklearn-like interface to scipy.optimize.nnls

__init__(*args, **kwargs)

fit(X, y)

Fit the NonNegativeLeastSquares linear model to data

Parameters

PositiveDefiniteLeastSquares

class dipy.core.optimize.PositiveDefiniteLeastSquares(m, A=None, L=None)

Bases: object

__init__(m, A=None, L=None)

Regularized least squares with linear matrix inequality constraints Generate a CVXPY representation of a regularized least squares optimization problem subject to linear matrix inequality constraints. Parameters ———- m : int Positive int indicating the number of regressors. A : array (t = m + k + 1, p, p) (optional) Constraint matrices \(A\). L : array (m, m) (optional) Regularization matrix \(L\). Default: None. Notes —– The basic problem is to solve for \(h\) the minimization of \(c=\|X h - y\|^2 + \|L h\|^2\), where \(X\) is an (m, m) upper triangular design matrix and \(y\) is a set of m measurements, subject to the constraint that \(M=A_0+\sum_{i=0}^{m-1} h_i A_{i+1}+\sum_{j=0}^{k-1} s_j A_{m+j+1}>0\), where \(s_j\) are slack variables and where the inequality sign denotes positive definiteness of the matrix \(M\). The sparsity pattern and size of \(X\) and \(y\) are fixed, because every design matrix and set of measurements can be reduced to an equivalent (minimal) formulation of this type. This formulation is used here mainly to enforce polynomial sum-of-squares constraints on various models, as described in [1]_. References ———- .. [1] Dela Haije et al. “Enforcing necessary non-negativity constraints for common diffusion MRI models using sum of squares programming”. NeuroImage 209, 2020, 116405.

solve(design_matrix, measurements, check=False, **kwargs)

Solve CVXPY problem Solve a CVXPY problem instance for a given design matrix and a given set of observations, and return the optimum. Parameters ———- design_matrix : array (n, m) Design matrix. measurements : array (n) Measurements. check : boolean (optional) If True check whether the unconstrained optimization solution already satisfies the constraints, before running the constrained optimization. This adds overhead, but can avoid unnecessary constrained optimization calls. Default: False kwargs : keyword arguments Arguments passed to the CVXPY solve method. Returns ——- h : array (m) Estimated optimum for problem variables \(h\).

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)

Returns

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

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

y1-d array of shape (N)

The data. Needs to be dense.

Xndarray. May be either sparse or dense. Shape (N, M)

The regressors

momentumfloat, optional (default: 1).

The persistence of the gradient.

step_sizefloat, optional (default: 0.01).

The increment of parameter update in each iteration

non_negBoolean, optional (default: True)

Whether to enforce non-negativity of the solution.

check_error_iterint (default:10)

How many rounds to run between error evaluation for convergence-checking.

max_error_checksint (default: 10)

Don’t check errors more than this number of times if no improvement in r-squared is seen.

converge_on_ssefloat (default: 0.99)

a percentage improvement in SSE that is required each time to say that things are still going well.

Returns

h_best : The best estimate of the parameters.

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

Attributes

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

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’)

References

http://docs.cython.org/src/tutorial/profiling_tutorial.html http://docs.python.org/library/profile.html http://packages.python.org/line_profiler/

__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

WichmannHill2006

dipy.core.rng.WichmannHill2006(ix=100001, iy=200002, iz=300003, it=400004)

Wichmann Hill (2006) random number generator.

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

Parameters

ix: int

First seed value. Should not be null. (default 100001)

iy: int

Second seed value. Should not be null. (default 200002)

iz: int

Third seed value. Should not be null. (default 300003)

it: int

Fourth seed value. Should not be null. (default 400004)

Returns

r_numberfloat

pseudo-random number uniformly distributed between [0-1]

Examples

>>> from dipy.core import rng
>>> N = 1000
>>> a = [rng.WichmannHill2006() for i in range(N)]

WichmannHill1982

dipy.core.rng.WichmannHill1982(ix=100001, iy=200002, iz=300003)

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.

Parameters

ix: int

First seed value. Should not be null. (default 100001)

iy: int

Second seed value. Should not be null. (default 200002)

iz: int

Third seed value. Should not be null. (default 300003)

Returns

r_numberfloat

pseudo-random number uniformly distributed between [0-1]

Examples

>>> from dipy.core import rng
>>> N = 1000
>>> a = [rng.WichmannHill1982() for i in range(N)]

LEcuyer

dipy.core.rng.LEcuyer(s1=100001, s2=200002)

Return a LEcuyer random number generator.

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

Parameters

s1: int

First seed value. Should not be null. (default 100001)

s2: int

Second seed value. Should not be null. (default 200002)

Returns

r_numberfloat

pseudo-random number uniformly distributed between [0-1]

Examples

>>> from dipy.core import rng
>>> N = 1000
>>> a = [rng.LEcuyer() for i in range(N)]

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, z1-D array_like

Vertices as x-y-z coordinates.

theta, phi1-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.

__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

xyzarray-like, 3 elements

A unit vector

Returns

idxint

The index into the Sphere.vertices array that gives the closest vertex (in angle).

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/____\b
   /\    /\
  /  \  /  \
 /____\/____\
x      c     z

Parameters

nint, optional

The number of subdivisions to perform.

Returns

new_sphereSphere

The subdivided sphere.

vertices()

property x

property y

property z

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)

Bases: Sphere

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, z1-D array_like

Vertices as x-y-z coordinates.

theta, phi1-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.

tolfloat

Angle in degrees. Vertices that are less than tol degrees apart are treated as duplicates.

See Also

Sphere

__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

xyzarray-like, 3 elements

A unit vector

Returns

idxint

The index into the Sphere.vertices array that gives the closest vertex (in angle).

classmethod from_sphere(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.

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.

Returns

faces(N, 3) ndarray

Indices into vertices; forms triangular faces.

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_mappingbool

If true, a mapping to the edges of each face is returned.

Returns

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

setsarray (N, k)

N sets of size k.

return_inversebool

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).

Returns

unique_setsarray

Unique sets.

inversearray (N,)

The indices to reconstruct sets from unique_sets.

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

hemiHemiSphere

Points on a unit sphere.

itersint

Number of iterations to run.

constfloat

Using a smaller const could provide a more accurate result, but will need more iterations to converge.

Returns

hemiHemiSphere

Distributed points on a unit sphere.

potentialndarray

The electrostatic potential at each iteration. This can be useful to check if the repulsion converged to a minimum.

Notes

This function is meant to be used with diffusion imaging so antipodal symmetry is assumed. Therefore, each charge must not only be unique, but if there is a charge at +x, there cannot be a charge at -x. These are treated as the same location and because the distance between the two charges will be zero, the result will be unstable.

disperse_charges_alt

dipy.core.sphere.disperse_charges_alt(init_pointset, iters, tol=0.001)

Reimplementation of disperse_charges making use of scipy.optimize.fmin_slsqp.

Parameters

init_pointset(N, 3) ndarray

Points on a unit sphere.

itersint

Number of iterations to run.

tolfloat

Tolerance for the optimization.

Returns

array-like (N, 3)

Distributed points on a unit sphere.