sims

Add noise of specified distribution to a 4D array.

Parameters

volarray, shape (X,Y,Z,W)

Diffusion measurements in W directions at each (X, Y, Z) voxel position.

snrfloat, optional

The desired signal-to-noise ratio. (See notes below.)

S0float, optional

Reference signal for specifying snr (defaults to 1).

noise_typestring, optional

The distribution of noise added. Can be either ‘gaussian’ for Gaussian distributed noise, ‘rician’ for Rice-distributed noise (default) or ‘rayleigh’ for a Rayleigh distribution.

Returns

volarray, same shape as vol

Volume with added noise.

Notes

SNR is defined here, following [1], as S0 / sigma, where sigma is the standard deviation of the two Gaussian distributions forming the real and imaginary components of the Rician noise distribution (see [2]).

References

1

Descoteaux, Angelino, Fitzgibbons and Deriche (2007) Regularized, fast and robust q-ball imaging. MRM, 58: 497-510

2

Gudbjartson and Patz (2008). The Rician distribution of noisy MRI data. MRM 34: 910-914.

Examples

>>> signal = np.arange(800).reshape(2, 2, 2, 100)
>>> signal_w_noise = add_noise(signal, snr=10, noise_type='rician')

numerical derivatives 2 eigenvectors

Provide full paths to example or test datasets.

Parameters

namestr

the filename/s of which dataset to return, one of: - ‘small_64D’ small region of interest nifti,bvecs,bvals 64 directions - ‘small_101D’ small region of interest nifti, bvecs, bvals

101 directions

• ‘aniso_vox’ volume with anisotropic voxel size as Nifti

• ‘fornix’ 300 tracks in Trackvis format (from Pittsburgh Brain Competition)

• ‘gqi_vectors’ the scanner wave vectors needed for a GQI acquisitions of 101 directions tested on Siemens 3T Trio

• ‘small_25’ small ROI (10x8x2) DTI data (b value 2000, 25 directions)

• ‘test_piesno’ slice of N=8, K=14 diffusion data

• ‘reg_c’ small 2D image used for validating registration

• ‘reg_o’ small 2D image used for validation registration

• ‘cb_2’ two vectorized cingulum bundles

Returns

fnamestuple

filenames for dataset

Examples

>>> import numpy as np
>>> from dipy.io.image import load_nifti
>>> from dipy.data import get_fnames
>>> fimg, fbvals, fbvecs = get_fnames('small_101D')
>>> bvals=np.loadtxt(fbvals)
>>> bvecs=np.loadtxt(fbvecs).T
>>> data, affine = load_nifti(fimg)
>>> data.shape == (6, 10, 10, 102)
True
>>> bvals.shape == (102,)
True
>>> bvecs.shape == (102, 3)
True

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

bvalscan 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).

bvecscan 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.

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.

Returns

gradientsGradientTable

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

Create a phantom based on a 3-D orbit f(t) -> (x,y,z).

Parameters

gtabGradientTable

Gradient table of measurement directions.

evalsarray, shape (3,)

Tensor eigenvalues.

funcuser defined function f(t)->(x,y,z)

It could be desirable for -1=<x,y,z <=1. If None creates a circular orbit.

tarray, shape (K,)

Represents time for the orbit. Default is np.linspace(0, 2 * np.pi, 1000).

datashapearray, shape (X,Y,Z,W)

Size of the output simulated data

origintuple, shape (3,)

Define the center for the volume

scaletuple, shape (3,)

Scale the function before applying to the grid

anglesarray, shape (L,)

Density angle points, always perpendicular to the first eigen vector Default np.linspace(0, 2 * np.pi, 32).

radiiarray, shape (M,)

Thickness radii. Default np.linspace(0.2, 2, 6). angles and radii define the total thickness options

S0double, optional

Maximum simulated signal. Default 100.

snrfloat, optional

The signal to noise ratio set to apply Rician noise to the data. Default is to not add noise at all.

Returns

dataarray, shape (datashape)

Examples

>>> def f(t):
...    x = np.sin(t)
...    y = np.cos(t)
...    z = np.linspace(-1, 1, len(x))
...    return x, y, z

>>> data = orbital_phantom(func=f)

Simulate diffusion-weighted signals with a single tensor.

Parameters

gtabGradientTable

Table with information of b-values and gradient directions g. Note that if gtab has a btens attribute, simulations will be performed according to the given b-tensor B information.

S0double,

Strength of signal in the presence of no diffusion gradient (also called the b=0 value).

evals(3,) ndarray

Eigenvalues of the diffusion tensor. By default, values typical for prolate white matter are used.

evecs(3, 3) ndarray

Eigenvectors of the tensor. You can also think of this as a rotation matrix that transforms the direction of the tensor. The eigenvectors need to be column wise.

snrfloat

Signal to noise ratio, assuming Rician noise. None implies no noise.

Returns

S(N,) ndarray

Simulated signal:

S(b, g) = S_0 e^(-b g^T R D R.T g), if gtab.tens=None S(B) = S_0 e^(-B:D), if gtab.tens information is given

References

1

M. Descoteaux, “High Angular Resolution Diffusion MRI: from Local Estimation to Segmentation and Tractography”, PhD thesis, University of Nice-Sophia Antipolis, p. 42, 2008.

2

E. Stejskal and J. Tanner, “Spin diffusion measurements: spin echos in the presence of a time-dependent field gradient”, Journal of Chemical Physics, nr. 42, pp. 288–292, 1965.

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

uarray, shape(3,)

varray, shape(3,)

Returns

Rarray, 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.])

Add noise of specified distribution to the signal from a single voxel.

Parameters

signal1-d ndarray

The signal in the voxel.

snrfloat

The desired signal-to-noise ratio. (See notes below.) If snr is None, return the signal as-is.

S0float

Reference signal for specifying snr.

noise_typestring, optional

The distribution of noise added. Can be either ‘gaussian’ for Gaussian distributed noise, ‘rician’ for Rice-distributed noise (default) or ‘rayleigh’ for a Rayleigh distribution.

Returns

signalarray, same shape as the input

Signal with added noise.

Notes

SNR is defined here, following [1], as S0 / sigma, where sigma is the standard deviation of the two Gaussian distributions forming the real and imaginary components of the Rician noise distribution (see [2]).

References

1

Descoteaux, Angelino, Fitzgibbons and Deriche (2007) Regularized, fast and robust q-ball imaging. MRM, 58: 497-510

2

Gudbjartson and Patz (2008). The Rician distribution of noisy MRI data. MRM 34: 910-914.

Examples

>>> signal = np.arange(800).reshape(2, 2, 2, 100)
>>> signal_w_noise = add_noise(signal, 10., 100., noise_type='rician')

Given the principle tensor axis, return the array of all eigenvectors column-wise (or, the rotation matrix that orientates the tensor).

Parameters

e0(3,) ndarray

Principle tensor axis.

Returns

evecs(3,3) ndarray

Tensor eigenvectors, arranged column-wise.

Calculates the perpendicular diffusion signal E(q) in a cylinder of radius R using the Soderman model [1]. Assumes that the pulse length is infinitely short and the diffusion time is infinitely long.

Parameters

qarray, shape (N,)

q-space value in 1/mm

radiusfloat

cylinder radius in mm

Returns

Earray, shape (N,)

signal attenuation

References

1(1,2)

Söderman, Olle, and Bengt Jönsson. “Restricted diffusion in cylindrical geometry.” Journal of Magnetic Resonance, Series A 117.1 (1995): 94-97.

Calculates the three-dimensional signal attenuation E(q) originating from within a cylinder of radius R using the Soderman approximation [1]. The diffusion signal is assumed to be separable perpendicular and parallel to the cylinder axis [2]. This function is basically an extension of the ball and stick model. Setting the radius to zero makes them equivalent.

Parameters

gtabGradientTable

Signal measurement directions.

taufloat

diffusion time in s

radiifloat

cylinder radius in mm

Dfloat

diffusion constant

S0float

Unweighted signal value.

anglesarray (K,2) or (K, 3)

List of K polar angles (in degrees) for the sticks or array of K sticks as unit vectors.

fractions[float]

Percentage of each stick. Remainder to 100 specifies isotropic component.

snrfloat

Signal to noise ratio, assuming Rician noise. If set to None, no noise is added.

Returns

Earray, shape (N,)

signal attenuation

References

1(1,2)

Söderman, Olle, and Bengt Jönsson. “Restricted diffusion in cylindrical geometry.” Journal of Magnetic Resonance, Series A 117.1 (1995): 94-97.

2

Assaf, Yaniv, et al. “New modeling and experimental framework to characterize hindered and restricted water diffusion in brain white matter.” Magnetic Resonance in Medicine 52.5 (2004): 965-978.

Construct B design matrix for DKI.

Parameters

gtabGradientTable

Measurement directions.

Returns

Barray (N, 22)

Design matrix or B matrix for the DKI model B[j, :] = (Bxx, Bxy, Bzz, Bxz, Byz, Bzz,

Bxxxx, Byyyy, Bzzzz, Bxxxy, Bxxxz, Bxyyy, Byyyz, Bxzzz, Byzzz, Bxxyy, Bxxzz, Byyzz, Bxxyz, Bxyyz, Bxyzz, BlogS0)

Simulated signal based on the diffusion and diffusion kurtosis tensors of a single voxel. Simulations are preformed assuming the DKI model.

Parameters

gtabGradientTable

Measurement directions.

dt(6,) ndarray

Elements of the diffusion tensor.

kt(15, ) ndarray

Elements of the diffusion kurtosis tensor.

S0float (optional)

Strength of signal in the presence of no diffusion gradient.

snrfloat (optional)

Signal to noise ratio, assuming Rician noise. None implies no noise.

Returns

S(N,) ndarray

Simulated signal based on the DKI model:

\[S=S_{0}e^{-bD+\frac{1}{6}b^{2}D^{2}K}\]

References

1

R. Neto Henriques et al., “Exploring the 3D geometry of the diffusion kurtosis tensor - Impact on the development of robust tractography procedures and novel biomarkers”, NeuroImage (2015) 111, 85-99.

Dot product of two arrays. Specifically,

• If both a and b are 1-D arrays, it is inner product of vectors (without complex conjugation).

• If both a and b are 2-D arrays, it is matrix multiplication, but using matmul() or a @ b is preferred.

• If either a or b is 0-D (scalar), it is equivalent to multiply() and using numpy.multiply(a, b) or a * b is preferred.

• If a is an N-D array and b is a 1-D array, it is a sum product over the last axis of a and b.

• If a is an N-D array and b is an M-D array (where M>=2), it is a sum product over the last axis of a and the second-to-last axis of b:

  dot(a, b)[i,j,k,m] = sum(a[i,j,:] * b[k,:,m])
  

Parameters

aarray_like

First argument.

barray_like

Second argument.

outndarray, optional

Output argument. This must have the exact kind that would be returned if it was not used. In particular, it must have the right type, must be C-contiguous, and its dtype must be the dtype that would be returned for dot(a,b). This is a performance feature. Therefore, if these conditions are not met, an exception is raised, instead of attempting to be flexible.

Returns

outputndarray

Returns the dot product of a and b. If a and b are both scalars or both 1-D arrays then a scalar is returned; otherwise an array is returned. If out is given, then it is returned.

Raises

ValueError

If the last dimension of a is not the same size as the second-to-last dimension of b.

Examples

>>> np.dot(3, 4)
12

Neither argument is complex-conjugated:

>>> np.dot([2j, 3j], [2j, 3j])
(-13+0j)

For 2-D arrays it is the matrix product:

>>> a = [[1, 0], [0, 1]]
>>> b = [[4, 1], [2, 2]]
>>> np.dot(a, b)
array([[4, 1],
       [2, 2]])

>>> a = np.arange(3*4*5*6).reshape((3,4,5,6))
>>> b = np.arange(3*4*5*6)[::-1].reshape((5,4,6,3))
>>> np.dot(a, b)[2,3,2,1,2,2]
499128
>>> sum(a[2,3,2,:] * b[1,2,:,2])
499128

Calculates the parallel Gaussian diffusion signal.

Parameters

qarray, shape (N,)

q-space value in 1/mm

taufloat

diffusion time in s

Dfloat

diffusion constant

Returns

Earray, shape (N,)

signal attenuation

Computes the diffusion kurtosis tensor element (with indexes i, j, k and l) based on the individual diffusion tensor components of a multicompartmental model.

Parameters

D_comps(K,3,3) ndarray

Diffusion tensors for all K individual compartment of the multicompartmental model.

frac[float]

Percentage of the contribution of each tensor. The sum of fractions should be equal to 100%.

ind_iint

Element’s index i (0 for x, 1 for y, 2 for z)

ind_jint

Element’s index j (0 for x, 1 for y, 2 for z)

ind_kint

Element’s index k (0 for x, 1 for y, 2 for z)

ind_l: int

Elements index l (0 for x, 1 for y, 2 for z)

DT(3,3) ndarray (optional)

Voxel’s global diffusion tensor.

MDfloat (optional)

Voxel’s global mean diffusivity.

Returns

wijklfloat

kurtosis tensor element of index i, j, k, l

Notes

wijkl is calculated using equation 8 given in [1]

References

1

R. Neto Henriques et al., “Exploring the 3D geometry of the diffusion kurtosis tensor - Impact on the development of robust tractography procedures and novel biomarkers”, NeuroImage (2015) 111, 85-99.

Simulate a Multi-Tensor signal.

Parameters

gtabGradientTable

Table with information of b-values and gradient directions. Note that if gtab has a btens attribute, simulations will be performed according to the given b-tensor information.

mevalsarray (K, 3)

each tensor’s eigenvalues in each row

S0float

Unweighted signal value (b0 signal).

anglesarray (K,2) or (K,3)

List of K tensor directions in polar angles (in degrees) or unit vectors

fractionsfloat

Percentage of the contribution of each tensor. The sum of fractions should be equal to 100%.

snrfloat

Signal to noise ratio, assuming Rician noise. If set to None, no noise is added.

Returns

S(N,) ndarray

Simulated signal.

sticks(M,3)

Sticks in cartesian coordinates.

Examples

>>> import numpy as np
>>> from dipy.sims.voxel import multi_tensor
>>> from dipy.data import get_fnames
>>> from dipy.core.gradients import gradient_table
>>> from dipy.io.gradients import read_bvals_bvecs
>>> fimg, fbvals, fbvecs = get_fnames('small_101D')
>>> bvals, bvecs = read_bvals_bvecs(fbvals, fbvecs)
>>> gtab = gradient_table(bvals, bvecs)
>>> mevals=np.array(([0.0015, 0.0003, 0.0003],[0.0015, 0.0003, 0.0003]))
>>> e0 = np.array([1, 0, 0.])
>>> e1 = np.array([0., 1, 0])
>>> S = multi_tensor(gtab, mevals)

Simulate the diffusion-weight signal, diffusion and kurtosis tensors based on the DKI model

Parameters

gtabGradientTable

mevalsarray (K, 3)

eigenvalues of the diffusion tensor for each individual compartment

S0float (optional)

Unweighted signal value (b0 signal).

anglesarray (K,2) or (K,3) (optional)

List of K tensor directions of the diffusion tensor of each compartment in polar angles (in degrees) or unit vectors

fractionsfloat (K,) (optional)

Percentage of the contribution of each tensor. The sum of fractions should be equal to 100%.

snrfloat (optional)

Signal to noise ratio, assuming Rician noise. If set to None, no noise is added.

Returns

S(N,) ndarray

Simulated signal based on the DKI model.

dt(6,)

elements of the diffusion tensor.

kt(15,)

elements of the kurtosis tensor.

Notes

Simulations are based on multicompartmental models which assumes that tissue is well described by impermeable diffusion compartments characterized by their only diffusion tensor. Since simulations are based on the DKI model, coefficients larger than the fourth order of the signal’s taylor expansion approximation are neglected.

References

1

R. Neto Henriques et al., “Exploring the 3D geometry of the diffusion kurtosis tensor - Impact on the development of robust tractography procedures and novel biomarkers”, NeuroImage (2015) 111, 85-99.

Examples

>>> import numpy as np
>>> from dipy.sims.voxel import multi_tensor_dki
>>> from dipy.data import get_fnames
>>> from dipy.core.gradients import gradient_table
>>> from dipy.io.gradients import read_bvals_bvecs
>>> fimg, fbvals, fbvecs = get_fnames('small_64D')
>>> bvals, bvecs = read_bvals_bvecs(fbvals, fbvecs)
>>> bvals_2s = np.concatenate((bvals, bvals * 2), axis=0)
>>> bvecs_2s = np.concatenate((bvecs, bvecs), axis=0)
>>> gtab = gradient_table(bvals_2s, bvecs_2s)
>>> mevals = np.array([[0.00099, 0, 0],[0.00226, 0.00087, 0.00087]])
>>> S, dt, kt =  multi_tensor_dki(gtab, mevals)

Simulate a Multi-Tensor rtop.

Parameters

mfsequence of floats, bounded [0,1]

Percentages of the fractions for each tensor.

mevalssequence of 1D arrays,

Eigen-values for each tensor. By default, values typical for prolate white matter are used.

taufloat,

diffusion time. By default the value that makes q=sqrt(b).

Returns

msdfloat,

Mean square displacement.

References

1

Cheng J., “Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI”, PhD Thesis, 2012.

Simulate a Multi-Tensor ODF.

Parameters

odf_verts(N,3) ndarray

Vertices of the reconstruction sphere.

mevalssequence of 1D arrays,

Eigen-values for each tensor.

anglessequence of 2d tuples,

Sequence of principal directions for each tensor in polar angles or cartesian unit coordinates.

fractionssequence of floats,

Percentages of the fractions for each tensor.

Returns

ODF(N,) ndarray

Orientation distribution function.

Examples

Simulate a MultiTensor ODF with two peaks and calculate its exact ODF.

>>> import numpy as np
>>> from dipy.sims.voxel import multi_tensor_odf, all_tensor_evecs
>>> from dipy.data import default_sphere
>>> vertices, faces = default_sphere.vertices, default_sphere.faces
>>> mevals = np.array(([0.0015, 0.0003, 0.0003],[0.0015, 0.0003, 0.0003]))
>>> angles = [(0, 0), (90, 0)]
>>> odf = multi_tensor_odf(vertices, mevals, angles, [50, 50])

Simulate a Multi-Tensor ODF.

Parameters

pdf_points(N, 3) ndarray

Points to evaluate the PDF.

mevalssequence of 1D arrays,

Eigen-values for each tensor. By default, values typical for prolate white matter are used.

anglessequence,

Sequence of principal directions for each tensor in polar angles or cartesian unit coordinates.

fractionssequence of floats,

Percentages of the fractions for each tensor.

taufloat,

diffusion time. By default the value that makes q=sqrt(b).

Returns

pdf(N,) ndarray,

Probability density function of the water displacement.

References

1

Cheng J., “Estimation and Processing of Ensemble Average Propagator and its Features in Diffusion MRI”, PhD Thesis, 2012.

Simulate a Multi-Tensor rtop.

Parameters

mfsequence of floats, bounded [0,1]

Percentages of the fractions for each tensor.

mevalssequence of 1D arrays,

Eigen-values for each tensor. By default, values typical for prolate white matter are used.

taufloat,

diffusion time. By default the value that makes q=sqrt(b).

Returns

rtopfloat,

Return to origin probability.

References

1

Cheng J., “Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI”, PhD Thesis, 2012.

Simulate diffusion-weighted signals with a single tensor.

Parameters

gtabGradientTable

Table with information of b-values and gradient directions g. Note that if gtab has a btens attribute, simulations will be performed according to the given b-tensor B information.

S0double,

Strength of signal in the presence of no diffusion gradient (also called the b=0 value).

evals(3,) ndarray

Eigenvalues of the diffusion tensor. By default, values typical for prolate white matter are used.

evecs(3, 3) ndarray

Eigenvectors of the tensor. You can also think of this as a rotation matrix that transforms the direction of the tensor. The eigenvectors need to be column wise.

snrfloat

Signal to noise ratio, assuming Rician noise. None implies no noise.

Returns

S(N,) ndarray

Simulated signal:

S(b, g) = S_0 e^(-b g^T R D R.T g), if gtab.tens=None S(B) = S_0 e^(-B:D), if gtab.tens information is given

References

1

M. Descoteaux, “High Angular Resolution Diffusion MRI: from Local Estimation to Segmentation and Tractography”, PhD thesis, University of Nice-Sophia Antipolis, p. 42, 2008.

2

E. Stejskal and J. Tanner, “Spin diffusion measurements: spin echos in the presence of a time-dependent field gradient”, Journal of Chemical Physics, nr. 42, pp. 288–292, 1965.

Simulate a Multi-Tensor rtop.

Parameters

evals1D arrays,

Eigen-values for the tensor. By default, values typical for prolate white matter are used.

taufloat,

diffusion time. By default the value that makes q=sqrt(b).

Returns

msdfloat,

Mean square displacement.

References

1

Cheng J., “Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI”, PhD Thesis, 2012.

Simulated ODF with a single tensor.

Parameters

r(N,3) or (M,N,3) ndarray

Measurement positions in (x, y, z), either as a list or on a grid.

evals(3,)

Eigenvalues of diffusion tensor. By default, use values typical for prolate white matter.

evecs(3, 3) ndarray

Eigenvectors of the tensor, written column-wise. You can also think of these as the rotation matrix that determines the orientation of the diffusion tensor.

Returns

ODF(N,) ndarray

The diffusion probability at r after time tau.

References

1

Aganj et al., “Reconstruction of the Orientation Distribution Function in Single- and Multiple-Shell q-Ball Imaging Within Constant Solid Angle”, Magnetic Resonance in Medicine, nr. 64, pp. 554–566, 2010.

Simulated ODF with a single tensor.

Parameters

r(N,3) or (M,N,3) ndarray

Measurement positions in (x, y, z), either as a list or on a grid.

evals(3,)

Eigenvalues of diffusion tensor. By default, use values typical for prolate white matter.

evecs(3, 3) ndarray

Eigenvectors of the tensor. You can also think of these as the rotation matrix that determines the orientation of the diffusion tensor.

taufloat,

diffusion time. By default the value that makes q=sqrt(b).

Returns

pdf(N,) ndarray

The diffusion probability at r after time tau.

References

1

Cheng J., “Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI”, PhD Thesis, 2012.

Simulate a Single-Tensor rtop.

Parameters

evals1D arrays,

Eigen-values for the tensor. By default, values typical for prolate white matter are used.

taufloat,

diffusion time. By default the value that makes q=sqrt(b).

Returns

rtopfloat,

Return to origin probability.

References

1

Cheng J., “Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI”, PhD Thesis, 2012.

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.

Simulate the signal for a Sticks & Ball model.

Parameters

gtabGradientTable

Signal measurement directions.

dfloat

Diffusivity value.

S0float

Unweighted signal value.

anglesarray (K,2) or (K, 3)

List of K polar angles (in degrees) for the sticks or array of K sticks as unit vectors.

fractionsfloat

Percentage of each stick. Remainder to 100 specifies isotropic component.

snrfloat

Signal to noise ratio, assuming Rician noise. If set to None, no noise is added.

Returns

S(N,) ndarray

Simulated signal.

sticks(M,3)

Sticks in cartesian coordinates.

References

1

Behrens et al., “Probabilistic diffusion tractography with multiple fiber orientations: what can we gain?”, Neuroimage, 2007.

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

uarray, shape(3,)

varray, shape(3,)

Returns

Rarray, 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.])