Curve Functions

functions for SRVF curve manipulations

moduleauthor:: J. Derek Tucker <jdtuck@sandia.gov>

curve_functions.Basis_Normal_A(q)

Find Normal Basis

Parameters:

q – numpy ndarray (n,T) defining T points on n dimensional SRVF

:rtype list :return delg: basis

curve_functions.calc_j(basis)

Calculates Jacobian matrix from normal basis

Parameters:

basis – list of numpy ndarray of shape (2,M) of M samples basis

Return type:

numpy ndarray

Return j:

Jacobian

curve_functions.calculate_variance(beta)

This function calculates variance of curve beta

Parameters:

beta – numpy ndarray of shape (2,M) of M samples

Return type:

numpy ndarray

Return variance:

variance

curve_functions.calculatecentroid(beta)

This function calculates centroid of a parameterized curve

Parameters:

beta – numpy ndarray of shape (2,M) of M samples

Return type:

numpy ndarray

Return centroid:

center coordinates

curve_functions.curve_to_q(beta, mode='O')

This function converts curve beta to srvf q

Parameters:

• beta – numpy ndarray of shape (2,M) of M samples

• mode – Open (‘O’) or closed curve (‘C’) (default ‘O’)

Return type:

numpy ndarray

Return q:

srvf of curve

Return lenb:

length of curve

Return lenq:

length of srvf

curve_functions.curve_zero_crossing(Y, q, bt, y_max, y_min, gmax, gmin)

finds zero-crossing of optimal gamma, gam = s*gmax + (1-s)*gmin from elastic curve regression model

Parameters:

• Y – response

• beta – predicitve function

• bt – basis function

• y_max – maximum repsonse for warping function gmax

• y_min – minimum response for warping function gmin

• gmax – max warping function

• gmin – min warping fucntion

Return type:

numpy array

Return gamma:

optimal warping function

Return O_hat:

rotation matrix

curve_functions.elastic_distance_curve(beta1, beta2, closed=0, rotation=True, scale=False, method='DP')

Calculates the two elastic distances between two curves in R^M :param beta1: numpy ndarray of shape (M,N) of N samples :param beta2: numpy ndarray of shape (M,N) of N samples :param closed: open (0) or closed (1) curve (default=0) :param rotation: compute optimal rotation (default=True) :param scale: include scale (default=False) :param method: method to apply optimization (default=”DP”) options are “DP” or “RBFGS”

Return type:

tuple

Return dist:

shape distance

Return dx:

phase distance

curve_functions.elastic_shooting(q1, v, mode=0)

Calculates shooting vector from v to q1

Parameters:

• q1 – vector of srvf

• v – shooting vector

• mode – closed or open (1/0)

:rtype numpy ndarray :return q2n: vector of srvf

curve_functions.elastic_shooting_vector(q1, q2, mode=0)

Calculates shooting between two srvfs

Parameters:

• q1 – vector of srvf

• q2 – vector of srvf

• mode – closed or open (1/0)

:rtype numpy ndarray :return v: shooting vector :return d: distance :return q2n: aligned srvf

curve_functions.find_basis_normal(q)

Finds the basis normal to the srvf

Parameters:

q1 – numpy ndarray of shape (2,M) of M samples

Return type:

list of numpy ndarray

Return basis:

list containing basis vectors

curve_functions.find_best_rotation(q1, q2, allow_reflection=False, only_xy=False)

This function calculates the best rotation between two srvfs using procustes rigid alignment

Parameters:

• q1 – numpy ndarray of shape (2,M) of M samples

• q2 – numpy ndarray of shape (2,M) of M samples

• allow_reflection – bool indicating if reflection is allowed (i.e. if the determinant of the optimal rotation can be -1)

• only_xy – bool indicating if rotation should only be allowed in the first two dimensions of the space

Return type:

numpy ndarray

Return q2new:

optimal rotated q2 to q1

Return R:

rotation matrix

curve_functions.find_rotation_and_seed_coord(beta1, beta2, closed=0, rotation=True, method='DP')

This function returns a candidate list of optimally oriented and registered (seed) shapes w.r.t. beta1

Parameters:

• beta1 – numpy ndarray of shape (2,M) of M samples

• beta2 – numpy ndarray of shape (2,M) of M samples

• closed – Open (0) or Closed (1)

• rotation – find rotation (default=True)

• method – method to apply optimization (default=”DP”) options are “DP” or “RBFGS”

Return type:

numpy ndarray

Return beta2new:

optimal aligned beta2 to beta1

Return q2best:

optimal aligned q2 to q1

Return Rbest:

rotation matrix

Return gamIbest:

warping function

curve_functions.find_rotation_and_seed_q(q1, q2, closed=0, rotation=True, method='DP')

This function returns a candidate list of optimally oriented and registered (seed) srvs w.r.t. q1

Parameters:

• q1 – numpy ndarray of shape (2,M) of M samples

• q2 – numpy ndarray of shape (2,M) of M samples

• closed – Open (0) or Closed (1)

• rotation – find rotation (default=True)

• method – method to apply optimization (default=”DP”) options are “DP” or “RBFGS”

Return type:

numpy ndarray

Return q2best:

optimal aligned q2 to q1

Return Rbest:

rotation matrix

Return gamIbest:

warping function

curve_functions.find_rotation_and_seed_unique(q1, q2, closed=0, lam=0.0, rotation=True, method='DP')

This function returns a candidate list of optimally oriented and registered (seed) shapes w.r.t. beta1

Parameters:

• beta1 – numpy ndarray of shape (2,M) of M samples

• beta2 – numpy ndarray of shape (2,M) of M samples

• closed – Open (0) or Closed (1)

• rotation – find rotation (default=True)

• lam – controls the elasticity (default = 0)

• method – method to apply optimization (default=”DP”) options are “DP” or “RBFGS”

Return type:

numpy ndarray

Return beta2new:

optimal rotated beta2 to beta1

Return O:

rotation matrix

Return tau:

seed

curve_functions.gram_schmidt(basis)

Performs Gram Schmidt Orthogonlization of a basis_o

param basis:

list of numpy ndarray of shape (2,M) of M samples

rtype:

list of numpy ndarray

return basis_o:

orthogonlized basis

curve_functions.group_action_by_gamma(q, gamma)

This function reparamerized srvf q by gamma

Parameters:

• f – numpy ndarray of shape (2,M) of M samples

• gamma – numpy ndarray of shape (2,M) of M samples

Return type:

numpy ndarray

Return qn:

reparatermized srvf

curve_functions.group_action_by_gamma_coord(f, gamma)

This function reparamerized curve f by gamma

Parameters:

• f – numpy ndarray of shape (2,M) of M samples

• gamma – numpy ndarray of shape (2,M) of M samples

Return type:

numpy ndarray

Return fn:

reparatermized curve

curve_functions.innerprod_q2(q1, q2)

This function calculates the inner product in srvf space

Parameters:

• q1 – numpy ndarray of shape (2,M) of M samples

• q2 – numpy ndarray of shape (2,M) of M samples

Return type:

numpy ndarray

Return val:

inner product

curve_functions.inverse_exp(q1, q2, beta2)

Calculate the inverse exponential to obtain a shooting vector from q1 to q2 in shape space of open curves

Parameters:

• q1 – numpy ndarray of shape (2,M) of M samples

• q2 – numpy ndarray of shape (2,M) of M samples

• beta2 – numpy ndarray of shape (2,M) of M samples

Return type:

numpy ndarray

Return v:

shooting vectors

curve_functions.inverse_exp_coord(beta1, beta2, closed=0, method='DP')

Calculate the inverse exponential to obtain a shooting vector from beta1 to beta2 in shape space of open curves

Parameters:

• beta1 – numpy ndarray of shape (2,M) of M samples

• beta2 – numpy ndarray of shape (2,M) of M samples

• closed – open (0) or closed (1) curve

• method – method to apply optimization (default=”DP”) options are “DP” or “RBFGS”

Return type:

numpy ndarray

Return v:

shooting vectors

Return dist:

distance

curve_functions.optimum_reparam_curve(q1, q2, lam=0.0, method='DP')

calculates the warping to align srsf q2 to q1

Parameters:

• q1 – matrix of size nxN or array of NxM samples of first SRVF

• time – vector of size N describing the sample points

• q2 – matrix of size nxN or array of NxM samples samples of second SRVF

• lam – controls the amount of elasticity (default = 0.0)

• method – method to apply optimization (default=”DP”) options are “DP” or “RBFGS”

Return type:

vector

Return gam:

describing the warping function used to align q2 with q1

curve_functions.parallel_translate(w, q1, q2, basis, mode=0)

parallel translates q1 and q2 along manifold

Parameters:

• w – numpy ndarray of shape (2,M) of M samples

• q1 – numpy ndarray of shape (2,M) of M samples

• q2 – numpy ndarray of shape (2,M) of M samples

• basis – list of numpy ndarray of shape (2,M) of M samples

• mode – open 0 or closed curves 1 (default 0)

Return type:

numpy ndarray

Return wbar:

translated vector

curve_functions.pre_proc_curve(beta, T=100)

This function prepcoessed a curve beta to set of closed curves

Parameters:

• beta – numpy ndarray of shape (2,M) of M samples

• T – number of samples (default = 100)

Return type:

numpy ndarray

Return betanew:

projected beta

Return qnew:

projected srvf

Return A:

alignment matrix (not used currently)

curve_functions.project_curve(q)

This function projects srvf q to set of close curves

Parameters:

q – numpy ndarray of shape (2,M) of M samples

Return type:

numpy ndarray

Return qproj:

project srvf

curve_functions.project_tangent(w, q, basis)

projects srvf to tangent space w using basis

Parameters:

• w – numpy ndarray of shape (2,M) of M samples

• q – numpy ndarray of shape (2,M) of M samples

• basis – list of numpy ndarray of shape (2,M) of M samples

Return type:

numpy ndarray

Return wproj:

projected q

curve_functions.psi(x, a, q)

This function formats variance output

Parameters:

• x – numpy ndarray of shape (2,M) of M samples curve

• a – numpy ndarray of shape (2,1) mean

• q – numpy ndarray of shape (2,M) of M samples srvf

Return type:

numpy ndarray

Return psi1:

variance

Return psi2:

cross variance

Return psi3:

curve end

Return psi4:

curve end

curve_functions.q_to_curve(q, scale=1)

This function converts srvf to beta

Parameters:

• q – numpy ndarray of shape (n,M) of M samples

• scale – scale of curve

Return type:

numpy ndarray

Return beta:

parameterized curve

curve_functions.resamplecurve(x, N=100, time=None, mode='O')

This function resamples a curve to have N samples

Parameters:

• x – numpy ndarray of shape (2,M) of M samples

• N – Number of samples for new curve (default = 100)

• time – timing vector (Default=None)

• mode – Open (‘O’) or closed curve (‘C’) (default ‘O’)

Return type:

numpy ndarray

Return xn:

resampled curve

curve_functions.scale_curve(beta)

scales curve to length 1

Parameters:

beta – numpy ndarray of shape (2,M) of M samples

Return type:

numpy ndarray

Return beta_scaled:

scaled curve

Return scale:

scale factor used

curve_functions.shift_f(f, tau)

shifts a curve f by tau

Parameters:

• f – numpy ndarray of shape (2,M) of M samples

• tau – scalar

Return type:

numpy ndarray

Return fn:

shifted curve