Solving inverse electromagnetic scattering problems via domain derivatives

This notebook provides the example codes for the paper Solving inverse electromagnetic scattering problems via domain derivatives by Felix Hagemann, Tilo Arens, Timo Betcke and Frank Hettlich. Please note that the notebook is very long running on a standard workstation.

Below we define the grid and the far field evaluation points required for this notebook.

In [ ]:
far_field_points_str =  """
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"""

We start with the usual imports and also set a smaller value for the H-Matrix tolerance. Then we convert the above strings representing the far field points and the grid into actual objects.

In [ ]:
import numpy as np
import bempp.api
import scipy.io as sio
import io

from scipy.linalg import lstsq
from matplotlib import pyplot as plt

# Enable plotting in Notebook
bempp.api.set_ipython_notebook_viewer()

bempp.api.global_parameters.hmat.eps = 1E-5

f_far_field_points = io.StringIO(far_field_points_str)
f_vertices = io.StringIO(vertices_str)
f_elements = io.StringIO(elements_str)

far_field_points = np.loadtxt(f_far_field_points)
vertices = np.loadtxt(f_vertices)
elements = np.loadtxt(f_elements).astype(np.int, casting='unsafe')
grid = bempp.api.grid_from_element_data(vertices, elements)

The following defines the incident wave.

In [ ]:
def plane_wave_electric(polarization, direction, wavenumber, point):
    return polarization * np.exp(1j * wavenumber * np.dot(point, direction))

def plane_wave_magnetic(polarization, direction, wavenumber, point):
    return np.cross(direction, polarization) * np.exp(1j * wavenumber * np.dot(point, direction))

We require a number of helper routines, namely the surface gradient, surface divergence, trace transformation operation and the product of functions. These are defined as follows.

In [ ]:
def surface_gradient(domain, range_, dual_to_range, label ="SURFACE_GRADIENT", parameters=None):
    from bempp.api.operators.boundary.sparse import operator_from_functors
    from bempp.api.assembly.functors import simple_test_trial_integrand_functor
    from bempp.api.assembly.functors import surface_divergence_functor
    from bempp.api.assembly.functors import scalar_function_value_functor
    return -operator_from_functors(
        domain, range_, dual_to_range,
        surface_divergence_functor(),
        scalar_function_value_functor(),
        simple_test_trial_integrand_functor(),
        label=label, parameters=parameters)

def surface_divergence(domain, range_, dual_to_range, label="SURFACE_DIVERGENCE", parameters=None):
    from bempp.api.operators.boundary.sparse import operator_from_functors
    from bempp.api.assembly.functors import hdiv_function_value_functor
    from bempp.api.assembly.functors import simple_test_trial_integrand_functor
    from bempp.api.assembly.functors import maxwell_test_trial_integrand_functor
    from bempp.api.assembly.functors import surface_gradient_functor
    return -operator_from_functors(
        domain, range_, dual_to_range,
        surface_gradient_functor(),
        hdiv_function_value_functor(),
        simple_test_trial_integrand_functor(),
        label=label, parameters=parameters)

def trace_transformation(domain, range_, dual_to_range, label="TRACE_TRANSFORMATION", parameters=None):
    return bempp.api.operators.boundary.sparse._maxwell_identity(
        domain, range_, dual_to_range,
        label=label, parameters=parameters)

def function_product(f, g, trial_space, test_space):
    """Compute the product of two functions"""
    from bempp.api.integration import gauss_triangle_points_and_weights
    from bempp.api.utils import combined_type

    accuracy_order = bempp.api.global_parameters.quadrature.far.single_order
    points, weights = gauss_triangle_points_and_weights(accuracy_order)
    if f.space.grid != g.space.grid:
        raise ValueError('f and g must be defined on the same grid')
    element_list = list(test_space.grid.leaf_view.entity_iterator(0))
    dtype = combined_type(f.dtype, g.dtype)
    result = np.zeros(test_space.global_dof_count, dtype=dtype)

    for element in element_list:
        dofs, multipliers = test_space.get_global_dofs(element, dof_weights=True)
        n_local_basis_funs = len(multipliers)
        for index in range(n_local_basis_funs):
            coeffs = np.zeros(n_local_basis_funs)
            coeffs[index] = 1
            integration_elements = element.geometry.integration_elements(
                points)
            basis_values = test_space.evaluate_local_basis(element, points, coeffs)
            f_prod_g = f.evaluate(element, points) * g.evaluate(element, points)
            prod_times_basis = np.sum(f_prod_g * basis_values, axis=0)
            local_res = np.sum(prod_times_basis * weights * integration_elements)
            result[dofs[index]] += multipliers[index] * local_res
    return bempp.api.GridFunction(trial_space, dual_space=test_space, projections=result)

We now set the various simulation parameters.

In [ ]:
### Degree of spherical harmonics expansion
degree = 2
### Number of spherical harmonics functions
number_of_functions = (1 + degree)**2
### Number of iterations
number_of_iterations = 3

# Thikonov alpha
alpha = 3.

polarization = np.array([1.0 + 1j, 2.0, -1.0 - 1.0 / 3.0 * 1j])
direction = np.array([1.0 / np.sqrt(14), 2.0 / np.sqrt(14), 3.0 / np.sqrt(14)], dtype='float64')

eps_rel = 1.9
mu_rel = 1.

k_ext = 1.047197551196598
k_int = k_ext * np.sqrt(eps_rel * mu_rel)

print("The exterior wavenumber is: {0}".format(k_ext))
print("The interior wavenumber is: {0}".format(k_int))

h = .3
base_grid = bempp.api.shapes.sphere(h=h)

The shape is defined by spherical harmonic coefficients. The strategy is to discretise a unit sphere and then to deform it according to the given spherical harmonics expansion. When the coefficients change in the iteration we can easily update the grid accordingly.

In [ ]:
from scipy.special import sph_harm

def spherical_to_cartesian(r, theta, phi):
    """Transform spherical to cartesian coordinates."""
    x = np.zeros(3, dtype=np.float64)
    x[0] = r * np.sin(theta) * np.cos(phi)
    x[1] = r * np.sin(theta) * np.sin(phi)
    x[2] = r * np.cos(theta)
    return x

def normalized_spherical_harmonics(m, n, theta, phi):
    """Normalized spherical harmonics
    See
    https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.sph_harm.html#scipy.special.sph_harm
    for their definition.
    Here, theta is the polar angle and phi the azumthal angle,
    which is reversed to the Scipy convention.
    """
    return sph_harm(m, n, phi, theta)

def normalized_spherical_harmonics_dtheta(m, n, theta, phi):
    """The theta derivative of a spherical harmonic.
    See
    http://functions.wolfram.com/Polynomials/SphericalHarmonicY/20/01/01/
    """
    y1 = normalized_spherical_harmonics(m, n, theta, phi)
    y2 = normalized_spherical_harmonics(m + 1, n, theta, phi)
    return m / np.tan(theta) * y1 + \
            np.sqrt((n - m) * (n + m + 1)) * np.exp(-1j * phi) * y2

def normalized_spherical_harmonics_dphi(m, n, theta, phi):
    """The phi derivative of a spherical harmonic
    See
    http://functions.wolfram.com/Polynomials/SphericalHarmonicY/20/01/02/
    """
    return 1j * m * normalized_spherical_harmonics(m, n, theta, phi)

def evaluate_spherical_harmonics_on_points(degree, points):
    """Evaluate spherical harmonics on a given set of points on the unit sphere."""

    r = np.linalg.norm(points, axis=0)
    theta = np.arccos(points[2] / r)
    phi = np.arctan2(points[1], points[0])

    spherical_basis = np.empty((points.shape[1], (degree + 1)**2), dtype='float64')
    count = 0
    for n in range(degree + 1):
        spherical_basis[:, count] = np.real(normalized_spherical_harmonics(0, n, theta, phi))
        count += 1
        for m in range(1, n + 1):
            spherical_basis[:, count] = np.real(normalized_spherical_harmonics(m, n, theta, phi))
            spherical_basis[:, count + 1] = np.imag(normalized_spherical_harmonics(m, n, theta, phi))
            count += 2
    return spherical_basis


def grid_from_spherical_harmonics_coefficients(degree, coefficients):
    """Return a grid based on spherical harmonics coefficients."""
    vertices = base_grid.leaf_view.vertices.copy()
    elements = base_grid.leaf_view.elements.copy()
    spherical_basis = evaluate_spherical_harmonics_on_points(degree, vertices)
    new_radii = np.matmul(spherical_basis,  coefficients)
    return bempp.api.grid_from_element_data(vertices * new_radii, elements)

def sphere_coefficients(degree):
    """Return radius coefficients for the unit sphere."""
    return 2. * np.sqrt(np.pi) * np.eye((degree + 1)**2, 1, dtype='float64').flatten()

def project_grid_onto_spherical_harmonics(degree, grid):
    """Project a given grid onto spherical harmonics. Return coefficients and relative residual."""
    vertices = grid.leaf_view.vertices
    radii = np.linalg.norm(vertices, axis=0)
    spherical_basis = evaluate_spherical_harmonics_on_points(degree, vertices)
    coefficients, residual, _, _ = lstsq(spherical_basis, radii)
    return coefficients, np.sqrt(residual) / np.linalg.norm(radii)

We now import the exact grid and project it into spherical harmonics to create a reference solution. We use a coarse approximation to speed up running time of the notebook. To get better approximations increase the degree parameter above. We also plot the spherical harmonics approximation of the grid that is used as reference shape.

In [ ]:
exact_coefficients, residual = project_grid_onto_spherical_harmonics(degree, grid)
print(str(exact_coefficients))
print("Error of approximating exact vertex radius with spherical harmonics: {0}".format(residual))

The following routines assemble the boundary value problem and compute the far field.

In [ ]:
def plane_wave(point):
    return polarization * np.exp(1j * k_ext * np.dot(point, direction))

def scaled_plane_wave(point):
    return plane_wave(point)

def tangential_trace(point, n, domain_index, result):
    result[:] =  np.cross(scaled_plane_wave(point), n)

def scaled_plane_wave_curl(point):
    return np.cross(direction, polarization)  * np.exp(1j * k_ext * np.dot(point, direction))


def neumann_trace(point, n, domain_index, result):
    result[:] = np.cross(scaled_plane_wave_curl(point), n)

def rescale(A, d1, d2):
    """Rescale the 2x2 block operator matrix A"""
    B = bempp.api.BlockedOperator(2, 2)
    for i in range(2):
        for j in range(2):
            B[i, j] = A[i, j]
    B[0, 1] = B[0, 1] * (d2 / d1)
    B[1, 0] = B[1, 0] * (d1 / d2)

    return B


def assemble_operators(grid):
    """Assemble operators for transmission problem."""

    multitrace_int = bempp.api.operators.boundary.maxwell.multitrace_operator(grid, k_int)
    multitrace_ext = bempp.api.operators.boundary.maxwell.multitrace_operator(grid, k_ext)
    identity = bempp.api.operators.boundary.sparse.multitrace_identity(grid, spaces='maxwell')
    electric_far = bempp.api.operators.far_field.maxwell.electric_field(
        multitrace_ext.domain_spaces[1], far_field_points, k_ext)
    magnetic_far = bempp.api.operators.far_field.maxwell.magnetic_field(
        multitrace_ext.domain_spaces[0], far_field_points, k_ext)

    return multitrace_int, multitrace_ext, identity, electric_far, magnetic_far

def evaluate_far_field(transmission_operators):
    """Solve the penetrable scattering problem and evaluate far-field."""
    multitrace_int, multitrace_ext, identity, electric_far, magnetic_far = \
        transmission_operators
    rescaled_int_op = rescale(multitrace_int, np.sqrt(eps_rel), np.sqrt(mu_rel))
    lhs_op = rescaled_int_op + multitrace_ext
    rhs_op = .5 * identity - rescaled_int_op
    electric_incident = bempp.api.GridFunction(lhs_op.domain_spaces[0], fun=tangential_trace,
                                              dual_space=lhs_op.dual_to_range_spaces[0])
    magnetic_incident = bempp.api.GridFunction(lhs_op.domain_spaces[1], fun=neumann_trace,
                                              dual_space=lhs_op.dual_to_range_spaces[1])

    rhs = rhs_op * [electric_incident, magnetic_incident]
    print("Solve the scattering problem.")
    solution, _ = bempp.api.linalg.gmres(lhs_op, rhs, use_strong_form=True)
    print("Evaluate the far-field.")
    far_field = 4. * np.pi * ( -electric_far * solution[1] - magnetic_far * solution[0])
    return far_field, solution

The following code computes the far-field of the exact shape and perturbs it to obtain simulated input measurement data.

In [ ]:
transmission_operators = assemble_operators(grid)
far_field_exact, _ = evaluate_far_field(transmission_operators)

delta = 0.0
noise_factor = 1 + delta * np.random.rand(*far_field_exact.shape) * \
    np.exp(2 * np.pi * 1j * np.random.rand(*far_field_exact.shape))

noisy_far_field = far_field_exact * noise_factor

The following routine is the core of this Notebook. Based on the current far-field residual it computes a correction based on evaluating shape derivatives and returns this correction.

In [ ]:
def compute_correction(transmission_operators, current_far_field, current_solution, current_grid):
    """Update the spherical harmonics coefficients."""
    electric_space, magnetic_space = transmission_operators[0].domain_spaces
    electric_dual, magnetic_dual = transmission_operators[0].dual_to_range_spaces
    scalar_space = bempp.api.function_space(current_grid, "B-P", 1)
    electric_incident = bempp.api.GridFunction(electric_space, fun=tangential_trace,
                                              dual_space=electric_dual)
    magnetic_incident = bempp.api.GridFunction(magnetic_space, fun=neumann_trace,
                                              dual_space=magnetic_dual)

    surface_divergence_electric = surface_divergence(electric_space, scalar_space, scalar_space)
    surface_divergence_magnetic = surface_divergence(magnetic_space, scalar_space, scalar_space)

    trace_transformation_electric = trace_transformation(electric_space, electric_space, electric_dual)
    trace_transformation_magnetic = trace_transformation(magnetic_space, magnetic_space, magnetic_dual)

    total_trace_electric_ext = current_solution[0] + electric_incident
    total_trace_magnetic_ext = current_solution[1] + magnetic_incident

    total_trace_electric_int = np.sqrt(eps_rel) * total_trace_electric_ext
    total_trace_magnetic_int = np.sqrt(mu_rel) * total_trace_magnetic_ext

    e_nu_ext = -1. / (1j * k_ext) * (surface_divergence_magnetic * total_trace_magnetic_ext)
    e_nu_int = -1. / (1j * k_int) * (surface_divergence_magnetic * total_trace_magnetic_int)
    h_nu_ext =  1. / (1j * k_ext) * (surface_divergence_electric * total_trace_electric_ext)
    h_nu_int =  1. / (1j * k_int) * (surface_divergence_electric * total_trace_electric_int)

    tangential_trace_electric_ext = -trace_transformation_electric * total_trace_electric_ext
    tangential_trace_electric_int = -trace_transformation_electric * total_trace_electric_int

    tangential_trace_magnetic_ext = -trace_transformation_magnetic * total_trace_magnetic_ext
    tangential_trace_magnetic_int = -trace_transformation_magnetic * total_trace_magnetic_int

    gradient_operator_electric = surface_gradient(scalar_space, electric_space, electric_space)
    gradient_operator_magnetic = surface_gradient(scalar_space, magnetic_space, magnetic_space)

    def bempp_fun_from_spherical_harmonics(m, n, transformation):

        def grid_fun(point, normal, domain_index, res):
            """Grid function from spherical harmonics."""
            r = np.linalg.norm(point)
            theta = np.arccos(point[2] / r)
            phi = np.arctan2(point[1], point[0])
            res[0] = transformation(
                normalized_spherical_harmonics(m, n, theta, phi)
            ) * np.dot(point, normal) / r

        return grid_fun

    bempp_functions = []

    for n in range(degree + 1):
        bempp_functions.append(
            bempp_fun_from_spherical_harmonics(0, n, np.real))
        for m in range(1, n + 1):
            bempp_functions.append(
                bempp_fun_from_spherical_harmonics(m, n, np.real))
            bempp_functions.append(
                bempp_fun_from_spherical_harmonics(m, n, np.imag))

    all_far_fields = []

    for fun in bempp_functions:
        # Now iterate through all possible spherical harmonics to compute the corrections.

        grid_fun = bempp.api.GridFunction(scalar_space, fun=fun)

        perturb_tangential_electric_ext = function_product(
            grid_fun, tangential_trace_electric_ext, magnetic_space, magnetic_dual)
        perturb_tangential_electric_int = function_product(
            grid_fun, tangential_trace_electric_int, magnetic_space, magnetic_dual)
        perturb_tangential_magnetic_ext = function_product(
            grid_fun, tangential_trace_magnetic_ext, electric_space, electric_dual)
        perturb_tangential_magnetic_int = function_product(
            grid_fun, tangential_trace_magnetic_int, electric_space, electric_dual)

        perturb_enu_ext = function_product(grid_fun, e_nu_ext, scalar_space, scalar_space)
        perturb_hnu_ext = function_product(grid_fun, h_nu_ext, scalar_space, scalar_space)
        perturb_enu_int = function_product(grid_fun, e_nu_int, scalar_space, scalar_space)
        perturb_hnu_int = function_product(grid_fun, h_nu_int, scalar_space, scalar_space)

        trace_grad_perturb_enu_ext = trace_transformation_electric * gradient_operator_electric * perturb_enu_ext
        trace_grad_perturb_enu_int = trace_transformation_electric * gradient_operator_electric * perturb_enu_int
        trace_grad_perturb_hnu_ext = trace_transformation_magnetic * gradient_operator_magnetic * perturb_hnu_ext
        trace_grad_perturb_hnu_int = trace_transformation_magnetic * gradient_operator_magnetic * perturb_hnu_int

        f_one_plus = trace_grad_perturb_enu_ext - 1j * k_ext * perturb_tangential_magnetic_ext
        f_two_plus = trace_grad_perturb_hnu_ext + 1j * k_ext * perturb_tangential_electric_ext

        f_one_minus = trace_grad_perturb_enu_int - 1j * k_int * perturb_tangential_magnetic_int
        f_two_minus = trace_grad_perturb_hnu_int + 1j * k_int * perturb_tangential_electric_int

        multitrace_int, multitrace_ext, identity, electric_far, magnetic_far = transmission_operators
        rescaled_int_op = rescale(multitrace_int, np.sqrt(eps_rel), np.sqrt(mu_rel))


        lhs_op = rescaled_int_op + multitrace_ext
        rhs_op = .5 * identity - rescaled_int_op
        rhs_plus = rhs_op * [f_one_plus, f_two_plus]
        rhs_minus = (multitrace_int - .5 * identity) * [f_one_minus, f_two_minus]
        rhs_minus[0] *= 1./np.sqrt(eps_rel)
        rhs_minus[1] *= 1./np.sqrt(mu_rel)
        rhs = [rhs_plus[0] + rhs_minus[0], rhs_plus[1] + rhs_minus[1]]
        sol, _ = bempp.api.linalg.gmres(lhs_op, rhs, use_strong_form=True)
        far_field = 4. * np.pi * (-electric_far * sol[1] - magnetic_far * sol[0])
        all_far_fields.append(far_field)

    far_field_correlation = np.empty((number_of_functions, number_of_functions), dtype='complex128')
    for i in range(number_of_functions):
        for j in range(number_of_functions):
            far_field_correlation[i, j] = np.sum(all_far_fields[j] * np.conjugate(all_far_fields[i]))

    penalty_vec = np.zeros(number_of_functions, dtype='complex128')
    penalty_vec[0] = 1
    count = 1
    for n in range(1, degree + 1):
        penalty_vec[count : count + 2 * n + 1] = n * (1 + n)
        count += 2 * n + 1
    tikhonov_matrix = far_field_correlation + alpha * np.diag(penalty_vec)
    residual = noisy_far_field - current_far_field
    tikhonov_rhs = np.empty(number_of_functions, dtype='complex128')
    for i in range(number_of_functions):
        tikhonov_rhs[i] = np.sum(np.conjugate(all_far_fields[i]) * residual)
    from scipy.linalg import solve
    correction = solve(tikhonov_matrix, tikhonov_rhs)
    print("Relative size of correction: {0}".format(np.linalg.norm(correction) / np.linalg.norm(residual)))
    return correction

We can now start the main simulation. The loop iteratively updates the shape approximation and displays the current residual as compared to the exact far-field. It also generates a number of grids that can be visualised.

In [ ]:
### Initialize coefficients using the unit sphere
coefficients = sphere_coefficients(degree)
print("Starting coefficients:")
print(str(coefficients))

for it in range(number_of_iterations):
    current_grid = grid_from_spherical_harmonics_coefficients(degree, coefficients)
    bempp.api.export(grid=current_grid, file_name="TimoDomDer_Grid_Iteration_"+str(it)+".msh")
    transmission_operators = assemble_operators(current_grid)
    current_far_field, current_solution = evaluate_far_field(transmission_operators)
    residual = np.linalg.norm(far_field_exact - current_far_field) / np.linalg.norm(far_field_exact)
    print("Current residual: {0}".format(residual))
    correction = compute_correction(transmission_operators, current_far_field, current_solution, current_grid)
    print(str(np.real(correction)))
    coefficients += np.real(correction)
    print("Coefficients = ")
    print(str(coefficients))


current_grid = grid_from_spherical_harmonics_coefficients(degree, coefficients)
bempp.api.export(grid=current_grid, file_name="Grid_Iteration_"+str(number_of_iterations)+".msh")
transmission_operators = assemble_operators(current_grid)
current_far_field, current_solution = evaluate_far_field(transmission_operators)
residual = np.linalg.norm(far_field_exact - current_far_field) / np.linalg.norm(far_field_exact)
print("Final residual: {0}".format(residual))
print("Final object approximation:")
current_grid.plot()