Synthetic Data & Noise

AbstractNoiseModel

Measurement noise added to simulated data: GaussianNoise, RelativeGaussianNoise, SourceMeterNoise. See add_noise and simulate_data.

GaussianNoise(s)

Additive white Gaussian noise with standard deviation s: a scalar, a vector with one standard deviation per measurement channel (data row), e.g. from an instrument specification, or a matrix with one per data entry (e.g. the noise of rotated data, see pattern_svd).

RelativeGaussianNoise(δ)

Additive Gaussian noise relative to the signal level: for every pattern (column) fᵢ with m entries, ηᵢ ~ N(0, (δ ‖fᵢ‖ / √m)² I), so that ‖ηᵢ‖ ≈ δ ‖fᵢ‖ (e.g. δ = 0.01 for 1 % noise).

SourceMeterNoise(source, meter)

Errors of the current source and of the voltmeter (or, for voltage-driven data, of the voltage source and the ammeter), each a GaussianNoise standard deviation (scalar or per channel): the applied inputs are inputs + source ε (projected back to zero net current for current patterns), the measurements F(applied) + meter ε. The reconstruction is given the nominal inputs. Needs the forward model, so it is applied by simulate_data.

add_noise(data, noise; rng = Random.default_rng())
add_noise!(data, noise; rng = Random.default_rng())

Noisy copy of data (or data overwritten) for an additive noise model (GaussianNoise, RelativeGaussianNoise). Pass a seeded rng for reproducible noise.

expected_squared_error(noise, data)

Expected squared norm E‖η‖² (summed over all entries) of the noise that noise adds to data.

discrepancy_target(obj::AdjointStateObjective, noise; τ = 1.1)

Target value of the objective for the discrepancy principle: τ² E[J(σ_true)], the expected misfit of exact data corrupted by noise, taken in the objective's metric (the whitening of its misfit and, for voltages, the removal of the weighted mean). Pass it as ftarget to minimize to stop when the data are explained to the noise level (τ slightly above 1). Relative noise levels are evaluated on the objective's (noisy) data.

discrepancy_target(obj::ApproximationErrorObjective; τ = 1.1)

τ² m / 2 for m residuals: the expected misfit of the whitened residual.

perturb_boundary_operator(R, s; rng = Random.default_rng())

Operator-level noise for a discrete boundary operator (e.g. an estimated Neumann-to-Dirichlet matrix): R + s (Ê + Êᵀ)/2 with i.i.d. standard normal Ê. Symmetry is preserved, positive semidefiniteness is not (for large s).

perturb_contact_impedance(model::CompleteElectrodeModel, s; rng = Random.default_rng())

Complete electrode model with log-normally perturbed contact impedances zₗ exp(s εₗ), a modelling error for data simulation (the reconstruction keeps the nominal model).

electrode_angles(L; offset = 0, jitter = 0, rng = Random.default_rng())

Centre angles offset + 2π(ℓ-1)/L + jitter εₗ of L electrodes, with Gaussian position errors of standard deviation jitter (radians); pass them as angles to angular_electrodes.

transfer_electrodes(src, electrodes, dst)

The electrodes of src as electrodes of the discretization dst of the same domain (e.g. a finer mesh for simulated data). Back end contract.

ApproximationError(R; noise, floor = :leave_one_out, rtol = 1e-10)

Gaussian model N(μ, Γ) of the modelling error from samples: the columns of R are residuals of the reconstruction model at the true parameters of sample conductivities, for noise-free data of the accurate model (e.g. a finer mesh). noise is the standard deviation η of the measurement noise per residual entry. Stores the mean μ and the low-rank factor of the whitening C^{-1/2}, C = (η² + ν²) I + Γ̂ (see whiten), with the sample covariance Γ̂ and a variance floor ν² for the directions the samples do not span: estimated by leave-one-out (floor = :leave_one_out: the mean squared part of a sample outside the span of the other samples, per remaining dimension; conservative, since the span of the other samples is itself perturbed), or given as a number (floor = 0: none). Singular values below rtol times the largest are dropped.

whiten(ae, r)

The whitened residual C^{-1/2} (r - μ) of the ApproximationError ae.

ApproximationErrorObjective(obj, ae)

The least-squares objective obj with the modelling error of ae accounted for: residual C^{-1/2}(r - μ), so that the misfit of the true conductivity is of the size of the noise again. Values, gradients, Jacobians (explicit and matrix-free, jacobian_operator), column norms and Gram matrices, so that all Gauss–Newton variants apply. Stop by the discrepancy principle with discrepancy_target(obj; τ) (the whitened residual has unit variance per entry).

discrepancy_target(obj::ApproximationErrorObjective; τ = 1.1)

τ² m / 2 for m residuals: the expected misfit of the whitened residual.

AbstractInclusion

A shape with a conductivity value for InclusionPhantom: CircleInclusion, EllipseInclusion, PolygonInclusion. Membership: x in inclusion.

CircleInclusion(center, radius, value)

Disc inclusion with conductivity value.

EllipseInclusion(center, (a, b), angle, value)

Elliptic inclusion with semi-axes a (along the direction angle) and b.

PolygonInclusion(vertices, value)

Polygonal inclusion (simple polygon, vertices in order) with conductivity value.

InclusionPhantom(background, inclusions)

Piecewise constant conductivity: background outside all inclusions, the value of the last inclusion containing x otherwise (later inclusions are painted on top). Callable: ph(x).

random_inclusions(rng = Random.default_rng(); count = 1:3, domain = :disk, center = (0, 0),
                  radius = 1, bbox = (-1, 1, -1, 1), sizes = (0.1, 0.35), values = (0.2, 5.0),
                  background = 1, shapes = (:circle, :ellipse, :polygon), margin = 0.05,
                  separation = 0.05, maxtries = 1000)

Random InclusionPhantom: rand(rng, count) non-overlapping inclusions of random shape (circles, ellipses with random orientation and aspect ratio, star-shaped polygons with 3–7 vertices) inside the domain (:disk with center and radius, or :box with bbox = (xmin, xmax, ymin, ymax)). Sizes (bounding radii) are drawn from sizes relative to the domain half-width, values log-uniformly from values. Every inclusion keeps the distance margin from the boundary and separation from the others; an inclusion that cannot be placed in maxtries attempts is dropped.

PixelFunction(values; bbox = (-1, 1, -1, 1), interpolation = :nearest)

Function on the rectangle bbox = (xmin, xmax, ymin, ymax) given by pixel values (row 1 at the top, as for to_image (Ferrite back end)), evaluated by :nearest pixel or :bilinear interpolation between pixel centres. Points outside the box take the value of the nearest border pixel.

image_phantom(img, σmin, σmax; bbox = (-1, 1, -1, 1), interpolation = :nearest)

Conductivity from a grayscale image with intensities in [0, 1], mapped affinely to [σmin, σmax] (0 < σmin < σmax, so the forward problem stays elliptic), as a PixelFunction on bbox.

gaussian_random_field(rng = Random.default_rng(); bbox = (-1, 1, -1, 1), ℓ = 0.2, order = 2,
                      pixel = ℓ / 10, pad = 3ℓ)

Sample of a Gaussian random field with covariance (1 - ℓ²Δ)^(-order) (Matérn type: length scale ℓ, smoothness ν = order - 1 in 2D; order = 2 is the Whittle field, order = 1.5 the exponential covariance), scaled to unit marginal variance on average over bbox. Sampled on a pixel grid (size pixel) over bbox enlarged by pad in the cosine basis of the Neumann Laplacian, which keeps the reflecting boundary away from the box. Returns a bilinear PixelFunction; combine with lognormal_phantom or levelset_phantom.

TransformedPhantom(f, g)

The function x ↦ g(f(x)), e.g. a conductivity derived from a random field; see lognormal_phantom and levelset_phantom.

lognormal_phantom(field; σ0 = 1, s = 0.5)

Log-normal conductivity σ0 exp(s f(x)) from a (Gaussian random) field f.

levelset_phantom(field; level = 0, inside = 2, outside = 1)

Two-phase conductivity: inside where f(x) > level, outside elsewhere. For a Gaussian random field this gives random inclusions with smooth, irregular boundaries.

conductivity(par::AbstractParametrization, θ)

Conductivity coefficients σ = P θ of the parameters θ.

conductivity(disc, phantom; method = :l2, quadrature_order = 4)
conductivity(disc, σ::AbstractVector)

Coefficients of a conductivity given as a function x ↦ σ(x) (a phantom, e.g. InclusionPhantom, PixelFunction, or any callable) in the σ space of disc: the L² projection (method = :l2, cell averages for piecewise constants, sampled with a rule of quadrature_order) or nodal interpolation (:interpolate). Coefficient vectors are passed through.

simulate_data(disc, fm, conductivity, inputs; mode = :neumann, noise = nothing,
              rng = Random.default_rng(), solver = DirectSolver(), quadrature_order = 4)

Simulated measurements for the forward model fm on disc: conductivity is a phantom (projected with conductivity) or a coefficient vector; inputs are current patterns (mode = :neumann, data = voltages) or voltage patterns (mode = :dirichlet, data = currents). noise: nothing, an additive AbstractNoiseModel or SourceMeterNoise.

Returns a named tuple (data, clean, σ, inputs, applied): noisy and exact data, the conductivity coefficients, the nominal inputs (to give to the reconstruction) and the inputs actually applied (different under source noise).

To avoid the inverse crime, simulate on a finer (or different) mesh than the reconstruction mesh: electrode patterns and electrode voltages do not depend on the mesh.

corrupt_image(img; rng = Random.default_rng(), steps = 1, spatial_noise = 0.05,
              spectral_noise = 0.01, damping = 1e-3, exponent = 1)

Synthetic degradation of an image for training denoisers (wiki: Spectral Image Corruption). Repeats steps times: add white noise (spatial_noise), transform to the orthonormal DCT-II basis, add frequency-dependent noise spectral_noise (1 + |ω|²) ξ, damp by exp(-damping |ω|^(2 exponent)), transform back. |ω|² are the eigenvalues of the discrete Neumann Laplacian of the pixel grid, scaled to ≈ k² + l² for low frequencies (k, l), so damping alone (exponent = 1) is the discrete heat semigroup: it preserves the mean and satisfies the maximum principle.