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Parametrizations

The unknowns of a reconstruction need not be the finite element coefficients of the conductivity. A parametrization $\sigma = P\theta$ maps parameters to coefficients, and ParametrizedObjective turns any objective into a function of $\theta$ (gradient $P^\top\nabla_\sigma J$, Jacobian $(\partial r/\partial\sigma) P$), so every optimizer applies.

  • PixelParametrization: the parameters are the pixels of an image on a rectangle that contains the domain, independent of the mesh. The mesh can be refined anywhere (e.g. towards the boundary) while the unknowns stay pixels, e.g. for image priors.
  • SubspaceParametrization: pixels restricted to a subspace, e.g. low-frequency DCT modes (dct_basis) for a stable first stage of a coarse-to-fine reconstruction, or DCT modes plus free pixels along the boundary (boundary_band_basis).
pp  = PixelParametrization(disc, 64, 64)                     # disc: any 2D mesh, e.g. refined at the boundary
obj = RegularizedObjective(ParametrizedObjective(data, pp),
                           1e-3 => TotalVariationRegularizer(pp.pixel_disc; ε = 1e-2))
res = minimize(obj, ones(parameter_count(pp)), GaussNewton(); lower = 0.05)
img = pixel_image(pp, res.σ)                                  # 64 × 64, NaN outside the domain

C   = dct_basis(pp, 8)                                        # coarse stage: 64 DCT modes
sp  = SubspaceParametrization(pp, C)
res0 = minimize(ParametrizedObjective(data, sp), [64.0; zeros(63)], GaussNewton())
θ0  = C * res0.σ                                              # pixel start for the fine stage

Bounds on pixel parameters bound the conductivity (the weights of $P$ are non-negative and sum to one). Subspace parameters cannot be bounded this way: trial steps with a non-positive conductivity are rejected (InfeasibleConductivityError).

Theory: wiki article Parametrizations of the Conductivity.

ModularEITFerrite.PixelParametrization — Type
PixelParametrization(disc, n, m; bbox = nothing, pixels = :square, quadrature_order = nothing)

Conductivity on disc parametrized by the values θ of an n × m pixel grid on the rectangle bbox (default: the bounding box of the mesh, widened to square pixels unless pixels = :stretch, as in unit_image): σ = P θ with P the lumped L² projection of the pixel function onto the σ space (weighted pixel averages; positive, constants preserved, exact on pixel-aligned meshes).

Fields: P (ndofs_σ × n m), pixel_disc (P0 discretization of the pixel grid; θ are its coefficients, e.g. for TotalVariationRegularizer), active (pixels that influence σ), centres (pixel centres in parameter order), bbox, n, m.

See conductivity, pixel_image, pixel_parameters, ParametrizedObjective, SubspaceParametrization.

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ModularEIT.SubspaceParametrization — Type
SubspaceParametrization(pixels::PixelParametrization, B)

Pixel values restricted to the span of the columns of B (n m × k): pixels = B θ, σ = P_pixels B θ. Bases: dct_basis (Ferrite back end) (smooth, low-dimensional), boundary_band_basis (Ferrite back end) (free pixels along the boundary), or combinations [B₁ B₂].

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ModularEIT.ParametrizedObjective — Type
ParametrizedObjective(obj, par)

The objective obj as a function of the parameters θ of par (σ = P θ): values, gradients Pᵀ ∇_σ J and, for least-squares objectives, residuals and Jacobians (∂r/∂σ) P, so that all optimizers apply (bounds on pixel parameters bound σ, see PixelParametrization (Ferrite back end)). Regularizers on the parameters act on the pixel discretization, e.g. RegularizedObjective(ParametrizedObjective(data, pp), α => TotalVariationRegularizer(pp.pixel_disc)).

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ModularEIT.pixel_image — Function
pixel_image(par, θ)

The parameters θ of a pixel-based parametrization as an image (row 1 at the top). Back end contract for pixel parametrizations.

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ModularEITFerrite.dct_basis — Function
dct_basis(pp::PixelParametrization, K)

The K × K lowest-frequency orthonormal DCT-II modes of the pixel grid as the columns of an n m × K² matrix (parameter order), sorted by total frequency (the constant mode first). For SubspaceParametrization: smooth, low-dimensional conductivities, e.g. as the first stage of a coarse-to-fine reconstruction.

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ModularEITFerrite.boundary_band_basis — Function
boundary_band_basis(pp::PixelParametrization, width)

Indicator vectors (sparse columns) of the active pixels whose centre lies within width of the domain boundary: free pixel values along the boundary, where the data determine the conductivity best. Combined with dct_basis ([C Bb]) the interior is smooth and low-dimensional and the boundary band fully resolved.

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ModularEIT.InfeasibleConductivityError — Type
InfeasibleConductivityError(msg)

Thrown when a conductivity is not admissible for the forward problem (e.g. non-positive values seen by a preconditioner). Optimizers treat it like a failed factorization: the trial point is rejected and the step shortened.

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Wiki articles

Theory behind this page in the theory wiki: