Regularisation and Parameter Choice Rules for Composite Linear Inverse Problems

1 Introduction

• Ill-posed inversions of noisy data require regularisation that balances data error against approximation error.Choosing the regularisation parameter appropriately in dependence on the noise level is also important for high-quality reconstructions.Source: §1 · p. 1
• For a composite operator, the proposed strategy uses regularisations for F and G instead of one complicated scheme for both operators simultaneously.The component regularisations may be easier to compute.Source: §1 · p. 1
• The paper extends filter-based parameter-choice results to composite operators, deriving optimal rules with smoothness information and an adapted discrepancy principle without it.The adapted principle is motivated by the absence of an error estimate for the second operator equation.Source: §1 · p. 2
• The method is tested on simulated magnetic particle imaging data and a deconvolution task, with applications of operator splitting also including image zooming and measurement preprocessing.The introduction also describes examples involving segmentation and reduced or limited measurements.Source: §1 · p. 2

2.1 Regularisation Property of the double regularisation

• The proposed double regularisation composes approximations to the inverses of G and F, assuming A† = F† ◦ G† because this equality need not hold generally.The paper gives sufficient and necessary conditions for that equality, citing.Source: §2.1 · p. 3

Theorem 1

• Under the stated existence and parameter-choice assumptions, the concatenation RF ◦ RG is a regularisation for A†.The theorem also relates the error to regularisation of F† and the noise-free right-hand side in Fx = y†.Source: Theorem 1 · p. 3

2.2 Analysis of the Convergence Speed

• The convergence-speed analysis assumes compact, simultaneously diagonalisable F and G, while restricting the true solution to a suitable subspace allows speed results and potentially optimal rates.The text notes that compactness of both operators simplifies the analysis and that arbitrary true data do not generally admit the needed convergence-speed result for ill-posed G.Source: Theorem 1 · p. 3, §2.2 · p. 4

2.2.1 Trivial Cases

• When F is well-posed, the composite method is order optimal, and with F = id and an optimal regularisation for G, it is optimal.The section introduces these one-well-posed-operator cases before treating the case where both operators are ill-posed.Source: §2.2.1 · p. 4

Case 1: F well-posed

• When F is well-posed, its continuous pseudoinverse permits choosing RF as F†, and the composite regularisation is order optimal.The passage states that each parameter-choice rule for the other regularisation then fulfils Theorem 1's conditions.Source: Case 1: F well-posed · p. 4
• If F = id and RG is an optimal regularisation, the composite method is optimal.The passage notes that the pseudoinverse may not always be available for use.Source: Case 1: F well-posed · p. 4
• When the pseudoinverse is unavailable, the analysis considers an a-priori rule for a regularisation of F† along sequences of noise levels tending to zero.The argument uses the regularisation property and the Banach–Steinhaus theorem.Source: Case 1: F well-posed · p. 4

Case 2: G well-posed

• When G is well-posed, choosing RG = G† and an a-priori parameter rule for F that tends to zero as δ tends to zero yields an order-optimal regularisation.The stated parameter choices fulfil condition (3).Source: Case 2: G well-posed · p. 5

2.2.2 Non-trival Case: both F and G ill-posed

• When both F and G are ill-posed, the paper derives both a-priori and a-posteriori parameter-choice rules.The a-posteriori rule is identified elsewhere in the paper as the double discrepancy principle.Source: §2.2.2 · p. 5

A-Priori Parameter Choice Rule

• The a-priori analysis assumes properties of RG, selects γ as a function of δ for RF, and considers asymptotic operator-norm behaviour.These are the stated ingredients of the parameter-choice analysis.Source: A-Priori Parameter Choice Rule · p. 5

A-Posteriori Parameter Choice Rule

• When the smoothness parameter ν is unknown, the authors propose a modified discrepancy principle for parameter choice.The classical Morozov principle uses a monotonically decreasing sequence of parameters.Source: A-Posteriori Parameter Choice Rule · p. 5

Further assume

• Classical Morozov discrepancy cannot be applied directly to the composite problem because the split inverse problems require separate parameter choices.The paper names rules (13) and (14) the double discrepancy principle.Source: Further assume · pp. 6–7

Remark 3

• The first regularisation is expected to amplify the noise, and the resulting intermediate error estimate is unavailable without knowing ν.The second inverse problem therefore receives an increased noise level after the first stage.Source: Further assume · p. 6

Further assume

• Under the stated assumptions, rule (13) coincides with Morozov and gives an order-optimal regularisation; finite qualification yields a suboptimal result.The result concerns the regularisation associated with α.Source: Further assume · pp. 8–9

3.1 Basics for Model-based Reconstruction in MPI

• For MPI, the model links measurements to the Radon transform, and changing the reconstruction order permits deconvolution in image space instead of sinogram space.The authors describe the image-space approach as computationally beneficial.Source: §3.1 · pp. 10, 12

3.2 Experiment 1: Model-based MPI Reconstruction

• The BPF method backprojects measurements before filtering; its filter bandwidth determines the regularisation parameter, and the subsequent filtering and deconvolution are computationally cheap.The backprojected result lies on a regular grid, unlike the measurements sampled along a curve in sinogram space.Source: §3.2 · p. 13
• The MPI experiments compare a Kaczmarz system-matrix benchmark with Radon TSVD and BPF regularisations followed by deconvolution; Gaussian noise has standard deviation 10% of signal amplitude.For the double-regularisation tests, Tikhonov regularisation with fixed parameter 0.05 is also applied to address the time-dependent factor s(t).Source: §3.2 · pp. 12–14
• The MPI double-discrepancy reconstructions look better than the system-matrix benchmark in phantom shape and background noise; BPF-based images also have more uniform intensity.The authors report that this holds across the tested regularisation choices, including BPF, which does not meet Assumptions 5.Source: §3.2 · pp. 14–15, Figure 3 · p. 15

3.3 Experiment 2: Convergence speed with the DDP

• The convolution experiment tests DDP convergence under source conditions because smoothness information is unavailable for the MPI example.The operators are discretised with the fast Fourier transform using 2^14 uniformly spaced samples on (−100, 100).Source: §3.3 · p. 15, Consider now the two functions · p. 16

Example 1

• Figure 4 analyses DDP on the convolution problem, including error behaviour as δ tends to zero and a representative TSVD reconstruction.The example reconstruction captures the peak at t = 0, with minor oscillations outside it; reconstruction from yδ amplifies oscillations along the flat part of x.Source: Figure 4 · p. 16, Consider now the two functions · p. 16

Consider now the two functions

• Under source conditions, the DDP’s numerical convergence rates were O(δ^0.4) for Tikhonov and O(δ^0.8) for TSVD; it performed better than expected for Tikhonov and worse for TSVD.Source: §3.3 · p. 15, Consider now the two functions · p. 17

4 Conclusion

• The method combines existing regularisations for structured inverse problems, enabling use of optimised methods, and supplies a-priori rules when source conditions are known and the DDP when they are unknown.The paper analyses convergence speed for all parameter-choice rules.Source: §4 · p. 17
• The a-priori rules have a suboptimal O(·) convergence result, and whether this speed is optimal for the setting remains open.For finite-qualification regularisations, the DDP also has a suboptimal convergence result.Source: §4 · p. 17
• The DDP gave visually strong simulated MPI results across regularisation methods, with the second regularisation parameter apparently stable across choices of the preceding method.The method also performed well with BPF, which is not covered by the assumptions used in the proof.Source: §4 · p. 17
• Convergence tests mostly confirmed the DDP analysis, but its better-than-expected Tikhonov performance requires further investigation.The conclusion identifies δ-free parameter-choice rules for composite problems as a future research direction.Source: §4 · p. 17

A.1 Basics from Regularisation Theory

• The Picard condition connects a source condition with an operator’s singular value decomposition.The cited theorem states a range condition for a compact linear operator in terms of its singular system.Source: §A.1 · p. 20
• An interpolation inequality is applied in the proof of Theorem 6, and a lemma links filter-function properties to the norm of the associated regularisation.The appendix attributes both results to EHN96 and Rie03, respectively.Source: §A.1 · p. 20

A.2 Basics from Computer Tomography

• The Fourier-Slice theorem connects the Radon transform with the ground truth and supports the reconstruction formulas.The theorem is attributed to Nat86.Source: §A.2 · p. 20
• The appendix uses weighted Hilbert spaces to state the Radon transform’s explicitly known singular value decomposition.It cites Dav83 and Lou84 for the decomposition and identifies Chebyshev and Jacobi polynomials in the associated basis.Source: §A.2 · p. 20

• The convolution setup fixes an even, real function h in L1(R) ∩ L2(R).Source: B Functional Calculus for Convolution · p. 21
• For non-compact A, the section restates functional calculus and defines σ(A), B(K), id, and 1.B(K) is the space of measurable functions K → R with the supremum norm, for compact K.Source: B Functional Calculus for Convolution · p. 21
• The stated functional-calculus theorem concerns self-adjoint A on a Hilbert space and includes an involutary algebra-homomorphism property.Source: B Functional Calculus for Convolution · p. 21
• The theorem also states a property for uniformly bounded measurable functions converging pointwise to f.Source: B Functional Calculus for Convolution · p. 21

• The convolution operator’s spectrum is σ(A) = F(h)(R) ⊂ R, and it is represented using the multiplication operator M.Source: We set · p. 21
• The section formulates the functional calculus of A and checks that the conditions of Theorem 15 hold.Source: We set · p. 21
• The proof uses the operator representation and the linearity of the Fourier transform to establish stated properties.The Fourier transform is described as an isometric isomorphism up to a constant.Source: We set · pp. 21–22
• For a uniformly bounded pointwise-convergent family, the proof bounds the relevant functions by N|x(t)y(t)| in L1(R) and invokes Lebesgue convergence.Source: We set · p. 22