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REVIEW 3 major objections 5 minor 64 references

Deep regularization networks for inverse problems with noisy operators

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A two-step neural network learns regularization maps for inverse problems with noisy operators and improves image contrast.

desk verdict A worthwhile two-step learning idea for noisy-operator regularization, but the Step 2 quality gains are carried by early stopping, not by the stated objective. read the letter →

arxiv 2506.07008 v2 pith:4WJO4EIC submitted 2025-06-08 math.NA cs.AIcs.NAeess.SP

Fatemeh Pourahmadian , Yang Xu This is my paper
classification math.NAcs.AIcs.NAeess.SP MSC 65F2265J2068T07
keywords linearsamplingmethodTikhonovregularizationMorozovdiscrepancyprinciplenetworksinversescatteringultrasonicimagingnoisyoperatorlossbalancing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to show that the expensive per-pattern choice of a regularization parameter in linear sampling method imaging can be learned, even when the forward operator itself is noisy. The proposed scheme trains a small MLP on a low-resolution subset of right-hand side patterns to imitate Morozov discrepancy maps made with a deliberately rough threshold, then refines the network by minimizing the Tikhonov imaging objective under an early-stopping rule. The authors claim this two-step procedure accelerates regularization while improving image contrast, with the largest gains appearing in 25-percent-noise reconstructions of a complex evolving crack field. A sympathetic reader would care because, if the claim holds, real-time ultrasonic imaging of damage could bypass manual regularization tuning and use only a tiny fraction of the right-hand side data.

What carries the argument

The load-bearing object is the regularization network (R-Net), an MLP whose input is the right-hand side pattern projected onto the left singular vectors of the noisy scattering operator, $U^*u_L$, and whose output is a Tikhonov parameter $\alpha_{NN}$ for that pattern. The spectral filter factors $D_{jj}/(\alpha_{NN}+D_{jj}^2)$ then define the regularized solution $g_{NN}=VD^\dagger U^*u_L$, which is what the linear sampling method indicator uses. The argument is carried by two training mechanisms: (1) in Step 1, the informed loss combines a normalized misfit to the Morozov $\alpha_M$ with the squared Morozov discrepancy functional $J_2^t$ evaluated at a rough threshold $\eta_\circ$, and the chosen weights in (19) make both gradient terms $O(\partial \alpha_{NN}/\partial w)$, so the discrepancy logic is learned without an inner optimization for loss balancing; (2) in Step 2, minimizing $J_{img}$ tailors $\alpha$ toward a min-norm solution, and Algorithm 2 stops training when the rms of normalized training or validation loss variations crosses a threshold or their ratio leaves an interval, which is the mechanism claimed to convert extra training into higher image contrast without overfitting.

What would settle it

In the 25-percent-noise cases, re-run Step 2 with the stopping thresholds $\sigma_a$ and $\sigma_r$ removed so training continues for many more epochs, and record the defect-to-background contrast at every epoch: if contrast keeps improving after the rule would have stopped, or never peaks near the rule's stop point, then the image enhancement is not produced by the loss-trajectory regulation that carries the claim.

Watch

Extended reading notes

Core claim

The central discovery claimed is that a network trained first to imitate cheap, low-resolution Morozov discrepancy maps and then fine-tuned on the Tikhonov imaging loss can produce linear sampling method reconstructions whose contrast matches or beats the Morozov maps obtained by per-time-step manual tuning, even when the scattering operator carries 25 percent noise. The network takes the projected right-hand side $U^*u_L$ as input and outputs the Tikhonov parameter $\alpha$; with that $\alpha$ the spectral filter factors $D_{jj}/(\alpha + D_{jj}^2)$ supply the regularized solution. Step 1 uses an informed loss that includes the discrepancy principle itself, with adaptive weights that normalize the gradients without a second optimization loop, and Step 2 minimizes the Bayes-risk Tikhonov loss with an early-stopping rule based on normalized training-versus-validation loss trajectories. The evidence is a synthetic elastic-plate experiment in which a single crack grows into a cloud of fifteen cracks over five sensing steps; at 25 percent noise the Step-2 images show higher defect-to-background contrast than the manually tuned Morozov baseline at almost every sensing step.

Load-bearing premise

The paper's claimed contrast gain in Step 2 rests on a heuristic early-stopping rule: the relative trajectories of training and validation Tikhonov losses are assumed to reveal when further minimization stops helping the image, and if that rule fails on a new geometry or noise level, the refinement step can make reconstructions worse rather than better.

Editorial extensions

If this is right

  • Dense regularization maps for the full 100 by 100 search grid can be produced from a network trained on only 2,500 of 720,000 right-hand side patterns.
  • A discrepancy-informed Step 1 generalizes better than direct regression on Morozov maps, so the choice of training loss determines whether the network can be refined in Step 2.
  • Step 2 improves the mean and maximum contrast of LSM images relative to manually tuned Morozov reconstructions, with the largest gains appearing at 25 percent noise and late sensing steps.
  • Because Step 2 needs no labeled optimal regularization maps, the network can adapt to test data online, removing the need to tune the Morozov threshold by hand.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: Apply the same scheme to generalized linear sampling method or time-domain imaging, where per-pattern root-finding is even more expensive; the discrepancy functional changes but the projection-plus-early-stopping logic should transfer, and this is untested in the paper.
  • Extension: The only guard against $\alpha$ collapsing toward zero in Step 2 is the stopping rule, so a deployment should monitor a ground-truth-free contrast metric alongside the loss trajectories; the paper reports no correlation analysis between loss rms and contrast.
  • Extension: If low-resolution-to-dense generalization is the mechanism, substituting a resolution-independent network for the fixed-size MLP could let one trained model serve arbitrarily fine grids, which the paper flags as possible but does not demonstrate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a two-step deep-learning method for choosing per-right-hand-side Tikhonov parameters in linear sampling method (LSM) reconstructions when the scattering operator itself is noisy. In Step 1, an MLP is trained on low-resolution regularization maps produced by the Morozov discrepancy principle with a rough threshold η0; two modes are considered, a basic squared-error mode and a 'discrepancy-informed' mode that adds the discrepancy functional to the loss with adaptive weights. In Step 2, the network is fine-tuned by minimizing the Tikhonov imaging loss on the training right-hand sides, with an early-stopping rule based on training and validation loss trajectories. The method is tested on synthetic elastodynamic data for a plate with an evolving damage zone at 0%, 10%, and 25% noise, and the resulting LSM images are compared with manually tuned Morozov reconstructions through contrast metrics. The authors report that discrepancy-informed R-Nets generalize from 50×50 to 100×100 grids and that Step 2 increases contrast, especially at high noise.

Significance. The engineering problem is relevant: per-RHS Tikhonov regularization for LSM is a genuine bottleneck for real-time imaging, and the idea of learning a map from projected right-hand sides to α from coarse Morozov maps is attractive. The paper's strengths include complete algorithmic pseudocode, an explicit SVD-based architecture, a substantial synthetic testbed with five damage configurations and three noise levels, and an honest comparison with manually tuned Morozov baselines. If the Step-2 claim were supported, the practical contribution would be valuable. However, the current manuscript does not establish that Step 2 improves images as a consequence of the stated objective; the objective is degenerate in α, and the claimed gains appear to be selected by early-stopping thresholds. The discrepancy-functional equations also contain an internal inconsistency that affects the Step-1 labels and the loss-balancing derivation.

major comments (3)
  1. [3.2, Eqs. (22)–(24)] The Step 2 objective is degenerate in α and does not, as stated, drive the network toward higher-quality images. For a fixed right-hand side u_L, substituting (23) into (22) gives J_t(α)=Σ_j α |(u_j^*,u_L)|^2/(α+d_j^2), which is strictly increasing for α>0 and tends to 0 as α→0; the unconstrained minimization therefore pushes α_NN toward 0 and ||g_NN|| toward infinity, and no interior optimum exists. Because V_img in (24) is the same functional evaluated on a validation set, its trajectory cannot independently detect this norm blow-up. As written, the reported Step 2 contrast gains are selected by the particular thresholds σ_r=5, σ_a=10^{-4}, N_rms=10^4 and the learning rate 5×10^{-8}, not by a proven property of the objective. Please either add a penalty or constraint that gives (22) a nontrivial minimizer, or reframe Step 2 as an empirically early-stopped heuristic and provide a sensitivity analysis over these thresholds plus an independent quality measure on held-out configurations.
  2. [3.1.2 and 4.4, Eqs. (15), (19), (32)] The discrepancy functional is internally inconsistent. For the Tikhonov filter in (23), the Morozov condition ||F^δ g−u_L||=η||g|| is Σ_j (α²−η²D_jj²)/(α+D_jj²)² |(u_j^*,u_L)|²=0, but Eqs. (15) and (32) write the numerator as α−η²D_jj². The derivative in (19), 2D_jj²(α+η²)/(α+D_jj²)³, is the derivative of the α² version, not of the printed α version. This matters because (32) generates the Morozov labels used for Step 1 training and for the manually tuned baselines, while (19) is the basis of the proposed loss-balancing weights. The equations should be reconciled and the numerical results re-checked with the corrected Morozov functional.
  3. [4.4, Figs. 18, 22, 26 and Tables 4–6] The evaluation of Step 2 is transductive: the network is trained on a subset of the RHS patterns of the same damage configuration, and the reported LSM images are then reconstructed from the full RHS dictionary of that same configuration, including the patterns used in Step 2. The claimed 'remarkable enhancement' therefore does not yet demonstrate generalization to unseen configurations. Please report results on a genuinely held-out configuration, or at least on RHS patterns excluded from both training steps, and state whether the contrast gains persist there.
minor comments (5)
  1. [Eq. (24)] The summation upper limit is written as Ntrn, although the validation set is indexed by ν=1,…,Nν; this should be Nν.
  2. [4.4, informed R-Net Step 1] The text states that the training runs for t∈[1,1000], while Step 2 is said to start at t=10001 and the caption of Fig. 17 says Step 1 ends at t=10^4; please reconcile the epoch counts.
  3. [3.2, sentence after Eq. (23)] The statement that minimizing (22) 'minimizes the residual ... while maximizing the LSM indicator functional' is not a direct consequence of the displayed objective, since it trades residual against ||g||² and does not explicitly maximize 1/||g||; please rephrase.
  4. [Algorithm 2] The stopping rule uses OR conditions, so a flattening of either the training or the validation loss can stop training; a patience parameter or a more explicit combination of the conditions may better match the stated goal of detecting overfitting.
  5. [Table 7] Reporting the time per epoch is useful, but the total wall-clock time until the stopping criterion is met, including the validation-loss evaluation in Step 2, would better support the real-time imaging claim.

Circularity Check

2 steps flagged · score 6.0 of 10

Step 2 'enhancement' reduces to a stopping-time artifact: the Tikhonov loss (22) with g_NN from (23) is strictly increasing in alpha and vanishes at alpha->0, so no interior optimum exists and reported gains are selected by Algorithm 2 thresholds, not by the stated imaging objective.

  1. self definitional [Section 3.2, Eqs. (22)-(23), Algorithm 2, and Section 4.4 around Eq. (34)]
    "In this step, the learning objective is to furnish regularization maps {αtNN}t=1,2,...,Ntrn that optimize the imaging (i.e., Tikhonov) loss function within the Bayes risk minimization framework, Jimg = ... wherein ∀t ∈ 1,2,...,Ntrn, gtNN(αtNN) = V D†U∗utL, D†(D, αtNN) = diag { Djj/(αtNN + D2jj) }."

    Substituting Eq. (23) into Eq. (22) gives, per RHS pattern, Jimg(α) = Σ_j α |(u_j*,u_L)|^2 / (α + d_j^2) with d_j = D_jj. This is strictly increasing for α > 0 and tends to 0 as α → 0, so the unconstrained minimum of the Step-2 objective is at α → 0, where ||g_NN|| diverges and the LSM indicator 1/||g|| collapses. Hence the claimed 'optimization' cannot select an interior regularizer or enhance contrast; every finite-α result is effectively chosen by the early-stopping thresholds in Algorithm 2 (σ_r = 5, σ_a = 10^-4, N_rms = 10000). The derivation 'minimizing the imaging loss improves the image' reduces by construction to 'a chosen stopping time selects the image', so the enhancement claim is not independently derived from the objective.

  2. fitted input called prediction [Abstract and Section 3.2, Eqs. (22)-(24), with results in Tables 4-6]
    "This approach enables direct learning from test data and dispenses with the need for a-priori knowledge of the optimal regularization maps (or labeled datasets) for training."

    Step 2 optimizes the network on a subset {U∗ut_L} of the same right-hand-side library {U∗un_L} that also defines the validation set and the final LSM reconstructions, with the validation set being the complement of the training subset within that same library. The reported contrast gains in Tables 4-6 are therefore measured on the very data distribution used to fit the Step-2 parameters, so the 'enhancement' is partly a measure of fitting to the evaluation data rather than an independent prediction. The paper explicitly advertises learning from 'test data' as an advantage, which makes the quality claim self-referential rather than externally validated.

full rationale

Step 1 is a legitimate supervised learning stage: low-resolution Morozov maps provide an independent anchor, and validation is performed on held-out RHS patterns. The self-citations to [41,42] establish operator properties and [47] only inspires loss balancing; none of these are load-bearing for the central claim. The circularity is concentrated in Step 2. Its Tikhonov objective is degenerate: because g_NN(α) is by construction the minimizer of the same loss, substituting it into Eq. (22) yields a function that is minimized at α → 0, so no finite interior optimum exists. The advertised 'optimization' therefore cannot be the mechanism of contrast enhancement; the reported images are selected by Algorithm 2's manually fixed stopping thresholds. In addition, the evaluation is performed on the same RHS library from which the Step-2 training and validation sets are drawn, so the headline claim of 'remarkable enhancement' rests on a partly self-referential fitting loop. Step 1 provides independent content, but the Step-2 'prediction' reduces by construction to the stopping rule and to fitting the evaluation library, giving a partial circularity score of 6.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The method introduces no new physical entity. Its load-bearing inputs are two hand-set scalars (η◦ and the baseline thresholds), a small set of network hyperparameters, and several unproven modeling assumptions about noise and Step 2 behavior. Correcting Eq (15)/(32) would change the training labels and loss.

free parameters (4)
  • Morozov threshold η◦ for training labels = 0.3 (recommended interval [0.3,0.5))
    Section 3.1: chosen based on the authors' synthetic experiments; every Step 1 label α_M is computed from this value.
  • Manually tuned baseline thresholds η_k^{np} = 0.08 to 0.30 depending on time step and noise level (Table 2)
    Used to define the 'optimal' Morozov maps that the R-Nets are compared against; selected by visual comparison of reconstructions.
  • Network training hyperparameters = learning rates 1e-5, 5e-6, 5e-8; epochs up to 10000; stop thresholds σa=1e-4, σr=5; Nrms=10000
    Section 4.4: chosen by hand for all experiments; the claimed performance depends on these values.
  • Downsampling factor and training grid size = m=2, 50x50 training grid, Ntrn=2500
    Section 4.4: one trial scatterer per point via Eq (10) plus factor-2 downsampling; generalization from this grid is load-bearing.
assumptions (7)
  • domain assumption The scattering operator F^δ is compact, injective, and has dense range.
    Section 2, after Eq (7); standard LSM hypothesis cited to [41,42,3].
  • domain assumption Classical LSM with B=I is an adequate imaging model; generalized LSM forms are deferred.
    Section 1 and Eq (8); the whole method is built on Tikhonov regularization of Fδg=u.
  • domain assumption Noise acts multiplicatively as Fδ=(I+Nδ)F with entries uniformly distributed in [-δ,δ].
    Section 4.3 Eq (30); the noisy-operator setting is modeled by this synthetic perturbation.
  • ad hoc to paper One trial scatterer per sampling point, selected by min-norm (Eq 10), and a 50x50 grid are representative for training the regularization map.
    Section 3.1 and Section 4.4; no analysis shows this selection preserves the dense-map structure.
  • ad hoc to paper Minimizing the Tikhonov loss in Step 2 improves LSM image contrast.
    Section 3.2; the objective is degenerate at α→0, so the claimed improvement relies on the empirical stop criterion.
  • ad hoc to paper The validation-loss trajectory criterion (Algorithm 2) detects overfitting and preserves image quality.
    Section 3.2, Eqs (25)-(28); introduced without theoretical guarantee.
  • standard math MLP with two ReLU hidden layers is sufficient to represent the regularization map.
    Section 3; justified only by universal approximation, with capacity selected heuristically.

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Cite this review

Pith. "Pith review of Deep regularization networks for inverse problems with noisy operators." pith.science (2026). https://pith.science/paper/4WJO4EIC

@misc{pith2026250607008,
  author       = {Pith},
  title        = {Pith review of: Deep regularization networks for inverse problems with noisy operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4WJO4EIC}},
  note         = {Machine review of arXiv:2506.07008}
}
read the original abstract

A supervised learning approach is proposed for regularization of large inverse problems where the main operator is built from noisy data. This is germane to superresolution imaging via the sampling indicators of the inverse scattering theory. We aim to accelerate the spatiotemporal regularization process for this class of inverse problems to enable real-time imaging. In this approach, a neural operator maps each pattern on the right-hand side of the scattering equation to its affiliated regularization parameter. The network is trained in two steps which entails: (1) training on low-resolution regularization maps furnished by the Morozov discrepancy principle with nonoptimal thresholds, and (2) optimizing network predictions through minimization of the Tikhonov loss function regulated by the validation loss. Step 2 allows for tailoring of the approximate maps of Step 1 toward construction of higher quality images. This approach enables direct learning from test data and dispenses with the need for a-priori knowledge of the optimal regularization maps. The network, trained on low-resolution data, quickly generates dense regularization maps for high-resolution imaging. We highlight the importance of the training loss function on the network's generalizability. In particular, we demonstrate that networks informed by the logic of discrepancy principle lead to images of higher contrast. In this case, the training process involves many-objective optimization. We propose a new method to adaptively select the appropriate loss weights during training without requiring an additional optimization process. The proposed approach is synthetically examined for imaging damage evolution in an elastic plate. The results indicate that the discrepancy-informed regularization networks not only accelerate the imaging process, but also remarkably enhance the image quality in complex environments.

Figures

Figures reproduced from arXiv: 2506.07008 by the authors.

Figure 1
Figure 1. The proposed learning logic for the Regularization Network. The network (i) takes the SVD of scattering operator F δ = UDV ∗ , (ii) projects the right-hand side u n L onto the eigenspace of measurements U, and (iii) outputs the affiliated regularization parameters α n NN. The output is then used to compute the spectral filter factors D † ii, i = 1, 2, . . . , Neig, germane to Tikhonov regularization and the scatteri… view at source ↗
Figure 2
Figure 2. Sensing configuration of synthetic experiments on an elastic plate (left) featuring a damage zone comprised of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Picard plots at sensing steps tk, k = 1, 2, . . . , 5, wherein Dii, i = 1, 2, . . . , 2000, represents the i th eigenvalue of scattering operator F k, constructed from noiseless data at every tk, while (U ∗u n L)i = (u ∗ i , u n L) with n = 1, 2, . . . , Ntrn is the projected RHS patterns onto the i th left eigenvector of F k. u n L (ξj ) over the observation grid ξj ∈ S obs , j = 1, 2, . . . , N, is computed separa… view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: LSM reconstructions from noiseless data by way of the Morozov discrepancy principle: (a) ground-truth [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: LSM reconstructions from 10% noisy data using the Morozov discrepancy principle: (a) manually optimized [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: LSM reconstructions from 25% noisy data using the Morozov discrepancy principle: (a) manually optimized [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Loss trajectories in Step 1 of training R-Nets in the basic mode using noiseless data: (a) training loss [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: R-Net’s performance in Step 1 of training in the basic mode using noiseless data: (a) network-predicted [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Convergence plots in Step 2 of training the [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Defect and background regions used to compute the contrast metrics for the reconstructions at time steps [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Performance of the basic R-Nets trained on noiseless data: (a) manually optimized Morozov regularization maps αM on the dense grid (of 100×100 sampling points) at every sensing step tk, k = 1, 2, . . . , 5, (b) network-predicted regularization maps α b1 NN by the end …
Figure 12
Figure 12. Figure 12: LSM reconstructions corresponding to the regularization maps of Fig. 11: (a) LSM images [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: Loss trajectories in Step 1 of training the [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Spatial distribution of loss weight w t 1 in (19) used for training discrepancy-informed R-Nets on a reduced grid (of 50×50 sampling points) when the training is conducted using: (a) noiseless data, (b) 10% noisy data, and (c) 25% noisy data [PITH_FULL_IMAGE:figures/…
Figure 15
Figure 15. Figure 15: Spatial distribution of loss weight w t 2 in (19) used for training discrepancy-informed R-Nets on a reduced grid (of 50×50 sampling points) when the training is conducted using: (a) noiseless data, (b) 10% noisy data, and (c) 25% noisy data. 25 [PITH_FULL_IMAGE:figu…
Figure 16
Figure 16. Figure 16: Loss trajectories in Step 2 of training the [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: Performance of the discrepancy-informed R-Nets trained on noiseless data: (a) manually optimized Morozov regularization maps αM on the dense grid (of 100×100 sampling points) at every sensing step tk, k = 1, 2, . . . , 5, (b) network-predicted regularization maps α i1…
Figure 18
Figure 18. Figure 18: LSM reconstructions corresponding to the regularization maps of Fig. 17: (a) LSM images [PITH_FULL_IMAGE:figures/full_fig_p027_18.png]
Figure 19
Figure 19. Figure 19: Loss trajectories in Step 1 of training the [PITH_FULL_IMAGE:figures/full_fig_p028_19.png]
Figure 20
Figure 20. Figure 20: Loss trajectories in Step 2 of training the [PITH_FULL_IMAGE:figures/full_fig_p029_20.png]
Figure 21
Figure 21. Figure 21: Performance of the discrepancy-informed R-Nets trained on 10% noisy data: (a) manually optimized Morozov regularization maps αM on the dense grid (of 100×100 sampling points) at every sensing step tk, k = 1, 2, . . . , 5, (b) network-predicted regularization maps α i1…
Figure 22
Figure 22. Figure 22: LSM reconstructions corresponding to the regularization maps of Fig. 21: (a) LSM images [PITH_FULL_IMAGE:figures/full_fig_p030_22.png]
Figure 23
Figure 23. Figure 23: Loss trajectories in Step 1 of training the [PITH_FULL_IMAGE:figures/full_fig_p031_23.png]
Figure 24
Figure 24. Figure 24: Loss trajectories in Step 2 of training the [PITH_FULL_IMAGE:figures/full_fig_p032_24.png]
Figure 25
Figure 25. Figure 25: Performance of the discrepancy-informed R-Nets trained on 25% noisy data: (a) manually optimized Morozov regularization maps αM on the dense grid (of 100×100 sampling points) at every sensing step tk, k = 1, 2, . . . , 5, (b) network-predicted regularization maps α i1…
Figure 26
Figure 26. Figure 26: LSM reconstructions corresponding to the regularization maps of Fig. 25: (a) LSM images [PITH_FULL_IMAGE:figures/full_fig_p033_26.png]

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Pith tools

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