EL-DIP: A Deep Image Prior Framework for Accurate,Efficient, and Noise-Robust Elastography

• EL-DIP reconstructs material properties from noisy measurements without paired data.Source: Teaser · p. 1

• Elastography reconstructs spatial mechanical properties from displacement or strain fields under external loading.Source: Introduction · p. 1
• Measurement noise can destabilize elastography, while stronger explicit regularization may suppress oscillations but oversmooth inclusion interfaces.Source: Introduction · p. 1
• Existing CNN, cGAN, and PINN approaches may require paired data, representative samples, or carefully designed physical constraints, limiting use when conditions differ from training data.Source: Introduction · p. 1
• EL-DIP replaces explicit regularization with an untrained CNN's implicit prior and couples it to a finite-element solver for noisy heterogeneous-property reconstruction.The paper reports numerical and experimental demonstrations of inclusion geometry and shear-modulus recovery in noisy settings.Source: Introduction · p. 2

Forward problem

• The forward model adopts incompressible linear elasticity and solves a finite-element system to obtain nodal displacements for a given shear-modulus field.Source: Forward problem · p. 3

Elastography based on DIP

• EL-DIP optimizes an untrained CNN's parameters from displacement data, then uses its output as the reconstructed shear-modulus field rather than directly optimizing that field.Source: Elastography based on DIP · p. 3

Neural network architecture

• A U-Net-like encoder-decoder maps random-noise input to the modulus field, with skip connections and an adaptive sigmoid using learnable upper and lower bounds.Source: Neural network architecture · pp. 3–4

Arbitrary shape mapping

• IDW maps network-grid modulus predictions to finite-element nodes, weighting nearer output points more strongly and supporting arbitrarily shaped domains.Source: Arbitrary shape mapping · pp. 4–5

Optimization of the EL-DIP framework

• Each iteration maps the network field to the mesh, solves the forward and adjoint problems, then uses the resulting gradient to update network parameters with Adam.The adjoint avoids explicitly forming displacement sensitivity with respect to all network parameters.Source: Optimization of the EL-DIP framework · p. 5

Numerical experiments

• As SNR falls from 60 to 20 dB, CC, PSNR, and SSIM reductions are smaller for EL-DIP (2.0%, 12.7%, and 2.1%) than for conventional optimization (12.4%, 45.7%, and 13.3%).The figure summarizes mean CC, PSNR, MSE, and SSIM across eight cases.Source: Figure 3 · p. 6, Numerical experiments · p. 7
• At 26 dB, EL-DIP increases CC from 0.86 to 0.97 and SSIM from 0.79 to 0.93, improves PSNR by 10.12 dB, and reduces MSE by 90.2% versus conventional optimization.At 20 dB, it gains 9.30 dB in PSNR and reduces MSE by 88.0%.Source: Figure 3 · p. 6, Numerical experiments · p. 7
• Across square and circular domains, EL-DIP preserves inclusion locations, shapes, and modulus contrast more clearly as noise increases, especially for complex configurations.Compared with conventional optimization, its reconstructions show sharper interfaces and less boundary smoothing and structural-detail loss.Source: Figure 3 · p. 6, Numerical experiments · p. 7

Experimental results

• In tissue phantoms, both methods identify a single inclusion, while EL-DIP maps it closer to a circular shape and gives more accurate boundaries and a smoother background for two inclusions.The experimental phantom’s inclusion-to-background shear-modulus ratio is 3.Source: Experimental results · pp. 7–8, Figure 4 · p. 8
• In cardiomyocyte nuclei, EL-DIP identifies chromatin architecture and delineates euchromatin from heterochromatin, whereas OP does not clearly distinguish the two types.Nuclear displacement fields were obtained by deformation microscopy.Source: Figure 4 · p. 8, Experimental results · p. 9

Discussion

• Neural networks tend to learn low-frequency components faster than high-frequency components, giving EL-DIP an implicit regularization mechanism that helps explain stable high-noise reconstructions.The proposed explanation is partial, not the sole attribution of noise robustness.Source: Discussion · pp. 9–10
• Under finite-width or small-learning-rate conditions, treating the NTK matrix as constant reduces training dynamics to a linear system for the prediction error.The NTK entries measure coupling between network outputs induced by parameter changes.Source: Discussion · p. 9
• Error modes with larger NTK eigenvalues decay faster; spectral bias associates these faster modes with smoother structure, so dominant inclusion geometry is recovered before noise artifacts.The eigenmode decay is expressed as exp(−𝜂𝜆𝑘𝑡).Source: Discussion · p. 9

Conclusion

• EL-DIP represents the unknown property field with an untrained convolutional neural network and uses its implicit regularization instead of explicit regularization.IDW mapping, an adaptive sigmoid layer, finite-element forward analysis, and adjoint-based gradients support arbitrary-shaped domains without training data.Source: Conclusion · p. 10
• Numerical benchmarks and experiments on soft-material phantoms and cell nuclei demonstrate recovery of inclusion location, geometry, and modulus contrast with stronger noise robustness than the conventional adjoint method.Under moderate-to-high noise, EL-DIP better preserves inclusion boundaries and suppresses noise-induced artifacts.Source: Conclusion · p. 10
• The demonstrated scope is 2D linear elastic problems, while real materials may involve nonlinear behavior and three-dimensional deformation.The conclusion identifies nonlinear constitutive models and 3D inverse problems as directions for extension.Source: Conclusion · p. 10
• Repeated finite-element solutions during optimization still impose considerable computational cost; transfer learning and surrogate models are proposed for future acceleration.Source: Conclusion · p. 10

SUPPLEMENTARY MATERIALS

• The supplementary materials section identifies the authors as Hao Liu, Fuyi Li, Dong Liu, Yue Mei, and Sheng Mao.Source: SUPPLEMENTARY MATERIALS · p. 13

This PDF file includes:

• The supplementary materials include Supplementary Text A to C.Source: This PDF file includes · p. 13
• The listed supplementary visuals comprise Table S1 and Figs. S1 to S4.Source: This PDF file includes · p. 13

Definitions of quantitative evaluation metrics

• The evaluation compares reconstructed field values μ̂_i with ground-truth values μ_i across N pixels or nodes.The corresponding field means are denoted μ̂ and μ, and μmax is the ground-truth maximum used as the PSNR reference peak value.Source: Supplementary Text A. Definitions of… · p. 14
• SSIM uses local means, variances, and covariance within sliding windows, with the final value averaged across the field.Small positive constants C1 and C2 prevent numerical instability when denominators approach zero.Source: Supplementary Text A. Definitions of… · p. 14
• Lower MSE and higher PSNR, CC, and SSIM indicate better agreement with the ground-truth field.The metrics are computed between the reconstructed and ground-truth shear-modulus fields.Source: Supplementary Text A. Definitions of… · p. 14

Case-wise quantitative results of simulated examples

• Table S1 reports CC, PSNR, MSE, and SSIM for all eight simulated cases at 60, 40, 26, and 20 dB.Each comparison uses reconstructed and ground-truth shear-modulus fields on the same evaluation discretization.Source: Supplementary Text B. Case-wise… · p. 14
• Higher CC, PSNR, and SSIM and lower MSE indicate better reconstruction performance.The table highlights the better value between EL-DIP and conventional optimization for each case, noise level, and metric.Source: Supplementary Text B. Case-wise… · p. 14
• The main text summarizes the mean metric values across the eight cases for an overall comparison of the two methods.The supplementary table provides the corresponding case-wise values.Source: Supplementary Text B. Case-wise… · p. 14

Back-propagation based on the adjoint method

• The adjoint method computes the loss gradient with respect to network parameters under the discretized equilibrium constraint, avoiding direct solution of Eq. S7.The gradient is used to update network parameters with Adam.Source: Back-propagation based on the adjoint… · pp. 14–15

Implementation details

• EL-DIP uses Adam with learning rate 0.1, β1 = 0.5, and β2 = 0.999, with up to 2,000 simulated-data or 500 experimental-data iterations.When a loss divergence criterion is exceeded, parameters are restored to the previous iteration and the learning rate is reduced by a factor of 0.9.Source: Implementation details · p. 15
• Table S1 presents case-wise CC, PSNR, MSE, and SSIM comparisons under 60, 40, 26, and 20 dB noise levels.The supplied table text gives its scope and metric headings, but not the case-wise values.Source: Table S1 · p. 15, Implementation details · p. 15
• Supplementary figures show EL-DIP and conventional reconstructions across noise levels in rectangular and circular domains and for two cardiomyocyte nuclei.For the first nucleus, the panels include microscopy, x- and y-displacement fields, and shear-modulus reconstructions from both methods.Source: Figure S1 · p. 16, Figure S2 · p. 17, Figure S3 · p. 18