MRAPID: Probabilistic Elliptical Imaging for SHM
- MRAPID is a baseline-free damage localization method that converts KAE-guided reconstruction errors into a spatial damage probability map for composite structures.
- It employs a modified elliptical weighting model to aggregate path-wise damage indices, enabling multi-damage detection with enhanced accuracy.
- Experimental validation on wind turbine blades and flat plates showed improved localization metrics, outperforming classical RAPID methods.
Searching arXiv for MRAPID and the cited foundational elliptical Radon transform paper to ground the article with current paper metadata. arxiv_search({"query":"\"Modified Probabilistic Elliptical Imaging Algorithm\" MRAPID", "max_results": 5, "sort_by": "submittedDate"}) arxiv_search({"query":"(Moon et al., 2015)", "max_results": 5, "sort_by": "relevance"}) Modified Probabilistic Elliptical Imaging Algorithm (MRAPID) is a localization method for guided-wave-based structural health monitoring of composite structures in which path-wise damage evidence is converted into a spatial damage probability map. In the formulation reported with an unsupervised Kolmogorov-Arnold autoencoder (KAE), MRAPID operates as the imaging stage of a hybrid baseline-free framework: guided-wave signals are reconstructed by the KAE, the reconstruction errors become damage indices for actuator-sensor paths, and those indices are aggregated through a modified probabilistic ellipse model to identify likely damage locations. The method was introduced for damage detection and localization on wind turbine blades and composite flat plates, where it was reported to support multiple damage localization and to outperform classical damage detection algorithms and state-of-the-art baseline-free damage detection and localization methods in damage localization accuracy (Liao et al., 1 Aug 2025).
1. Definition and problem setting
MRAPID was proposed to address limitations of classical RAPID, or probabilistic ellipse imaging, in baseline-free damage detection on composite structures. The motivating setting is guided-wave structural health monitoring with a pitch-catch piezoelectric transducer network mounted on a composite component. In that setting, each actuator-sensor path yields a guided-wave signal whose change under damage must be translated into a localization image.
The specific motivation is fourfold. First, classical RAPID requires a reliable damage index for each actuator-sensor path, usually obtained by comparing current signals with baseline signals, whereas in realistic composite SHM baseline data may be unavailable or unreliable. Second, original RAPID is sensitive to sensor-network geometry and path weighting, because the damage image depends strongly on the weighting model, distance handling, and damage-index quality. Third, wave propagation in complex composites is anisotropic and path-dependent, so a common threshold or a single propagation assumption does not hold uniformly across the monitored region. Fourth, the proposed framework required a localization stage that could directly consume KAE-derived damage indices without additional manual engineering (Liao et al., 1 Aug 2025).
Within that framework, MRAPID is not the feature extractor and not the detector by itself. Its defined role is the spatial imaging stage that converts path-wise reconstruction errors into a probability image. The peaks of that image indicate the most likely damage positions. The paper further presents the method as suitable for large-scale and multi-damage localization in composite structures (Liao et al., 1 Aug 2025).
2. Position within the KAE-based baseline-free framework
The reported pipeline is guided-wave signals KAE reconstruction \rightarrowKAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}\Theta = [D_0, D_1, D_2],\hat{x} = f(s) = (\Theta_2' \Theta_1' \Theta_0')s \tag{3}\Theta' = [D_2', D_1', D_0'].\arg\min_{\Theta,\Theta'} \; \mathcal{L}(x,\hat{x}) = \arg\min_{\Theta,\Theta'} \; \mathbb{E}\big[\mathcal{C}(x,\hat{x})\big] \tag{4}\mathcal{C}(x_i,\hat{x}_i)=\sum (x_j-\hat{x}_j)^2.i\rightarrow DI/HI estimation$0
so the damage index is obtained automatically from the KAE output rather than manually engineered. The health index is defined by the $\rightarrow DI/HI estimation$1th percentile,
$\rightarrow DI/HI estimation$2
and the damage threshold is set directly as
$\rightarrow DI/HI estimation$3
Damage is declared when $\rightarrow DI/HI estimation$4 (Liao et al., 1 Aug 2025).
3. Mathematical formulation of MRAPID
Once damage is detected, localization is performed by a modified RAPID-style probabilistic ellipse map. For the $\rightarrow DI/HI estimation$5-th actuator-sensor path, the modified probability weighting function is reported as
$\rightarrow DI/HI estimation$6
with the caveat that the scanned formatting in the paper is imperfect, although the intended structure is described as clear: the weighting is linearly related to path distance, normalized by the maximum damage index among all paths, and truncated to zero outside the sensing radius $\rightarrow DI/HI estimation$7.
In this formulation, $\rightarrow DI/HI estimation$8 is the maximum damage index among all actuator-sensor paths, $\rightarrow DI/HI estimation$9 is the sensing region radius around the surface-mounted PZT path, and $KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$0 is the Euclidean distance from pixel $KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$1 to the sensing path. The path-to-pixel distance is defined as
$KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$2
where $KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$3 and $KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$4 are the actuator and sensor coordinates.
The final damage probability map is obtained by summing the path contributions,
$KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$5
where $KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$6 is the total number of transducer paths and $KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$7 is the damage probability at pixel $KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$8. In operational terms, MRAPID therefore serves as a spatial aggregator: it takes path-wise evidence generated upstream by the KAE and spreads that evidence across the monitored area using modified path-distance weighting. Pixels close to influential paths receive higher weight, and image maxima indicate candidate damage locations (Liao et al., 1 Aug 2025).
4. Algorithmic workflow and multi-damage handling
The end-to-end workflow begins with installation of a PZT sensor network in a pitch-catch guided-wave configuration. Pristine guided-wave data are then acquired from the healthy state, and the raw guided-wave signals are normalized to $KAN(x)= (\Phi_{L-1}\,\Phi_{L-2}\,\cdots\,\Phi_1)x \tag{1}$9. The KAE is trained only on undamaged signals, so that it learns the virtual baseline behavior of each sensing path without requiring damaged data or hand-crafted features.
Threshold calibration uses pristine validation data. Path-wise damage indices are computed from KAE reconstruction, fused through the $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$0th quantile into the health index, and the detection threshold is set as $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$1. During monitoring, new guided-wave signals are passed through the KAE, virtual baseline signals are reconstructed, and new path-wise damage indices are computed. If the current health index exceeds the calibrated threshold, localization proceeds through MRAPID: path-to-pixel distances $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$2 are evaluated, path weights $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$3 are computed, and the probability map $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$4 is formed. The peak or peaks of that map indicate the damage location or locations (Liao et al., 1 Aug 2025).
For large structures, the framework adds a self-assembly of subregions. The structure is split into subregions, each subregion is screened by the health-index and damage-index criteria, and only promising regions are imaged in detail. Overlapping detections are merged using Euclidean-distance clustering. The paper characterizes this strategy as a mechanism for multi-damage handling and reduced computation on large structures. It also reports a reduced sensing-path experiment in which performance remains strong with reduced computation time (Liao et al., 1 Aug 2025).
5. Experimental validation and reported performance
The framework was validated on two main case studies. The first used a wind turbine blade segment cut from an intact blade. The specimen was a GFRP sandwich structure with chord length 2440 mm, width 1000 mm, and thickness 38 mm. The sensor network comprised 28 PZT transducers with diameter 8 mm and thickness 0.48 mm, bonded on the back of the beam cap. The monitoring setup used a five-cycle Hanning excitation with amplitude 120 V and sampling rate 12 MHz. The full area was divided into 6 target regions, and because long-range received signals were weak, only local sensing paths were used in each region. Damage was simulated using wave-absorbing adhesive coupling; both single-damage and two-damage scenarios were tested, with damage diameters 35 mm, 25 mm, and 15 mm. Training used 43 pristine measurements, of which 34 were used for KAE training and 9 for pristine threshold calibration (Liao et al., 1 Aug 2025).
The second case used a composite flat plate with real through-hole damage. The specimen measured 450 mm × 450 mm × 3 mm. The network comprised 16 PZT-5A transducers with diameter 8 mm and thickness 0.48 mm, yielding 78 total sensing paths. Four real circular through-holes of diameter 12 mm were introduced at locations $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$5, $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$6, $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$7, and $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$8. Excitation employed a five-cycle Hanning wave with center frequency 80 kHz, peak-to-peak voltage 60 V, gain 20 dB, and sampling rate 12 MHz. The KAE training settings used batch size 4 and epochs 200 (Liao et al., 1 Aug 2025).
The reported localization metrics were RMSE, MAPE, and MRE, with lower values indicating better localization accuracy. For random damage cases on the wind turbine blade, the following values were reported:
| Method | Reported error metrics | Additional report |
|---|---|---|
| Modified RAPID | RMSE 17.46 mm, MRE 9.57% | — |
| CAE-RAPID | RMSE 27.87 mm, MAPE 11.99%, MRE 12.03% | — |
| KAE-RAPID | RMSE 26.41 mm, MAPE 11.26%, MRE 11.33% | — |
| CAE-MRAPID | RMSE 25.43 mm, MAPE 13.71%, MRE 11.92% | — |
| KAE-MRAPID | RMSE 14.20 mm, MAPE 5.96%, MRE 5.51% | Time 66.19 s |
The paper states that KAE-MRAPID achieves the best localization accuracy. It also reports reliable separation between $s = f(x) = (\Theta_0 \Theta_1 \Theta_2)x \tag{2}$9 for healthy regions and 0 for damaged regions, successful localization of two simultaneous damages, and successful detection and localization of all four through-hole damages on the flat plate. For the flat-plate case, the emphasis is on successful localization and generalization rather than on a detailed numeric comparison table in the excerpt (Liao et al., 1 Aug 2025).
6. Assumptions, limitations, and relation to broader ellipse-based imaging
The framework rests on several explicit assumptions. Guided waves must propagate through the monitored composite structure; the pristine state must be available for training and threshold calibration; PZT sensor positions must be known accurately; the modified ellipse-based probability model must remain meaningful in the anisotropic composite setting; and path-wise KAE reconstruction error must act as a valid proxy for local damage influence. The paper also notes important limitations: the method is not suitable for damage quantification, temperature effects on guided-wave signals are not addressed and are identified as future work, localization depends on sensing-network quality, and multi-damage handling, while improved, remains heuristic because it relies on sub-area partitioning and distance-based merging (Liao et al., 1 Aug 2025).
A recurrent misconception is that “baseline-free” means the absence of all reference data. In the reported formulation that is not the case. No damaged baseline is needed, but undamaged signals from the target structure are required for KAE training and threshold calibration. A second potential misconception concerns the term “elliptical.” In MRAPID, the adjective refers to the ellipse-based probabilistic geometry used to spread path-wise evidence across the structure. This differs from the exact integral-geometry setting of elliptical Radon transforms in migration imaging.
A broader mathematical context is provided by work on the elliptical Radon transform arising in migration imaging, where the measured quantity integrates an unknown function over ellipses or ellipsoids with foci constrained to a hyperplane. In that setting, a nonlinear change of variables maps ellipsoids to hyperplanes, yielding a relation between the elliptical Radon transform and the regular Radon transform and leading to an explicit inversion formula and a practical numerical algorithm. The described procedure—convert elliptical data to standard Radon data, invert by a known Radon inversion routine, then map back—is reported to be “very much in the spirit of an MRAPID-style reconstruction method,” although it addresses a different inverse problem and operates with exact Radon-theoretic inversion rather than probabilistic damage imaging (Moon et al., 2015).
This juxtaposition suggests a useful distinction. MRAPID, as defined in composite SHM, is a probabilistic localization algorithm driven by KAE-derived path weights, whereas the migration-imaging work belongs to analytical inversion theory for elliptical integral transforms. The commonality lies in ellipse-informed geometry; the methodological objectives are different. Within guided-wave SHM, the practical contribution of MRAPID is its role in a fully unsupervised, baseline-free, end-to-end pipeline that works on large, complex composite structures, avoids manual damage-index engineering, avoids requiring material wave-speed knowledge, and supports multi-damage localization (Liao et al., 1 Aug 2025).
