From Fourier Amplitude to Persistence Images: A New Approach to Impact-Echo Defect Detection

Impact-echo testing is a non-destructive technique used to detect internal defects in concrete structures, but boundary reflections, structural interference, and measurement noise often complicate the Fourier amplitude spectrum, making defect classification difficult. A recent study proposes converting the Fourier amplitude spectrum into a two-dimensional frequency–amplitude persistence image (FA-PI), offering a new way to address this challenge.

From 1D Spectrum to 2D Image: The Construction Logic of FA-PI

Traditional impact-echo signal analysis relies on the Fourier amplitude spectrum, identifying peak positions along the frequency axis to determine defects. However, when a concrete structure has multiple reflective interfaces or external interference, peaks in the spectrum become blurred or even overlap, making simple peak detection unreliable for distinguishing defect types.

The core innovation of this study is that it no longer treats Fourier components as isolated points on a one-dimensional frequency axis. Instead, each retained Fourier component is mapped into a two-dimensional space—one coordinate is the frequency bin position, and the other is amplitude. On this basis, the researchers apply a Gaussian kernel function to smooth the data in this 2D space, ultimately generating a Persistence Image.

This construction has a direct analytical connection to the one-dimensional Gaussian-smoothed Fourier amplitude spectrum. In other words, FA-PI does not completely discard traditional spectral information; rather, it preserves its core features while adding a structured representation along the amplitude dimension. The fixed parameterization method of Persistent Homology ensures that each Fourier component can be precisely represented, avoiding information loss.

Persistent Homology is a core tool in Topological Data Analysis (TDA), used to extract multi-scale topological features from data. Its basic idea is to examine the shape of data under different scale parameters, record the "birth" and "death" of topological features (such as connected components, loops, and voids), and encode this information as a Persistence Diagram or Persistence Barcode. A Persistence Image is a vectorized representation that projects points in the persistence diagram onto a fixed grid weighted by a Gaussian kernel, making it directly usable by machine learning classifiers. The innovation of this paper lies in treating each component of the Fourier amplitude spectrum as a point in the persistence diagram, thereby achieving topological encoding of spectral information.

Experimental Validation: Systematic Evaluation on Two Datasets

The research team evaluated FA-PI on two types of datasets: one is an experimental backfill void dataset, and the other is a public bridge deck impact-echo dataset. These two datasets represent controlled laboratory conditions and real-world engineering scenarios, respectively, covering different levels of defect detection needs.

The evaluation used a linear classifier with Softmax output, and ensured that the signal-level data splits were completely consistent. The key experimental design is that testing includes not only original signals but also signals contaminated by Gaussian noise, sinusoidal interference, or mixed noise. More importantly, the training, validation, and testing stages used matching noise types and signal-to-noise ratios, ensuring the fairness and reproducibility of the evaluation results.

Performance metrics include accuracy as well as macro-averaged precision, recall, and F1 score. The use of macro-averaging means each class is given equal weight, avoiding evaluation bias caused by class imbalance.

Macro-averaging is a commonly used metric aggregation method in multi-class evaluation. Unlike micro-averaging, macro-averaging first calculates precision, recall, and F1 score separately for each class, then takes the arithmetic mean across all classes, giving each class equal weight. This is especially important in class-imbalanced datasets: if a class has very few samples, micro-averaging will be dominated by the majority class and mask the performance on the minority class, whereas macro-averaging can expose the model's shortcomings on minority classes. This paper's use of macro-averaged precision, recall, and F1 score indicates that the researchers are concerned with whether various defect types (such as voids, delamination, and intact regions) are identified in a balanced way, rather than merely pursuing overall accuracy.

Performance Advantages Under Noisy Conditions

In the summary of noise conditions across both datasets, FA-PI consistently outperformed the raw Fourier amplitude spectrum and the Gaussian-smoothed Fourier amplitude spectrum baselines. This result has clear practical significance: impact-echo testing in real engineering often faces various noise interferences, and only methods that can maintain stable performance under noisy conditions have genuine application value.

From a methodological perspective, this finding indicates that representing Fourier spectral components as structures in a two-dimensional frequency–amplitude space can improve impact-echo defect classification, and this improvement goes beyond what can be achieved by simply smoothing along the frequency axis. In other words, the introduction of the amplitude dimension is not just an embellishment; it provides additional discriminative information that frequency-axis smoothing cannot capture.

Signal-to-Noise Ratio (SNR) is a key parameter for measuring signal strength, defined as the ratio of signal power to noise power, usually expressed in decibels (dB). In impact-echo testing, SNR determines whether the characteristic defect peaks in the spectrum can be reliably identified: under high SNR, peaks are clearly distinguishable; under low SNR, noise may submerge or shift the peaks. In the experiments in this paper, matching noise types and SNR were used in the training, validation, and testing stages, meaning the model learns and is evaluated under the same noise conditions. This avoids artificially inflated or deflated performance caused by a "clean" training set and a "noisy" test set, ensuring fair comparison. This rigorous design also allows FA-PI's advantages to be attributed to its representation method itself, rather than to incidental factors in the data split.

Methodological Implications and Limitations

The value of this work is not limited to impact-echo testing itself. It provides a general idea for the fields of signal processing and defect detection: when one-dimensional spectral analysis hits a bottleneck, embedding spectral components into a higher-dimensional space may bring new breakthroughs. As a tool of topological data analysis, persistence images have found a concrete engineering application scenario here.

However, the researchers also maintained a cautious attitude. The paper points out that the construction of FA-PI relies on fixed-parameterization exact multi-parameter persistent homology, which means parameter selection may affect final performance. In addition, the current evaluation is limited to linear classifiers, and whether more complex classifiers can further amplify FA-PI's advantages remains to be explored.

From an engineering deployment perspective, FA-PI's computational complexity, real-time performance, and difficulty of integration with existing impact-echo testing equipment are all issues that require further consideration. Nevertheless, this study provides a noteworthy new direction for the field of non-destructive testing of concrete structures.

Fixed Parameterization in persistent homology refers to presetting the resolution or grid size of scale parameters, rather than adaptively selecting parameters. In the construction of persistence images, fixed parameterization determines the bandwidth of the Gaussian kernel and the resolution of the image: too small a bandwidth leads to a sparse image that is sensitive to noise; too large a bandwidth may blur the boundaries between different topological features. This paper emphasizes "fixed-parameterization exact multi-parameter persistent homology," meaning each Fourier component is deterministically mapped to a fixed position in the two-dimensional space, without changing coordinates as the signal varies. This ensures consistency and reproducibility of the representation, but it also means that parameter choices (such as Gaussian kernel width and the granularity of frequency bin partitioning) need to be tuned for specific tasks and may affect final classification performance.

One-line Summary

This study proposes the FA-PI method, which maps the Fourier amplitude spectrum into two-dimensional persistence images, significantly improving impact-echo defect classification under noisy conditions.

Paragraph Summary

This article introduces a new method for impact-echo defect detection—Fourier Amplitude Persistence Image (FA-PI). To address the problem that traditional Fourier amplitude spectra suffer from blurred peaks and difficult defect classification under boundary reflections and noise interference, FA-PI maps each Fourier component into a two-dimensional space (frequency bin position and amplitude) and applies a Gaussian kernel to generate persistence images, which has an analytical connection to the one-dimensional Gaussian-smoothed spectrum. The study systematically evaluates the method on two datasets: an experimental backfill void dataset and a public bridge deck dataset, covering Gaussian noise, sinusoidal interference, and mixed noise, with matched noise conditions used in training/validation/testing. Results show that FA-PI consistently outperforms the raw and Gaussian-smoothed Fourier amplitude spectrum baselines under noisy conditions, with better macro-averaged precision, recall, and F1 score. The method demonstrates that a two-dimensional frequency–amplitude representation can provide additional discriminative information that frequency-axis smoothing cannot capture, opening a new direction for impact-echo defect classification.