Accurate fully automatic horizontal-position estimation of buried pipes from GPR B-scans under asymmetric crescent signals

When ground-penetrating radar (GPR) is scanned across a buried pipe, the reflected echoes trace a crescent-shaped (hyperbolic) arch in the B-scan image. Kobayashi, Uchikado, and Nakano (ICWAPR 2012) built a pipe-position detection pipeline that follows this crescent with a particle filter whose likelihood comes from two-dimensional Gabor wavelet angle information, and showed that it locates most pipes in single-layer and multi-layer test environments. However, its horizontal-position estimates are biased toward the strong side of the crescent whenever the weak side fluctuates - a consequence of unknown, spatially non-uniform soil permittivity - producing residuals of up to 0.266 m against a pipe radius of 0.05 m. The open problem, stated in the source as a requirement for fully automatic detection, is to remove this asymmetric-crescent bias so that automatic estimation locates every pipe accurately, eliminating the inaccuracy of some detections that the source identifies as the obstacle to fully automatic detection; the authors suggested reducing the asymmetric-crescent effect and improving the robustness of the trajectory estimation as possible directions. No later work resolves it: the authors' own 2014 Dirichlet-process replacement deferred precise estimation to future research, and subsequent hyperbola-fitting, Kalman-filter, and deep-learning approaches evaluate different criteria on different data.

Background

Ground-penetrating radar (GPR) is a non-destructive geophysical tool that emits short electromagnetic pulses into the ground and records the reflected waves. A single reflected waveform sampled as amplitude versus time is called an A-scan; the two-dimensional array of A-scans recorded while the antenna is moved horizontally across the surface is a B-scan, denoted $B(t,x)$, with $t$ the reflection delay time and $x$ the horizontal antenna position. The pulse travels at speed $v = c_0/\sqrt{\varepsilon_r}$, where $c_0$ is the speed of light in vacuum and $\varepsilon_r$ is the relative permittivity (dielectric constant) of the medium; because real soils are non-uniformly wet and water has high permittivity, $\varepsilon_r$ underground is unknown and spatially varying, which makes depth estimates from delay times unreliable and motivates focusing on horizontal position. When the GPR crosses a buried pipe, the echoes form a crescent-shaped arch in the B-scan, called a crescent signal; the slope of this arch changes with the permittivity structure between the antenna and the pipe, so a large variety of crescent patterns arises and simple template-based pattern matching is unsuitable for recognizing them.

Building on their earlier work on denoising GPR signals with a two-dimensional Gabor wavelet transform and on emphasizing reflections using Markov chain Monte Carlo sampling with infinite Gaussian mixtures (a Bayesian mixture model whose number of components is not fixed in advance), Kobayashi, Uchikado, and Nakano (ICWAPR 2012) proposed the following pipeline for estimating pipe horizontal positions. A two-dimensional Gabor wavelet transform (2D GWT) - the convolution of the B-scan with a Gaussian-windowed complex exponential rotated to analysis angle $\tau$ - is computed at integer angles from -30 degrees to +30 degrees, so the squared transform magnitude $W_{\tau}$ measures how strongly the image content is aligned with direction $\tau$. Delaunay triangulation (a triangulation of a point set in which no point lies inside any triangle's circumcircle, avoiding skinny triangles) is applied to random points in a region of interest; connected components of triangles whose integral of $W_{30}$ exceeds the mean are retained, and a weighted mean and covariance over each component seed the particles. A particle filter (a sequential Monte Carlo estimator that propagates a cloud of weighted random samples, resampling them by weight and perturbing them at each step) then follows the crescent slope: its likelihood function is $W_{\tau}$, and its prediction function advances each particle by $\nu\,[|c|, |c|\tan\tau]^{\top}$ with $c$ drawn from a Cauchy distribution and direction flag $\nu = +1$ for a forward pass over angles 30 to -30 degrees and $\nu = -1$ for a backward pass back to 30 degrees. The expectation (weighted mean) of the particle cloud at each backward-pass angle forms an expectation trajectory, to which a second-order polynomial is fitted by polynomial regression (least-squares fitting of a quadratic in the horizontal coordinate); if the quadratic opens upward (is convex in the $(x, t)$ plane), the trajectory is accepted as tracing a crescent and the horizontal coordinate of its vertex (apex) is taken as the pipe's horizontal position.

The method was evaluated on three environments: a single layer (soil with four metal pipes of 0.1 m diameter - radius 0.05 m - at depths 0.5, 1.0, 1.5, and 2.0 m, spaced 2 m apart), a multi-layer environment 1 (the same pipes beneath 0.3 m of pavement, simulating a sidewalk), and a multi-layer environment 2 (two pipes beneath 0.5 m of pavement, simulating a roadway). Regions of interest were indicated manually, and each environment was run 1,000 times with independent random seeds, reporting per-pipe mean estimated positions, residuals, and standard deviations. Most pipes were located, but residuals reached 0.266 m for the deepest single-layer pipe and -0.163 m in multi-layer environment 1, exceeding the 0.05 m pipe radius. The authors identified the cause: the weak side of the crescent signal fluctuates (the reflection time from the pipe varies with position because permittivity is non-uniform), which pulls the particle-filter trajectory toward the strong side of the crescent and biases the polynomial apex with it. They stated that this inaccuracy must be resolved to achieve fully automatic detection, suggesting reduction of the asymmetric-crescent effect and improved robustness of the particle-filter trajectory estimation as future work.

Later literature has not closed this question. The same authors' follow-up (IECON 2014) replaced the particle filter with a Dirichlet process mixture model (a Bayesian nonparametric mixture whose number of components is inferred rather than fixed, based on the Dirichlet process) over samples drawn from the 2D GWT result, reported only that some estimated positions were close to the true ones, found the constant-permittivity assumption erroneous, and deferred precise estimation to future research; neither author published further GPR work after 2014. The Zhou-Chen group (2018-2020) developed automatic B-scan interpretation by hyperbola segmentation and constrained parabolic fitting, GPR combined with electric-field sensing for plastic pipes, and a Dirichlet Process Pipeline Mapping Model combining GPR with GPS for survey-level pipeline mapping. Afrasiabi et al. (2023, 2025) built Kalman-filter-based frameworks (the Kalman filter is a linear-Gaussian recursive state estimator; theirs was tuned by a genetic algorithm and combined with discrete wavelet transforms) for detecting and locating buried utilities from GPR data, evaluated by receiver operating characteristic (ROC) detection statistics. The 2025 review by Liu et al. documents the field's shift toward deep-learning detectors and states persisting challenges in annotated-data dependence, generalization, and computational cost; hyperbola asymmetry itself appears in the literature mainly as a feature for saliency mapping (Yuan and Fan 2018), not as a corrected source of position bias.

Problem Statement

Resolve the asymmetric-crescent position bias that prevents fully automatic buried-pipe detection from GPR B-scans, as stated in the 2012 ICWAPR paper and left unresolved by later literature.

Specifically: given B-scan data $B(t,x)$ acquired over ground under which metal pipes of 0.1 m diameter (radius 0.05 m) are buried at depths between 0.5 m and 2.0 m - in a single-layer soil, or beneath 0.3 m to 0.5 m pavement layers over soil, with unknown and spatially non-uniform relative permittivity - estimate the horizontal position of every pipe such that:

1. Accuracy: every pipe is located and the inaccuracy of some detections that the source states must be enhanced for automatic detection is eliminated. This must hold in particular for pipes whose crescent signals are asymmetric because of weak-side fluctuation, which in the 2012 pipeline coincided with the worst reported residuals - 0.266 m (deepest pipe, single layer) and -0.163 m (multi-layer environment 1) against a pipe radius of 0.05 m - while the remaining pipes achieved per-pipe mean residuals between -0.029 m and 0.093 m (single layer excluding the deepest pipe: -0.024 m to 0.049 m; multi-layer environment 2: -0.029 m to 0.044 m). The large asymmetric-crescent residuals must be removed: the per-pipe mean residual of every tested pipe, taken against known ground truth over repeated independent trials, must have magnitude at most the 0.05 m pipe radius, and no tested pipe's per-pipe mean residual may be larger than its value under the uncorrected baseline on the same testbed.
2. Automation: the pipeline operates fully automatically, without manual region-of-interest selection or per-scan expert parameter tuning.

The specific estimator is not prescribed. The evaluation environments must exhibit the phenomenon under study: crescent signals made asymmetric by the spatially non-uniform permittivity, of the kind that biased the source's pipeline. A testbed whose permittivity is so benign that crescents are effectively symmetric and the uncorrected baseline is already accurate for every pipe does not exercise or resolve the problem.

References

1. Makoto Kobayashi, Tomohiro Uchikado, Kazushi Nakano (2012). Wavelet-based position detection of buried pipes from GPR signals by use of angle information. 2012 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR), Xian, China, pp. 390-396. DOI: 10.1109/icwapr.2012.6294813. https://doi.org/10.1109/icwapr.2012.6294813
2. Makoto Kobayashi, Kazushi Nakano (2014). Dirichlet process crescent-signal mixture model for ground-penetrating radar signals. IECON 2014 - 40th Annual Conference of the IEEE Industrial Electronics Society, Dallas, TX, USA, pp. 3431-3437. DOI: 10.1109/IECON.2014.7049007. https://doi.org/10.1109/IECON.2014.7049007
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