Numerical reconstruction of polygonal obstacles by diffracted waves

1.1. Overview.

• A single pulsed point source and receiver curve provide diffracted-wave data from which the visible portion of a polygonal obstacle can be uniquely determined.The constructive theory assigns tip locations to arrival times and corner orientations and opening angles to diffraction coefficients.Source: §1.1 · p. 1, §1.2 · p. 2

1.2. Inverse obstacle scattering by diffracted waves.

• A vertex is visible when unobstructed line segments connect it to the source and receiver, defining which part of the obstacle the inverse problem targets.The paper formulates scattering by a rigid polygon in its exterior domain.Source: §1.2 · pp. 1–2

1.2.1. Corner identification.

• For a single corner, the opening parameter β determines the exterior angle 2πβ, and the principal diffracted wave uniquely identifies a diffractive corner under the stated receiver-curve condition.Convex corners have β > 1/2 and concave corners β < 1/2; non-diffractive corners have β equal to the reciprocal of a positive integer.Source: §1.2.1 · pp. 2–3

1.2.2. Polygon identification.

• Polygon recovery localizes temporally separated primary diffracted waves, distinguishes multiple diffraction by Sobolev regularity, and assembles recovered wedges into the visible boundary.The uniqueness theorem is stated for compact convex polygons under geometric conditions; the paper notes the local corner analysis also applies to non-convex polygons when grazing rays are avoided.Source: §1.2.2 · p. 3

1.3. Related work.

• Classical polygon-scattering methods primarily use reflected or transmitted information and typically require substantial-aperture far-field or boundary data, while related work also studies diffraction-based imaging and crack characterization.The paper draws on diffraction theory and numerical wave-solver methods to supply synthetic data for a diffraction-adapted inversion.Source: §1.3 · p. 4

1.4. Contributions.

• The numerical scheme uses diffracted arrival times to locate visible vertices and amplitudes to recover corner angles, enabling local imaging in the reflection shadow.It is tested on convex and concave corners, a triangle, a pentagon, and a non-convex star-shaped obstacle.Source: §1.4 · p. 4

1.5. Structure of the paper.

• The paper proceeds from the diffraction formula and reconstruction scheme to forward-solver and signal analyses, then reports corner, polygon, and noisy-data inversions before summarizing results and limitations.The numerical cases include convex and concave corners, a triangle, a pentagon, and a star-shaped obstacle.Source: §1.5 · p. 4

2.1. Diffraction by a single corner.

• For a single corner, the principal diffracted wave has a Heaviside jump at the tip-dependent arrival time, while its amplitude depends on angular parameters.The diffraction coefficient uniquely determines the angular parameters for diffractive corners in the stated range.Source: §2 · p. 5, §2.1 · p. 6

2.2. Two-step reconstruction for a single corner.

• Each visible corner is recovered in two stages: arrival times locate its tip, then amplitudes determine its angular parameters.The angular least-squares minimizer is unique in the noise-free case.Source: §2.1 · p. 6, §2.2 · p. 7

2.3. Recovery procedure for polygonal obstacles.

• The reconstruction uses the local single-corner analysis at each visible vertex, while the measured field also contains geometric waves and weaker multiple diffracted waves.Invisible vertices and vertices reached only by grazing rays are excluded.Source: §2.3 · p. 7
• The numerical pipeline isolates primary diffracted signals by time separation and Sobolev regularity, then localizes and reconstructs visible corners individually.Assembling the recovered corners yields visible vertices and directions of their adjacent edges.Source: §2.3 · p. 7

• The section proceeds from describing the forward solver to examining simulated diffracted signals and reconstruction experiments.Source: §3 · p. 7
• The wave equation is rewritten as a first-order velocity–pressure system for the forward problem.Source: §3 · p. 7

3.1. Numerical method for the forward problem.

• The forward solver combines a staggered-grid leapfrog scheme with a classical split-field PML and is second-order accurate in space and time.Source: §3.1 · pp. 7–8

3.1.1. Staggered grid and time stepping.

• Pressure is stored at cell centres and velocity components on staggered faces, with pressure at integer time steps and velocity at half-integer steps.The leapfrog update applies away from PML and obstacles and is conditionally stable under a standard CFL restriction.Source: §3.1.1 · p. 8

3.1.2. Initialisation.

• Pulse experiments set φ = 0 and approximate ψ with a narrow Gaussian centred at the source.Source: §3.1.2 · p. 8

3.1.3. Standard split-field PML.

• Rigid-obstacle Neumann conditions are imposed by setting normal velocity to zero, retaining tangential velocity on slanted edges, and reflecting ghost pressures across interfaces.The stated combination yields a stable second-order treatment on the staggered grid used in the experiments.Source: §3.1.5 · pp. 8–9
• The forward solver is checked against the predicted qualitative structure of geometric and diffracted wavefronts before inversion.Source: §3.1.5 · p. 9

3.2. Numerical simulation of the diffracted wave.

• The full-circle receiver configuration visualizes wave propagation and signal structure; this subsection performs no parameter recovery.The simulation uses a Gaussian pulse of width σ = 0.02.Source: §3.2 · p. 9

3.2.1. Convex corner: configuration and snapshots.

• Snapshots show the incident pulse expanding, specular reflections on illuminated faces, and then a circular diffracted wavefront emanating from tip P.Source: §3.2.1 · p. 9, Figure 3.2 · p. 10

3.2.2. Regions I–III and angular Regions A, B, C.

• Region A combines strong specular reflection with weaker diffraction, while Region B lacks reflection and Region C lacks incident signals.The sector pattern explains why diffracted signals remain informative when receivers do not record reflected signals.Source: §3.2.2 · pp. 9–10

3.2.3. Full-circle space–time records and pointwise signals.

• The extracted diffracted signal arrives later than geometric signals, near t = r + r1, and its amplitude already shows the angular modulation predicted by S(θ1; ϕ, β).Source: §3.2.3 · p. 10
• In Regions B and C, diffraction dominates where reflected signals are absent or incident signals do not reach receivers.At representative stations, Region B's diffracted signal arrives at approximately r + r1; Region C records no incident signal.Source: §3.2.2 · p. 10, §3.2.3 · p. 11

3.3. Numerical inversion.

• Arrival times and signed jump amplitudes are extracted from isolated primary diffracted signals; two least-squares stages locate visible tips and fit their angular parameters.The experiments use a single pulsed source and exterior receiver arcs; no wave equations are solved during inversion.Source: §3.3 · pp. 12–13, §3.3.1 · p. 14

3.3.1. Convex single corner.

• The noise-free convex-corner inversion places the tip within about 2 × 10^-3 and recovers β ≈ 0.825 versus 5/6 ≈ 0.833, about 1% relative error.The tip error is measured in the computational window [−1, 1]^2.Source: §3.3.1 · p. 14

3.3.2. Convex corner with noisy data.

• At 0.2% additive noise on L1, the convex-corner tip error is about 3.3%, while the opening parameter β has about 0.67% relative error.The time derivative is more irregular than in the noise-free record, but the diffracted front remains identifiable.Source: §3.3.2 · pp. 14–15

3.3.3. Concave single corner.

• The concave-corner reconstruction recovers the tip and opening parameter β within the mesh resolution, showing the two-step procedure also works for a visible diffractive concave tip.For a concave corner, β ∈ (0, 1/2), and the same inverse map applies when the tip is visible and diffractive.Source: §3.3.3 · pp. 15–16

3.3.4. Triangular obstacle.

• For the triangle, visible vertices are reconstructed one at a time and their local wedges are joined along visible edges; invisible vertices remain undetermined.The inversion uses primary diffracted signals recorded on L1.Source: §3.3.4 · p. 17

3.3.5. Pentagonal obstacle.

• For the pentagon, arrival-time windows and a jump filter recover visible corners separately from L1; primary diffracted waves are more singular than multiple diffracted waves.The angular fit is well conditioned on a single arc when its receiver-angle range makes S(θ1; ϕ, β) vary non-trivially.Source: §3.3.5 · p. 18

3.3.6. Star-shaped obstacle.

• The star-shaped test reconstructs visible vertices from L1 despite mixed convex and concave corners; later multiple-diffraction signals are excluded by arrival-time windows.The result applies when each reconstructed vertex is visible and diffractive and its signal can be separated in time.Source: §3.3.6 · pp. 18–19

4. Conclusions

• A two-stage least-squares method uses diffracted arrival times to locate visible vertices and amplitudes to recover their corner angles.It uses explicit diffraction formulas and requires no wave-equation solves during inversion.Source: §4 · p. 19
• Experiments on convex and concave corners, a triangle, a pentagon, and a non-convex star support Theorems 1.1 and 1.2 at the tested resolutions.For the convex corner, the reported tip error is of order 10^-3 and the relative opening error is about one percent.Source: §4 · p. 19
• The noise test is limited to one convex geometry, one receiver arc, and a known pulse shape; SNR sweeps, model mismatch, and calibration errors remain untested.Vertices close enough for arrival-time windows to overlap would require a sharper microlocal filter; three-dimensional, penetrable-scatterer, and multi-source extensions remain open.Source: §4 · p. 19