P-Spline Curvature Estimator | Mo Talebi-Kalaleh

Data

Arc length [m] Sample spacing Δs [m] Number of bends Bend width σ [m] Noise std [deg] Random seed

Each bend is a smooth step in pitch angle modelled as an error function. The displayed curvature is the analytical derivative; the smoother is asked to recover it from the noisy angle signal alone.

Drop or choose a CSV / TXT One column = y, uniform spacing · or two columns = x, y

Sample step (single column)

Smoother

B-spline degree p Number of segments K Penalty order d log₁₀(λ) λ = 10^2.0

Smoothed signal

Noisy measurements (gray), penalized B-spline fit (bronze). For synthetic data, the true underlying signal is shown in black for comparison.

First derivative (curvature)

Closed-form analytical derivative of the B-spline fit (no numerical differentiation of noisy data). Vertical lines mark detected bends from peak-finding on the absolute curvature; for synthetic data, the true bend locations are shown for reference.

Second derivative

Closed-form second derivative of the B-spline fit, useful for identifying bend start / end and inflection points.

GCV / AIC scan

Generalized cross-validation and Akaike-information-criterion curves over a log-spaced grid of λ. The minimum is the data-driven choice used by the auto button.

Bend table

Method

The fit is a penalized B-spline (Eilers and Marx, 1996). Given noisy observations y_i at locations x_i, we represent the unknown function as a linear combination of M = K + p evenly-spaced B-splines of degree p: f(x) = Σ α_j B_j(x). The coefficients α are found by minimizing

||y - Bα||² + λ · ||D α||²

where D is the d-th order finite-difference operator and λ ≥ 0 controls the trade-off between fidelity and smoothness. The solution is α = (B^TB + λ D^TD)^-1 B^Ty and the smoothed values are ŷ = Bα.

Derivatives of the fit are obtained analytically using the B-spline derivative recursion dB_i,p/dx = (p/h)(B_i,p-1 - B_i+1,p-1), which is much more reliable than finite-differencing the noisy data. The first derivative is the local curvature for an IMU pitch / yaw signal along a pipeline.

Optimal λ is chosen by Generalized Cross-Validation: GCV(λ) = (n · RSS) / (n - tr(H))², with H = B (B^TB + λ D^TD)^-1 B^T the hat matrix. AIC = n · log(RSS/n) + 2·tr(H) is shown alongside for reference.

Bend detection: peak finding on the absolute first derivative with a height threshold equal to a multiple of the residual standard deviation propagated through the derivative basis.

Related publications

Talebi-Kalaleh, M.; Tan, L.; Shirzad, N.; Adeeb, S.; Mei, Q. (2026). Penalized B-Spline Curvature Estimation for Automated Pipeline Bend Detection from Inertial Measurement Unit Data. ASME IPC 2026, Paper IPC2026-186408.
Eilers, P.H.C.; Marx, B.D. (1996). Flexible Smoothing with B-splines and Penalties. Statistical Science, 11(2), 89-121.
de Boor, C. (2001). A Practical Guide to Splines, revised ed., Springer.