Comparing Parametric Curves: Fourier Series-Based Similarity Metric

External question — Cross Validated Stack Exchange Author: E Fresher Original post: https://stats.stackexchange.com/questions/659553 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Developing a Similarity Metric for Parametric Curves Using Fourier Series I'm exploring ways to compare parametric curves on the xy-plane using their Fourier series representations. My goal is to develop a similarity metric that best captures the 'shape' of the curves. Here are some ideas I'm considering: Background: A parametric curve on the xy-plane is represented as a complex function: $z(t) = x(t) + iy(t)$ where $x(t)$ and $y(t)$ are the parametric equations of the curve, and $t \in (0, 1)$ . This complex function can be expressed as a Fourier series: $z(t) = \sum_{n=-\infty}^{\infty} c_n e^{2\pi int}$ The Fourier coefficients $c_n$ are calculated as: $c_n = \int_0^1 z(t) e^{-2\pi int} dt$ These coefficients contain both magnitude and phase information: $c_n = |c_n|e^{i\phi_n}$ where $|c_n|$ is the magnitude and $\phi_n$ is the phase. To start, we normalize the Fourier series to have energy 1: $\sum_{n=-N}^{N} |c_n|^2 = 1$ Current Considerations: Phase Information : I believe phase is crucial in determining curve shape. Is there evidence suggesting otherwise? Related Work : Some existing methods compare power spectral densities (ignoring phase) These approaches often treat the PSD as a probability density Similarity of two discrete Fourier transforms Comparing two distributions in Fourier space Potential Approaches: Separate Magnitude and Phase Comparison : Compare power spectral density and phase spectral density separately Treat both as probability distributions Use methods from the related work mentioned above Cosine Similarity in Cartesian Form : Write Fourier coefficients in flattened Cartesian form (which naturally includes both magnitude and phase information) Apply cosine similarity Pros: Naturally bounded between -1 and 1 Cons: Unclear interpretation for scores near 0 or negative L1/L2 Distance in Cartesian Form : Similar to approach 2, but use L1 or L2 distance instead Question: Is there a principled way to develop a similarity metric for parametric curves on the xy-plane using their Fourier series representations that accounts for both magnitude and phase information? I'm particularly interested in understanding the trade-offs between these approaches and any other methods I might have overlooked.