GFRS: Generalized Fast Radial Symmetry

Overview

GFRS extends FRST to detect elliptical symmetry under affine distortion. When a circular target is viewed under perspective, it projects to an ellipse. Standard FRST misses such targets because gradient directions no longer converge to a single center at a single radius. GFRS recovers the detection by warping gradient directions through a set of affine maps before voting.

Algorithm

For a sampled set of 2D affine transformations ${A_k}$:

1. At each pixel $p$ with gradient direction $\hat{\mathbf{g}}(p)$, compute the warped direction $A_k, \hat{\mathbf{g}}(p)$.
2. Compute the affected pixel offset using the warped direction and vote into a per-map accumulator.
3. Smooth each accumulator and record its peak response.

The map $A_k$ that produces the strongest peak gives both the center location and an estimate of the affine distortion. The distortion matrix directly yields the ellipse semi-axes and orientation.

Configuration

#![allow(unused)]
fn main() {
pub struct AffineFrstConfig {
    /// Single voting radius (pixels).
    pub radius: u32,
    /// Minimum gradient magnitude to vote.
    pub gradient_threshold: f32,
    /// Gaussian sigma = smoothing_factor * radius.
    pub smoothing_factor: f32,
    /// Set of affine maps to evaluate.
    pub affine_maps: Vec<AffineMap>,
}
}

The number of affine maps controls the trade-off between angular/scale resolution and computation time. Typical usage samples 20–50 maps covering the expected range of perspective distortion.

Feature gate

GFRS is gated behind the affine feature flag:

radsym = { version = "...", features = ["affine"] }

Reference

Ni, K., Singh, M., Bahlmann, C. “Fast radial symmetry detection under affine transformations.” IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 932–939 (2012). doi:10.1109/CVPR.2012.6247767