Pipeline Overview

Architecture

The radsym detection pipeline has three stages, each independently usable:

                   ┌──────────┐    ┌───────┐    ┌────────┐
  Image ──Gradient──▶ Propose ├───▶│ Score ├───▶│ Refine ├──▶ Detections
                   └──────────┘    └───────┘    └────────┘

The design deliberately avoids a monolithic detector. Each stage is a pure function: it reads immutable inputs and returns a new value. You can swap, skip, or repeat stages as your application requires.

Stage 1: Propose

Goal. Convert the gradient field into a small set of candidate center locations.

The primary algorithm is the Fast Radial Symmetry Transform (FRST) (Loy & Zelinsky, ECCV 2002). For every pixel whose gradient magnitude exceeds a threshold, FRST casts a vote at the position offset by $\pm n$ pixels along the gradient direction, where $n$ is a candidate radius. Votes accumulate into an orientation image $O_n$ and a magnitude image $M_n$, which are combined as:

$$F_n = \lvert \tilde{O}_n \rvert^{\alpha} \cdot \tilde{M}_n$$

where $\alpha$ controls radial strictness and $\tilde{\cdot}$ denotes Gaussian-smoothed, clamped accumulators. The per-radius responses are summed into a final ResponseMap.

An alternative proposer, RSD (Barnes et al. 2008), uses a faster single-radius voting scheme optimized for real-time applications.

Non-maximum suppression (extract_proposals) then picks local peaks from the response map, returning a ranked list of Proposal values.

Key types

Stage 2: Score

Goal. Quantify how strongly image evidence supports each proposed circle or ellipse.

Scoring samples the gradient field inside an annular region around the hypothesis. For a circle of radius $r$, the annulus spans $[(1 - m) \cdot r,; (1 + m) \cdot r]$, where $m$ is the configurable margin.

Two complementary signals are extracted:

• Ringness — the mean alignment between each sampled gradient and the radial direction from the hypothesized center. A perfect circle gives ringness $\approx 1$.
• Angular coverage — the fraction of angular bins (around the center) that contain at least one supporting gradient sample. Full coverage means evidence is spread around the entire circumference, not concentrated in one arc.

These are combined into a single SupportScore:

$$\text{total} = w_r \cdot \text{ringness} + w_c \cdot \text{coverage}$$

where the weights $w_r$ and $w_c$ are set in ScoringConfig. Proposals whose total falls below a threshold are discarded.

Key types

Stage 3: Refine

Goal. Improve center position and shape parameters to subpixel accuracy.

Refinement operates in two layers:

1. Center refinement via the Parthasarathy radial center algorithm (Nature Methods, 2012). Each pixel in a local patch defines a line through its position along its gradient direction. The subpixel center is the least-squares intersection of all such lines — a closed-form weighted linear solve, non-iterative and fast.

2. Shape refinement via iterative least-squares. Given the refined center, gradient-weighted points on the annulus are fit to a circle (Kasa, 1976) or an ellipse (Fitzgibbon et al., 1999). The fit updates the shape parameters, a new annulus is sampled, and the process repeats until convergence or the iteration limit.

The output is a RefinementResult<H> parameterized by the hypothesis type (Circle or Ellipse). It contains the refined hypothesis, an RMS residual, the iteration count, and a RefinementStatus flag (Converged, MaxIterations, Degenerate, or OutOfBounds).

Key types

Using Stages Independently

Each stage is a free function with no hidden state. Common patterns:

• Skip proposal generation when centers are known from an external source (e.g., a coarse detector or user annotation). Jump straight to scoring or refinement with a manually constructed Circle or Ellipse.
• Use scoring without refinement for fast accept/reject decisions on a large batch of candidates.
• Run refinement without scoring when you trust the initial hypotheses and only need subpixel geometry.

Homography-Aware Variants

When the target circle is viewed under perspective, it projects to an ellipse in the image. radsym provides homography-aware versions of the propose and refine stages:

• frst_response_homography runs FRST voting in a rectified coordinate frame, so that circles remain circular during voting even when the original image shows a perspective-distorted view.
• refine_ellipse_homography fits a circle in rectified space and maps the result back to an image-space ellipse via the homography.

These functions accept a Homography (a 3x3 projective transform) and a RectifiedGrid that defines the resampled workspace.