Projective Center Recovery

This chapter derives the algorithm that recovers the true projected center of a circle from two concentric circle projections, without requiring camera intrinsics.

The Problem

Under perspective projection, a circle in 3D projects to an ellipse in the image. The center of the projected ellipse is not the projection of the circle’s 3D center. This projective bias is systematic — it pushes the apparent center away from the image center — and grows with the viewing angle and distance from the optical axis.

For calibration applications where subpixel center accuracy is required, this bias must be corrected.

Concentric Circles and the Conic Pencil

Consider two concentric circles in the target plane with radii r_inner and r_outer. Under perspective projection, they map to two conics (ellipses) Q_inner and Q_outer in the image plane.

The conic pencil spanned by these two conics is the family:

Q(λ) = Q_outer - λ · Q_inner

where λ is a scalar parameter. Each member of the pencil is a 3×3 symmetric matrix representing a conic.

Key Insight

The pencil contains degenerate members — conics with determinant zero — that factor into pairs of lines. These degenerate conics correspond to eigenvalues λ of the generalized eigenvalue problem:

Q_outer · v = λ · Q_inner · v

equivalently:

(Q_outer · Q_inner⁻¹) · v = λ · v

The true projected center lies at the intersection of the line pairs from the degenerate pencil members.

Algorithm

The algorithm as implemented in ringgrid:

Step 1: Normalize Conics

Both conic matrices are normalized to unit Frobenius norm. This improves numerical stability for subsequent eigenvalue computation.

Step 2: Compute Eigenvalues

Form the matrix A = Q_outer · Q_inner⁻¹ and compute its three eigenvalues λ₁, λ₂, λ₃ (which may be complex).

Step 3: Find Candidate Centers

For each eigenvalue λᵢ, compute the candidate center using two methods:

Method A (Wang et al.): Find the null vector u of (A - λᵢI), then compute p = Q_inner⁻¹ · u. The candidate center in image coordinates is (p₁/p₃, p₂/p₃).

Method B: Find the null vector of (Q_outer - λᵢ · Q_inner) directly, and dehomogenize.

Both methods are algebraically equivalent but may differ numerically; ringgrid tries both and selects the best.

Step 4: Score Candidates

Each candidate center p is scored by combining several criteria:

Geometric residual: Measures how well p lies on the pole-polar relationship with both conics. Computed as the normalized cross-product of Q₁·p and Q₂·p:

residual = ||( Q₁·p ) × ( Q₂·p )|| / (||Q₁·p|| · ||Q₂·p||)

A true projective center yields residual ≈ 0.

Eigenvalue separation: The gap between λᵢ and its nearest neighbor. Well-separated eigenvalues indicate a stable solution; degenerate (repeated) eigenvalues are numerically unstable.

Imaginary-part penalty: Small weights penalize complex eigenvalues and eigenvectors, since the true solution should be real.

Ratio prior: When the expected radius ratio k = r_inner/r_outer is known, the eigenvalue should be close to k². The penalty |λ - k²| biases selection toward the physically expected solution.

The total score combines these terms:

score = residual + w_imag_λ · |Im(λ)| + w_imag_v · ||Im(v)|| + w_ratio · |λ - k²|

Step 5: Select Best Candidate

Candidates are compared by eigenvalue separation first (preferring well-separated eigenvalues), then by score. The candidate with the best combined criterion is selected.

Gates in the Detection Pipeline

The detector applies additional gates via ProjectiveCenterParams:

When any gate rejects the correction, the original ellipse-fit center is preserved.

Accuracy

On synthetic data with clean conics (no noise), the algorithm recovers the true projected center to machine precision (~1e-8 px). With noisy ellipse fits from real edge points, typical corrections are on the order of 0.01–0.5 px, depending on the perspective distortion.

The algorithm is scale-invariant: scaling either conic by a constant does not affect the result.

References

• Wang, Y., et al. “Projective Correction of Circular Targets.” 2019.

Source: ring/projective_center.rs