Conic and Quadric Estimators

A general conic has 6 homogeneous coefficients, i.e. 5 degrees of freedom once the overall scale is divided out — which is exactly why ConicRobustEstimator.MINIMUM_SIZE is 5 (next section). Fixing a=c and b=0 removes 2 of those degrees of freedom and removes the scale ambiguity at the same time: once is forced non-zero, the whole equation can be divided by it, leaving the non-homogeneous circle equation with exactly 3 free real numbers (equivalently: center and radius ). That is why CircleRobustEstimator.MINIMUM_SIZE = 3 (CircleRobustEstimator.java:36): 3 points, 3 unknowns, one equation per point. The identical argument one dimension up — a quadric’s 10 homogeneous coefficients (9 DOF) reduced by the 5 constraints , to the 4 free reals in  — is why SphereRobustEstimator.MINIMUM_SIZE = 4 (SphereRobustEstimator.java:36).

Because fixing the shape also removes the scale ambiguity, the minimal-sample system is not the homogeneous A·x=0 this library solves elsewhere via SVD null-space (Points, Lines and Planes, Conics and Quadrics) — it is an ordinary square, inhomogeneous system A·x=b with a unique solution. Circle.setParametersFromPoints (Circle.java:176-249) builds, for each of the 3 points, the row of A and the entry (each row normalized first to improve conditioning), then solves the 3×3 system directly with Utils.solve(m, b) — no SVD, because there is nothing left to disambiguate. Sphere.setParametersFromPoints (Sphere.java:181-276) does the identical thing one dimension up with a 4×4 system. Both robust estimators simply call the three-argument/four-argument constructor on the minimal sample and catch the degenerate case:

Sphere’s constructor throws `CoplanarPointsException in the equivalent degenerate case (4 coplanar points cannot pin down a unique sphere).

Unlike the other four hierarchies below, CircleRobustEstimator.residual/SphereRobustEstimator.residual score a candidate with the actual Euclidean distance from the point to the circle/sphere boundary, via Circle.getDistance/Sphere.getDistance (CircleRobustEstimator.java:728-732, SphereRobustEstimator.java:723-727):

(Circle.signedDistance/Sphere.signedDistance, Circle.java:369-371). This is possible only because a circle/sphere’s inhomogeneous, scale-free parameterization already carries real physical units (the same units as the input points) — there is no matrix to normalize and no arbitrary scale to divide out first, so the natural residual is just the point-to-boundary distance. That is also why both robust estimators default their inlier threshold to 1.0 (RANSACCircleRobustEstimator.DEFAULT_THRESHOLD, RANSACSphereRobustEstimator.DEFAULT_THRESHOLD, both 1.0): one unit of distance, matching a "typical resolution of 1 pixel/voxel" per their own javadoc. Contrast this with the algebraic residual and correspondingly tiny default thresholds used by Conic/DualConic/Quadric/DualQuadric below.

Both hierarchies expose the same five concrete robust subclasses named after the pattern established in estimators.adoc#shared-architecture: RANSACCircleRobustEstimator, LMedSCircleRobustEstimator, MSACCircleRobustEstimator, PROSACCircleRobustEstimator, PROMedSCircleRobustEstimator (and the Sphere equivalents), created through CircleRobustEstimator.create(…​)/SphereRobustEstimator.create(…​) factory overloads exactly like every other hierarchy in this package.

A conic has 6 homogeneous coefficients, hence 5 degrees of freedom once scale is factored out — which is exactly ConicRobustEstimator.MINIMUM_SIZE = 5 (ConicRobustEstimator.java:39). This is also worked out explicitly in Hartley & Zisserman §2.2, "Five points define a conic" (pp. 30-31): each point places one linear constraint on the 6-vector of coefficients, so 5 points give a 5×6 homogeneous system whose 1-dimensional null-space is the conic, "determined uniquely (up to scale) by five points in general position" — HZ’s own words. By the point/tangent-line duality of the previous section, a dual conic (an envelope of tangent lines) is likewise fixed by 5 lines, hence DualConicRobustEstimator.MINIMUM_SIZE = 5 (DualConicRobustEstimator.java:39).

Unlike Circle/Sphere, a conic has no way to remove the scale ambiguity, so Conic.setParametersFromPoints (Conic.java:313-409) builds exactly the homogeneous 5×6 system HZ describes — for each point the row (row-normalized for numerical accuracy) — and finds its null-space via SingularValueDecomposer (Conic.java:387-408), the same general technique this library uses for every other homogeneous join/meet problem (Points, Lines and Planes, Numerical Recipes §2.6, "Singular Value Decomposition", pp. 65-67). If the decomposed rank is below 5, the 5 points are coincident or otherwise degenerate and CoincidentPointsException is thrown instead of a spurious solution (Conic.java:390-391). DualConic.setParametersFromLines (DualConic.java:291-390) is the line-dual mirror of the same 5×6 SVD null-space computation. RANSACConicRobustEstimator.estimatePreliminarSolutions simply calls the 5-point constructor on the minimal sample:

RANSACDualConicRobustEstimator is the identical pattern with new DualConic(line1..line5) and CoincidentLinesException.

ConicRobustEstimator.residual (ConicRobustEstimator.java:731-759) and its dual counterpart DualConicRobustEstimator.residual (DualConicRobustEstimator.java:730-758) both score a candidate with the absolute value of the conic’s own defining bilinear form — an algebraic residual, not a true geometric distance to the curve:

computed by normalizing / and the point/line first, then literally multiplying the three matrices (ConicRobustEstimator.java:748-757). This is the same quantity Conic.isLocus/ DualConic.isLocus threshold against (Conics and Quadrics) — residual zero means exactly "on the locus" — but it is not in the same units as a Euclidean distance, and it scales non-linearly with how far a point actually is from the curve. That mismatch in scale is directly visible in the library’s own defaults: RANSACCircleRobustEstimator.DEFAULT_THRESHOLD is 1.0, while RANSACConicRobustEstimator.DEFAULT_THRESHOLD is 1e-6 (RANSACConicRobustEstimator.java:45) and RANSACDualConicRobustEstimator.DEFAULT_THRESHOLD is 1e-7 (RANSACDualConicRobustEstimator.java:44) — three to seven orders of magnitude smaller, because is a genuinely different (and much smaller, for normalized inputs near the locus) quantity than a pixel distance.

Both hierarchies again expose the standard five concrete subclasses — {RANSAC,LMedS,MSAC,PROSAC, PROMedS}ConicRobustEstimator and the DualConic equivalents, 10 classes total — built through ConicRobustEstimator.create(…​)/DualConicRobustEstimator.create(…​).

A quadric has 10 homogeneous coefficients, hence 9 degrees of freedom once scale is factored out — exactly QuadricRobustEstimator.MINIMUM_SIZE = 9 (QuadricRobustEstimator.java:36) and, by the same point/tangent-plane duality argument as conics, DualQuadricRobustEstimator.MINIMUM_SIZE = 9 (DualQuadricRobustEstimator.java:38). This is the direct 3D analogue of HZ’s "five points define a conic" argument above, generalized to a 9×10 homogeneous system — the source PDFs used for this documentation pass do not contain an explicit "nine points define a quadric" passage the way HZ spells out the conic case, but the linear-algebra argument (one constraint row per point, minimal sample size = degrees of freedom) is identical and is the one the code itself implements.

Quadric.setParametersFromPoints (Quadric.java:317-506) builds, for each of the 9 points, the row of a 9×10 matrix (row-normalized), then takes its null-space via SingularValueDecomposer, exactly like Conic (Quadric.java:478-502), throwing CoincidentPointsException if the decomposed rank is below 9 (Quadric.java:481-482). DualQuadric.setParametersFromPlanes (DualQuadric.java:325-513) is the identical construction on plane coefficients instead of point coordinates. Both concrete RANSAC estimators are again thin wrappers around the 9-argument constructor:

RANSACDualQuadricRobustEstimator mirrors this with 9 Plane arguments and CoincidentPlanesException.

QuadricRobustEstimator.residual (QuadricRobustEstimator.java:737-762) and DualQuadricRobustEstimator.residual (DualQuadricRobustEstimator.java:735-760) score exactly the same kind of algebraic quantity as the conic case, one dimension up:

and the same order-of-magnitude gap in default thresholds shows up again: RANSACQuadricRobustEstimator.DEFAULT_THRESHOLD is 1e-6 (RANSACQuadricRobustEstimator.java:43) and RANSACDualQuadricRobustEstimator.DEFAULT_THRESHOLD is 1e-7 (RANSACDualQuadricRobustEstimator.java:44) — both far smaller than the 1.0 used by the geometric-distance-based Circle/Sphere estimators, for exactly the same reason as the conic/dual-conic case above.

Again 10 concrete subclasses total: {RANSAC,LMedS,MSAC,PROSAC,PROMedS}QuadricRobustEstimator and the DualQuadric equivalents, built through QuadricRobustEstimator.create(…​)/ DualQuadricRobustEstimator.create(…​).