BUG: Detect non-positive-definite covariance in GaussianMembershipFunction

[hjmjohnson]

Summary

Fixes #3589.

GaussianMembershipFunction::SetCovariance() previously checked

const vnl_matrix_inverse<double> inv_cov(m_Covariance.GetVnlMatrix());
const double det = inv_cov.determinant_magnitude();
if (det < 0.) { itkExceptionStringMacro(\"det( m_Covariance ) < 0\"); }

vnl_matrix_inverse::determinant_magnitude() returns the product of singular values, which is always non-negative, so the det < 0. check is unreachable dead code. A singular or negative-definite covariance matrix would silently slip through and propagate NaNs into the multivariate Gaussian normalization constant downstream (m_PreFactor = 1.0 / (std::sqrt(det) * ...)).

This PR replaces the check with vnl_determinant() on the original (non-inverted) covariance and rejects det <= 0, which catches both singular (det == 0) and non-positive-definite (det < 0) cases. The cost is O(n^3) but n is the measurement-vector dimension (typically 1..16), negligible compared to the SVD already computed for the inverse on the next line.

Diff

+#include \"vnl/algo/vnl_determinant.h\"
 ...
-  // the determinant is then costless this way
-  const double det = inv_cov.determinant_magnitude();
-
-  if (det < 0.)
-  {
-    itkExceptionStringMacro(\"det( m_Covariance ) < 0\");
-  }
+  // Compute the *signed* determinant of the covariance matrix.  See
+  // commit message for the rationale (determinant_magnitude is the
+  // product of singular values and is never negative).
+  const double det = vnl_determinant(m_Covariance.GetVnlMatrix());
+
+  if (det <= 0.0)
+  {
+    itkExceptionMacro(\"Covariance matrix must be positive definite (det = \" << det << \").\");
+  }

Tests

Three new regression cases added to itkGaussianMembershipFunctionTest.cxx:

1. Singular covariance [[1,1],[1,1]] (det = 0) — must throw.
2. Negative-definite (indefinite) covariance [[1,2],[2,1]] (det = −3) — must throw.
3. Well-formed covariance [[2,0.5],[0.5,2]] (det = 3.75, symmetric positive-definite) — must succeed.

AI Assistance

Generated by Claude Code as part of the 2026-04-10 open-issue triage cleanup.

Test plan

• Local build (ITKStatisticsTestDriver) compiles cleanly
• itkGaussianMembershipFunctionTest passes locally with the new singular / negative-definite / positive-definite assertions
• pre-commit run clean (clang-format, kw-pre-commit)
• CI: Linux, Windows, macOS, Python, ARM

[hjmjohnson]

@greptileai review this draft before I make it official