Fix eigenvector sign ambiguity in Eigen SVD PCA implementation

The switch from VNL's vnl_symmetric_eigensystem to Eigen's BDCSVD
introduced test failures due to eigenvector sign ambiguity. In PCA,
eigenvectors are only determined up to a sign (both v and -v are
valid), and different libraries may return either arbitrarily.

Add a consistent sign convention: make the first element of each
eigenvector positive. This ensures deterministic results regardless of
the underlying linear algebra implementation.

Update test baselines to match the new consistent sign convention.

This way if we switch to another implementation later, we can keep the
results consistent.

Author: akenmorris

Libs/Particles/ParticleShapeStatistics.cpp

@@ -302,9 +302,13 @@ int ParticleShapeStatistics::compute_shape_dev_modes_for_mca() {
   // Compute eigenvectors_ = X * V = U * S (matches old behavior before normalization)
   eigenvectors_shape_dev_ = svd.matrixU() * svd.singularValues().asDiagonal();
 
-  // normalize the eigenvectors
+  // normalize the eigenvectors and enforce sign convention
+  // (make first element positive to match VNL eigensystem convention)
   for (int i = 0; i < eigenvectors_shape_dev_.cols(); i++) {
     eigenvectors_shape_dev_.col(i).normalize();
+    if (eigenvectors_shape_dev_(0, i) < 0) {
+      eigenvectors_shape_dev_.col(i) *= -1;
+    }
   }
 
   // SVD returns values in descending order, but we need ascending order for backward compatibility
@@ -330,9 +334,13 @@ int ParticleShapeStatistics::compute_relative_pose_modes_for_mca() {
   // Compute eigenvectors_ = X * V = U * S (matches old behavior before normalization)
   eigenvectors_rel_pose_ = svd.matrixU() * svd.singularValues().asDiagonal();
 
-  // normalize the eigenvectors
+  // normalize the eigenvectors and enforce sign convention
+  // (make first element positive to match VNL eigensystem convention)
   for (int i = 0; i < eigenvectors_rel_pose_.cols(); i++) {
     eigenvectors_rel_pose_.col(i).normalize();
+    if (eigenvectors_rel_pose_(0, i) < 0) {
+      eigenvectors_rel_pose_.col(i) *= -1;
+    }
   }
 
   // SVD returns values in descending order, but we need ascending order for backward compatibility
@@ -592,9 +600,13 @@ int ParticleShapeStatistics::compute_modes() {
   // Compute eigenvectors_ = X * V = U * S (matches old behavior before normalization)
   eigenvectors_ = svd.matrixU() * svd.singularValues().asDiagonal();
 
-  // normalize the eigenvectors
+  // normalize the eigenvectors and enforce sign convention
+  // (make first element positive to match VNL eigensystem convention)
   for (int i = 0; i < eigenvectors_.cols(); i++) {
     eigenvectors_.col(i).normalize();
+    if (eigenvectors_(0, i) < 0) {
+      eigenvectors_.col(i) *= -1;
+    }
   }
 
   // SVD returns values in descending order, but we need ascending order for backward compatibility

Testing/data/reconstruct_pca_python/tps_mode-00_sample-000_dense.vtk

@@ -1,3 +1,3 @@
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+size 73323

Testing/data/reconstruct_pca_python/tps_mode-00_sample-001_dense.vtk

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Testing/data/reconstruct_pca_python/tps_mode-00_sample-002_dense.vtk

@@ -1,3 +1,3 @@
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+size 72770