Algorithm Overview

Algorithm Overview

DCReg is organized around three core modules.

1. Spectral Degeneracy Detection

Starting from the ICP normal equation H = J^T J, DCReg forms the Schur complements for rotation and translation:

• S_R = H_RR - H_Rt H_tt^{-1} H_tR
• S_t = H_tt - H_tR H_RR^{-1} H_tR

This removes misleading coupling effects and exposes subspace-level observability directly.

2. Physical-Axis Degeneracy Characterization

The raw Schur eigenvectors are aligned to physical motion axes:

• rotation: roll / pitch / yaw
• translation: x / y / z

This produces:

• raw Schur eigenvalues
• aligned eigenvalues in physical coordinates
• degeneracy mask
• axis contribution ratios
• clamped eigenvalues used by the solver

Detection	Characterization

3. Preconditioned Linear Solve

DCReg clamps only the weak aligned eigenvalues, builds a block-diagonal preconditioner, and solves the normal equation with PCG. This stabilizes the weak directions without over-regularizing the fully observable ones.

Source Entry Points

• DCReg/include/dcreg.hpp
• DCReg/include/utils.hpp
• DCReg/src/dcreg_minimal_example.cpp