Algorithm Overview

Algorithm Overview

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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 observability directly in the decoupled subspaces.

2. Physical-Axis Degeneracy Characterization

The raw Schur eigenvectors are aligned to physical motion axes:

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

This step produces:

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

Schur-Based Detection	Physical-Axis 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 improves stability without over-regularizing the fully observable directions.

Source Entry Points

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