Low-rank matrix fitting based on subspace perturbation analysis with applications to structure from motion

Area of Science:

• Computer Vision
• Machine Learning
• Data Science

Background:

• Low-rank matrix approximation is crucial in science and engineering.
• Missing data and noise complicate matrix approximation.
• Existing methods struggle with selecting optimal submatrices for noise reduction.

Purpose of the Study:

• Develop a criterion to select submatrices least affected by noise.
• Improve the accuracy of low-rank matrix recovery from incomplete and noisy data.
• Enhance structure-from-motion (SFM) algorithms.

Main Methods:

• Proposing a novel criterion based on the distinctness of r-column matrices.
• Formally proving that greater vector distinctness reduces noise influence.
• Developing a noise model to bound noise and occlusion effects.
• Deriving affine and projective structure-from-motion algorithms.

Main Results:

• Demonstrated that distinct r vectors yield solutions less susceptible to noise.
• Derived an upper bound for noise and occlusion effects.
• Successfully recovered noise-free matrices of rank r.
• Showcased superior performance of the proposed SFM algorithms on synthetic and real data.

Conclusions:

• The proposed criterion effectively identifies robust submatrices for low-rank approximation.
• The method significantly enhances the recovery of noise-free matrices.
• The derived SFM algorithms outperform existing state-of-the-art approaches.