3D Reconstruction by Fitting Low-Rank Matrices with Missing Data

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Abstract
A technique for building consistent 3D reconstructions from many views based on fitting a low rank matrix to a matrix with missing data is presented. Rank-four submatrices of minimal, or slightly larger, size are sampled and spans of their columns are combined to constrain a basis of the fitted matrix. The error minimized is expressed in terms of the original subspaces which leads to a better resistance to noise compared to previous methods. More than 90% of the missing data can be handled while finding an acceptable solution efficiently. Applications to 3D reconstruction using both affine and perspective camera models are shown. For the perspective model, a new linear method based on logarithms of positive depths from chirality is introduced to make the depths consistent with an overdetermined set of epipolar geometries. Results are shown for scenes and sequences of various types. Many images in open and closed sequences in narrow and wide base-line setups are reconstructed with reprojection errors around one pixel. It is shown that reconstructed cameras can be used to obtain dense reconstructions from epipolarly aligned images.

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