Perturbation Estimation of the Subspaces for Structure from Motion with Noisy and Missing Data

It is common when analyzing experimental data to encounter matrices that have been contaminated by noise and have missing elements. Missing data can be recovered with imputation methods if the measurement data matrix is of low rank and the data is noise-free. However, iterative imputation can produce poor results for cases of large noise or a large proportion of missing data. Non-imputing methods rely on the use of existent data and require a selection of complete submatrices. Jacobs introduced a non-imputing method which can produce good results, but the randomness in selecting the submatrices cannot guarantee a consistently accurate recovery of missing data. Chen and Suter's method chooses the most reliable submatrix based on the number of missing data elements only, which fails to consider the effect of noise on the selected data. Herein, a new criterion based on an estimate of the sensitivity of submatrices to perturbation is introduced which takes into consideration that in some cases a column with more missing data could provide more useful information than one with less missing data. Experimental results for the problem of structure from motion with noisy point correspondences and missing data show that our criterion can sort submatrices properly in terms of their possible perturbation and recover the 3D structure of the scene more accurately than other non-imputing methods.