Recovering the motion of a non-rigid body from#R##N#a set of monocular images permits the analysis of dynamic#R##N#scenes in uncontrolled environments. However, the extension of#R##N#factorisation algorithms for rigid structure from motion to the#R##N#low-rank non-rigid case has proved challenging. This stems from#R##N#the comparatively hard problem of finding a linear “corrective#R##N#transform” which recovers the projection and structure matrices#R##N#from an ambiguous factorisation.#R##N##R##N#We elucidate that this greater difficulty is due to the need#R##N#to find multiple solutions to a non-trivial problem, casting a#R##N#number of previous approaches as alleviating this issue by either#R##N#a) introducing constraints on the basis, making the problems nonidentical,#R##N#or b) incorporating heuristics to encourage a diverse#R##N#set of solutions, making the problems inter-dependent. While it#R##N#has previously been recognised that finding a single solution to#R##N#this problem is sufficient to estimate cameras, we show that it is#R##N#possible to bootstrap this partial solution to find the complete#R##N#transform in closed-form. However, we acknowledge that our#R##N#method minimises an algebraic error and is thus inherently#R##N#sensitive to deviation from the low-rank model.#R##N##R##N#We compare our closed-form solution for non-rigid structure#R##N#with known cameras to the closed-form solution of Dai et al. [1],#R##N#which we find to produce only coplanar reconstructions. We#R##N#therefore make the recommendation that 3D reconstruction error#R##N#always be measured relative to a trivial reconstruction such as a#R##N#planar one.
['Jack Valmadre', 'Sridha Sridharan', 'Simon Denman', 'Clinton Fookes', 'Simon Lucey']
Closed-form solutions for low-rank non-rigid reconstruction