Structure from Motion with Missing Data is NP-Hard

This paper shows that structure from motion is NP-hard for most sensible cost functions when missing data is allowed. The result provides a fundamental limitation of what is possible to achieve with any structure from motion algorithm. Even though there are recent, promising attempts to compute globally optimal solutions, there is no hope of obtaining a polynomial time algorithm unless P=NP. The proof proceeds by encoding an arbitrary Boolean formula as a structure from motion problem of polynomial size, such that the structure from motion problem has a zero cost solution if and only if the Boolean formula is satisfiable. Hence, if there was a guaranteed way to minimize the error of the relevant family of structure from motion problems in polynomial time, the NP-complete problem 3SAT could be solved in polynomial time, which would imply that P=NP The proof relies heavily on results from both structure from motion and complexity theory.