Unique Recovery of Motion and Optic Flow via Lie Algebras

Abstract

Wc use "ome ideas from thc thcory of Lic groups and Lic algcbras to study the problcm of rccovcring rigid motion from a timc-varyiog picturc. Wc are ablc to avoid thc prohlcm of finding corrcsponding points hy considcring only what can bc dctcrrnincd from picturc point valucs and thcir time derivatives. We do not assumc that we can track individual point.• in tbc imagc, nor that wc are given any of their velocit.ics (i.c., thc optic flow) . Among our rcsults are: Thc 6 point df I dt theorem, showing that generically* •• the valucs of df I dt at Ii points of thc monochrome image I arc neccssary aud sufncicut to spccify the motion of a given ohject. The 2-co\or theorem for optic flow, which states that the optic flow vcctor is uniquely spccificd at a gencric point of the image if there arc 2 or morc color dirncnsions. Also, wc gct the color vcrsion of the 6 point theorem, thc 2 colors, 3 points corollary, which reduccs the numbcr of points rcquired to 3, if there arc at least 2 color dimensions.