January 2005 Timed Fast Exact Euclidean Distance t

January 2005 Timed Fast Exact Euclidean Distance (t. FEED) Maps Theo Schouten Harco Kuppens Egon van den Broek

t. FEED Distance transformation • distance map D(p) = min { dist(p, q), q O }

t. FEED Euclidean distance • not by local operations using scans – approximations (city-block, chamfer) • disconnected Voronoi tile • semi-exact ED often wright sometimes wrong

t. FEED Principle of FEED • D(p) = if (p O) then 0 else for each q O for each p: D(p) = min ( D(p), ED(q, p)) • inverse of definition • reduce number of q O to feed distances: only the border pixels of O, not the “inside” pixels • ED( (xq, yq), (xp, yp)) = M(|xq-xp|, |yq-yp|) M can contain any non-decreasing f(ED) square (ED), floating point, rounded integer

t. FEED Speed up, bisection lines • reduce number of p to update per B • search and bookkeeping < time gained

t. FEED Search optimization • 76800 pixels 13942 object 1725 border 86487 updates, 8. 4 ms • 290771 updates, 5. 7 ms • 179373 updates, 4. 5 ms •

t. FEED Results • FEED is about factor 2 faster than Shih & Wu 2 -scan ED (CVIU 2004) – few % wrong, error 50% of chamfer 3, 4 – FEED uses less memory • FEED is about factor 2 slower than Borgefors chamfer 3, 4 (CVGIP, 1986) • FEED time depends more on content of image than the scan methods

t. FEED Video generation • generated with Macromedia Flash – vector oriented – preserve color maps

t. FEED video distance maps • Dfixed+moving = min { Dfixed, Dmoving } • FEED on fixed objects • per frame original FEED, but: – initialize with Dfixed – B Omoving – up to dmax in Dfixed additional object does not increase max distance

t. FEED Scan methods video distance maps • the scan methods need a rectangle: – bounding box of moving object, extended with dmax – moving object has no influence outside rectangle • in rectangle Dfixed+moving • copy with min operator into Dfixed

t. FEED Video results • t. FEED factor 6 faster than FEED/frame • factor 3 - 4 faster than adapted Shih & Wu (semi) ED • 20 -50% faster than adapted Borgefors chamfer 3, 4 • which is often faster than the city-block which gets a larger rectangle

t. FEED Video example • further developments: – encoding fixed objects for faster search in FEED – faster locating the moving object • more effect on t. FEED

t. FEED conclusions • DT’s (FEED, scan methods) adapted for fast generation of distance maps for video • t. FEED: – gives exact ED – faster than city-block, chamfer 3, 4 other (semi) ED – more complicated to implement

t. FEED The End