Fast Exact Euclidean Distance FEED Transformation Theo Schouten

Fast Exact Euclidean Distance (FEED) Transformation Theo Schouten Egon van den Broek Radboud University Nijmegen August 2004 FEED

Distance transformation • distance map D(p) = min { dist(p, q), q O } • approximation of Euclidean • Rosenfeld & Pfaltz – local, parallel or sequential • Borgefors – chamfer, weighted distances August 2004 FEED

Euclidean distance • not by local operations • disconnected Voronoi tile • often right, sometimes wrong ED • correction August 2004 Cuisenaire & Macq CVIU 76 (1999) FEED

Principle of FEED • D(p) = if (p O) then 0 else for each q O for each p O D(p) = min ( D(p), ED(q, p)) • inverse of definition • correct, terrible slow August 2004 FEED

Speed up, step 1 • reduce q O to consider • only the border pixels of O x x x August 2004 Border: q O at least 1 4 -conn p O FEED

Speed up, step 2 • pre-computation of ED(q, p) • matrix, size of image translation, reflection invariant • M = fnon-decr( ED), like square • size can be reduced – in case max. dist. is known – only up to a maximum is interesting August 2004 FEED

Speed up, step 3 • reduce p O to update per B August 2004 FEED

Balance • time lost: – searching object pixels – administration bisection line • against time gained: – not updating certain p O • optimum, distribution object pixels August 2004 FEED

Results • Shih & Liu 4 -scan ED (PR 31, 1998) – not their correction method • test images, object-like images • FEED is faster, up to 2. 7 – up to 4. 5 reduced M • random dot images, faster < 15% • FEED uses less memory August 2004 FEED

Applications • human color categories – black, white, gray, red, green, blue, yellow, brown, purple, pink, orange • 216 web-safe colors • classify 2563 colors • RGB->HSI, SI: 3 /8, 3, HI: 8 • content based image retieval, texture a August 2004 FEED

Further developments • step 3: faster, simpler • formal proofs • partial maps, fixed objects + moving objects in video • color space applications August 2004 FEED

FEED conclusions • • • EDT inverse definition simple, correct, slow 3 speed up approaches faster than 4 -scan method up to maximum, partial maps human-centered color space August 2004 FEED