Fast Exact Euclidean Distance (FEED) Transformation

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

2. 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 FEED

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

4. 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 FEED

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

6. 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 FEED

7. Speed up, step 3 • reduce p  O to update per B FEED

8. Balance • time lost: • searching object pixels • administration bisection line • against time gained: • not updating certain p  O • optimum, distribution object pixels FEED

9. 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 FEED

10. 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 FEED

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

12. 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 FEED