Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps January 2006 Theo Schouten Harco...

Three Dimensional Fast Exact Euclidean

Distance (3D-FEED) Maps

January 2006

Theo SchoutenHarco Kuppens

Egon van den Broek

Three Dimensional Fast Exact EuclideanDistance (3D-FEED) Maps

Distance Maps

• D(p) = min { dist(p,q), q O}

Euclidean distance City-block distance

easy to calculate, 4 line program, slow


Calculation

• forward + backward raster scan, local distances– 1966 Rosenfeld and Pfaltz: city-block distance– 1986 Borgefors: chamfer distances

• region growing methods• euclidean distances can not be done these ways• semi-exact ED’s• ordered propagation+ corrections at tile boundaries• intersection Voronoi diagram with rows +

dimensionality reduction: O(N) algorithms• recently one with low time constant

C. Maurer, R. Qi, V. Raghaven


Fast Exact Euclidean Distance (FEED) Maps

• D(p) = if (p O) then 0 else each q O

feeds its ED to each p:D(p) = min ( D(p), ED(q,p))

1. restrict q : only border pixels of O– having one of its neighbors O

2. restrict feed distance: only p’s which are closer to B than to another q O

3. do something smart with the ED calculation:– use ED2, pre-calculation


Bisection hyperplane’s• time:

– search for q’s– administrate

bisection planes, area to feed

• should be smaller than the time– saved by doing

less updates


Search and administration process

• keep bounding box • local around B• radial search lines,

only first qup to a maximum

• stop when small• special cases• many parameters• rather easy to

adjust


3D-2D implementation comparison

• more (type) neighbors: 6-plane, 12-line, 8-point• effectively non-uniform memory access time

m(l); x+1: m(l+1); y+1: m(l+width); z+1: m(l+width*height)

• 2D: pre-calculated ED’s stored in a matrix3D: recalculated ED2 30% faster

• 2D: bounding box + filling per quadrant around B3D: single bounding box, 1 loop over Z for filling

• 3D: special searches to reduce bb in Z


Images• generated• 64x64x64• 128x128x128• “or” and “xor”• roughening

surfaces• 1024 images


Timing and accuracy comparison3D-FEED

City Block

CH3,4,5 i

CH3,4,5 f

SemiExEDT

Linear Time

AMD ms 320.3 96.2 210.2 299.8 531.2

Intel ms 209.5 72.4 155.8 230.9 371.0 278.3

wrong voxels (%) 69.67 67.88 80.45 4.41

av abs error 3.79 0.44 0.40

max abs error 46.46 9.31 6.88 4.72

av rel error (%) 26.38 2.93 2.89

max rel error (%) 73.21 10.55 8.80 9.82

AMD rel time 3.3 1.0 2.2 3.1 5.5

Intel rel time 2.9 1.0 2.2 3.2 5.2


Time vs % object pixels


Conclusion• principles Fast Exact Euclidean Distance (FEED)• 3-D implementation, 5 other DT, 1024 images• fast ; less easy to implement• 2D video with stationary and moving objects

adding influence of moving objects per frameFEED faster than CH3,4 faster than City-Block

• future:– more dependent on image content -> faster– adaptable to anisotropic voxels– dimension independent: 2D, 3D… 4D, 5D


The End


slices through the shown images


Disjunct Voronoi tiles


Random dot images


OLDTiming and accuracy comparison3D-FEED

City Block

CH3,4,5 i

CH3,4,5 f

SemiExEDT

Linear Time

AMD ms 320.3 96.2 210.2 299.8 531.2

Intel ms 209.5 72.4 155.8 230.9 371.0 278.3

wrong voxels (%) 69.67 67.88 80.45 4.41

av abs error 3.79 0.44 0.40

max abs error 46.46 9.31 6.88 4.72

av rel error (%) 26.38 2.93 2.89

max rel error (%) 73.21 10.55 8.80 9.82

AMD rel time 3.3 1.0 2.2 3.1 5.5

Intel rel time 2.9 1.0 2.2 3.2 5.2


OLDConclusion• principles Fast Exact Euclidean Distance (FEED)• 3-D implementation, 4 other DT, 1024 images• up to 2x faster semi-exact EDT, 3x slower City-Block• same speed as CH3,4,5 ; less easy to implement• future:

– more dependent on image content -> faster– adaptable to non-square voxels– dimension independent: 2D, 3D… 4D, 5D– 2D video with stationary and moving objects

adding influence of moving objects per frameFEED faster than CH3,4 faster than City-Block

Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps January 2006 Theo Schouten Harco...

Documents

Transcript of Three Dimensional Fast Exact Euclidean Distance (3D-FEED) Maps January 2006 Theo Schouten Harco...