Continuous Morphology and Distance Maps

Ron Kimmel

www.cs.technion.ac.il/~ron

Computer Science Department Technion-Israel Institute of Technology

Geometric Image Processing Lab

Given a closed planar curve

Define the distance map

Distance map21:)( RS pC

yxpCyxTp

C ,)(minarg),(

),( 11 yx

),( 11 yxTC

Distance Map Properties

Almost everywhereThe level sets of , given by

are the offsets of C

1T

),( yxT

c),(:),( (c) 1 yxTyxT

Distance Map Properties

By Huygens principle a level sets of , is given by the envelope of all disks of radius c centered on the curve C.

The new shape is also known as `dilation’ with a circular `structuring element’ of the shape.

),( yxT

Distance Map Properties

The distance map represents the set , generated by the curve

evolution

with the right `entropy condition’

),( tpC

NCt

N

),( yxT

Distance Map Properties

The vector is pointing to the closestpoint on the zero set

TT 0),(:),( yxTyx

TT

How to Compute?

Accuracy/EfficiencyQ1: How to compute the distance from a

single`source point’?Given T(k,l)=0, find ),( ),,(),( , jiTlkji lk k

l

i

j

Solution:22

, )()(),( ljkijiT lk So ???

),(, jiT lk

How to Compute?

Q2: How to compute the distance from two`source points’?Given T(k,l)=0, find

),( ),,(),(, },{ jiTlkjip lkpp1k

1l

i

j

Solution:22 )()(minarg),( pp

pljkijiT

),(, jiT lk

2k

2lComplexity:

points source ofnumber

points grid ofnumber where)(

P

NPNO

How to Compute?

Q3: can it be computed in O(N) ?Solution: Danielson algorithm.

k

l

4 scans algorithm with alternating directions (up/down left/right)

Ask your 4 neighbors their coordinate offset to the closest detected source point.

Compute your offset to that point and decide if you change your choice of closest source point

Initial offset is ,

0,0 , , ,

, , , ,

, , , ,

Danielson algorithm O(N)

1,0 2,0 3,0

1,1 2,1 3,1 0,1

1,2 2,2 3,2 0,2

k

l

Alternative solution

We can also use a numeric scheme to solve the ‘eikonal’ equation

Initialize all source points and all non-source points Solve the quadratic equation for given by the

upwind monotone approximation of the eikonal equation

Again, use the 4 scans with alternating directions

ij T

10,,max0,,max22 ij

yij

yij

xij

x TDTDTDTD

1T

ij T0kl T

3D/4D/…nD

All these methods can be extended to higher dimensions.

1D->2 2D->4 3D->8 4D->16

Continuous Morphology

Structuring element

NNvN

Bv

,sup

v NvN CBv

tBv

t ,sup ,sup

?

Morphology and dual spaces

1T

1 1

{ : , }

( ) ( ) { : , }

= { : , }

= { ( ) : , }

= ( )

( ) ( ) ( )

A B a b a A b B

T A T B a b a T A b T B

T a T b a A b B

T a b a A b B

T A B

T T A T B T T A B A B

T

1311 12

21 22 23

aa ax xT

a ay y a

Gray Scale Erosion and Dilation

Level set by level set:E.g. Cylinder (level set circle)

vI IBv

t ,sup

I I t

1. Segment at local curvature maxima2. Compute distance map from each segment3. Find intersection sets of the distance functions4. Prune the tails

Skeletons and level sets