Continuous Morphology and Distance Maps Ron Kimmel Computer Science Department Technion-Israel Institute of Technology Geometric Image Processing Lab

Given a closed planar curve Define the distance map Distance map

Distance Map Properties Almost everywhere The level sets of, given by are the offsets of C

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.

Distance Map Properties The distance map represents the set, generated by the curve evolution with the right `entropy condition’

Distance Map Properties The vector is pointing to the closest point on the zero set

How to Compute? Accuracy/Efficiency Q1: How to compute the distance from a single `source point’? Given T(k,l)=0, find Solution: So ???

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

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

Danielson algorithm O(N)

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

3D/4D/…nD All these methods can be extended to higher dimensions. 1D->2 2D->4 3D->8 4D->16

Continuous Morphology Structuring element

? Morphology and dual spaces

Gray Scale Erosion and Dilation Level set by level set: E.g. Cylinder (level set circle)

1.Segment at local curvature maxima 2.Compute distance map from each segment 3.Find intersection sets of the distance functions 4.Prune the tails Skeletons and level sets