1. Extended Grassfire Transform on Medial Axes of 2D Shapes
Tao Ju, Lu Liu
Washington University in St. Louis
Erin Chambers, David Letscher
St. Louis University

2. Medial axis
The set of interior points with two or more closest points on the boundary
A graph that captures the protrusions and topology of a 2D shape
First introduced by H. Blum in 1967
A widely-used shape descriptor
Object recognition
Shape matching
Skeletal animation

3. Grassfire transform An erosion process that creates the medial axis
Imagine that the shape is filled with grass. A fire is ignited at the border and propagates inward at constant speed.
Medial axis is where the fire fronts meet.

4. Medial axis significance
The medial axis is sensitive to perturbations on the boundary
Some measure is needed to identify significant subsets of medial axis

5. Medial axis significance
A mathematically defined significance function that captures global shape property and resists boundary noise is lacking
Local measures
Does not capture global feature
Potential Residue (PR) [Ogniewicz 92], Medial Geodesic Function (MGF) [Dey 06]
Discontinuous at junctions
Sensitive to boundary perturbations
Erosion Thickness (ET) [Shaked 98]
Lacking explicit formulation

6. Medial axis significance
A mathematically defined significance function that captures global shape property and resists boundary noise is lacking
Local measures
Does not capture global feature
Potential Residue (PR) [Ogniewicz 92], Medial Geodesic Function (MGF) [Dey 06]
Discontinuous at junctions
Sensitive to boundary perturbations
Erosion Thickness (ET) [Shaked 98]
Lacking explicit formulation

7. Medial axis significance
A mathematically defined significance function that captures global shape property and resists boundary noise is lacking
Local measures
Does not capture global feature
Potential Residue (PR) [Ogniewicz 92], Medial Geodesic Function (MGF) [Dey 06]
Discontinuous at junctions
Sensitive to boundary perturbations
Erosion Thickness (ET) [Shaked 98]
Lacking explicit formulation

8. Medial axis significance
A mathematically defined significance function that captures global shape property and resists boundary noise is lacking
Local measures
Does not capture global feature
Potential Residue (PR) [Ogniewicz 92], Medial Geodesic Function (MGF) [Dey 06]
Discontinuous at junctions
Sensitive to boundary perturbations
Erosion Thickness (ET) [Shaked 98]
Lacking explicit formulation

9. Shape center A center point is needed in various applications
Shape alignment
Motion tracking
Map annotation

10. Shape center
Definition of an interior, unique, and stable center point does not exist so far
Centroid
not always interior
Geodesic center [Pollack 89]
may lie at the boundary
Geographical center
not unique

11. Shape center
Definition of an interior, unique, and stable center point does not exist so far
Centroid
not always interior
Geodesic center [Pollack 89]
may lie at the boundary
Geographical center
not unique
Centroid

12. Shape center
Definition of an interior, unique, and stable center point does not exist so far
Centroid
not always interior
Geodesic center [Pollack 89]
may lie at the boundary
Geographical center
not unique
Centroid
Geodesic center

13. Shape center
Definition of an interior, unique, and stable center point does not exist so far
Centroid
not always interior
Geodesic center [Pollack 89]
may lie at the boundary
Geographical center
not unique
Centroid
Geodesic center
Geographic center

14. Contributions
Unified definitions of a significance function and a center point on the 2D medial axis
The function: capturing global shape, continuous, and stable
The center point: interior, unique, and stable
A simple computing algorithm
Extends Blum’s grassfire transform
Applications

15. Intuition Measure the shape elongation around a medial axis point
By the length of the longest “tube” that fits inside the shape and is centered at that point

16. Tubes
Union of largest inscribed circles centered along a segment of the medial axis
The segment is called the axis of the tube
The radius of the tube w.r.t. a point on the axis is its distance to the nearer end of the tube
𝑟 𝑡 𝑥 =min⁡(𝑑 𝑥, 𝑦 1 +𝑅 𝑦 1 ,
𝑑 𝑥, 𝑦 2 +𝑅 𝑦 2 )
𝑑: geodesic distance
𝑅: distance to boundary 𝑥
𝑦 1
𝑦 2
𝑅(𝑦 1 )
𝑅(𝑦 2 )
𝑟 𝑡 (𝑥)

17. Tubes
Union of largest inscribed circles centered along a segment of the medial axis
The segment is called the axis of the tube
The radius of the tube w.r.t. a point on the axis is its distance to the nearer end of the tube
Infinite on loop parts of axis
(there are no “ends”) 𝑥

18. EDF Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝 𝑡 𝑟 𝑡 (𝑥)
Simply connected shape

19. EDF Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝 𝑡 𝑟 𝑡 (𝑥)
𝐸𝐷𝐹(𝑥) 𝑥
Simply connected shape

20. EDF Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝 𝑡 𝑟 𝑡 (𝑥)
𝐸𝐷𝐹(𝑥)
Simply connected shape 𝑥

21. EDF Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝 𝑡 𝑟 𝑡 (𝑥)
Simply connected shape 𝑥
𝐸𝐷𝐹(𝑥)

22. EDF Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝 𝑡 𝑟 𝑡 (𝑥)
Shape with a hole

23. EDF Properties No smaller than distance to boundary
Equal at the ends of the medial axis
Continuous everywhere
Along two branches at each junction
Constant gradient (1) away from maxima
Distance to boundary

24. EDF Properties No smaller than distance to boundary
Equal at the ends of the medial axis
Continuous everywhere
Along two branches at each junction
Constant gradient (1) away from maxima
Loci of maxima preserves topology
Single point (for a simply connected shape)
System of loops (for shape with holes)
EDF
Distance to boundary

25. EDF Properties No smaller than distance to boundary
Equal at the ends of the medial axis
Continuous everywhere
Along two branches at each junction
Constant gradient (1) away from maxima
Loci of maxima preserves topology
Single point (for a simply connected shape)
System of loops (for shape with holes)
EDF
Distance to boundary

26. EMA Extended Medial Axis (EMA): loci of maxima of EDF
Intuitively, where the longest fitting tubes are centered

27. EMA Extended Medial Axis (EMA): loci of maxima of EDF Properties
Intuitively, where the longest fitting tubes are centered
Properties
Interior
Unique point
(For simply connected shapes)

28. Extended grassfire transform
An erosion process that creates EDF and EMA
Fire is ignited at each end 𝑥 of medial axis at time 𝑅(𝑥), and propagates geodesically at constant speed. Fire front dies out when coming to a junction, and quenches as it meets another front.
EDF is the burning time
EMA consists of
Quench sites
Unburned parts

29. Extended grassfire transform
An erosion process that creates EDF and EMA
Fire is ignited at each end 𝑥 of medial axis at time 𝑅(𝑥), and propagates geodesically at constant speed. Fire front dies out when coming to a junction, and quenches as it meets another front.
EDF is the burning time
EMA consists of
Quench sites
Unburned parts
A simple discrete algorithm

30. Extended grassfire transform
Can be combined with Blum’s grassfire
Fire “continues” onto the medial axis at its ends

31. Comparison with PR/MGF
EDF and EMA are more stable under boundary perturbation
PR and its maxima

32. Comparison with PR/MGF
EDF and EMA are more stable under boundary perturbation
EDF and EMA

33. Relation to ET Erosion Thickness (ET) [Shaked 98] New definition
The burning time of a fire that starts simultaneously at all ends and runs at non-uniform speed 1/(1−𝑅′(𝑥))
No explicit definition exists
New definition
𝐸𝑇 𝑥 =𝐸𝐷𝐹 𝑥 −𝑅(𝑥)
Simpler to compute
More intuitive: length of the tube minus its thickness
EDF ET

34. Application: Pruning Medial Axis
Observation
The difference between EDF and the distance-to-boundary gives a robust measure of shape elongation relative to its thickness
EDF and boundary distance
EDF

35. Application: Pruning Medial Axis
Two significance measures: relative and absolute difference of EDF and boundary distance (R)
Absolute diff (ET): “scale” of elongation
Relative diff: “sharpness” of elongation
Preserving medial axis parts that are high in both measures
𝐸𝐷𝐹 𝑥 −𝑅(𝑥)
1−𝑅(𝑥)/𝐸𝐷𝐹(𝑥)

36. Application: Pruning Medial Axis
Preserving medial axis parts that score high in both measures

37. Application: Pruning Medial Axis
Preserving medial axis parts that score high in both measures

38. Application: Shape alignment
Stable shape centers for alignment
Centroid
Maxima of PR
EMA

39. Application: Shape alignment
Stable shape centers for alignment
Centroid
Maxima of PR
EMA

40. Application: Boundary Signature
Boundary Eccentricity (BE): “travel” distance to the EMA
Travel is restricted to be on the medial axis 𝑝
𝐵𝐸 𝑃 =𝑑 𝑥,𝐸𝑀𝐴 +𝑅(𝑥) 𝑥
EMA

41. Application: Boundary Signature
Boundary Eccentricity (BE): “travel” distance to the EMA
Highlights protrusions and is invariant under isometry
Shape 1
Shape 2
Matching

42. Summary
New definitions of significant function and medial point over the medial axis in 2D
EDF(x): length of the longest tube centered at x
EMA: the center of the longest tube
Extending Blum’s grassfire transform to compute them
Future work: 3D?
New global significance function on medial surfaces
New definition of center curve (or curve skeleton)