Extended Grassfire Transform on Medial Axes of 2D Shapes

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 geodesic distance distance to boundary

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”)

18. EDF • Extended Distance Function (EDF): radius of the longest tube Simply connected shape

19. EDF • Extended Distance Function (EDF): radius of the longest tube Simply connected shape

20. EDF • Extended Distance Function (EDF): radius of the longest tube Simply connected shape

21. EDF • Extended Distance Function (EDF): radius of the longest tube Simply connected shape

22. EDF • Extended Distance Function (EDF): radius of the longest tube 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 • 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 of medial axis at time , and propagates geodesicallyat 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 of medial axis at time , and propagates geodesicallyat 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] • The burning time of a fire that starts simultaneously at all ends and runs at non-uniform speed • No explicit definition exists • New definition • 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

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 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)