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Extended Grassfire Transform on Medial Axes of 2D Shapes

This research paper explores the grassfire transform and medial axis significance in analyzing 2D shapes. It introduces a unified definition of a significance function and a center point on the medial axis, along with a simple computing algorithm. The paper also discusses applications and introduces the concept of measuring shape elongation using tubes and geodesic distances.

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Extended Grassfire Transform on Medial Axes of 2D Shapes

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  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)

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