Extended Grassfire Transform on Medial Axes of 2D Shapes
Tao Ju, Lu LiuWashington University in St. Louis
Erin Chambers, David LetscherSt. Louis University
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
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.
Medial axis significance
• The medial axis is sensitive to perturbations on the boundary
– Some measure is needed to identify significant subsets of medial axis
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
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
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
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
Shape center
• A center point is needed in various applications
– Shape alignment
– Motion tracking
– Map annotation
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
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
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
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
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
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
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
𝑥𝑦 1 𝑦 2𝑅 (𝑦¿¿1)¿
𝑅 (𝑦¿¿2)¿𝑟 𝑡 (𝑥)
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
EDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
Simply connected shape
EDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
𝑥𝐸𝐷𝐹 (𝑥)
Simply connected shape
EDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
𝑥
𝐸𝐷𝐹 (𝑥)
Simply connected shape
EDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
𝑥𝐸𝐷𝐹 (𝑥)
Simply connected shape
EDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
Shape with a hole
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
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)
Distance to boundary
EDF
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)
Distance to boundary
EDF
EMA
• Extended Medial Axis (EMA): loci of maxima of EDF
– Intuitively, where the longest fitting tubes are centered
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)
Extended grassfire transform
• An erosion process that creates EDF and EMA
– Fire is ignited at each end of medial axis at time , 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
Extended grassfire transform
• An erosion process that creates EDF and EMA
– Fire is ignited at each end of medial axis at time , 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
Extended grassfire transform
• Can be combined with Blum’s grassfire
– Fire “continues” onto the medial axis at its ends
Comparison with PR/MGF
• EDF and EMA are more stable under boundary perturbation
PR and its maxima
Comparison with PR/MGF
• EDF and EMA are more stable under boundary perturbation
EDF and EMA
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
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 EDF and boundary distance
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
1−𝑅(𝑥 )/𝐸𝐷𝐹 (𝑥)
𝐸𝐷𝐹 (𝑥 )−𝑅 (𝑥)
Application: Pruning Medial Axis
• Preserving medial axis parts that score high in both measures
Application: Pruning Medial Axis
• Preserving medial axis parts that score high in both measures
Application: Shape alignment
• Stable shape centers for alignment
Centroid Maxima of PR EMA
Application: Shape alignment
• Stable shape centers for alignment
Centroid Maxima of PR EMA
Application: Boundary Signature
• Boundary Eccentricity (BE): “travel” distance to the EMA
– Travel is restricted to be on the medial axis
𝑥EMA
𝑝𝐵𝐸 (𝑃 )=𝑑 (𝑥 ,𝐸𝑀𝐴 )+𝑅(𝑥 )
Application: Boundary Signature
• Boundary Eccentricity (BE): “travel” distance to the EMA
– Highlights protrusions and is invariant under isometry
Shape 1 Shape 2 Matching
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)