Extended Grassfire Transform on Medial Axes of 2D ShapesExtended Grassfire Transform on Medial Axes of 2D Shapes
Tao Ju, Lu Liu
Washington University in St. Louis
Erin Chambers, David Letscher
St. Louis University
Medial axisMedial 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 transformGrassfire 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 significanceMedial 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 significanceMedial 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 significanceMedial 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 significanceMedial 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 significanceMedial 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 centerShape center
• A center point is needed in various applications
– Shape alignment
– Motion tracking
– Map annotation
Shape centerShape 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 centerShape 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 centerShape 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 centerShape 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
ContributionsContributions
• 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
IntuitionIntuition
• 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
TubesTubes
• 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)
TubesTubes
• 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
EDFEDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
Simply connected shape
EDFEDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
𝑥𝐸𝐷𝐹 (𝑥)
Simply connected shape
EDFEDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
𝑥
𝐸𝐷𝐹 (𝑥)
Simply connected shape
EDFEDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
𝑥𝐸𝐷𝐹 (𝑥)
Simply connected shape
EDFEDF
• Extended Distance Function (EDF): radius of the longest tube
𝐸𝐷𝐹 (𝑥 )=𝑠𝑢𝑝𝑡 𝑟 𝑡(𝑥 )
Shape with a hole
EDFEDF
• 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
EDFEDF
• 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
EDFEDF
• 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
EMAEMA
• Extended Medial Axis (EMA): loci of maxima of EDF
– Intuitively, where the longest fitting tubes are centered
EMAEMA
• 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 transformExtended 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 transformExtended 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 transformExtended grassfire transform
• Can be combined with Blum’s grassfire
– Fire “continues” onto the medial axis at its ends
Comparison with PR/MGFComparison with PR/MGF
• EDF and EMA are more stable under boundary perturbation
PR and its maxima
Comparison with PR/MGFComparison with PR/MGF
• EDF and EMA are more stable under boundary perturbation
EDF and EMA
Relation to ETRelation 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 AxisApplication: Pruning Medial Axis
• Observation
– The difference between EDF and the distance-to-boundary gives a robust measure of shape elongation relative to its thickness
EDFEDF and boundary distance
Application: Pruning Medial AxisApplication: 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 AxisApplication: Pruning Medial Axis
• Preserving medial axis parts that score high in both measures
Application: Pruning Medial AxisApplication: Pruning Medial Axis
• Preserving medial axis parts that score high in both measures
Application: Shape alignmentApplication: Shape alignment
• Stable shape centers for alignment
Centroid Maxima of PR EMA
Application: Shape alignmentApplication: Shape alignment
• Stable shape centers for alignment
Centroid Maxima of PR EMA
Application: Boundary SignatureApplication: 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 SignatureApplication: Boundary Signature
• Boundary Eccentricity (BE): “travel” distance to the EMA
– Highlights protrusions and is invariant under isometry
Shape 1 Shape 2 Matching
SummarySummary
• 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)