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)