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 axis The set of interior points with two or more closest points on the boundary ID: 327603
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Slide1
Extended Grassfire Transform on Medial Axes of 2D Shapes
Tao
Ju
, Lu Liu
Washington University in St. Louis
Erin Chambers, David
Letscher
St. Louis UniversitySlide2
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 1967A widely-used shape descriptorObject recognitionShape matchingSkeletal animationSlide3
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.Slide4
Medial axis significance
The medial axis is sensitive to perturbations on the boundary
Some measure is needed to identify
significant subsets of medial axisSlide5
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 formulationSlide6
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 formulationSlide7
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 formulationSlide8
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 formulationSlide9
Shape center
A center
point
is needed in various applicationsShape alignmentMotion trackingMap annotationSlide10
Shape center
Definition of an interior, unique, and stable center point does not exist so far
Centroid
not always interiorGeodesic center [Pollack
89]may lie at the boundary
Geographical center
not uniqueSlide11
Shape center
Definition of an interior, unique, and stable center point does not exist so far
Centroid
not always interiorGeodesic center [Pollack 89]
may lie at the boundary
Geographical center
not unique
CentroidSlide12
Shape center
Definition of an interior, unique, and stable center point does not exist so far
Centroid
not always interiorGeodesic center [Pollack 89]
may lie at the boundaryGeographical center
not
unique
Centroid
Geodesic centerSlide13
Shape center
Definition of an interior, unique, and stable center point does not exist so far
Centroid
not always interiorGeodesic center [Pollack 89]
may lie at the boundaryGeographical centernot
unique
Centroid
Geodesic center
Geographic centerSlide14
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 stableA simple computing algorithmExtends Blum’s grassfire transformApplicationsSlide15
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 pointSlide16
Tubes
Union of largest inscribed circles centered along a segment of the medial axis
The segment is called the
axis of the tubeThe 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
Slide17
Tubes
Union of largest inscribed circles centered along a segment of the medial axis
The segment is called the
axis of the tubeThe radius of the tube w.r.t. a point on the axis is its distance to the nearer end of the tubeInfinite on loop parts of axis (there are no “ends”)
Slide18
EDF
Extended Distance Function (EDF): radius of the longest tube
Simply connected shapeSlide19
EDF
Extended Distance Function (EDF): radius of the longest tube
Simply connected shapeSlide20
EDF
Extended Distance Function (EDF): radius of the longest tube
Simply connected shapeSlide21
EDF
Extended Distance Function (EDF): radius of the longest tube
Simply connected shapeSlide22
EDF
Extended Distance Function (EDF): radius of the longest tube
Shape with a holeSlide23
EDF
Properties
No smaller than distance to boundary
Equal at the ends of the medial axisContinuous everywhereAlong two branches at each junctionConstant gradient (1) away from maxima
Distance to boundarySlide24
EDF
Properties
No smaller than distance to boundary
Equal at the ends of the medial axis
Continuous everywhereAlong two branches at each junctionConstant gradient (1) away from maximaLoci of maxima preserves topology
Single point (for a simply connected shape)System of loops (for shape with holes)
Distance to boundary
EDFSlide25
EDF
Properties
No smaller than distance to boundary
Equal at the ends of the medial axis
Continuous everywhereAlong two branches at each junctionConstant gradient (1) away from maximaLoci of maxima preserves topology
Single point (for a simply connected shape)
System of loops (for shape with holes)
Distance to boundary
EDFSlide26
EMA
Extended Medial Axis (EMA): loci of maxima of EDF
Intuitively, where the longest fitting tubes are centeredSlide27
EMA
Extended Medial Axis (EMA): loci of maxima of EDF
Intuitively, where the longest fitting tubes are centered
PropertiesInteriorUnique point (For simply connected shapes)Slide28
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 timeEMA consists of Quench sites
Unburned parts Slide29
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 timeEMA consists of Quench sites
Unburned partsA simple discrete algorithm
Slide30
Extended grassfire transform
Can be combined with Blum’s grassfire
Fire “continues” onto the medial axis at its endsSlide31
Comparison with PR/MGF
EDF and EMA are more stable under boundary perturbation
PR and its maximaSlide32
Comparison with PR/MGF
EDF and EMA are more stable under boundary perturbation
EDF and EMASlide33
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
ETSlide34
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 distanceSlide35
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 elongationPreserving medial axis parts that are high in both measures
Slide36
Application: Pruning Medial Axis
Preserving medial axis parts that score high in both measuresSlide37
Application: Pruning Medial Axis
Preserving medial axis parts that score high in both measuresSlide38
Application: Shape alignment
Stable shape centers for alignment
Centroid
Maxima of PR
EMASlide39
Application: Shape alignment
Stable shape centers for alignment
Centroid
Maxima of PR
EMASlide40
Application: Boundary Signature
Boundary Eccentricity (BE):
“travel” distance to the EMA
Travel is restricted to be on the medial axis
EMA
Slide41
Application: Boundary Signature
Boundary Eccentricity (BE): “travel” distance to the EMA
Highlights protrusions and is invariant under
isometry
Shape 1
Shape 2
MatchingSlide42
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 tubeExtending Blum’s grassfire transform to compute themFuture work: 3D?New global significance function on medial surfacesNew definition of center curve (or curve skeleton)