Local Symmetry - 2D
Ribbons, SATs and Smoothed Local Symmetries
Asaf Yaffe
Image Processing Seminar, Haifa University, March 2005
Outline

Symmetry and Shape Description

Ribbons

Symmetry Axis Transform (SAT)

Smoothed Local Symmetries (SLS)
Symmetry and Shape Description
Global Symmetry
Every symmetry element concerns the
whole image or shape
 All points in the object contribute to
determining the symmetry
 Behind the scope of this presentation…

Local Symmetry
Symmetry elements are local to a subset
of the image or shape
 The subset is a continuous section of the
shape’s contour
 Generally used for shape description

 Compact
coding
 Shape recognition
Motivation for Local Symmetry

In many vision systems (e.g., robotics),
shape is represented in terms of global
features:
 Centers
of area/mass, number of holes,
aspect ration of the principal axes
Global features can be computed
efficiently
 But…

Motivation for Local Symmetry

Global features cannot be used to
describe occluded objects
 A feature’s
value of the visible portion has no
relationship to the value of the whole object
Therefore, it is nearly impossible to
recognize occluded parts using global
features
 Hence, the need for local features

Shape Description

Contour-based Representations
 Chain-code,

Region-based Representations
 Axial

Fourier descriptors…
representations (MAT)…
Shape descriptor properties
 Generative:
reconstruct the shape from its
descriptor
 Recoverable: create a unique descriptor for a
shape
General Terms and Definitions









Normal - ‫אנך‬
Tangent - ‫משיק‬
Curvature - ‫עקמומיות‬
Perpendicular – ‫מאונך‬/‫ניצב‬
Oblique - ‫אלכסוני‬
Concave - ‫קעור‬
Convex – ‫קמור‬
Contour – ‫מתאר‬
Planar – ‫מישורי‬
Ribbons
What is a Ribbon?
A planar shape
 Locally symmetric around an arc called
“axis” or “spine”

What is a Ribbon?
S
GO
O




S – Spine. Assume S is a simple, continuous arc with a
tangent at every point
G – Generator. A simply connected set. May be of any
shape
O – Center. Generator’s reference point (center)
GO – Generator centered at O.
What is a Ribbon?
S
R
GO



rO
G’s are geometrically similar and may differ only in size
rO – Radius. The size of GO.
R – Ribbon. The union of all GO for all O  S
What is a Ribbon?




Let O’ and O’’ be the endpoints for S
bR – the border of R. The border is smooth
Ribbon ends – parts of the border that are in GO’ or GO’’
but not in any other GO
Ribbon sides – the remaining parts of the border of R.
Requirements




GO moves along S
S is a simple arc
G’s should not intersect (well… sort of… hard to
define…)
G’s must be maximal. Otherwise R may not follow the
shape of its spine.
Requirements
In all cases which follow, G is symmetric
about its center O.
 The symmetry of G tends to make R
“locally symmetric”.
 This, however, does not imply global
symmetry

Ribbon Classes

“Blum” Ribbons (Blum, 1967, 1978)

“L-Ribbons”
 “Brooks”
 “Brady”
Ribbons (Brooks, 1981)
Ribbons (Brady, 1984)
Blum Ribbons

Ribbons generated by disks centered on
the spine

The disks are circles with varying radii
Blum Ribbons are Recoverable
Theorem: “If R is a Blum ribbon, the spine
and generators of R are uniquely
determined”
 Proof:

1: “If R is simply connected and its
border bR smooth, then any maximal disk D
contained in R is tangent to bR”
 Proposition
Proof (cont.)
2: “If R is a Blum ribbon, every
maximal disk D contained in R is one of the
G’s (and has its center on S)”
 Proposition

Corollary: “The set of maximal disks is the same as
the set of G’s”
{DP | P  bR} be the set of all maximal
disks tangent to the border of R
 By proposition 1, A contains all maximal disks
 By proposition 2, A is identical to the set of all
G’s. The spine S is the locus of their centers
 Let A =
Blum Ribbons Limitations

A Thick Blum ribbon cannot have points of high
positive curvature on its border

A Thick Blum ribbon cannot turn rapidly

The “non-self-intersection” requirement is hard
to define
L-Ribbons
Ribbons generated by a line segment with
its midpoint on the spine
 The length and orientation of the line may
vary as it moves along the spine



The sides of R are the loci of the lines’ endpoints
The ends of R are the lines at the ends of the spine
L-Ribbon Properties

The “non-self-intersection” requirement is
easily defined
 Generators

may not intersect
More flexible than Blum ribbons
 Thick
ribbons can make sharp turns
 Can have points of high positive (or negative)
curvature on their borders
L-Ribbon Properties

L-Ribbons may have long protuberances on their borders as long as every
point is visible from the spine
L-Ribbons Difficulties

Highly ambiguous
 Same

shape can be generated in many different ways
Need to apply constraints on the definition…
Brooks Ribbons

The generator is required to make a fixed
angle with the spine
 We

assume that the angle is 90 degrees
This limits the ability of Brooks ribbons to
make sharp turns
 The
thickness cannot exceed twice the radius
of the curvature of the spine
S
Brooks Ribbons


If the sides of the ribbon are straight and
parallel, its spine and generators are uniquely
determined
If the sides are not parallel, the spine need not
be a straight line, and thus may not be unique
Brady Ribbons

The generator always makes equal angles
with the sides of the ribbon
Brady Ribbons

If the ribbon has just one straight side, its
spine and generators are uniquely
determined
 Theorem:
if both sides are straight, the spine
is a segment of the angle bisector and the
generators are perpendicular to the spine

In the general case, the spine and
generators are not unique
Brady Ribbons

Theorem proof:
1

2
 - 1 = 2 -  =>  = (2 -  1) / 2
  is constant. Hence, all G’s are parallel

Brady Ribbons

Thick Brady ribbons can make sharp turns
 Thus,
there are Brady ribbons which are not Brooks
ribbons

Every Blum ribbon is a Brady ribbon (ignoring
the ends)
O
G
Special Cases

If the spine is straight, and we ignore the
ends then
 Every
Blum ribbon is a Brooks ribbon
 Every
Brooks ribbon is a Brady ribbon
Blum  Brooks  Brady
Special Cases

Even if the spine is straight…
 There
are Brady ribbons which are not Brooks
 There
are Brooks ribbons which are not Blum
Blum  Brooks  Brady
Symmetry Axis Transform (SAT)
Symmetry Axis Transform
The loci of the centers of all maximal disks
entirely contained within the shape
 The disks must touch the border of the
shape (at least in one section)
 Also known as Medial Axis Transform
(MAT)

Symmetry Axis Transform
Captures the major axis of the shape and
its orientation
 Reflects local boundary formations (e.g,
corners) of the shape

SAT “Skeleton” Points

The centers of maximal disks can be classified
into 3 classes:
 End
points: disks touching the border in one section
 Normal points: disks touching the border in 2 sections
 Branch points: disks touching the border in 3 or more
sections

Major cause for problems, such as losing the symmetry axes
of rectangular shapes
SAT Properties

Piecewise smooth
 Comprised

Recoverable
 The

of one or more smooth spines
SAT of a shape is uniquely determined
Generative
 A shape
its SAT
can be perfectly reconstructed from
SAT Weaknesses

Very sensitive to noise

May lose the symmetry axes of the shape
Smooth Local Symmetries (SLS)
Smooth Local Symmetries

Defined in two parts
 Determination
of the local symmetry
 Formation of maximal smooth loci of these
symmetries
Determining Local Symmetry




Let A, B be points on the shape’s border
Let nA be the outward normal at A
Let nB be the inward normal at B
A and B are locally symmetric if both angles of the
segment AB and the normals are the same
Determining Local Symmetry

A point may have a local symmetry with
several points
Point A has local symmetry with both B and C
Forming the “Skeleton”

The shape’s “skeleton” is the union of
symmetry axes
 An
axis is the formation of maximal smooth
loci of local symmetries
 The symmetry locus is the midpoint of the
segment connecting the local symmetry
points
Smooth Local Symmetry Axes

An axis describes some piece of the
contour and the region
 This
portion is called a Cover
Some covers are wholly contained in other
covers (subsumed)
 Subsumed covers are of less importance
but still convey useful information

Smooth Local Symmetry Axes

The short diagonal axes are subsumed

The diagonal axes are not subsumed
SLS Difficulties

May generate redundant spines

Difficult to compute
O(n2) algorithm exists which tests all pairs of
border points for local symmetry
 A faster algorithm exists which calculates an
approximation of the SLS
 An
Comparing SLS and SAT
SLS
SAT
Summary
Local symmetry can be used to describe
parts of shapes
 Local symmetry can be described in
various ways

 Ribbons
 SAT
 SLS
References



A. Rosenfeld. “Axial Representation of Shape”.
Computer Vision, Graphics and Image Processing, Vol. 33, pp. 156173. 1986
M.J. Brady, and H. Asada. “Smooth Local Symmetries and Their
Implementations”.
Int. J. of Robotics Reg. 3(3). 1984
J.Ponce, "On Characterizing Ribbons and Finding Skewed
Symmetries,"
Computer Vision, Graphics, and Image Processing, vol. 52, pp. 328340, 1990

H. Zabrodsky, “Computational Aspects of Pattern Characterization –
Continuous Symmetry”.
pp. 13 – 21. 1993