Local Symmetry - 2D

Ribbons, SATs and Smoothed Local Symmetries

Asaf YaffeImage 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 codingShape 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 objectsA 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, Fourier descriptors… Region-based Representations

Axial 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 – ניצב/מאונך 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 – 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.

S

O

GO

What is a Ribbon?

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

SR

rO

GO

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” Ribbons (Brooks, 1981)

“Brady” 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:Proposition 1: “If R is simply connected and its

border bR smooth, then any maximal disk D contained in R is tangent to bR”

Proof (cont.)Proposition 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)”

Corollary: “The set of maximal disks is the same as the set of G’s”

Let A = {DP | P bR} be the set of all maximal disks tangent to the border of R

By proposition 1, A contains all maximal disksBy proposition 2, A is identical to the set of all

G’s. The spine S is the locus of their centers

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 definedGenerators may not intersect

More flexible than Blum ribbonsThick ribbons can make sharp turnsCan 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

Highly ambiguous Same shape can be generated in many different ways

Need to apply constraints on the definition…

L-Ribbons Difficulties

Brooks Ribbons The generator is required to make a fixed

angle with the spineWe assume that the angle is 90 degrees

This limits the ability of Brooks ribbons to make sharp turnsThe thickness cannot exceed twice the radius

of the curvature of the spineS

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 determinedTheorem: 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

Theorem proof:

- 1 = 2 - => = (2 - 1 ) / 2 is constant. Hence, all G’s are parallel

Brady Ribbons

1

2

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)

G

O

Special Cases If the spine is straight, and we ignore the

ends thenEvery 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 of one or more smooth spines Recoverable

The SAT of a shape is uniquely determined Generative

A shape can be perfectly reconstructed from its SAT

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 symmetryFormation 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 axesAn axis is the formation of maximal smooth

loci of local symmetriesThe 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 regionThis 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 An 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

Comparing SLS and SAT

SLS SAT

Summary Local symmetry can be used to describe

parts of shapes Local symmetry can be described in

various waysRibbonsSATSLS

References A. Rosenfeld. “Axial Representation of Shape”.

Computer Vision, Graphics and Image Processing, Vol. 33, pp. 156-173. 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. 328-

340, 1990 H. Zabrodsky, “Computational Aspects of Pattern Characterization –

Continuous Symmetry”.pp. 13 – 21. 1993