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