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### 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)

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, 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

What is a Ribbon?

- A planar shape
- Locally symmetric around an arc called “axis” or “spine”

S

O

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.

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

S

R

GO

rO

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 disks
- By 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 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

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

G

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)

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

- 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

- 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
- 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

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. 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

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