Local Symmetry - 2D

1. Local Symmetry - 2D Ribbons, SATs and Smoothed Local Symmetries Asaf Yaffe Image Processing Seminar, Haifa University, March 2005

2. Outline • Symmetry and Shape Description • Ribbons • Symmetry Axis Transform (SAT) • Smoothed Local Symmetries (SLS)

3. Symmetry and Shape Description

4. 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…

5. 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

6. 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…

7. 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

8. 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

9. General Terms and Definitions • Normal - אנך • Tangent - משיק • Curvature - עקמומיות • Perpendicular – ניצב/מאונך • Oblique - אלכסוני • Concave - קעור • Convex – קמור • Contour – מתאר • Planar – מישורי

10. Ribbons

11. What is a Ribbon? • A planar shape • Locally symmetric around an arc called “axis” or “spine”

12. GO 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.

13. 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

14. 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.

15. 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.

16. 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

17. Ribbon Classes • “Blum” Ribbons (Blum, 1967, 1978) • “L-Ribbons” • “Brooks” Ribbons (Brooks, 1981) • “Brady” Ribbons (Brady, 1984)

18. Blum Ribbons • Ribbons generated by disks centered on the spine • The disks are circles with varying radii

19. 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”

20. 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

21. 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

22. 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

23. 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

24. L-Ribbon Properties • L-Ribbons may have long protuberances on their borders as long as every point is visible from the spine

25. L-Ribbons Difficulties • Highly ambiguous • Same shape can be generated in many different ways • Need to apply constraints on the definition…

26. S 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

27. 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

28. Brady Ribbons • The generator always makes equal angles with the sides of the ribbon

29. 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

30. Brady Ribbons • Theorem proof: •  - 1 = 2 -  =>  = (2 -1 ) / 2 •  is constant. Hence, all G’s are parallel

31. O 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)

32. 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

33. 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

34. Symmetry Axis Transform (SAT)

35. 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)

36. Symmetry Axis Transform • Captures the major axis of the shape and its orientation • Reflects local boundary formations (e.g, corners) of the shape

37. 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

38. 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

39. SAT Weaknesses • Very sensitive to noise • May lose the symmetry axes of the shape

40. Smooth Local Symmetries (SLS)

41. Smooth Local Symmetries • Defined in two parts • Determination of the local symmetry • Formation of maximal smooth loci of these symmetries

42. 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

43. Determining Local Symmetry • A point may have a local symmetry with several points Point A has local symmetry with both B and C

44. 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

45. 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

46. Smooth Local Symmetry Axes • The short diagonal axes are subsumed • The diagonal axes are not subsumed

47. 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

48. Comparing SLS and SAT SLS SAT

49. Summary • Local symmetry can be used to describe parts of shapes • Local symmetry can be described in various ways • Ribbons • SAT • SLS

50. 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