Embed Size (px)
Citation preview
Symmetry Detection via Symmetry Detection via String MatchingString Matching
Sergey Kiselev
Symmetry Seminar
Haifa University, Spring 2005
MotivationMotivation
Knowledge of the symmetry of object is useful for solving problems in image shape analysis, robotic applications, computer graphics, etc.
We will discus several algorithms that detect symmetry of objects by reducing object from 2D or 3D to string representation while saving symmetry information and check symmetry of result string.
Definitions - SymmetryDefinitions - Symmetry
Transform T of an object P = {p1, p2,…} is
the object T(P) = {T(p1), T(p2), ... }.
P is symmetrical under the transform T,
if T(P) = P.
A line/plane is an axis of symmetry of an object if the object is invariant to symmetry transform with respect to this axis.
Rotational SymmetryRotational Symmetry
Rotational transform: Ra,
rotation of degrees about d-2 dimensional axis a
Rotational symmetry: symmetry under rotational transform
Rotational symmetry transform: Ca,k = Ra,360/
If some object P is symmetrical under Ca,k then a is called “k-fold axis of rotational symmetry”
Rotational SymmetryRotational Symmetry
3-fold rotational symmetry
5-fold rotational symmetry
Reflectional SymmetryReflectional Symmetry
Involutional transform: Zb,k
rotation 360/k degrees around line b’ reflection about d-1 dimensional axis b only Zb,1 is defined in 2D space in 3D most interesting are Zb,1 and Zb,2
Reflectional symmetry: symmetry under involutional transform
If some object P is symmetrical under Zb,k then b is called “k-fold axis of reflectional symmetry”
Reflectional SymmetryReflectional Symmetry
ReflectionalSymmetryZb,1
ReflectionalSymmetry
Zb,2
Snow CrystalsSnow Crystals
6-fold rotation symmetry + reflection symmetry
CentroidCentroid
m
i
iiCpw1
0
m
i
i
m
i
i OpwOC1
1
1
)(
Coordinates of C:
The centroid of set of points p1,…,pm such that every pi has a weight wi > 0, is the unique point C such that
C can be computed in O(m) time
String matchingString matching
String matching algorithm.KMP – finds in string with length n, substring
with length m in O(n+m) time. Infinite alphabet can be used in strings.
Symmetry DetectionSymmetry Detection
Steps of symmetry detection algorithm:
ORDER: Sort the points of object into cycles.
ENCODE: Encode each cycle into a string.
CHECK: Test the symmetry of the encoded string.
Symmetry DetectionSymmetry Detection
Cycle property of ORDER: If P is symmetrical under any
transform T, such that T(ci) = cj,
then for all k, T(ci + k) = cj + k
Uniqueness property of ENCODE:Encoded strings of two objects P1 and P2 have
cyclic permutations of each other if, and only if, for some T, T(P1) = P2.
Polygon - ORDERPolygon - ORDER
ORDER step is unnecessary for a polygon, since vertices of polygon are already form a valid cycle.
Polygon - ENCODEPolygon - ENCODE
ENCODE step generates a two-tuple of measures for each vertex which describes the location of that vertex.
Measures should be invariant under symmetrical transform.
For example:- Distances between adjacent vertices- Angles formed by edges at each vertex
So each two-tuple will be:
si = <dist (pi, pi+1), angle (pi-1, pi, pi+1)>
Polygon - ENCODEPolygon - ENCODEOther possibilities: Distances from centroid,
angles formed at centroid by two adjacent vertices
Not all combinations give unique encoding.Example: distances between adjacent vertices
and distances from centroid:
Polygon - ENCODEPolygon - ENCODE
ENCODE Example:
Encoded string:
S = <<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>>
Polygon - CHECKPolygon - CHECK
Finding rotational symmetryLet S = <s0, s1, … ,sn-1> be the encoded cycle of
polygon. We search for S in string A:
A = <s1, … ,sn-1, s0, s1, … ,sn-1>
If S first occurs in A at offset k-1 then the polygon must have n/k-fold rotational symmetry.
At least a one-fold symmetry will be found for any polygon – if S is found nowhere else in A it will be found at offset n-1.
Polygon - CHECKPolygon - CHECK
Finding rotational symmetry – Example
S = <<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>>
A = <<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>, <a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>>
S first occurs in A at offset 1 =>
Polygon has n/2 = 4-fold
rotational symmetry
Polygon - CHECKPolygon - CHECK
Finding reflectional symmetryLet R = <sn-1, sn-2, … ,so> be the reverse encoded
cycle of polygon. We search for R in string B:
B = <s0, s1, … ,sn-1, s0, s1, … ,sn-1>
If a match found at offset j, and n–j is odd, then there is a axis of symmetry bisecting the angle at p(n-j-1)/2
If n–j is even, axis of symmetry bisects the edge connecting p(n-j-2)/2 and p(n-j)/2
Polygon - CHECKPolygon - CHECKFinding reflectional symmetry – Example
R = <<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>>
B = <<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>, <a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>,<a,>>
Match occurs at offsets: 1, 3, 5, 7Polygon has 4 axes of reflectional symmetry that
bisect vertices P3, P2, P1, P0 respectively.Note: since we found that polygon has n/k-fold
rotational symmetry, its enough to check only k first symbols of R.
Polygon - ComplexityPolygon - Complexity
ORDER: not needed
ENCODE: O(n)O(1) for each vertex
CHECK: O(n)KMP, string and substring length is order of n.
Total: O(n)
2D Point Set - ORDER2D Point Set - ORDER
It possible to produce a polygon by connecting points sorted by they polar coordinates, taking angle as primary sort key and omitting points at centroid, and using previous polygon symmetry detection algorithm on resulting polygon.
This algorithm has O(nlogn) complexity since it requires sorting.
2D Point Set - ORDER2D Point Set - ORDER
This makes the algorithm extremely sensitive to round-off errors.
To avoid this problem we’ll partition points to several cycles such that all points in same cycle are at same distance from centroid and sorted by angle.
In practice such algorithm may have serious problem
2D Point Set - ENCODE2D Point Set - ENCODEIn encoded string each point is represented by angle
formed at centroid by this point and it’s successor.Its not needed to include distance from centroid,
since it constant within each cycle
Encoded string: A = <,,,>
2D Point Set – CHECK2D Point Set – CHECK
The tests to check a cycle for rotational and reflectional symmetry are the same as these for polygons. However they must be applied for all cycles of the point set.
The degree of rotational symmetry for the point set is a greatest common divisor of the orders of the cycles:
k = GCD (k1, k2, … , km)
2D Point Set – Complexity2D Point Set – Complexity
ORDER: O(nlogn)Requires generic number sorting algorithm. Such
algorithms need O(nlogn) operations.
ENCODE: O(n)Computing an angle for each point in the point set
CHECK: O(n)String matching requires O(n).
Finding GCD of m numbers requires O(m) operations, were m is number of cycles, m << n
Total: O(nlogn)
Segments - ORDERSegments - ORDER
Find centroid of set of midpoints of the segments, when every midpoint has a weight equal to the length of the its segment.
Partition segments to several cycles such that segments in same cycle have equal length, midpoints of all segments are at same distance from centroid and segments are sorted by angle.
Segments - ENCODESegments - ENCODE
Each segment is encoded as a two-tuple containing angle formed at centroid by midpoint of this segment and midpoint it’s successor, and angle between segment and line passing through centroid and midpoint of segment.
Segments - CHECKSegments - CHECK
The tests to check a cycle for rotational and reflectional symmetry are the same as these for polygons and point sets.
Like with point sets, tests must be applied for all cycles, and degree of rotational symmetry equals to a greatest common divisor of the orders of the cycles:
k = GCD (k1, k2, … , km)
Segments symmetry test has same complexity like point set – O(nlogn)
Circles and Other 2D ObjectsCircles and Other 2D Objects
Same symmetry check technique can be applied to other 2D objects.
As we saw above it’s needed to implement correct (such that hold described properties) ORDER and ENCODE steps corresponding to particular object.
For example circles can be partitioned according to distance from they center to centroid, sorted by angle between two circles. Each circle should be encoded by this angle and diameter.
3D Point Set – Axial Symmetry3D Point Set – Axial Symmetry
Problem: Given an axis and a 3D point set find the
rotational symmetry about that axis, and find all planes of reflectional symmetry containing that axis.
This problem is a direct extension of 2D point set problem. Now ORDER step will partition points to cycles not only by their distance from centroid (axis in this case), but also by their Z coordinate. (Z-axis is parallel to axis of rotation).
PolyhedronPolyhedron
Problem: Given an axis and a polyhedron with
connected surface graph find the rotational symmetry about that axis, and find all planes of reflectional symmetry containing that axis.
Observation: A (nontrivial) axis of symmetry can intersect
the surface of a polyhedron in only one of three ways: It may intersect a vertex, the midpoint of an edge, or the centroid of a face.
Polyhedron - ORDERPolyhedron - ORDER
C1 – cycle containing vertices topologically adjacent to point of intersection with axis.
Each point in Ck+1 is edge-connected to some point in Ck. A point can be edge-connected to several points in Ck. To distinguish one of these edges for each point in Ck+1 we’ll use function (pj, pi) whose value is three-tuple of Cartesian coordinates of the point p i in the coordinate system whose origin is at p j, whose Z-axis is parallel to axis of rotation, and whose Y-axis intersects the axis of rotation.
Polyhedron - ORDERPolyhedron - ORDERValue of function is unique for all edges
adjacent to pj, and symmetrical points have exactly the same set of values for their adjacent points. Thus for each point p j in Ck+1, we distinguish the adjacent point p i in Ck, which has lexicographical minimum value for (pj, pi). This defines mapping under which each point in Ck+1 maps exactly to one point in Ck.
Exact algorithm which constructs m cycles from vertex set with size n of a polyhedron is given in the literature [1] and has O(n) time complexity.
Polyhedron - ENCODEPolyhedron - ENCODE
Each vertex of polyhedron can be encoded as tuple containing:
Coordinates of this vertex (pj)
– Cylindrical angle coordinate– Radius coordinate– Z-coordinate
List of adjacent vertices (pi)– Sorted in clockwise order
– Given by (pj, pi) value
Polyhedron – CHECKPolyhedron – CHECK
While previous step uses tuples of variable size to represent different vertices, it still possible to run CHECK algorithm in linear time.
Let vi, vi,1, vi,2, … vi,ki be the element of i-th tuple.
Encoded string is:<v1, v1,1, v1,2, … v1,k1
,v2, v2,1, v2,2, … v2,ki ,vn, vn,1, vn,2, … vn,kn
>
For each vertex string contains three point coordinate and for each edge two three-tuple, one for vertex on each end. Total length of string is order of |V| + |E|, which is O(n).
Polyhedron – Axes of SymmetryPolyhedron – Axes of Symmetry So far, we have considered only polyhedron
symmetry about a given axis
The possible arrangements of nontrivial axes of symmetry is 3D space are fairly restricted
It possible to find symmetry group by reducing surface graph of polyhedron to either a ring, a skein, or one of the graphs of Platonic solids
Such reduction doesn’t destroy existing symmetry, but may create new symmetries
Polyhedron – Axes of SymmetryPolyhedron – Axes of Symmetry
(k) - One k-fold line of symmetry, as in a regular k-sided cone
There only one axis of symmetry, which must intersect the polyhedron surface in same place it intersects surface graph.
Polyhedron – Axes of SymmetryPolyhedron – Axes of Symmetry
(2,3,3) - Four 3-fold lines and three 2-fold lines, arranged as in regular tetrahedron
Polyhedron – Axes of SymmetryPolyhedron – Axes of Symmetry
(2,3,4) - Three 4-fold lines, four 3-fold lines and six 3-fold lines, as in a regular octahedron or hexahedron
Polyhedron – Axes of SymmetryPolyhedron – Axes of Symmetry
(2,3,5) - Six 5-fold lines, ten 3-fold lines and fifteen 2-fold lines, as in a regular dodecahedron or icosahedron
Polyhedron – Axes of SymmetryPolyhedron – Axes of Symmetry
(2,2,k) - One k-fold line of symmetry and k 2-fold lines of symmetry uniformly spaced in the plane perpendicular to the first line, as in a k-sided regular prism
This case requires more difficult algorithm, since number of symmetry axes not limited
““Near” SymmetryNear” Symmetry Next figures appear to have some symmetry,
but according to our definition have none. In some cases we’ll want to find degree of
symmetry.
““Near” SymmetryNear” Symmetry
Maximal Symmetric Subset Problem.
- For each three distinct points p, q, r only one rotation transform maps p to q and q to r.
- For n points there are
n(n-1)(n-2) possible
transform candidates.- O(nlogn) to check each
candidate.- Total: O(n4logn).
““Near” SymmetryNear” SymmetryMinimal Symmetric Decomposition
ProblemMinimal Symmetric Partition Problem
Strip patterns
SummarySummary
Reduction 2D or 3D objects to strings
Symmetry checking and finding symmetry axes for 2D objects
Symmetry checking of 3D objects
Finding axes of symmetry for 3D objects
“Near” symmetry problem
Questions?Questions?
ReferencesReferences[1] J. D. Wolter, T. C. Woo, and R. A. Volz,“Optimal algorithms for symmetry detection in two and three dimensions,”Visual Computer, vol. 1, pp. 37-48, 1985.
[2] P. Eades,“Symmetry finding algorithms,”Computational Morphology (G. T. Toussaint, Ed.).Amsterdam: North-Holland, 1988, pp. 41-51.
[3] M.Atallah, “On Symmetry Detection,”IEEE Trans. Computers, vol. 34, no. 7, 1985
[4] D.E. Knuth, J.H. Morris, and V.R. Pratt,“Fast pattern matching in strings,”SIAM J. Comput., vol. 6, no. 2, pp. 323-350, 1977.
