SYMMETRY
Safi Msallam
Uses:
• Recognition.
• Reconstruction.
• physical and chemical processes.
• medical diagnosis.
S(obj)0 : 1
• Symmetry has been treated as a binary feature.
a. b. c.
Amount of Symmetry
• Symmetry Distance (SD) : quantifier of the minimum effort required to turn a given shape into a symmetric shape.
Symmetry Transform ST of a shape P, as the symmetric shape closest to P .
Definitions of Symmetry
• mirror-symmetry.
• Cn-Symmetry/rotational-symmetry.
What affects Symmetry?
• Even when objects are symmetrical, their projection onto a digital image is not necessarily so.
• Resolution.
more symmetric at low resolution than at high resolution.
Symmetry Transforms
Evaluating the Symmetry Transform
Example
Point Selection for Shape Representation
• Selection of points influences the value of SD.
• If a shape is inherently created from points the job is done.
• If not so There are several ways to select a sequence of points to represent continuous shapes.
Examples
Don’t worry
contour length is not a meaningful measure, as in noisy or occluded shapes.
In such cases we propose to select points on a smoothed version of the contour and then project them back onto the original contour.
Center of Symmetry
• selection of points about a point other than the centroid will give a different symmetry distance value.
• Center of symmetry: point about which selection at equal angles gives the minimum SD.
Like this:
Multiresolution Scheme
• Reflection plane did not converge to the correct one.
due to the sensitivity of the symmetry
value to noise and digitization errors.
Finding Locally symmetric Regions
• Looking at an image that contains a collection of symmetric patterns.
• We aim to locate local symmetries.
How !
• We use the Quad Tree structure which is a hierarchical representation of an image.
• all quadrants of the image of the quad tree are tested with a function of the Symmetry Distance.
Measuring symmetry of point sets.
• Grunbaum, B. “Measures of symmetry for convex sets”. Proc. Sympos. Pure Math., vol. 7. Providence, USA, pp. 233-270. 1963
• A.B. Buda, AB and K. Mislow, “A Hausdorff Chirality Measure”. J. Am. Chem. Soc. 1992, 114, 6006-6012
• H.Zabrodsky, S.Peleg and D.Avnir, “Symmetry as a Continuous Feature” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 17(12), 1995, pp.1154-1165
• E. Yodogawa, "Symmetropy, an entropy-like measure of visual symmetry,"Perception Psychophys., vol. 32, no. 3, pp. 230-240, 1982.