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.