CONTINUOUS SYMMETRY FOR SHAPES

Symmetry is treated as a continuous feature and the Symmetry Distance SD of a shape is de ned to be the minimum distance required to move points of the original shape in order to obtain a symmetrical shape This general de nition of a symmetry measure enables a comparison of the amount of symmetry of di erent shapes and the amount of di erent symmetries of a single shape The measure is applicable to any type of symmetry in any dimension The Symmetry Distance gives rise to a method of reconstructing symmetry of occluded shapes We extend the method to deal with symmetries of noisy and fuzzy data Introduction One of the basic features of shapes and objects is symmetry Symmetry is consid ered a pre attentive feature which enhances recognition and reconstruction of shapes and objects Symmetry is also an important parameter in physical and chemical processes and is an important criterion in medical diagnosis However the exact a b c Figure Faces are not perfectly symmetrical a Original image b Right half of original image and its re ection c Left half of original image and its re ection mathematical de nition of symmetry is inadequate to describe and quantify the symmetries found in the natural world nor those found in the visual world a classic example is that of faces see Fig Furthermore even perfectly symmetric objects loose their exact symmetry when projected onto the image plane or the retina due to occlusion perspective transformations digitization etc Thus although symmetry is usually considered a binary feature i e an object is either symmetric or it is not symmetric we view symmetry as a continuous feature where intermediate values of symmetry denote some intermediate amount of symmetry We introduce a Sym metry Distance that can measure and quantify all types of symmetries of objects This measure will enable a comparison of the amount of symmetry of di erent shapes and the amount of di erent symmetries of a single shape In Sect we de ne the Symmetry Distance and in Sect we describe a method for evaluting this measure We then describe features of the symmetry distance including its use in dealing with occluded objects and with noisy data A Continuous Symmetry Measure De nition We de ne the Symmetry Distance SD as a the minimum e ort required to turn a given shape into a symmetric shape This is measured by the mean of the square distances each point is moved from its location in the original shape to its location in the symmetric shape No a priori symmetric reference shape is assumed Denote by the space of all shapes of a given dimension where each shape P is represented by a sequence of n points fPig n i We de ne a metric d on this space as follows d R d P Q d fPig fQig n n X i kPi Qik This metric de nes a distance function between every two shapes in We de ne the Symmetry Transform ST of a shape P as the symmetric shape closest to P in terms of the metric d The Symmetry Distance SD of a shape P is now de ned as the distance between P and it s Symmetry Transform SD d P ST P The SD of a shape P fPig n i is evaluated by nding the symmetry transform P of P Fig and computing SD n n i kPi Pik Figure The symmetry transform of fP P P g is f P P P g SD i kPi Pik C Symmetry = 12.80 3 P1 P2 P2 ^ P1 P 0 P