SYMMETRY TRANSFORM

N. Kiryati, Y. Gofman, "Detecting symmetry in grey-level images: the global optimization approach"
D. Reisfeld, H. Wolfson, Y. Yeshurun, "Context free attentional operators: the generalized symmetry transform"

Symmetry Measure A one dimensional real function can be decomposed in a symmetric and an antisymmetric part,

f(x) = fs(x) + fa(x)

where fs(x)=½( f(x) + f(-1) ) and fa(x)=½( f(x) - f(-1) ). Therefore a measure of symmetry, for 1D functions, is

S(f) = |fs|² / |f|²

It is easily seen that S(f) is relate to the correlation of f(x) with f(-x),

2 S(f) - 1 = C(f) = ∫ f(x) f(-x) dx / |f|²

For 2D functions we define the symmetry at orientation t using the frame of reference

u = x cos(t) + y sin(t)
v = y cos(t) - x sin(t)

St(f) = ∫ | fs(u,v) |² du / ∫ | f(u,v) |² du

where the norms are evaluated with respect to v. It is easily seen that this is related to the correlation with the function mirrored about the u axis,

2 St(f) - 1 = Ct(f) = ∫ f(u,v) f(u,-v) dv du / ∫ | f(u,v) |² dv du

This symetry measure is scale independent. To avoid emphasizing the small scales, a scale dependent factor should be included.

Generalized Symmetry Transform

Given an image I we compute the gradient (dxI, dyI) and write it in polar representation, ie, with modulus g and phase t. To define th e symmetry transform, at each point we evaluate the contribution from neighboring symmetric pairs. More explicictly, at any point p=(x,y) we define the set G(p) of symmetric points as the pairs of points p1 p2 such that &fract12;(p1 + p2) = p. Each pair contribution depends on the strength of the edge at the two points, ie, to the modulus of the gradient. Actually we use R(g) = log( 1 + g ). Each pair contribution is weighted with a distance function D. This is a monotonically decreasing function that goes to zero at infinity, for example a gaussian, D(x) = 1/(sqrt(2 Pi) s) exp( - x²/2s² ). Here s determines the scale of the symmetry, ie, the size of the neighbor. Finally each pair contribution is weighted by

P(p1, p2) = ( 1 - cos( t1 + t2 - 2 a ) ( 1 - cos( t1 - t2 ) )

where a is the angle made by the line between the two points and the x-axis. The first term is maximum when t1 - a + t2 - a = Pi, i.e., the two gradients are in the same direction towards each other. The second term reduced the contribution of perpendicular straight edges ( ti-a = Pi/2 ).

Putting all together, the symmetry at the point p is

S(p) = Sum D(|p1 - p2|) P(p1, p2) R(g1) R(g2)

The symmetry map is invariant under rotations and remains effective when the image is skewed.

Marco Corvi - Page hosted by geocities.com.