Circular-centred top-hat
Opening by adjunction (erosion, dilation): ?B(x) = sup { inf [ f (y) , y ? Bi ] , i ? I } where {Bi, i ? I} is the family of structuring elements which contain point x
The top-hat is therefore: f (x) - ?B(x) = - sup { inf [ f (y) - f (x), y ? Bi ], i ? I }
As there are only increments of the function f around point x, we can transpose to functions of circular values a: (th a)(x) = - sup { inf [- (a(x) ? a(y)) , y ? Bi ], i ? I }