Why multiple openings/closing with a same kernel does not have effect?

Question

I know about closing and opening, but there is still one question to me! According to the "Digital Image Processing, 3rd edition", by Gonzales, The multiple application of opening/closing doesn't have any effect after the first time you apply it! I couldn't figure it out? Can anyone help?

Answer 1

This is expected behavior since openings and closings are idempotent operations. An operation is idempotent if, whenever it is applied twice to any value, it gives the same result as if it were applied once: f(f(x)) = f(x). Openings are operators on lattice L that are idempotent, increasing, and anti-extensive while closings are operators on L that are idempotent, increasing, and extensive. One can find a discussion on what is idempotence here.

In a more intuitive sense, an opening on set X is a erosion followed by dilation by the same structuring function. Once the first iteration is done the set X does not change since the erosion and dilation remove and add the same '1's in the set X. The product of opening and closing is also an idempotent operation - which is very interesting. One the other hand if at each iteration one changes the radius of the structuring element for the openings/closings, one would obtain an Alternated Sequential Filter, which produces multiscale simplification of the image, thus producing a scale-space.

I would refer you to the book by Jean Serra on mathematical morphology or better understanding.