Prove the validity of the following gray-scale morphology expressions. You may assume that...

Prove the validity of the following gray-scale morphology expressions. You may assume that b is a flat structuring element. Recall that fc(x, y) = −f(x, y)and that .

(a) Duality of erosion and dilation: .

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(b) .

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(c) and . Assume a symmetric structuring element.

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(d) .

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(e) , where (f ⊕ nb)indicates n erosions of f by b. Also, .

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(f) and . Assume that the structuring element is symmetric with respect to its origin.