12.7 Mathematical morphology[2CQ]
Let be in the following \(X\) be a vector space normed with norm \(\| ⋅\| \).
[124]For \(A,B⊆ X\) arbitrary subsets, we recall the definition of Minkowski sum \(A ⊕ B=\{ x+y : x∈ A, y∈ B\} \) defined in 345.
Having now fixed a set \(B\), we define
the erosion of a set \(A⊆ X\) as
\[ A ⊖ B=\{ z∈ X:(B+z)⊆ A\} \quad ; \]the closing \(A ∙ B = ( A ⊕ B ) ⊖ B\);
the opening \( A ∘ B = ( A ⊖ B ) ⊕ B\).
Where, given \(B⊆ X,z∈ X\), we have indicated with \(B+z=\{ b+z:b∈ B\} \) the translation of \(B\) in the direction \(z\). In previous operations \(B\) it is known as ”structural element”, And in applications often \(B\) it’s a puck or a ball.
Let in the following \(A,B,C⊆ X,w,z∈ X\). Some of the following exercises are taken from [ 27 ] .
- E346
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Show the following identities:
\begin{eqnarray*} A ⊕ B & =& ⋃_{y∈ B}( A + y) \\ A ⊖ B & =& ⋂_{y∈ B}( A - y) \end{eqnarray*}Hidden solution: [UNACCESSIBLE UUID ’126’]
- E346
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Let \(\tilde B=\{ -b:b∈ B\} \): show that \((A ⊕ B)^ c=A^ c ⊖ \tilde B\), where \(A^ c= X⧵ A\) is the complementary. Hidden solution: [UNACCESSIBLE UUID ’128’]
- E346
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Show that the four operations are monotonic: if \(A⊆ C\) then \(A ⊕ B⊆ C ⊕ B\), \(A ⊖ B⊆ C ⊖ B\), \(A ∙ B⊆ C ∙ B\) and \( A ∘ B ⊆ C ∘ B \). Hidden solution: [UNACCESSIBLE UUID ’12B’]
- E346
[12C]Prerequisites:346,2,2.If \(A\) is closed, show that \(A ⊖ B\) is closed. Hidden solution: [UNACCESSIBLE UUID ’12D’]
- E346
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Show that erosion has the invariant property in this sense:
\[ (A + z) ⊖ (B + z) = A ⊖ B~ . \] - E346
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Moreover, the erosion satisfies \(( A ⊖ B ) ⊖ C = A ⊖ ( B ⊕ C )\). Hidden solution: [UNACCESSIBLE UUID ’12H’]
- E346
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Show that the expansion enjoys the distributive property with respect to union:
\[ (A \cup C) ⊕ B = (A ⊕ B) \cup (C ⊕ B). \]Hidden solution: [UNACCESSIBLE UUID ’12K’]
- E346
[12M]Prerequisites:346,7,2.Show that erosion has the distributive property with respect to the intersection:
\[ (A \cap C) ⊖ B = (A ⊖ B) \cap (C ⊖ B). \]Hidden solution: [UNACCESSIBLE UUID ’12N’]
- E346
[12P]Prerequisites:346,2.Sia \(\tilde B=\{ -b:b∈ B\} \). Show that
\[ (A ∙ B)^ c= ( A ^ c ∘ \tilde B )~ . \]Hidden solution: [UNACCESSIBLE UUID ’12Q’]
- E346
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Show that
\[ A ⊆ ( C ⊖ B ) \]if and only if
\[ ( A ⊕ B ) ⊆ C ~ ~ . \]Hidden solution: [UNACCESSIBLE UUID ’12S’]
- E346
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Recall that the operation \(A ∙ B=(A⊕B)⊖B\) is called ”closing”.
Show that \(A ⊆ A ∙ B\).
Let \(X=ℝ^ n\), \(B=B_ r=\{ \| x\| {\lt} r\} \) a ball, find an example of a set \(A\) that is open non-empty bounded, and \(A ∙ B=A\).
Setting \(X=ℝ^ n\), \(B=B_ r\) a ball, find an example where \(A ∙ B≠ A\).
Hidden solution: [UNACCESSIBLE UUID ’12V’]
- E346
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The opening is also given by \(A ∘ B = ⋃_{x∈ X,B+x ⊆ A} (B+ x)\), which means that it is the locus of translations of the structuring element \(B\) inside the set \(A\). Hidden solution: [UNACCESSIBLE UUID ’12X’]
- E346
[12Y] Prerequisites:346.In the following \(A,B,\hat B⊆ℝ^ n\).
Find an example where \(B\stackrel{≠}{⊂}\hat B\) and \(A ∘ B\stackrel{≠}{⊂} A ∘ \hat B\).
Find an example where \(B\stackrel{≠}{⊂}\hat B\) and \(A ∘ \hat B\stackrel{≠}{⊂} A ∘ B\).
Hidden solution: [UNACCESSIBLE UUID ’12Z’]
- E346
[130] Prerequisites:346,12.If \(A\) is convex and \(\hat B\) is the convex envelope (see 365 of \(B\), show that \(A ∘ B⊆ A ∘ \hat B\). Show with an example that equality may not apply. Hidden solution: [UNACCESSIBLE UUID ’131’]
- E346
[132] Prerequisites:346,12.If \(A,B\) are convex, show that \(A ∘ B\) is convex. Hidden solution: [UNACCESSIBLE UUID ’133’]