By Remco C. Veltkamp

ISBN-10: 0387588086

ISBN-13: 9780387588087

ISBN-10: 3540588086

ISBN-13: 9783540588085

This monograph is dedicated to computational morphology, really to the development of a two-dimensional or a 3-dimensional closed item boundary via a suite of issues in arbitrary position.

By making use of innovations from computational geometry and CAGD, new effects are constructed in 4 levels of the development strategy: (a) the gamma-neighborhood graph for describing the constitution of a collection of issues; (b) an set of rules for developing a polygonal or polyhedral boundary (based on (a)); (c) the flintstone scheme as a hierarchy for polygonal and polyhedral approximation and localization; (d) and a Bezier-triangle dependent scheme for the development of a delicate piecewise cubic boundary.

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**Extra resources for Closed Object Boundaries from Scattered Points**

**Example text**

42) We know that cj = uj ∨ vj cj−1 . 44) This looks like the preﬁx problem. We have to prove that G = ({A(0 0) A(0 1) A(1 0)} ◦) is a monoid. 47) The operation ◦ on sets of functions is always associative. Therefore the conditions for the application of the preﬁx algorithms are fulﬁlled. 50 We only have to design a circuit for the operation ◦ . Let A(u v) = A(u2 v2) ◦ A(u1 v1) . 48) Here we ﬁnd again the characteristic computation of triangles and rectangles as in Krapchenko’s adder. 48) a subcircuit for the operation ◦ has size 3 and depth 2 .

1 : We save the ﬁrst step and deﬁne f ∈ B4 by Q4 , the set of implicants of length 4 . Q4: Q4 4 = ◦ , Q4 3 = {a b c d a b c d} , Q4 2 = {a b c d a b c d a b c d} , Q4 1 = {a b c d a b c d a b c d} , Q4 0 = {a b c d} . Q3: Q3 3 = ◦ , Q3 2 = {a b c a b c b c d} , Q3 1 = {a b d a c d a b c a c d b c d} , Q3 0 = {a b c a c d b c d} . P4 = ◦ . Q2: Q2 2 = ◦ , Q2 1 = {b c} , Q2 0 = {c d a c} . P3 = {a b c a b d} . Q1 = ◦ . P2 = Q2 . PI(f) = {a b c a b d b c c d a c} The PI-table of f 0010 0011 0100 0101 0111 1010 1011 1110 1111 abc abd bc cd ac 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 1 c1 c3 and c8 have a single one.

M has rows for xi xn+1 xn+2 (1 ≤ i ≤ n) and columns for (a 0 0) and some a ∈ S . The columns (a 0 1) and (a 1 0) have all been eliminated. Column (a 0 0) has been eliminated either during the elimination of row ma xn+1 iﬀ a ∈ S and |a| = 1 or during the elimination of row ma xn+2 iﬀ a ∈ S and |a| = 0 . Furthermore xi xn+1 xn+2(a 0 0) = ai . Therefore the partially reduced PI-table M is equal to the given matrix M . Since M is reduced, M is reduced too. All prime implicants have length 3 . 6 Discussion As we have shown the minimization of a Boolean function is (probably) a hard problem.

### Closed Object Boundaries from Scattered Points by Remco C. Veltkamp

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