Hexagonal parallel thinning algorithms based on sufficient conditions for topology preservation

An (m, n)-simple 1 in a binary image I has the property that its deletion “preserves topology” when m-adjacency is used on the 1’s and n-adjacency on the 0’s of I. This paper presents new, easily visualized, necessary and sufficient conditions for a 1 in I to be (m, n)-simple, for (m, n)=(26, 6), (18, 6), (6, 26) or (6, 18) when I is a 3-d image and (m, n)=(8, 4) or (4, 8) when I is a 2-d image. Systematic and fairly general methods of verifying that a given parallel thinning algorithm always preserves topology are described, for the cases where 8-/26-adjacency is used on the 1’s and 4-/6-adjacency on the 0’s, or vice versa. The verification methods for 2-d algorithms are mainly due to Ronse and Hall; the methods for 3-d algorithms were found by Ma and Kong. New proofs are given of the correctness of these verification methods, using the characterizations of simple 1’s presented in this paper.

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Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

Journal of Applied Probability ◽

10.1017/s0021900200003119 ◽

2007 ◽

Vol 44 (02) ◽

pp. 492-505

Author(s):

M. Molina ◽

M. Mota ◽

A. Ramos

Keyword(s):

Branching Process ◽

Branching Processes ◽

Sufficient Conditions ◽

Positive Probability ◽

Limit Laws ◽

Bisexual Branching Processes ◽

Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

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The extinction time of a general birth and death process with catastrophes

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

Author(s):

P. J. Brockwell

Keyword(s):

Laplace Transform ◽

Size Distribution ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Death Process ◽

Extinction Time ◽

Birth And Death Process ◽

Different Types ◽

The Laplace Transform ◽

Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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Normal approximations to discrete unimodal distributions

Journal of Applied Probability ◽

10.1017/s0021900200115931 ◽

1986 ◽

Vol 23 (04) ◽

pp. 1013-1018

Author(s):

B. G. Quinn ◽

H. L. MacGillivray

Keyword(s):

Stationary Distribution ◽

Sufficient Conditions ◽

Random Variables ◽

Hypergeometric Distribution ◽

Death Process ◽

Normal Approximations ◽

Discrete Random Variables ◽

Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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