Optimal Grouping of Line Segments into Convex Sets.

A Decomposition Theorem for m-Convex Sets in Rd

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-102-5 ◽

1976 ◽

Vol 28 (5) ◽

pp. 1051-1057

Author(s):

Marilyn Breen

Keyword(s):

Topological Space ◽

Convex Sets ◽

Decomposition Theorem ◽

Linear Topological Space ◽

Local Convexity ◽

Line Segments ◽

Local Nonconvexity ◽

Locally Convex

Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of S, at least one of the line segments determined by these points lies in S. A point x in S is said to be a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z Є N ⌒ S, then [y, z] ⊆ S. If S fails to be locally convex at some point a in S, then q is called a point of local nonconvexity (lnc point) of S.

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Intersections of m-Convex Sets

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-141-0 ◽

1975 ◽

Vol 27 (6) ◽

pp. 1384-1391 ◽

Cited By ~ 1

Author(s):

Marilyn Breen

Keyword(s):

Topological Space ◽

Convex Sets ◽

Linear Topological Space ◽

Local Convexity ◽

Line Segments ◽

Local Nonconvexity ◽

Locally Convex

Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of line segments determined by these points lies in S. A point x in S is called a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z ∈ N⋂ S, then [y, z] ⊆ S. If S fails to be locally convex at some point q in S, then q is called a point of local nonconvexity (lnc point) of S.

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A Decomposition for Sets Having a Segment Convexity Property

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-002-7 ◽

1980 ◽

Vol 32 (1) ◽

pp. 21-26

Author(s):

Marilyn Breen

Keyword(s):

Euclidean Space ◽

Convex Sets ◽

Convex Set ◽

Finite Union ◽

Combinatorial Property ◽

Convexity Property ◽

Convexity Condition ◽

Line Segments ◽

Isolated Points ◽

Decomposition Theorems

Let 5 be a subset of Euclidean space. The set 5 is said to be m-convex, m ≥ 2, if and only if for every m distinct points of S, at least one of the line segments determined by these points lies in 5. Clearly any union of m mdash — 1 convex sets will be m-convex, yet the converse is false. However, several decomposition theorems have been proved which allow us to write any closed planar m-convex set as a finite union of convex sets, and actual bounds for the decomposition in terms of m have been obtained ([6], [4], [3]). Moreover, with the restriction that (int cl S) ∼ S contain no isolated points, an arbitrary planar m-convex set S may be decomposed into a finite union of convex sets ([1].Here we strengthen the m-convexity condition to define an analogous combinatorial property for segments.

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On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010018135x ◽

1990 ◽

Vol 48 (1) ◽

pp. 520-521

Author(s):

Neng-Yu Zhang ◽

Bruce F. McEwen ◽

Joachim Frank

Keyword(s):

Function Space ◽

Convex Sets ◽

Electron Tomography ◽

Spinal Ganglion ◽

Fourier Space ◽

Missing Information ◽

Voltage Electron ◽

High Voltage Electron Microscope ◽

Energy Bound ◽

Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

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