T. Yung Kong

Algorithms for Fuzzy Segmentation

Fuzzy segmentation is an effective way of segmenting out objects in pictures containing both random noise and shading. This is illustrated both on mathematically created pictures and on some obtained from medical imaging. A theory of fuzzy segmentation is presented. To perform fuzzy segmentation, a ‘connectedness map’ needs to be produced. It is demonstrated that greedy algorithms for creating such a connectedness map are faster than the previously used dynamic programming technique. Once the connectedness map is created, segmentation is completed by a simple thresholding of the connectedness map. This approach is efficacious in instances where simple thresholding of the original picture fails.

Simultaneous fuzzy segmentation of multiple objects

Fuzzy segmentation is a technique that assigns to each element in an image (which may have been corrupted by noise and/or shading) a grade of membership in an object (which is believed to be contained in the image). In an earlier work, the first two authors extended this concept by presenting and illustrating an algorithm which simultaneously assigns to each element in an image a grade of membership in each one of a number of objects (which are believed to be contained in the image). In this paper, we prove the existence of such a fuzzy segmentation that is uniquely specified by a desirable mathematical property, show further examples of its use in medical imaging, and illustrate that on several biomedical examples a new implementation of the algorithm that produces the segmentation is approximately seven times faster than the previously used implementation. We also compare our method with two recently published related methods.

Speeding up stochastic reconstructions of binary images from limited projection directions

Abstract In earlier work, a stochastic method for reconstructing certain classes of two-dimensional binary images from limited projection directions was presented. In the present study, we experiment with different implementations of this method to minimize the running time. Our fastest implementation is based on a look-up table and pre-generated arrays of random integers. This is more than 40 times faster than the implementation used in the earlier work. This speedup makes it practical to conduct extensive searches to find the optimal values of the methods parameters for each class of images to be reconstructed.

Tomographic Equivalence and Switching Operations

A binary picture on an arbitrary grid is a mapping f from the set of all grid points to {0,1} such that f (x) = 1 for only finitely many grid points x. If two binary pictures f 1 and f 2 on the same grid have the property that for every grid line P the sets {p ∈ l| f 1(p)= 1} and{ p ∈ l|f 2(p)= 1} contain exactly the same number of grid points, then we say that f 1 and f 2 are tomographically equivalent. Given a binary picture f on the usual 2-dimensional square grid, there may exist an upright rectangle R (of any size) whose sides are grid lines, such that f = 1 at two diagonally opposite corner points of R and f= 0 at the other two corner points. If so, then we call the process of changing the value of the picture f from 1 to 0 and 0 to 1 at the four corner points of R (without changing the value of f at any other grid point) a rectangular 4-switch. Ryser showed in the 1950s that two binary pictures on the square grid are tomographically equivalent if and only if one picture can be transformed to the other by a finite sequence of rectangular 4-switches. We present a few different versions of this theorem, describe an application, and also give a proof of the result. We then show that the result has no analog on grids that have grid lines in three or more directions (such as the 3-dimensional cubic grid),because on such grids one can find for every integer L two tomographically equivalent binary pictures that differ at more than L grid points and are not tomographically equivalent to any other binary picture.

On which grids can tomographic equivalence of binary pictures be characterized in terms of elementary switching operations

It is a well‐known result, due to Ryser, that if a binary picture on the square grid has the same x and y projections as another such picture, then the first picture can be transformed into the second by a series of switching operations, each of which changes the picture at just four grid points and preserves both projections. In this article, we show that if a grid [such as a two‐dimensional (2D) hexagonal grid or the 3D cubic grid] has three or more directions of projection, then Rysers result has no analog for that grid. Specifically, we show that on any grid with three or more directions of projection there cannot exist any constant L such that every binary picture can be transformed to any other binary picture with the same projections by a series of projection‐preserving changes, each of which involves at most L grid points. This is proved for a very general concept of “grid” that encompasses virtually all practical grids in Euclidean n‐space, and even some grids in higher‐dimensional analogs of cylindrical and toroidal surfaces. (In fact, the set of grid points can be any finitely generated Abelian group of rank ≥ 2.)

Minimal non-simple sets in 4-dimensional binary images with (8,80)-adjacency

We first give a definition of simple sets of 1’s in 4D binary images that is consistent with “(8,80)-adjacency”—i.e., the use of 8-adjacency to define connectedness of sets of 1’s and 80-adjacency to define connectedness of sets of 0’s. Using this definition, it is shown that in any 4D binary image every minimal non-simple set of 1’s must be isometric to one of eight sets, the largest of which has just four elements. Our result provides the basis for a fairly general method of verifying that proposed 4D parallel thinning algorithms preserve topology in our “(8,80)” sense. This work complements the authors’ earlier work on 4D minimal non-simple sets, which essentially used “(80,8)-adjacency”—80-adjacency on 1’s and 8-adjacency on 0’s.

Simultaneous Fuzzy Segmentation of Multiple Objects

Abstract Fuzzy segmentation is a technique that assigns to each element in an image (corrupted by noise and/or shading) a grade of membership in an object (which is believed to be contained in the image). In an earlier work the first two authors extended this concept by presenting and illustrating an algorithm which simultaneously assigns to each element in an image a grade of membership in each one of a number of objects (which are believed to be contained in the image). In this paper we establish the correctness of this algorithm (in the sense of producing an output that is uniquely specified by a desirable mathematical property) and present a further example of its use in medical imaging. We also compare our method with two recently published related methods.

History trees as descriptors of macromolecular structures

High-level structural information about macromolecules is now being organized into databases. One of the common ways of storing information in such databases is in the form of three-dimensional (3D) electron microscopic (EM) maps, which are 3D arrays of real numbers obtained by a reconstruction algorithm from EM projection data. We propose and demonstrate a method of automatically constructing, from any 3D EM map, a topological descriptor (which we call a history tree) that is amenable to automatic comparison.

Minimal non-simple and minimal non-cosimple sets in binary images on cell complexes

The concepts of weak component and simple 1 are generalizations, to binary images on the n-cells of n-dimensional cell complexes, of the standard concepts of “26-component” and “26-simple” 1 in binary images on the 3-cells of a 3D cubical complex; the concepts of strong component and cosimple 1 are generalizations of the concepts of “6-component” and “6-simple” 1 Over the past 20 years, the problems of determining just which sets of 1s can be minimal non-simple, just which sets can be minimal non-cosimple, and just which sets can be minimal non-simple (minimal non-cosimple) without being a weak (strong) foreground component have been solved for the 2D cubical and hexagonal, 3D cubical and face-centered-cubical, and 4D cubical complexes This paper solves these problems in much greater generality, for a very large class of cell complexes of dimension ≤4.

Strongly Normal Sets of Tiles in N Dimensions

Abstract The first and third authors and others [2,8,9,10,11,12] have studied sets of “tiles” (a generalization of pixels or voxels) in two and three dimensions that have a property called strong normality (SN): For any tile P, only finitely many tiles intersect P, and any nonempty intersection of these tiles must also intersect P. This paper presents extensions of the basic results about SN sets of tiles to n dimensions. One of our results is that if SN holds for every n + 1 or fewer tiles in a locally finite set of tiles in Rn, then the entire set of tiles is SN. Other results are that SN is equivalent to hereditary local contractibility, that simpleness of a tile in an SN set of tiles is equivalent to contractibility of its shared subset, and that deletion of a simple tile in an SN set of tiles preserves the homotopy type of the union of all the tiles.