Elliptical Objects Research Articles

Model-based reconstruction of multiple circular and elliptical objects from a limited number of projections

We consider tomographic image reconstruction from a limited number of noisy projections. An efficient algorithm based on maximum likelihood estimation (MLE) is developed to reconstruct images of multiple discs with unknown locations and radii. The algorithm is successfully applied to images with signal-to-noise ratio (SNR) as low as 0 dB, using as few as 16 projections, and containing as many as twelve discs with widely varying radii. Experimental results show that our approach significantly outperforms conventional convolution back projection. The algorithm is successfully extended to the multiple ellipse case.

Novel detection of conics using 2-D Hough planes

The authors present a new approach to the use of the Hough transform for the detection of ellipses in a 2-D image. In the proposed algorithm, the conventional 5-D Hough voting space is replaced by four 2-D Hough planes which require only 90 kbytes of memory for a 384/spl times/256 image. One of the main differences between the proposed transform and other techniques is the way to extract feature points from the image under question. For the accumulation process in the Hough domain, an inherent property of the suggested algorithm is its capability to effect verification. Experimental results from the authors' work on real and synthetic images show that a significant improvement of the recognition is achieved as compared to other algorithms. Furthermore, the proposed algorithm is applicable to the detection of both circular and elliptical objects concurrently.

ELLIROT—A program to view and analyze spatial distributions of ellipses

Abstract The program ELLIROT (ELLIpse ROTation), written in TurboPascal for IBM-compatible microcomputers, uses computer graphics to examine the three-dimensional distribution of elliptical objects, if the data distribution can be represented by an ellipsoid. In strain analysis, the final strain ellipsoid can be determined with ELLIROT from elliptical markers with arbitrary orientations in space.

R f/ø f strain analysis using an orientation net

Since the classical work of Cloos, deformed distributions of elliptical objects such as ooids or pebbles have been recognized as an extremely important category of geological strain marker. However, elliptical objects are not easily analyzed, especially where primary sedimentary fabrics are tectonically imbricated. This paper demonstrates that previously published analytical techniques generally address only specific aspects of deformed ellipse distributions; as research tools, they are like a stereonet with great circles only or small circles only. All of the above methods can be combined with the aid of a new orientation net which is as convenient to use in the field as a standard stereonet. Uniform and imbricate fabrics are evaluated with equal ease and assumptions are subjected to statistical testing.

Delineating Elliptical Objects with an Application to Cardiac Scintigrams

To delineate the myocardium in planar thallium-201 scintigrams of the left ventricle, a method, based on the Hough transformation, is presented. The method maps feature points (X, Y, Y')-where Y' reflects the direction of the tangent in edge point (X,Y)-into the two-dimensional space of the axis lengths of the ellipse. Within this space, a probability density function (pdf) can be estimated. When the center of the ellipse or its orientation are unknown, the 2-D pdf of the lengths of the axes is extended to a 5-D pdf of all parameters of the ellipse (lengths of the axes, coordinates of the center, and the orientation). It is shown that the variance of the edge-point-based estimates of the axis lengths increases when the location error of the center of the supposed ellipse or its orientation error increases. The likelihood of the estimates is expected to decrease with increasing variance. Therefore, local search algorithms can be applied to find the maximum likelihood estimate of the parameters of the ellipse. Curves describing the convergency of the algorithm are presented, as well as an example of the application of the algorithm to real scintigrams. The method is able to detect contours even if they are only partly visualized, as in thallium scintigrams of the myocardium of patients with ischemic heart disease. As long as the number of parameters describing the contour is relatively low, such an algorithm is also suitable for application to differently curved contours.

Compact Elliptical Galaxies

We summarize our present knowledge on low-mass high-surface brightness elliptical objects near massive galaxies that are often called M32-type or compact elliptical galaxies. The origin of the low mass of these objects is a controversial matter: is it intrinsic to their formation or produced, as classically believed, by tidal stripping from the massive neighbor? We present new observational data allowing to define better the characteristics of these objects and a simple theoretical model whose consequences support the idea that the precursors of compact ellipticals are related to the low-mass end of the luminosity function of elliptical galaxies.

Elastic moduli of two-dimensional composite continua with elliptical inclusions

The elastic field around an elliptical inclusion in two dimensions is obtained. This result is then used to compute the effective moduli of a composite medium containing many randomly positioned and oriented elliptical objects. Two different self-consistent methods are described and the special cases of voids and rigid reinforcement are considered in detail. The asymmetric self-consistent method shows that the Young’s modulus E goes to zero as E=E1(p−pc)/(1−pc), where 1−p is the concentration of the voids and subscript one denotes the void free material. The Poisson ratio σ is also linear in p and goes to a value σc at pc that is independent of the initial value of Poisson’s ratio σ1. Unlike the corresponding three-dimensional case, the two elastic moduli decouple in this special case. The corresponding elastic threshold is pc=[1+ab/(a2+b2)], where a and b are the major semiaxes. The symmetric self-consistent method yields a different pc =2[1+(2(a+b)2/(a2+b1))1/2]−1 and a different concentration dependence of the effective moduli. Both methods give the same result for circular inclusions and both methods give pc+σc=1 for all aspect ratios. Similar results are presented for the case of rigid reinforcements.

Domain structure and ambiguities in the dipole direction

A self-consistent mathematical formulation of the magnetization distribution in ideal soft-magnitude media, which is based on the constitutive equation previously derived from micromagnetic principles by the author, is presented. We confine ourselves to two-dimensional solenoidal distributions in thin-film objects. In particular, we focus on the magnetization distribution in elliptical objects, and Lagrange's method is used to solve the boundary-value problem. A function of the fourth degree in the magnetization direction results that gives rise to more than one real root for each location. We make a systematic inventory of the meaning of the various roots and construct magnetization distributions that satisfy the boundary conditions. We prove that the requirement of continuity of the magnetization distributions leads to the conclusion that the magnetization direction is multiple-valued in some regions and that the domain structure is required to prevent these ambiguities from occurring. We demonstrate both theoretically and experimentally that the simplest domain structure in an ellipse is formed by a simple, straight domain wall along the longitudinal symmetry axis of the ellipse.

A comparison of methods used in measuring finite strains from ellipsoidal objects

The available methods for measuring strains in two and three dimensions from passive ellipsoidal objects are compared in an attempt to determine the most useful and precise procedure for the structural geologist. A comparison of data collection techniques showed that the use of thin orthogonally cut slabs or mylar overlays used with a monocomparater provides the most reproducible data. Lisle's theta-curve and Shimamoto and Ikeda's algebraic methods provided the most precise, and probably most accurate, two-dimensional data while Miller and Oertel's procedure and possibly Dunnet's PHASE 5 program gave the best three-dimensional results. An examination of errors encountered in strain analyses suggests that all of the available methods give accurate orientations of the finite strain ellipsoid. However, the magnitudes of strain ratios show large variations that are dependent on the sample size and procedure used. Shimamoto and Ikeda's method again proved to be the most precise, giving reproducible results with as few as ten elliptical objects. The samples used in the above comparison are part of a larger analysis of strains occurring in southeastern Maine. Structural elements observed in four selected areas of Avalonian belt rocks and the strain data collected suggest that the region has undergone at least three non-coaxial deformations with D 1 > D 2 > D 3.

A simple algebraic method for strain estimation from deformed ellipsoidal objects. 1. Basic theory

A new algebraic method is developed to determine the shape and orientation of the strain ellipsoid by using deformed ellipsoidal objects as strain markers. It is assumed that objects are of truly ellipsoidal shape with random orientation in the undeformed state, and that they deform homogeneously with their matrix. This part presents basic theories for: 1. (1) the determination of the strain ellipse on a plane section from deformed elliptical objects; 2. (2) determination of the strain ellipsoid from measurements of lengths and orientations of all principal axes of deformed ellipsoidal objects; and 3. (3) construction of the strain ellipsoid from the two-dimensional analysis on three mutually orthogonal planes. General, finite, homogeneous deformation is treated, and the analysis gives the deviatoric natural strains in the principal directions of the Eulerian finite-strain tensor. The simplicity and usefulness of our method is demonstrated by the two-dimensional, pure-shear deformation of model objects. Evaluation of error and optimum sample size, and geological applications of the method will be discussed in detail in subsequent papers.

Preliminary Survey of Freshwater Early Tertiary Invertebrates from Trans-Pecos Texas: ABSTRACT

Freshwater invertebrates of early Tertiary age have been collected from eight scattered locations in Presidio and Brewster Counties in Trans-Pecos Texas. Most specimens are internal End_Page 2211------------------------------ molds of several gastropod genera of Eocene and Oligocene ages. Several specimens of a Paleocene pelecypod genus have been collected. Some small (10-25 mm in length), elliptical objects of uncertain paleontologic classification were found in Eocene, Oligocene, and lower Miocene. End_of_Article - Last_Page 2212------------

Light scattering by anisotropic elliptical objects

AbstractThe small‐angle light scattering in Hv and Vv modes is calculated for elliptical disks with the use of an elliptical coordinate system. The method is general for all degrees of ellipticity, from a circular disk to rodlike extensions, and permits the definition of any desired dipole orientations. The solution is obtained by computer‐assisted numeric integration. Two models are considered, an “elliptical” one an “affine deformation” one differing in the orientation of scattering dipoles. The calculated patterns show a significant dependence of the distribution of the scattered intensity on the size and the elliptical axial ratio in both models, permitting the determination of both the size and the degree of ellipticity of the disk from its patterns. In addition, the differences between the calculated results for the two models are sufficiently large to permit the selection of the experimentally appropriate model, at least within the range of moderate degree of ellipticity.

Determination of Finite Strain and Initial Shape from Deformed Elliptical Objects

To determine finite strain from material such as deformed pebbles and oolites, it is essential to consider the initial shape and orientation of the particles. The ellipsoidal objects are studied as ellipses on plane sections. The distribution of axial ratios and orientations of undeformed particles is used as the basis for a classification of initial distributions. Simple and general equations relating the initial and deformed shape and orientation of the particles are derived by considering the matrix algebra. A shape factor grid enables accurate calculation of the magnitude and orientation of the strain ellipse. A variety of important initial distributions can be determined and used. The sudden onset of continuous cleavage and oriented textures observed in some orogenic areas is partially due to the effect of the initial shape of crystals, grains, and clastic particles.

A technique of finite strain analysis using elliptical particles

When initial elliptical objects are deformed homogeneously with their matrix they change both their ellipticity and axial orientation. Mathematical expressions are derived which relate the final elliptical ratio and orientation of particles to their initial form and the magnitude and orientation of the finite rotational or irrotational strain. Curves are constructed from these expressions and examples given of their use to determine finite strain and initial shape of deformed aggregates.