Showing papers on "Symmetry (geometry) published in 1994"

Euclidean symmetry and the dynamics of rotating spiral waves.

Abstract: It is shown that the dynamics of spiral waves in excitable media are organized around a codimension-two point where a Hopf bifurcation from rotating waves interacts with Euclidean symmetry. A simple ordinary-differential-equation model of this bifurcation generates dynamics like the ``meandering'' of spiral waves.

Symmetries, isometries and length spectra of closed hyperbolic three-manifolds

Abstract: Previously known algorithms to compute the symmetry group of a cusped hyperbolic three-manifold and to test whether two cusped hyperbolic three-manifolds are isometric do not apply directly to closed manifolds. But by drilling out geodesics from closed manifolds one may compute their symmetry groups and test for isometries using the cusped manifold techniques. To do so, one must know precisely how many geodesics of a given length the closed manifold has. Here we prove the propositions needed to rigorously compute a length spectrum, with multiplicities. We also tabulate the symmetry groups of the smallest known closed hyperbolic three-manifolds.

Parallel computation of symmetry but not repetition within single visual shapes

Abstract: Visual symmetry is highly salient to human observers. Indeed, Mach (1885) observed that symmetry is more salient than repetition in the contours of a shape. We find that symmetry within a single shape can be detected in parallel, whereas repetition is apparently detected by a serial process. Subjects were required to judge whether a pseudorandom block-shape was symmetrical (Experiment I) or had repeated contours (Experiment 2). When present, these relations arose around either vertical or horizontal axes of elongation, which were unpredictably intermingled. In both cases, symmetry judgements were scarcely affected by the number of discontinuities along the contours to be compared, whereas repetition judgements showed substantial delays when there were more discontinuities. These results are consistent with parallel encoding of a part-description for shapes, in accordance with Hoffman and Richards' (1984) curvature-minima rule.

Detection of bilateral symmetry using spatial filters

Abstract: When bilaterally symmetric images are spatially filtered and thresholded, a subset of the resultant 'blobs' cluster around the axis of symmetry. Consequently, a quantitative measure of blob alignment can be used to code the degree of symmetry and to locate the axis of symmetry. Four alternative models were tested to examine which components of this scheme might be involved in human detection of symmetry. Two used a blob-alignment measure, operating on the output of either isotropic or oriented filters. The other two used similar filtering schemes, but measured symmetry by calculating the correlation of one half of the pattern with a reflection of the other. Simulations compared the effect of spatial jitter, proportion of matched to unmatched dots and width or location of embedded symmetrical regions, on models' detection of symmetry. Only the performance of the oriented filter + blob-alignment model was consistent with human performance in all conditions. It is concluded that the degree of feature co-alignment in the output of oriented filters is the cue used by human vision to perform these tasks. The broader computational role that feature alignment detection could play in early vision is discussed, particularly for object detection and image segmentation. In this framework, symmetry is a consequence of a more general-purpose grouping scheme.

Symmetries of islamic geometrical patterns

Abstract: Islamic Patterns and Their Structures In Praise of Pattern Symmetry and Unity The Gateway from Islamic Patterns to Invariance and Groups Classification, Recognition and Construction of the 17 Types of Two-Dimensional Periodic Patterns Islamic Patterns and Their Symmetries.

Symmetries of plane partitions and the permanent-determinant method

Abstract: In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives formulas for the number of plane partitions in each of ten symmetry classes. This paper together with results by Andrews [J. Combin. Theory Ser. A 66 (1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the project of proving all ten formulas. We enumerate cyclically symmetric, self-complementary plane partitions. We first convert plane partitions to tilings of a hexagon in the plane by rhombuses, or equivalently to matchings in a certain planar graph. We can then use the permanent-determinant method or a variant, the Hafnian-Pfaffian method, to obtain the answer as the determinant or Pfaffian of a matrix in each of the ten cases. We row-reduce the resulting matrix in the case under consideration to prove the formula. A similar row-reduction process can be carried out in many of the other cases, and we analyze three other symmetry classes of plane partitions for comparison.

Symmetric 3D objects are an easy case for 2D object recognition

Abstract: According to the 1.5-views theorem (Poggio, Technical Report #9005-03, IRST, Povo, 1990; Ullman and Basri, IEEE Trans. PAMI 13, 992-1006, 1991) recognition of a specific 3D object (defined in terms of pointwise features) from a novel 2D view can be achieved from at least two 2D model views (for each object, for orthographic projection). This note considers how recognition can be achieved from a single 2D model view by exploiting prior knowledge of an object's symmetry. It is proved that, for any bilaterally symmetric 3D object, one non-accidental 2D model view is sufficient for recognition since it can be used to generate additional 'virtual' views. It is also proved that, for bilaterally symmetric objects, the correspondence of four points between two views determines the correspondence of all other points. Symmetries of higher order allow the recovery of Euclidean structure from a single 2D view.1

Method and apparatus for locating an acquisition target in two-dimensional images by detecting symmetry in two different directions

Abstract: A system and method are disclosed for locating an acquisition target consisting of a plurality of concentric rings of alternating levels of reflectivity (bull's-eye pattern) in two-dimensional images such as optically encoded symbologies. Such targets may be found even if they vary in size and tilt angle with respect to an imaging camera. Symmetry characteristics of the targets are used to locate the targets independent of the threshold level selected for determining transitions between rings of different reflectivity.

Symmetry transformations in Batalin-Vilkovisky formalism

Abstract: Let us suppose that the functionalS on an odd symplectic manifold satisfies the quantum master equation Δ ρ es = 0. We prove that in some sense every quantum observable (i.e. every functionH obeying Δ p (Hes) = 0) determines a symmetry of the theory with the action functionalS.

Computation by symmetry operations in a structured model of the brain: Recognition of rotational invariance and time reversal.

Abstract: Symmetries have long been recognized as a vital component of physical and biological systems. What we propose here is that symmetry operations are an important feature of higher brain function and result from the spatial and temporal modularity of the cortex. These symmetry operations arise naturally in the trion model of the cortex. The trion model is a highly structured mathematical realization of the Mountcastle organizational principle [Mountcastle, in The Mindful Brain (MIT, Cambridge, 1978)] in which the cortical column is the basic neural network of the cortex and is comprised of subunit minicolumns, which are idealized as trions with three levels of firing. A columnar network of a small number of trions has a large repertoire of quasistable, periodic spatial-temporal firing magic patterns (MP's), which can be excited. The MP's are related by specific symmetries: Spatial rotation, parity, ``spin'' reversal, and time reversal as well as other ``global'' symmetry operations in this abstract internal language of the brain. These MP's can be readily enhanced (as well as inherent categories of MP's) by only a small change in connection strengths via a Hebb learning rule. Learning introduces small breaking of the symmetries in the connectivities which enables a symmetry in the patterns to be recognized in the Monte Carlo evolution of the MP's. Examples of the recognition of rotational invariance and of a time-reversed pattern are presented. We propose the possibility of building a logic device from the hardware implementation of a higher level architecture of trion cortical columns.

Continuous symmetry: a model for human figural perception

Abstract: Symmetry is usually viewed as a discrete feature: an object is either symmetric or non-symmetric. In this presentation, symmetry is treated as a continuous feature and a continuous measure of symmetry (the Symmetry Distance) is defined. This measure can be easily evaluated for any shape or pattern in any dimension. A preliminary study presented here shows that the Symmetry Distance is commensurate with human perceptual experience. Good correlation is found between the continuous symmetry values and the perceived goodness of figures.

Symmetry and isomeric structures of giant fullerenes and polyhedral clusters

Abstract: In order to illustrate the extension of cage cluster of given symmetry to a possible larger giant cluster of the same symmetry type, we have designed methods for drawing Schlegel (or quasi-Schlegel) diagrams that show clearly all (or almost all) the faces, vertices and edges of a three-dimensional cluster in a two-dimensional plane. The symmetries and structural details were used to select unit cells which serve as “abbreviations” for huge (fullerene) clusters, and these unit cells were then used as basis to derive mathematical principles for extending the structures up to infinitely large clusters. Possible isomers and various fullerene clusters (up to C150 of D5h symmetry) of different point-group symmetries are presented (the point groups involved for different sizes are Ih, D6d, D5d, D3d, D6h, D5h, D3h, D3, Td, C3, C3v, and D6). Simple illustrations are given to show how such clearly visible detailed structures can be used to derive the possible magic numbers of (van der Waals) clusters and to formulate the mechanism and reaction coordinates of isomeric transformations. For mixed clusters with different sets of atoms (molecules), the common (minimum) subgroup symmetry is illustrated, which can be used to determine the total structures. An example of A8B12 (e.g. Ti8C+12) is given to show the possibility of Th or D3d symmetry.

Mirror symmetry and parallelism: two opposite rules for the identity transform in space perception and their unified treatment by the Great Circle Model.

Abstract: Two opposite rules control the contributions of individual lines to the perceptual processing of two different spatial dimensions of egocentric localization and orientation. For lines restricted to the frontal plane, a tilted line on one side of the median plane induces a rotation of the orientation visually perceived as vertical (VPV) identical to that induced by the same tilt on the other side of the median plane, but the influences exerted on the elevation of visually perceived eye level (VPEL) are mirror symmetric. The rule for VPV fits our intuitions; the rule for VPEL does not. However, the reverse peculiarity holds when the inducing lines are rotated within sagittal planes (pitched): Two parallel, pitched-from-vertical lines on opposite sides of the median plane generate identical effects on VPEL but mirror symmetric effects on VPV. These counterintuitive symmetry reversals are reconciled by the Great Circle Model of spatial orientation (GCM), in which line orientations are represented by the great circle coordinates of their images on a sphere centered at the nodal point of the eye via central projection.

Quantification of local symmetry: application to texture discrimination

Abstract: Symmetry is one of the most prominent cues in visual perception as well as in computer vision. We have recently presented a Generalized Symmetry Transform that receives as input an edge map, and outputs a symmetry map, where every point marks the intensity and orientation of the local generalized symmetry. In the context of computer vision, this map emphasizes points of high symmetry, which, in turn, are used to detect regions of interest for active vision systems. Many psychophysical experiments in texture discrimination use images that consist of various micro-patterns. Since the Generalized Symmetry Transform captures local spatial relations between image edges, it has been used here to predict human performance in discrimination tasks. Applying the transform to micro-patterns in some well-studied quantitative experiments of human texture discrimination, it is shown that symmetry, as characterized by the present computational scheme, can account for most of them.

Explicit expressions of the Barnett-Lothe tensors for anisotropic materials

Abstract: The three Barnett-Lothe tensorsS, H, andL appear frequently in the real form solutions of two-dimensional anisotropic elasticity problems. Explicit expressions for the components of these tensors are presented for general anisotropic materials. The special cases of monoclinic materials with the plane of material symmetry at x3=0, x2=0, and x1=0 are then deduced. For monoclinic materials with the symmetry plane at x2=0 or x1=0, the locations of image singularities for the Green's functions for a half-space have a special geometry.

Documentation note query-document symmetry and dual models

Abstract: The idea that there is some natural symmetry between queries and documents is explored. If symmetry can be assumed, then it leads to a conception of ‘dual’ models in information retrieval (given a model, we can construct a dual model in which the roles of documents and queries are reversed). But symmetry breaks down in various ways, which may invalidate this construction. If we can construct a dual, it is not obvious that it can be combined with the original.