Invariant Shape Research Articles

Affine Curvature Scale Space with Affine Length Parametrisation

The maxima of Curvature Scale Space (CSS) image have been used to represent 2D shapes under affine transforms. The CSS image is expected to be in the MPEG-7 package of standards. Since the CSS image employs the arc length parametrisation which is not affine invariant, we expect some deviations in the maxima of the CSS image under general affine transforms. Affine length and affine curvature have already been introduced and used as alternatives to arc length and conventional curvature in affine transformed environments. The utility of using these parameters to enrich the CSS representation is addressed in this paper. We use arc length to parametrise the curve prior to computing its CSS image. The parametrisation has been proven to be invariant under affine transformation and has been used in many affine invariant shape recognition methods. Since the organisation of the CSS image is based on curvature zero crossings of the curve, in this paper, we also investigate the advantages and shortcomings of using affine curvature in computation of the CSS image. The enriched CSS representations are then used to find similar shapes from a very large prototype database, and also a small classified database, both consisting of original as well as affine transformed shapes. An improvement is observed over the conventional CSS image.

A sensitivity analysis of the Bongaarts-Feeney method for adjusting bias in observed period total fertility rates.

Our sensitivity analysis shows that the adjusted TFR'(t) using the formula of Bongaarts and Feeney (1998), which assumes an invariant shape for the fertility schedule, usually does not differ significantly from an adjusted TFR"(t) that allows the shape of the fertility schedule to change at a constant annual rate. Because annual changes in the shape of the fertility schedules often are approximately constant except in abnormal conditions, the Bongaarts-Feeney (B-F) method is generally robust for producing reasonable estimates of the adjusted TFR'(t). The adjusted TFR'(t) neither represents any real cohort experiences from the past nor forecasts any future trend. It merely provides an improved reading of the period fertility measure, which reduces the tempo distortion.

Hopfield neural networks for affine invariant matching

The affine transformation, which consists of rotation, translation, scaling, and shearing transformations, can be considered as an approximation to the perspective transformation. Therefore, it is very important to find an effective means for establishing point correspondences under affine transformation in many applications. In this paper, we consider the point correspondence problem as a subgraph matching problem and develop an energy formulation for affine invariant matching by a Hopfield type neural network. The fourth-order network is investigated first, then order reduction is done by incorporating the neighborhood information in the data. Thus we can use the second-order Hopfield network to perform subgraph isomorphism invariant to affine transformation, which can be applied to an affine invariant shape recognition problem. Experimental results show the effectiveness and efficiency of the proposed method.

Fabrication of Nanostructured Surfaces Using Self-Assembled Monolayers

We describe a new method for 3 dimensional nanostructuring on silicon surfaces using self-assembled monolayers. Partial multilayers were formed by repeated deposition of trichlorosilylheptadecanoic acid methyl ester (TSHEME) and subsequent reduction to yield a hydroxylic surface. These structures were afterwards oxidised using UV/ozone, yielding silicon oxide features. In this way both organic multilayered structures as well as ones comprised of silicon oxide have been produced with precise control of the height and invariant lateral shape of these structures. We have tried to apply samples prepared in this fashion to the calibration of AFM-scanners in vertical direction. Due to height artefacts caused by the tip-sample interaction a general calibration is not possible on the molecular scale. However, the structures produced can be used as model systems for the investigation of various sources of height artefacts and also for calibration purposes as long as samples with similar chemical and mechanical properties are to be investigated.

A fast and accurate method to register medical images using Wavelet Modulus Maxima

This paper presents a fast, accurate and automatic method to register images of rigid bodies. It uses wavelets to obtain control points. Wavelets are not shift invariant but the structures determined by the wavelet high pass image, the Modulus Maxima Image, provide the information necessary for a fast-rough convergence. These structures represent shapes from which we segment the invariant shapes for the images being registered. For example, the MRI and CT images of the brain can be considered as rigid bodies that do not undergo a change in shape over reasonable periods of time. By using a convex hull, we make the procedure insensitive to the internal changes in the object. Hence, even with the growth of tumors, the procedure registers brain images very accurately. The method uses the correlation coefficient to measure the similarity between images and to determine how well the images are registered. The method has been extensively tested with various types of images and in all cases the registration accuracy is very high. The correlation coefficient used to validate the registrations has deficiencies that occasionally required a visual inspection to terminate the algorithm after a point of marginal improvement.

Polymer–surfactant complexes: solubilization of polymeric globule by surfactants

The behavior of polymeric globule in a solution containing surfactants is analyzed within the framework of the Flory lattice theory. The amphiphilic structure of surfactant molecule is modeled as two adjacent lattice cells with opposite interaction parameters. Therefore, model surfactants prefer the globule–solvent interface rather than regions inside and outside the globule. It is found that at surfactant concentrations below some Critical Solubilization Concentration (CSC), surfactant molecules are adsorbed on the globular surface of almost invariable spherical shape. The coverage of the globular surface by surfactants takes place in rather broad region of concentrations. If the surfactant concentration exceeds the CSC, the globule sharply transforms to a coil. The transition is found to take place in a very narrow surfactant concentration region, i.e. polymer chain bypasses intermediate globular shapes and can be approximately regarded as a chain of spherical blobs. In this case, the solvent quality and the surfactant concentration are two main factors inducing globule–coil transition. Reducing of the transition point with increasing surfactant concentration as well as the steepness of the transition is in qualitative agreement with the experimental data.

Affine invariant deformation curves a tool for shape characterization

In this paper we describe a combination of two well-known concepts: morphological deformation curves and affine invariant evolution processes. A morphological deformation curve is a shape descriptor obtained by continuously deforming the shape and in the meantime measuring some geometric parameter. This measurement as a function of the deformation provides a characterization of the shape that can be used for shape classification and recognition. In classical morphology the deformation process is taken to be a morphological operator (erosions, dilations and combinations thereof). Most of the morphological operators are only invariant under the Euclidean symmetry group (translation and rotation) and (relative) invariant under isotropic rescalings. Only recently in the context of computer vision, an affine invariant shape deformation process has been identified. When we measure the area of the object while it is deformed in this way we obtain a function that provides a characterization of a shape in an affine invariant way. In this paper an affine invariant deformation curve is introduced and its application for shape characterization is illustrated.

Statistical models for granular materials

Granular materials, like sand or powders, have many peculiar properties, most of them not yet fully understood. These are due to the special physical properties of granular media like dissipation, friction, dilatancy and the possibilities of having grains of many different sizes and shapes. In the last 10 years much progress has been achieved in the basic understanding of granular media due to the use of tools from Statistical Physics, including disordered systems, critical phenomena, instabilities and chaos. We will present in this talk two examples: Models for compaction which very much resemble those of structural glasses and predict the logarithmic time dependence experimentally measured in Chicago. Models that describe the evolution of free surfaces giving the angle of repose, invariant shapes with logarithmic tails and stratification patterns in agreement with observations.

QCD studies in the Breit frame at HERA

Fragmentation functions, invariant energy spectra and differential events shape spectra in Deep Inelastic Scattering are studied in the current region of the Breit frame. The data, taken by the H1 and ZEUS experiments at HERA, are compared with results from e + e − collisions and fragmentation universality is tested.

Self-Similar Spatiotemporal Structure of Intermaterial Boundaries in Chaotic Flows

The evolution of macroscopic material closed filaments in a time-periodic chaotic 2D flow is simulated for cases with large, small, and very small islands of regular motion using an algorithm that preserves spatial continuity. The length of the stretched filament increases much faster than predicted by the Liapunov exponent. In chaotic regions, the filament asymptotically evolves into a self-similar structure with permanent spatial nonuniformities in density. Filament densities and local length scales corresponding to different times are described by families of frequency distributions with invariant shape that can be collapsed onto a single curve by means of a simple scaling.

Multiscale Contour Segmentation and Approximation: An Algorithm Based on the Geometry of Regular Inscribed Polygons

This paper describes an algorithm to segment and approximate a contour into circular arcs and straight line segments. This algorithm finds an adequate set of circular arcs and straight line segments to describe the contour shape. The resulting description respects a large number of shape representation criteria much better than any previous method. These criteria support efficient, general-purpose, unique, shape preserving, local, robust, stable, invariant, multiscale, coherent, hierarchical, and multipart shape representations. The adequacy of the algorithm is a direct consequence of the formal computational theory on which it is based. The resulting description may depend on sensor accuracy, sampling step size, minimum and maximum contour curvatures, and scale. The algorithm is thus parameterized using a small number of thresholds related to these data set and task properties. Predictions regarding the ground-truth performance of the algorithm can be made given some knowledge of the above properties. Experimental comparative results are presented for contour data of varying complexity.

Towards a unitary formulation for invariant image description: application to image coding

Here, we introduce a joint topology and harmonic analysis formulation for the extraction of global shape descriptors which are invariant under a given group of geometrical transformations. The topology approach allows the rigorous definition of the notions of shape, shape space, the invariant features space, and a metric between shapes. Therefore a new definition of completeness is given. A stability criterion is defined mathematically. Using harmonic analysis, a unitary operator that is able to separate the shape information and the geometric transformation, allows us to extract a relevant invariant shape descriptors under a given group of transformations. It also gives a robust method for the evaluation of the global object motion. In the closed curves, some three-dimensional surfaces and planar gray level image cases, such an operator becomes the Fourier transform on a given group. Therefore, under some assumptions, a complete convergent set of invariant features exists and can be constructed. We derived from this a shape metric. Recent developments in image coding domain for moving pictures offer new perspectives to the application of the image invariant representations of regions and contours. Therefore, we intend to illustrate the importance of our approach in image coding and indexing applications.

A theoretical investigation of reference frames for the planning of speech movements.

Does the speech motor control system use invariant vocal tract shape targets when producing vowels and semivowels? A 4-part theoretical treatment favoring models whose only invariant targets are regions in auditory perceptual space over models that posit invariant constriction targets is presented. Auditory target regions are hypothesized to arise during development as an emergent property of neural map formation in the auditory system. Furthermore, speech movements are planned as trajectories in auditory perceptual space. These trajectories are then mapped into articulator movements through a neural mapping that allows motor equivalent variability in constriction locations and degrees when needed. These hypotheses are illustrated using computer simulations of the DIVA model of speech acquisition and production. Finally, several difficult challenges to proponents of constriction theories based on this theoretical treatment are posed.

Affine Invariant Texture Segmentation and Shape from Texture by Variational Methods

We address the problem of texture segmentation by using a novel affine invariant model. The introduction of affine invariance as a requirement for texture analysis goes beyond what is known of the human performance and also beyond the psychophysical theories. We propose to compute texture features using affine invariant intrinsic neighborhoods and affine invariant intrinsic orientation matrices. We discuss several possibilities for the definition of the channels and give comparative experimental results where an affine invariant Mumford-Shah type energy functional is used to compute the multichannel affine invariant segmentation. We prove that the method is able to retrieve faithfully the texture regions and to recover the shape from texture information in images where several textures are present. The numerical algorithm is multiscale.

A genetic algorithm for affine invariant object shape recognition

In this paper, a novel technique for matching images of object shapes which have been subject to affine transformation caused by variations in the camera position is reported. The method is based on the genetic algorithm, and is more efficient and reliable than conventional approaches that rely on corresponding dominant point pairs to determine the best alignment between object boundaries. Experimental results are presented to demonstrate the feasibility of the approach and its capability in identifying object shapes that had been distorted with heavy noise contamination

The Redshift Evolution of the LY alpha Forest

We have analyzed the properties of low column density Lyman alpha forest clouds (12.8 < log N_HI < 16.0) toward 5 QSOs at different redshifts, using high signal-to-noise data from the HIRES spectrograph on the Keck I 10-m telescope. The results are used to study the evolution of these clouds in the interval 2.1 < z < 3.5. We find: 1) The differential density distribution of forest clouds, f(N_HI), fits an empirical power law with a typical slope of -1.35 to -1.55 for clouds with N_HI < 10^14.3 cm^{-2} and changes extremely slowly with redshift over this N_HI range. 2) The deviation of f(N_HI) from this power-law distribution at higher column densities depends on redshift: for higher redshifts, the departure from a power-law distribution occurs at higher N_HI, which means that higher column density clouds rapidly disappear with decreasing redshift. This may be consistent with an invariant shape for the different column density distributions, but with a shift to a lower column density normalization for systems at different redshifts driven by the overall expansion of the universe. 3) The line widths of the forest clouds increase as redshift decreases. 4) The correlation strength of the forest clouds seems to increase as redshift decreases. We discuss these results in terms of the evolution of the IGM comparing the results to models and analytic descriptions of the evolution of structure in the gas.

Stationary configurations of a vortex filament in background flows

We investigate the three-dimensional configurations of a thin vortex filament embedded in background flows of an inviscid incompressible fluid, based on the localized induction approximation. An analogy is found between stationary configurations of a vortex filament in a steady flow and trajectories of a charged particle in a steady magnetic field. An analogy with trajectories of a sound ray in a steady low-Mach number flow is drawn as well. These analogies allow us to use the Lagrangian and Hamiltonian formalism of classical mechanics for calculating fully nonlinear forms of a vortex filament. The equations for the filament curve are integrable when the background flow has two spatial symmetries. Only integrable cases are explored in some detail. To illustrate the advantages of use of the analogy, we re-examine the invariant shapes of a vortex filament moving through a still fluid obtained by Kida (1981). Analogies of the Kida class with the motions of a heavy symmetrical top and a charged spherical pendulum in the field of a magnetic monopole are discussed. As next examples, we consider a point source (sink) and a line source (sink) flows. A filament takes the form of a non-uniform helix wound around a cone. The last example is a circular jet of a parabolic velocity profile. We construct a closed filament with azimuthal waves, lying on a plane orthogonal to the flow direction. Among them, an elliptic vortex ring is found to be distinct from higher waves.

Crack nucleation on an elastic polycrystal surface in a corrosive environment: Low dimensional dynamical models

This paper analyses a process of crack nucleation on the surface of a ceramic subject to a stress parallel to the surface, in an environment where the ceramic evaporates slowly and deforms elastically. We formulate dynamical models of one or two degrees of freedom by combining an existing elasticity solution and a variational approach. Both grain boundary tension and elastic stress concentration cause the surface to groove along its intersections with the grain boundaries. Two behaviors are identified. If the applied stress is small, the grooves approach an invariant shape, and the ceramic erodes by gross mass loss. If the applied stress is large, the grooves sharpen into crack fronts, and the ceramic breaks by decohesion. The models relate crack nucleation threshold and time to the applied stress, surface and grain boundary tensions, chemical free energy, grain size, and kinetic parameters. Surface self-diffusion is also included in the analysis.

Cone angle radial profiles

Various methodologies have been reported in the literature for the evaluation of the size of a ligand or organic functional group, with the Tolman cone angle methodology proving to be most popular in inorganic chemistry. A procedure to extend the Tolman concept to the measurement of a ligand size, as a profile over a radial distance from the metal (apex), has been developed. The representation of the Tolman cone angle as a function of distance from the metal is termed a cone angle radial profile (CARP). CARPs for PH 3 , PMe 3 , PEt 3 and PPh 3 have been determined and are described. CARPs for molecules containing the constrained phosphite ligands P(OCH 2 ) 3 CMe, as determined in actual structures (Cambridge Structural Database), have also been evaluated and indicate the invariant shape of the ligand in the different metal environments. The influence of the choice of the H atom van der Waals radius in steric measurements is also discussed.

First passage times for threshold growth dynamics on ${\bf Z}\sp 2$

In the threshold growth model on an integer lattice, the occupied set grows according to a simple local rule: a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood. In this paper, we analyze the behavior of two-dimensional threshold growth dynamics started from a sparse Bernoulli density of occupied sites. We explain how nucleation of rare centers, invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case. We also briefly address scaling laws for the critical case.