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We show photonic responses of selected two-dimensional photonic crystals that are based on a combination of geometrical design and spatial self-similarity. Structures based on multi-layer circles, squares, and triangles are studied, finding a range of local band gaps that can be varied with the self-similarity of the photonic crystal patterns. Selected photonic modes are shown that couple to different geometrical features of the photonic crystal. The role of boundary sharpness between the two dielectric components is emphasized.

Acta mechanica (2021). doi:10.1007/s00707-021-03056-8, Published by Springer, Wien

This short paper highlights the importance of William Lawrence Bragg's first publication as the starting point of the use of X-rays in determining crystal structures. As such it marks the boundary between the traditional and modern eras of crystallography.

In 1968 Fattorini published in SIAM Journal on Control his well-known paper with the title Boundary Control Systems. In this paper he sets up the abstract theory for systems described by partial differential equations with a control at the boundary. The theory as presented is complete, meaning that it could be copied into textbooks and articles without major alternations. In his paper and in these textbooks one finds examples illustrating the use of this abstract concept. However, as we will show, these examples are much older, and are more or less standard within the theory of partial differential equations with an inhomogeneous boundary condition.

In Part I of this paper, we defined a focusing wave field as the time reversal of an observed point‐source response. We showed that emitting a time‐reversed field from a closed boundary yields a focal spot that acts as an isotropic virtual source. However, when emitting the field from an open boundary, the virtual source is highly directional and significant artefacts occur related to multiple scattering. The aim of this paper is to discuss a focusing wave field, which, when emitted into the medium from an open boundary, yields an isotropic virtual source and does not give rise to artefacts. We start the discussion from a horizontally layered medium and introduce the single‐sided focusing wave field in an intuitive way as an inverse filter. Next, we discuss single‐sided focusing in two‐dimensional and three‐dimensional inhomogeneous media and support the discussion with mathematical derivations. The focusing functions needed for single‐sided focusing can be retrieved from the single‐sided reflection response and an estimate of the direct arrivals between the focal point and the accessible boundary. The focal spot, obtained with this single‐sided data‐driven focusing method, acts as an isotropic virtual source, similar to that obtained by emitting a time‐reversed point‐source response from a closed boundary.

A common drawback of paper-based separation devices is their poor detection limit. In this study, we combined field-amplified sample stacking with moving reaction boundary electrophoresis on a paper chip with six array channels for the parallel separation and concentration of multiple samples. With a new hyphenated technique, the brown I2 from the Fe3+ /I- oxidation-reduction reaction emerged near the boundary between the dilute ethylene diamine tetraacetic acid and potassium iodide and highly concentrated KCl solutions. For the separation and concentration of three components, Cr3+ , Cu2+ , and Fe3+ , the Fe3+ detection limit was improved at least 266-fold by comparing the hyphenated technique with moving reaction boundary electrophoresis. The detection limit of Fe3+ was found to be as low as 0.34 ng (20 μM) on the paper chip. We also demonstrated the analysis of a real sample of four metal ions, with detection limits as follows: 0.16 μg Cr3+ , 1.5 μg Ni2+ , 0.64 μg Cu2+ , and 1.5 μg Co2+ . The synergy of field-amplified sample stacking and moving reaction boundary electrophoresis in the micron paper-based array channels dramatically improved the detection limit and throughput of paper-based electrophoresis.

We derive a one-dimensional model that describes pressing of water saturated paper in the press-section of the paper machine. The model involves two nonlinear diffusion equations which are coupled across an internal boundary. Existence and uniqueness as a number of qualitative properties are demonstrated. Further, computational results for a concrete case are discussed.

In (2012), Leonetti and Siepe [10] considered solutions to boundary value problems of some anisotropic elliptic equations of the type { ∑ i = 1 n D i ( a i ( x , D u ( x ) ) ) = 0 , x ∈ Ω , u ( x ) = θ ( x ) , x ∈ ∂ Ω . Under some suitable conditions, they obtained an integrability result, which shows that higher integrability of the boundary datum θ forces solutions u to have higher integrability as well. In the present paper, we consider K ψ , θ ( p i ) -obstacle problems of the nonhomogeneous anisotropic elliptic equations ∑ i = 1 n D i ( a i ( x , D u ( x ) ) ) = ∑ i = 1 n D i f i ( x ) under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe.

A numerical model based on the Fokker-Planck equation for studying the orientation of paper fibers in a suspension is presented. The model is implemented in a finite volume framework. The rotational operators are discretized on a triangulated half sphere and the spatial operators are discretized on a Cartesian octree grid, where internal walls are handled with immersed boundary methods. Transient solutions of the two dimensional fiber orientation distribution are computed in the entire fluid domain. This is in contrast to earlier approaches that either only compute along streamlines or one dimensional projected orientations. The fibers are assumed to have negligible inertia and thus to follow the fluid perfectly and to be evenly distributed. A head-box geometry relevant for paper forming is used to exemplify the usefulness of the method. The influence of translational and rotational diffusion is included in the presented method.

In this commentary on Steven H. Cooper’s paper, the author acknowledges that the term boundary violation presents a vexing problem when we try to integrate it with the traditional use of the term boundary in psychoanalytic thought. He suggests that the two discourses represent two separate but somewhat related levels of discourse and that overlap occurs in some areas but not in others. While the use of “boundary violations” to describe ethical misconduct presents problems, the author suggests that it may be useful for some analysts who have experienced a collapse of analytic space and can no longer “play” in a symbolic realm. He also suggests that we may be stuck with the term because of the unconscious function it serves for the analytic community.

Freehand sketching on a paper is commonly found to be used as it is a natural and intuitive communication channel for portraying ideas. Additional strokes are often drawn over the existing ones to make the sketch clearer. For its usefulness in subsequent steps, a sketch is transformed to be a 3D CAD model that is typically performed manually by a designer due to a lack of technology support. Geometrical reconstruction from an offline scanning of a sketch has been a challenge because enriched information obtained during sketching is reduced to a batch of data points. This paper presents an algorithm for identifying a single-line drawing from a paper-based overtraced sketch as it is the first important step towards geometrical reconstruction. Key activities in this approach are thick-line sketch creation, contour boundary extraction, segments identification, and line drawing creation. The algorithm has been successfully implemented on LabVIEW and tested with several samples.

In this paper, the two kinds of solutions to remove a boundary image sticking of an ac plasma display panel are introduced. One solution is to completely recover the boundary image sticking cells by means of full white aging process. It was observed that after the 100-hour full-white aging process of the PDP cells with boundary image sticking, the MgO surface morphology in the boundary region with image sticking region was similar to that in the region with no image sticking. The other solution is to prohibit inherently the production of the boundary image sticking by sealing the PDP panel under a vacuum condition. In the panel fabricated by the vacuum sealing method, no boundary image sticking phenomenon was observed.

With the rapid development of machine learning and document analysis techniques, the research on the character mosaic and paper scrap recovery based on simulated annealing algorithm is urgent needed. In this paper, we approach is proposed based on pieces of text line features, form line semi-auto stitching method, mainly introduced the fragments of text line direction, form line direction acquisition algorithm, and the debris semi-auto stitching algorithm based on boundary intersection distance, pieces of the algorithm does not depend on the geometric features, implementation is simple and reliability is better. Also analyses the algorithm of computing workload, analysis shows that for the majority of less than 200 of the total number of pieces of actual scraps of paper, joining together the computing workload within the scope of the permit. The experiment simulation proves the effectiveness and feasibility of our proposed method.

In this study we analyze the importance of new magnetostratigraphic data on the Nordvik section for solving the problem of detailed Tethyan–Boreal correlation around the Jurassic–Cretaceous boundary with a special emphasis on the aspects of interpretation of the paleomagnetic data in magnetostratigraphic studies and the need for the integrated (paleontological and paleomagnetic) approach to recognition of the base of the Berriasian.

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\Omega$, following an approach introduced by Filonov for this type of problems., Comment: 23 pages, added Remark 5.4

The ink diffusion on Xuan paper is essentially a particle diffusion process, and the diffusion result is mainly determined by paper structure and ink attributes. This paper introduces weighting fiber structure to model the Xuan paper, which regards the paper fiber as the hindrance to ink diffusion. Based on the paper model, we present a novel simulation method of ink diffusion using the diffusion equation with variable coefficient. The diffusion coefficient consists of several factors including the fiber’s weights and the ink quantity in the current diffusion location. To efficiently solve the ink diffusion equation, we also propose a new implicit difference method with high accuracy and linear time complexity. Compared with the previous similar methods, our method is able to describe the spontaneous gray evolution and generate a more natural diffusion boundary. The experimental results demonstrate that our approach can realistically simulate different diffusion effects on different kinds of Xuan paper.

This paper discusses a direct method for numerical computation of the local quadratic approximations of ( n − 1)-dimensional stable manifolds for the unstable equilibria on the stability boundary in large systems. The algorithm proposed here does not require time-consuming normal form or symbolic computations. The numerical procedure is illustrated on transient stability problems in simple power system models. © 1997 Elsevier Science Ltd.

A promising approach for treating tissue or organ failure involves the use of bioartificial tissue substitutes grown in scaffolds with appropriate structure and shape. Currently, however, the engineering of tissue substitutes is a long and costly process based exclusively on experimentation. Predictive computer models can greatly reduce the development costs of tissue-engineered therapies by enabling scientists to rapidly evaluate the effect of system parameters on the growth rates and quality of regenerated tissues. We report here the parallel implementation of a three-dimensional model that employs cellular automata to describe the dynamic behavior of a population of mammalian cells that migrate, interact and proliferate to generate new tissues. The simulator uses MPI for interprocessor communication and is suitable for distributed memory multi-computers. Three parallel algorithms are developed to approximate the sequential algorithm describing this dynamic process of tissue growth. The parallel algorithms progressively relax the correctness requirements using different approaches to handle the cells that either move/ divide in the boundary layers of processors or cross sub-domain boundaries. Finally, a systematic study is carried out to evaluate the accuracy and performance of these algorithms.

This work presents a numerical framework designed to simulate the early paper forming process. This process is complex and includes strong fluid-structure interaction and complex geometries. The fluid flow solver IBOFlow, employs immersed boundary methods to simulate the flow around the fibers without the necessity of a boundary conforming grid. The fibers are approximated as slender beams with an elliptic cross section and modeled with the Euler-Bernoulli beam equation. A penalty based contact model is implemented. Finally, the potential of the framework is illustrated with an example.

The range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and e>0 , an approximate range query treats the range as a fuzzy object, meaning that points lying within distance ew of the boundary of Q either may or may not be counted. We show that in any fixed dimension d , a set of n points in R d can be preprocessed in O (n+ log n) time and O (n) space, such that approximate queries can be answered in O ( log n(1/e) d ) time. The only assumption we make about ranges is that the intersection of a range and a d -dimensional cube can be answered in constant time (depending on dimension). For convex ranges, we tighten this to O ( log n+(1/e) d−1 ) time. We also present a lower bound for approximate range searching based on partition trees of Ω( log n+(1/e) d−1 ) , which implies optimality for convex ranges (assuming fixed dimensions). Finally, we give empirical evidence showing that allowing small relative errors can significantly improve query execution times.

Stripes are important features for scrap paper images mosaics. The stripe feature extraction can directly affect the effect of mosaics. A method of stripe feature extraction from scrap paper images is proposed in this paper. In order to obtain the background binary image, the original scrap paper images are firstly carried out by close operation and binarization. Secondly, the single pixel edges are extracted by using edge detection operator. Finally, the crossing points of stripes and the boundary of the background are detected by using the Progressive Probabilistic Hough Transformation. The length, angle and direction features of stripes are also extracted during this process. The experimental results show that the method of stripe feature extraction proposed in this paper is reasonable. By using this method the stripe features can be extracted from scrap paper images quickly and correctly. It provides features for the subsequent image mosaics.

Let S be a surface obtained from a plane polygon by identifying infinitely many pairs of segments along its boundary. A condition is given under which the complex structure in the interior of the polygon extends uniquely across the quotient of its boundary to make S into a closed Riemann surface. When this condition holds, a modulus of continuity is obtained for a uniformizing map on S., 36 pages, 11 figures

We developed a mathematical model to simulate the bending and creasing behavior of paper-like sheets. This model permits interactive deformation of a surface remaining isometric to a plane rectangular sheet. It exhibits some important characteristics of paper behavior, such as the appearance of creases for some shapes of the boundary. Accurately simulating the detailed behavior of paper creases would involve several complex tasks, like employing mechanical shell modeling with possible boundary layer phenomena and studying the dynamics of buckling, or considering paper's plasticity. We do not discuss these difficult problems, instead simply analyzing the creasing phenomenon as a transition between two kinds of applicable surfaces. >

In this discussion paper, the original authors reply to a comment on their 2003 study of the electrostatic repulsion between particles in cement suspensions. The original authors departed from the usual practice of treating pore solution as ideal and symmetrical electrolytes. Instead, they treated them as noninteger symmetric electrolytes. This has allowed the authors to approach the real world situation more closely. The commentator had asked four questions: What potential should be used to calculate the electrostatic repulsion at the initial stages of hydration? Where should the solid-liquid interface boundary be placed? And how is this interface located? Also, how will the number density of cement hydration products effect the electrostatic repulsion? In this discussion, Flatt and Bowen address each of these questions, emphasizing practical approaches to the concerns raised.

Freehand sketch on a piece of paper is a quick rough drawing for portraying ideas, and commonly used during a conversation to elaborate an explanation. Overlapping lines provide additional information to make the rough sketch clearer, but the overtraced sketch should be simplified first to a single line drawing to be useful for downstream processes (e.g., input for generating 2D tool paths or for constructing 3D model). Presented in this paper is a new approach for identifying a single line drawing from a paper-based overtraced freehand sketch. Key activities in this approach are thick line sketch creation, boundary extraction, contour expansion and shrink, and line drawing creation. The current stage of its implementation is also reported in this paper.

This note concerns the paper "On the controllability of a coupled system of two Korteweg–de Vries equations" by Micu et al. [2]. They study a nonlinear coupled system of two Korteweg–de Vries equations and prove that the system is controllable by using four boundary controls. Here, we prove that in some cases it is possible to get the controllablity of the system by using only two controls. This can be done depending on both the spatial domain and the control time.

We made an error in the numerical example simulating the unconfined compression of a cylindrical cartilage sample, given in §5.2. In articular cartilage, the collagen fibres are close to lying on the transverse plane in the superficial zone and close to aligned with the axial direction in the deep zone. These two opposite situations correspond to model (A) and model (B), respectively, as described in §3.4 and figure 1. In both cases, the limit for the concentration parameter b approaching zero yields isotropy, which is attained in the middle zone. In our example, we imposed b to vary linearly with the coordinate running from the bone – cartilage interface to the articular surface: from 5 to 0 (model (B)) in the deep zone, and from 0 to 5 (model (A)) in the superficial zone. Also, in order to reproduce the experiment of Park et al. [1], we ‘cut’ a 0.5 mm layer of the deep zone, so that the concentration parameter b was varying from 2.85 to 0 in the deep zone. Because of an error in a sign in our code, the fibre distribution in the cylindrical sample of §5.2 is upside-down, i.e. model (A) was used in place of model (B), and vice versa. As a result of this error, things went such that a 0.5 mm layer of superficial (rather than deep) zone was ‘cut’. This implies that the contour plot shown in figure 4 is only roughly but not exactly upside-down, and the force response in figure 3 is not accurate because the zonal subdivision (deep-middle-superficial) is not as desired. Furthermore, the comparison of the radial displacement of the lateral boundary in figure 4 with the experiments of Fortin et al. [2] was not appropriate, as the radial displacements in their tests were relative to a cartilage sample attached to a layer of subchondral bone, which constrains radial displacement at the bone – cartilage interface. While these errors do not affect the validity of the model nor the robustness of the numerical implementation, we felt it was important to correct them.

In this paper, a post-processing technique for PTV measurement of flow around a deforming object is proposed. In general PTV, a position of velocity vector is defined as the midpoint of a tracer particle’s displacement obtained by two sequential images. This is because this velocity vector is close to the tangent vector of the path line. In this case, however, position of the object boundary at the middle time between two sequential PIV images must be interpolated. In this paper, a technique using three-frame images and Eulerian averaging in order to obtain velocity distribution on the regular grid points is proposed and evaluated. As the result, it is found that smooth and correct velocity distribution on the boundary of a bubble is obtained by the present approach.

It is very common in mathematics to construct surfaces by identifying the sides of a polygon together in pairs: For example, identifying opposite sides of a square yields a torus. In this article the construction is considered in the case where infinitely many pairs of segments around the boundary of the polygon are identified. The topological, metric, and complex structures of the resulting surfaces are discussed: In particular, a condition is given under which the surface has a global complex structure (i.e., is a Riemann surface). In this case, a modulus of continuity for a uniformizing map is given. The motivation for considering this construction comes from dynamical systems theory: If the modulus of continuity is uniform across a family of such constructions, each with an iteration defined on it, then it is possible to take limits in the family and hence to complete it. Such an application is briefly discussed.

In this paper we present the meshless local Petrov-Galerkin (MLPG) formulation for the generalized plane strain problem with specific emphasis on micromechanics of composite materials containing material discontinuities. The problem requires the introduction of an extra discrete degree of freedom, the out-of- plane uniform normal strain. The treatment of material discontinuity at the interface between the two phases of the composite is presented by means of direct imposition of interface boundary conditions. The MLPG method is used in the micromechanical model for predicting the elastic constants of the composite. To our knowledge, this is the first study in which the MLPG method is formulated for the so-called MLPG method based micromechanical analysis. Examples are presented to illustrate the effectiveness of the current method, and it is validated by comparing the results with available analytical and numerical solutions. The current method has the potential for use in micromechanics, especially for textile composites, where the meshing of the unit cell has been quite difficult. = part of local boundary over which no boundary conditions are specified N = total number of nodes n = number of points in the neighborhood of x for which W(x-x )>0 i ni = unit outward normal to the boundary P P (x) = vector of the complete monomial basis of order m (m=3) ri = radius of the domain of influence of the weight function (ri =4ci) ro = radius of the local domain rs = part of the local boundary located on the global boundary

The technique discussed in this article proposes to distinguish between unattended and stolen objects by combining shape and appearance similarity measures of foreground objects observed in consecutive frames of a video. Static objects are detected by examining trajectories and people are removed from consideration. Object shape boundary is first refined using active contours. Shape similarity is then defined by computing gradient magnitudes along the object boundaries and counting how many boundary pixels have values higher/lower than predefined thresholds. Appearance similarity is based on differences between histograms using the foreground mask on the current and background images. Probabilities are defined assuming the shape and appearance measures follow the Gaussian distribution (with trained parameters). Final measures of unattended/stolen objects are produced by averaging the probabilities and used to classify static-nonhuman objects. Experiments show that combining the three measures gives better results than using each of the measures alone.

In his paper [10], Sasaki studied the holomorphic slice S of the space of punctured torus groups determined by the trace equation xy = 2z. He found a simply connected domain E contained in S by using his system of inequalities which characterizes some quasifuchsian punctured torus groups (c.f. [9]). Moreover decomposing the boundary of E into 3 pieces ∂E = el U e2 U e3 he showed that e1 U e2 is contained in S and e3 (consisting of two points) is in the boundary ∂S. In this paper we consider the slice S itself more precisely.

The level set method provides a means of segmenting images. Fundamentally, the level set method for image segmentation is a search algorithm that determines where an evolving curve's boundary pixels - who are meant to encompass an image segment's perimeter - should be placed according to some criteria. A method has been devised that utilizes the Mumford-Shah functional as a means of establishing that criteria. It has been shown that this method for image segmentation has limitations that, while mentioned in the original research, were not quantified with examples. We present evidence of these limitations, discuss how they occurred and describe our preliminary attempts at overcoming them. We conclude by offering a direction where this research might lead. This work is part of an ongoing research project.

This paper considers the effect of noise on the quality of convergence of genetic algorithms (GAs). A population-sizing equation is derived to ensure that minimum signal-to-noise ratios are favorable to the discrimination of the best building blocks required to solve a problem of bounded deception. In five test problems of varying degrees of nonlinearity, nonuniform scaling, and nondeterminism, the sizing relation proves to be a conservative predictor of average correct convergence. These results suggest how the sizing equation may be viewed as a coarse delineation of a boundary between two distinct types of GA behavior. Besides discussing a number of extensions of this work, the paper discusses how these results may one day lead to rigorous proofs of convergence for recombinative GAs operating on problems of bounded deception.

We consider by means of the optimal fluctuation method the initial stage of the evolution of the noise-induced escape through various types of boundaries, especially concentrating on two types of the boundary - the wall and the boundary of the basin of attraction. We show in both cases that, if the damping is small enough, then the escape flux evolution possesses a remarkable property: it is it stairs-like i.e. intervals of a nearly constant flux alternate with intervals of a sharply increasing flux. This property is related to the successive increase of the number of turning points in the most probable escape path as time increases. Our results are relevant both for the absorbing and transparent boundaries. The major results of the theory are verified in computer simulations.

A reply to a discussion of an earlier paper published in this journal (Volume 36, Number 5, May 2006), written by H.S. Wong and N.R. Buenfeld and commenting on work by S. Diamond, is presented. The authors of the reply posit that the micro-scale inhomogeneities described by the discusser are not considered to be what is termed patch microstructure (PM). PM refers instead to broad areas, in the order of several hundred microns or more of dense, almost nonporous paste, and dark, highly porous paste, separated by sharp and distinct boundaries. These patches are irregularly shaped and occur randomly in 'bulk' and classical 'ITZ' paste. The most striking feature of PM is the sharp boundary between the porous and dense paste. It is emphasized that the characteristics of PM given in this reply and in the authors' paper are identical to those originally reported in Refs. [1] and [2]. It is reiterated that the appearance of PM is an artefact of sample preparation and does not reflect the true nature of the hydrated cement paste. The appearance of dense patches comes from areas that have been ground beyond the epoxy intrusion depth. In a BSE image, non-resin filled pores are not visible as backscatter electrons are generated from the base or side walls of the pores.

In this discussion paper, the authors comment on a study of the electrostatic repulsion between particles in cement suspensions by Flatt and Bowen (2003). The commentator applauds the original authors for departing from the usual practice of treating pore solution as ideal and symmetrical electrolytes. Instead, they have treated them as noninteger symmetric electrolytes. This has allowed the authors to approach the real world situation more closely. The commentator describes some misgivings about the methodology and asks the authors to clear up some of these concerns. The author concludes by asking four questions: what potential should be used to calculate the electrostatic repulsion at the initial stages of hydration? Where should the solid-liquid interface boundary be placed? And how is this interface located? Also, how will the number density of cement hydration products effect the electrostatic repulsion.

1. RF values of 12 amino acids were determined in phenol-water and benzyl alcohol-acetic acid-water using unidimensional and circular techniques of paper chromatography. 2. It has been observed that the square of circular RF values (considering frontal boundary of solute) equals the linear RF values, and this relation is valid for the amino acids as well as the sugars studied. 3. The relation is found to be independent of the distance traversed by the solvent in the case of both amino acids and sugars. 4. A theoretical support has been given to the experimental observations.

An integral formulation for heat conduction problems in non-homogeneous media has recently been proposed by Kassab and Divo [Engineering Analysis with Boundary Elements 1996;18:273]. The goal of this communication is to revisit and clarify two key features of the formulation of Kassab and Divo. First, the contention that the integral equation formulation proposed by them does not possess the desired boundary-only character is made and substantiated; it is shown in particular that Eq. (10) therein does not hold owing to the fact that a crucial requirement for the fundamental solution, Eqs. (5c) and (5d) therein, is actually not met. Second, the limiting process associated with a vanishing neighbourhood in connection with the particular kernel function used therein is revisited.

Earlier studies of the human vision system suggest that a complex object may be recognized from the qualitative features associated with its boundary. Since the human vision system uses qualitative features for object recognition, it is extremely robust to deformation caused by noise, obstruction, scale change, or optical distortion. We considered various qualitative descriptors for digital images. The descriptors developed in this research are based on the estimated curvature function from digital boundaries and can discriminate features such as straightness of a contour segment, perpendicularity of two contour segments, parallelness of two straight contour segments, parallelness of two curved contour segments in a global sense, and the direction of convergence if two straight contour segments are not parallel. We demonstrated that the qualitative descriptors developed in this paper can be applied for identification of elementary shapes, such as cylinders, bricks, cones, etc. We also discussed the recognition of more complex shapes after decomposing them into simpler components.