A nonparametric approach to 3D shape analysis from digital camera images — I

This article for the first time develops a nonparametric methodology for the analysis of projective shapes of configurations of landmarks on real 3D objects from their regular camera pictures. A fundamental result in computer vision, emulating the principle of human vision in space, claims that, generically, a finite 3D configuration of points can be retrieved from corresponding configurations in a pair of camera images, up to a projective transformation. Consequently, the projective shape of a 3D configuration can be retrieved from two of its planar views, and a projective shape analysis can be pursued from a sample of images. Projective shapes are here regarded as points on projective shape manifolds. Using large sample and nonparametric bootstrap methodology for extrinsic means on manifolds, one gives confidence regions and tests for the mean projective shape of a 3D configuration from its 2D camera images. Two examples are given: an example of testing for accuracy of a simple manufactured object using mean projective shape analysis, and a face identification example. Both examples are data driven based on landmark registration in digital images.     

The Interplay Between Fairness and Randomness in a Spatial Computer Game

This article describes how children use an expressive microworld to articulate ideas about how to make a game seem fair with the use of randomness. Our aim in this study is to disentangle different flavours of fairness and to find out how children used each flavour to make sense of potentially complex behaviour. In order to achieve this, a spatial computer game was designed to enable children to examine the consequences of their attempts to make the game fair. The study investigates how 23 children, aged between 5.5 and 8 years, engaged in constructing a crucial part of a mechanism for a fair spatial lottery machine (microworld). In particular, the children tried to construct a fair game given a situation in which the key elements happened randomly. The children could select objects, determine their properties, and arrange their spatial layout in the machine. The study is based on task-based interviewing of children who were interacting with the computer game. The study shows that children have various cognitive resources for constructing a random fair environment. The spatial arrangement, the visualisation and the manipulations in the lottery machine allow us gain a view into the children’s thinking of the two central concepts, fairness and randomness. The paper reports on two main strategies by which the children attempted to achieve a balance in the lottery machine. One involves arranging the balls symmetrically and the other randomly. We characterize the nature of the thinking in these two strategies: the first we see as deterministic and the latter as stochastic, exploiting the random collisions of the ball. In this article we trace how the children’s thinking moved between these two perspectives.

Entorno geométrico eficiente para el modelado de imágenes en 3D

The objective of this paper is exploring implementation of a realistic images reconstruction 3D using geometric algebra (GA). We illustrate the suitability of GA for representing structures and developing algorithms in computer graphics, especially for engineering applications as 3D images modeling. A first consequence is to propose an efficient framework model to be implemented in hardware programmable. The obtained results showed that using GA, the computations are less complex and shows as simple computations geometrical operations. The obtained model to hardware can be implemented as a next step in 3D image reconstruction. We also include the potential of GA for optimizations and highly efficient implementations.     

Image edge detection with the use of the Tikhonov regularization method

The study considers application of the Tikhonov regularization method for smoothing of 1D signals, noise suppression, and determination of objects’ edges in images. An analytic solution to the Euler equation is obtained in the 1D case for certain stabilizers. Application of the considered method for determination of objects’ edges in noisy images is analyzed.     

Using FSBT technique with Rough Set Theory for personal investment portfolio analysis

This study proposes a novel Forward Search and Backward Trace (FSBT) technique based on Rough Set Theory to improve data analysis and extend the scope of observations made from sample data to solve personal investment portfolio problems. Rough Set Theory mathematically classifies data into class sets. The class set with the most objects may generate one decision rule. The rules generated from RST are rough and fragmented, that are very difficult to interpret the information. An empirical case is used to generate more than 85 rules by the RST method in comparison with FSBT method which only generated 14 rules. This result can show our proposed method is better than traditional RST method based on class sets that contain the most objects. Much of human knowledge is described in natural language. It is a very important thing to convert information from computer databases into normal human language. Sample data taken from features with the same backgrounds are used to compile different portfolios that investment companies and investment advisors can employ to satisfy the investor’ needs. The method not only can provide decision-making rules, but also can offer alternative strategies for better data analysis. We believe that the FSBT technique can be fully applied in research on investment marketing.     

Geodesic Embeddings and Planar Graphs

Schnyder labelings are known to have close links to order dimension and drawings of planar graphs. It was observed by Ezra Miller that geodesic embeddings of planar graphs are another class of combinatorial or geometric objects closely linked to Schnyder labelings. We aim to contribute to a better understanding of the connections between these objects. In this article we prove • a characterization of 3-connected planar graphs as those graphs admitting rigid geodesic embeddings, • a bijection between Schnyder labelings and rigid geodesic embeddings, • a strong version of the Brightwell–Trotter theorem. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimal planning of network structures within an exchange area

The cost-optimal planning and capacity extension of a given network structure is described by a mixed-integer program. The decision variables in this model are first discrete digital systems (PCM) set up on existing cable lines and secondly new cable links required in the future. Furthermore, this formulation takes account of the circuit capacity of the system and path diversification required for reliability reasons. This planning model describes a real-world problem that can be formulated not only mathematically as a true reproduction of reality but can also be solved in a closed form within reasonable computer time. Opposite to the operational planning methods, this model allows a global cost-optimal network to be obtained, which is subject to a certain set of constraints.     

Geometric Perception of Pose and Tracking

Object tracking and manipulation is an important process for many applications in robotics and computer vision. A novel 3D pose estimation of objects using reflectionally symmetry formulated in conformal geometric algebra (CGA) is proposed in this work. The synthesis of the kinematical model for robots using the CGA approach is described. Real time implementation results are presented for the pose estimation of object using a stereo vision system.     

CONSTRUCTION OF CONCEPTUAL KNOWLEDGE: THE CASE OF COMPUTER-AIDED EXPLORATION OF PERIOD DOUBLING

This research focuses on students using an experimental approach with computer software to give visual meaning to symbolic ideas and to provide a basis for further generalisation. They use computer software that draws orbits of x=f(x) iteration and are encouraged to investigate the iterations of f_λ(x)=λx(1-x) as λ increases. The iterations pass through successive acts of period-doubling as λ=λ₀, λ₁, λ₂, …,. students are invited to estimate the values of λ and to compare their experimental results with the theory of geometric convergence. The supervisor acts as a mentor, using various styles of questioning to provoke links between different ideas. A variety of data is collected to give evidence for the ways in which students develop conceptual links between symbolic theory and the visual and numeric aspects of computer experiment.     

Construction of stationary self-similar generalized fields by random wavelet expansion

Random wavelet expansion is introduced in the study of stationary self-similar generalized random fields. It is motivated by a model of natural images, in which 2D views of objects are randomly scaled and translated because the objects are randomly distributed in the 3D space. It is demonstrated that any stationary self-similar random field defined on the dual space of a Schwartz space of smooth rapidly decreasing functions has a random wavelet expansion representation. To explicitly construct stationary self-similar random fields, random wavelet expansion representations incorporating random functionals of the following three types are considered: (1) a multiple stochastic integral over the product domain of scale and translate, (2) an iterated one, first integrating over the scale domain, and (3) an iterated one, first integrating over the translate domain. We show that random wavelet expansion gives rise to a variety of stationary self-similar random fields, including such well-known processes as the linear fractional stable motions. Received: 11 December 1998 / Revised version: 31 January 2001 / Published online: 23 August 2001     

Two-view curve reconstruction based on the snake model

Reconstruction of 3D curves from their stereo images is an important issue in computer vision. Based on deformation of the snake model and NURBS representation, we evolve the curve in the view of inverse optimization to finish reconstruction. This manner can reduce the need of matching multi-view space curve projections, meanwhile improve the reconstruction precision. Considering that the 2D data reconstruction exists error inevitably, based on two cameras, a discussion on its influence to stereo reconstruction is given next. Finally, the proposed approach is experimented with artificial and real data, and gains a satisfying reconstruction effect.     

Competent Workplace Mathematics: How Signs Become Transparent in Use

Past research has shown that many scientists, when asked to interpret unfamiliar graphs that have nevertheless been culled from introductory undergraduate courses in their own field, experience problems and cannot give the standard answer accepted in the field. Yet, these same scientists turn out to be highly competent when it comes to graphs from their immediate domain of research. In this research, which is based on ethnographic studies among scientists and technicians, I show how graph interpretation in one biology laboratory initially required tool (computer)mediation. After scientists had become familiar with the phenomenon, data collection,and resulting graphs, they interpret the latter correctly without requiring prior transformation. Furthermore, in the course of their work, they established what they understood to be a one-to-one correspondence between graphs and some aspect of the natural world. As a result of scientists' embodied laboratory work, talk about the graphical representation and talk about the object represented are often indistinguishable. The process of developing competency in graph use is equivalent to that of a tool that becomes transparent to the consciousness of its user;I describe this process in terms of activity theory as a transition from tool-mediated action to operation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Two-view curve reconstruction based on the snake model

Reconstruction of 3D curves from their stereo images is an important issue in computer vision. Based on deformation of the snake model and NURBS representation, we evolve the curve in the view of inverse optimization to finish reconstruction. This manner can reduce the need of matching multi-view space curve projections, meanwhile improve the reconstruction precision. Considering that the 2D data reconstruction exists error inevitably, based on two cameras, a discussion on its influence to stereo reconstruction is given next. Finally, the proposed approach is experimented with artificial and real data, and gains a satisfying reconstruction effect.     

The Impact of Teacher Privileging on Learning Differentiation with Technology

This study examines how two teachers taught differentiation using a hand held computer algebra system, which made numerical,graphical and symbolic representations of the derivative readily available. The teachers planned the lessons together but taught their Year 11 classes in very different ways. They had fundamentally different conceptions of mathematics with associated teaching practices,innate ‘privileging’ of representations, and of technology use. This study links these instructional differences to the different differentiation competencies that the classes acquired. Students of the teacher who privileged conceptual understanding and student construction of meaning were more able to interpret derivatives. Students of the teacher who privileged performance of routines made better use of the CAS for solving routine problems. Comparison of the results with an earlier study showed that although each teacher's teaching approach was stable over two years, each used technology differently with further experience of CAS. The teacher who stressed understanding moved away from using CAS, whilst the teacher who stressed rules,adopted it more. The study highlights that within similar overall attainment on student tests, there can be substantial variations of what students know. New technologies provide more approaches to teaching and so greater variations between teaching and the consequent learning may become evident. This revised version was published online in July 2006 with corrections to the Cover Date.     

Ordering Cellular Spaces with Application to Curves and Knots

This paper concerns the problem of ordering geometrical objects, which have the structure of finite cellular spaces. We introduce cellular codes, cellular indices, and (k,l)-sizes and apply them to ordering 2-cells, plane curves, and projections of knots. Results of this paper can be applied (1) to ordering other geometrical objects with cellular space structures, (2) in psychological and computer tests for comparison of complexity of geometrical objects, and (3) to ordering objects with fractal and chaotic structure, which admit approximation by cellular spaces.     

Iterative methods for the reconstruction of astronomical images with high dynamic range

In most cases astronomical images contain objects with very different intensities such as bright stars combined with faint nebulae. Since the noise is mainly due to photon counting (Poisson noise), the signal-to-noise ratio may be very different in different regions of the image. Moreover, the bright and faint objects have, in general, different angular scales. These features imply that the iterative methods which are most frequently used for the reconstruction of astronomical images, namely the Richardson–Lucy Method (RLM), also known in tomography as Expectation Maximization (EM) method, and the Iterative Space Reconstruction Algorithm (ISRA) do not work well in these cases. Also standard regularization approaches do not provide satisfactory results since a kind of adaptive regularization is required, in the sense that one needs a different regularization for bright and faint objects. In this paper we analyze a number of regularization functionals with this particular kind of adaptivity and we propose a simple modification of RLM and ISRA which takes into account these regularization terms. The preliminary results on a test object are promising.     

An algorithm for finding an approximate solution to the Weber problem on a line with forbidden gaps

Under study is the problem of optimal location of interconnected objects on a line with forbidden gaps. The task is to minimize the total cost of links between objects and between objects and zones. The properties of the problem are found that allowed us to reduce the initial continuous problem to a discrete problem. Some algorithm for obtaining an approximate solution is developed, and the results of a computational experiment are given.     

Special representations forn-bridge links

We describe a simple algorithm to obtain a catalogue of 3-bridge links by computer, depending upon 6-tuples of positive integers. This permits us to represent the genus two 3-manifolds by standardly constructed graphs with colored edges. Finally, we prove some results about the topological structure of these manifolds and extend the combinatorial representation ton-bridge links This work was performed under the auspices of the GNSAGA of the CNR (National Research Council) of Italy and financially supported by the Ministero della Universitá e della Ricerca Scientifica e Tecnologica of Italy within the project “Geometria Reale e Complessa”.     

Young Children Do Not Hold the Classic Earth's Shadow Misconception to Explain Lunar Phases

This research explored young children's early thoughts about natural phenomena and investigated sources of influence toward their knowledge construction. Two Piagetian interviews were conducted with four children. Each child was questioned about two phenomena in particular: (a) the moon and its changing appearance (moon phases) and (b) the formation of shadows. The first Piagetian interview involved a lunar protocol, and the second (given a week later) was a shadow protocol. The external interest of this research study concerns when and why do children develop the commonly held Earth's shadow misconception as the cause of the moon's phases. Findings showed that none of the young children believed that various lunar phases were due to an Earthly shadow cast upon the moon. The shadow interview results revealed that young children had primitive understandings of the shadow concept. In fact, two of the four children indicated that no light source was needed to produce a shadow.     

Decision Making and STS Education: Exploring Scientific Knowledge and Social Responsibility in Schools and Science Centers Through an Issues-Based Approach

In an attempt to bridge scientific knowledge and social responsibility, an issues-based approach to learning in science, technology and society (STS) education was adopted by a classroom teacher in concert with a science center. This school and nonschool focus on a socioscientific issue (mining) provides a rich context from which students and teachers can explore the multiple perspectives and complexities of controversy. In particular, the case study examines (a) how fifth- and sixth-grade students interpret and reason through a controversial socioscientific issue (mining in this context), and (b) how school and nonschool environments can provide a supportive forum for dialogue and decision making around a contextualized issue. This research supports the claim that it is important for young children to develop skills of natural inquiry, critical thinking, and decision making about science and technology and the links to the world they encounter at an early age in their education.