Polyhedral object recognition by indexing

In computer vision, the indexing problem is the problem of recognizing a few objects in a large database of objects while avoiding the help of the classical image-feature-to-object-feature matching paradigm. In this paper we address the problem of recognizing three-dimensional (3-D) polyhedral objects from 2-D images by indexing. Both the objects to be recognized and the images are represented by weighted graphs. The indexing problem is therefore the problem of determining whether a graph extracted from the image is present or absent in a database of model graphs. We introduce a novel method for performing this graph indexing process which is based both on polynomial characterization of binary and weighted graphs and on hashing. We describe in detail this polynomial characterization and then we show how it can be used in the context of polyhedral object recognition. Next we describe a practical recognition-by-indexing system that includes the organization of the database, the representation of polyhedral objects in terms of 2-D characteristic views, the representation of this views in terms of weighted graphs and the associated image processing. Finally, some experimental results allow the evaluation of the system performance.     

An efficient parallel algorithm for geometrically characterising drawings of a class of 3-D objects

Labelling the lines of a planar line drawing of a 3-D object in a way that reflects the geometric properties of the object is a much studied problem in computer vision, considered to be an important step towards understanding the object from its 2-D drawing. Combinatorially, the labellability problem is a Constraint Satisfaction Problem and has been shown to be NP-complete even for images of polyhedral scenes. In this paper, we examine scenes that consist of a set of objects each obtained by rotating a polygon around an arbitrary axis. The objects are allowed to arbitrarily intersect or overlay. We show that for these scenes, there is a sequential lineartime labelling algorithm. Moreover, we show that the algorithm has a fast parallel version that executes inO(log³ n) time on an Exclusive-Read-Exclusive-Write Parallel Random Access Machine withO(n ³/log³ n) processors. The algorithm not only answers the decision problem of labellability, but also produces a legal labelling, if there is one. This parallel algorithm should be contrasted with the techniques of dealing with special cases of the constraint satisfaction problem. These techniques employ an effective, but inherently sequential, relaxation procedure in order to restrict the domains of the variables.This research was partially supported by the European Community ESPRIT Basic Research Program under contracts 7141 (project ALCOM II) and 6019 (project Insight II).     

Perceptual grouping of line features in 3-D space: a model-based framework

In this paper, we propose a novel model-based perceptual grouping algorithm for the line features of 3-D polyhedral objects. Given a 3-D polyhedral model, perceptual grouping is performed to extract a set of 3-D line segments which are geometrically consistent with the 3-D model. Unlike the conventional approaches, grouping is done in 3-D space in a model-based framework. In our unique approach, a decision tree classifier is employed for encoding and retrieving the geometric information of the 3-D model. A Gestalt graph is constructed by classifying input instances into proper Gestalt relations using the decision tree. The Gestalt graph is then decomposed into a few subgraphs, yielding appropriate groups of features. As an application, we suggest a 3-D object recognition system which can be accomplished by selecting a best-matched group. In order to evaluate the performance of the proposed algorithm, experiments are carried out on both synthetic and real scenes.     

Appearance-Based Structure from Motion Using Linear Classes of 3-D Models

In this paper, we address the problem of recovering 3-D models from sequences of partly calibrated images with unknown correspondence. To that end, we integrate tracking, structure from motion with geometric constraints (specifically in the form of linear class models) in a single framework. The key to making the proposed approach work is the use of appearance-based model matching and refinement which updates the estimated correspondences on each iteration of the algorithm. Another key feature is the matching of a 3-D model directly with the input images without the conventional 2-step approach of stereo data recovery and 3-D model fitting. Initialization of the linear class model to one of the input images (the reference image) is currently partly manual.This synthesis and refine approach, or appearance-based constrained structure from motion (AbCSfm), is especially useful in recovering shapes of objects whose general structureis known but which may have little discernable texture in significant parts of their surfaces. We applied the proposed approach to 3-D face modeling from multiple images to create new 3-D faces for DECface, a synthetic talking head developed at Cambridge Research Laboratory, Digital Equipment Corporation. The DECface model comprises a collection of 3-D triangular and rectangular facets, with nodes as vertices. In recovering the DECface model, we assume that the sequence of images is taken with a camera with unknown focal length and pose. The geometric constraints used are of the form of linear combination of prototypes of 3-D faces of real people. Results of this approach show its good convergence properties and its robustness against cluttered backgrounds.     

Pattern nesting on irregular-shaped stock using Genetic Algorithms

Pattern nesting aims to position 2-D shapes on a sheet so as to achieve maximum usage of a stock, or equivalently to minimise wastage. There are different methods used on computer to lay out the positions of the shapes on the stock, such as linear programming and heuristic method. A recent approach attempts to use Genetic Algorithms (GAs) to solve the problem of pattern nesting. The successful development of using GAs to nest 2-D shapes on regular-shaped stock has proved the feasibility of using GAs to solve pattern nesting problem. This work presents a new method of solving the pattern nesting problem on irregular-shaped stock using GAs, known as the evolutionary boundary nesting algorithm. This approach further generalises the scope of the pattern nesting problem by allowing nesting on stocks of any shapes and sizes. This implies that the nesting algorithm can be used universally in any industry, such as the garment, shipbuilding and aerospace industry. Basically, the shapes are nested sequentially in the stock and the evolutionary boundary nesting algorithm uses GAs to find the best position to nest each shape along the boundary of the stock.     

On the recognition of curved objects

The problem of determining the identity and pose of occluded objects from noisy data is examined. Previous work has shown that local measurements of the position and surface orientation of small patches of an object's surface may be used in a constrained search process to solve this problem, for the case of rigid polygonal objects using 2-D sensory data, or rigid polyhedral objects using 3-D data. The recognition system is extended to recognize and locate curved objects. The extension is done in two dimensions, and applies to the recognition of 2-D objects from 2-D data, or to the recognition of the 3-D objects in stable positions from 2-D data     

Simultaneous aligning and smoothing of surface triangulations

In this work we develop a procedure to deform a given surface triangulation to obtain its alignment with interior curves. These curves are defined by splines in a parametric space and, subsequently, mapped to the surface triangulation. We have restricted our study to orthogonal mapping, so we require the curves to be included in a patch of the surface that can be orthogonally projected onto a plane (our parametric space). For example, the curves can represent interfaces between different materials or boundary conditions, internal boundaries or feature lines. Another setting in which this procedure can be used is the adaption of a reference mesh to changing curves in the course of an evolutionary process. Specifically, we propose a new method that moves the nodes of the mesh, maintaining its topology, in order to achieve two objectives simultaneously: the piecewise approximation of the curves by edges of the surface triangulation and the optimization of the resulting mesh. We will designate this procedure as projecting/smoothing method and it is based on the smoothing technique that we have introduced for surface triangulations in previous works. The mesh quality improvement is obtained by an iterative process where each free node is moved to a new position that minimizes a certain objective function. The minimization process is done on the parametric plane attending to the surface piece-wise approximation and to an algebraic quality measure (mean ratio) of the set of triangles that are connected to the free node. So, the 3-D local projecting/smoothing problem is reduced to a 2-D optimization problem. Several applications of this method are presented.     

Multiple Translational Containment Part I: An Approximate Algorithm

We present an algorithm for finding a solution to the two-dimensional translational approximate multiple containment problem: find translations for k polygons which place them inside a polygonal container so that no point of any polygon is more than 2ε inside of the boundary of any other polygon. The polygons and container may be nonconvex. The value of ε is an input to the algorithm. In industrial applications the containment solution acts as a guide to a machine cutting out polygonal shapes from a sheet of material. If ε is chosen to be a fraction of the cutter's accuracy, then the solution to the approximate containment problem is sufficient for industrial purposes. Given a containment problem, we characterize its solution and create a collection of containment subproblems from this characterization. We solve each subproblem by first restricting certain two-dimensional configuration spaces until a steady state is reached, and then testing for a solution inside the configuration spaces. If necessary, we subdivide the configuration spaces to generate new subproblems. The running time of our algorithm is , where s is the largest number of vertices of any polygon generated by a restriction operation. In the worst case s can be exponential in the size of the input, but, in practice, it is usually not more than quadratic. Received June 24, 1994; revised August 22, 1995.     

Drawing graphs with nonuniform nodes using potential fields

Graphs with nonuniform nodes arise naturally in many real-world applications. Although graph drawing has been a very active research in the computer science community during the past decade, most of the graph drawing algorithms developed thus far have been designed for graphs whose nodes are represented as single points. As a result, it is of importance to develop drawing methods for graphs whose nodes are of different sizes and shapes, in order to meet the need of real-world applications. To this end, a potential field approach, coupled with an idea commonly found in force-directed methods, is proposed in this paper for drawing graphs with nonuniform nodes in 2-D and 3-D. In our framework, nonuniform nodes are uniformly charged, while edges are modelled by springs. Using certain techniques developed in the field of potential-based path planning, we are able to find analytically tractable procedures for computing the repulsive force and torque of a node in the potential field induced by the remaining nodes. The crucial feature of our approach is that the rotation of every nonuniform node and the multiple edges between two nonuniform nodes are taken into account. In comparison with the existing algorithms found in the literature, our experimental results suggest this new approach to be promising, as drawings of good quality for a variety of moderate-sized graphs in 2-D and 3-D can be produced reasonably efficiently. That is, our approach is suitable for moderate-sized interactive graphs or larger-sized static graphs. Furthermore, to illustrate the usefulness of our new drawing method for graphs with zero-sized nodes, we give an application to the visualization of hierarchical clustered graphs, for which our method offers a very efficient solution.     

Efficiently computing and representing aspect graphs of polyhedralobjects

An efficient algorithm and a data structure for computing and representing the aspect graph of polyhedral objects under orthographic projection are presented. The aspect graph is an approach to representing 3-D objects by a set of 2-D views, for the purpose of object recognition. In this approach the viewpoint space is partitioned into regions such that in each region the qualitative structure of the line drawing does not change. The viewing data of an object is the partition of the viewpoint space together with a representative view in each region. The algorithm computes the viewing data for line drawings of polyhedral objects under orthographic projection     

MPCA: Multilinear Principal Component Analysis of Tensor Objects

This paper introduces a multilinear principal component analysis (MPCA) framework for tensor object feature extraction. Objects of interest in many computer vision and pattern recognition applications, such as 2-D/3-D images and video sequences are naturally described as tensors or multilinear arrays. The proposed framework performs feature extraction by determining a multilinear projection that captures most of the original tensorial input variation. The solution is iterative in nature and it proceeds by decomposing the original problem to a series of multiple projection subproblems. As part of this work, methods for subspace dimensionality determination are proposed and analyzed. It is shown that the MPCA framework discussed in this work supplants existing heterogeneous solutions such as the classical principal component analysis (PCA) and its 2-D variant (2-D PCA). Finally, a tensor object recognition system is proposed with the introduction of a discriminative tensor feature selection mechanism and a novel classification strategy, and applied to the problem of gait recognition. Results presented here indicate MPCA's utility as a feature extraction tool. It is shown that even without a fully optimized design, an MPCA-based gait recognition module achieves highly competitive performance and compares favorably to the state-of-the-art gait recognizers.     

On The Lower Bound Of Edge Guards Of Polyhedral Terrains

Guarding polyhedral terrains is a relatively new problem in computational geometry. It is known as NP-hard problem. In 1997, P. Bose, T. Shermer, G. Toussaint and B. Zhu stated the bounds on the number of guards and proposed some algorithms for placing vertex and edge guards. In this contribution, we point to the inconsistency in the proof of the lower bound of the number of edge guards. We show that following the approach of Bose et al. for an n-vertex polyhedral terrain only a weaker lower bound of ?(2n?4)/7? can be achieved. Hence deriving the proof for the lower bound equal to ?(4n?4)/13? originally stated by Bose et al. remains an open issue.     

Optimal Algorithms for the Coverability, the Subword, the Containment, and the Equivalence Problems for Commutative Semigroups

In this paper, we present decision procedures for the coverability, the subword, the containment, and the equivalence problems for commutative semigroups. These procedures require at most space 2^c·n, where n is the size of the problem instance, and c is some problem independent constant. Furthermore, we show that the exponential space hardness of the above problems follows from the work of Mayr and Meyer. Thus, the presented algorithms are space optimal. Our results close the gap between the 2^{c′·n·log n} space upper bound, shown by Rackoff for the coverability problem and shown by Huynh for the containment and the equivalence problems, and the exponential space lower bound resulting from the corresponding bound for the uniform word problem established by Mayr and Meyer.     

A general procedure to check conjunctive query containment

In this paper, we present a general procedure to test conjunctive query containment. We divide the containment problem into four categories, taking into account the underlying semantics (set or bag theoretic) and the presence or absence of built-in predicates in the queries. After a brief review of previous work on conjunctive query containment, we present a new procedure, called QCC (Query Containment Checker), which we show to be a general and uniform procedure to check the containment among conjunctive queries under the four categories mentioned above. We briefly describe the use of QCC to check bag containment of conjunctive queries, and explain in detail how to use QCC to check set containment of conjunctive queries with built-in predicates. In our conclusions, we point out some uses of QCC for other types of containment. Received: 21 January 2000 / 19 November 2001     

Interpolating Polyhedral Models Using Intrinsic Shape Parameters

Metamorphosis, or morphing, is the gradual transformation of one shape into another. It generally consists of two subproblems: the correspondence problem and the interpolation problem. This paper presents a solution to the interpolation problem of transforming one polyhedral model into another. It is an extension of the intrinsic shape interpolation scheme (T. W. Sederberg, P. Gao, G. Wang and H. Mu, ‘2-D shape blending: an intrinsic solution to the vertex path problem, SIGGRAPH '93, pp. 15–18.) for 2D polygons. Rather than considering a polyhedron as a set of independent points or faces, our solution treats a polyhedron as a graph representing the interrelations between faces. Intrinsic shape parameters, such as dihedral angles and edge lengths that interrelate the vertices and faces in the two graphs, are used for interpolation. This approach produces more satisfactory results than the linear or cubic curve paths would, and is translation and rotation invariant. © 1997 by John Wiley & Sons, Ltd.     

A Multimodal Approach for Face Modeling and Recognition

In this paper, we present a fully automated multi- modal (3-D and 2-D) face recognition system. For the 3-D modality, we model the facial image as a 3-D binary ridge image that contains the ridge lines on the face. We use the principal curvature $kappa_{rm max}$ to extract the locations of the ridge lines around the important facial regions on the range image (i.e., the eyes, the nose, and the mouth.) For matching, we utilize a fast variant of the iterative closest point to match the ridge image of a given probe image to the archived ridge images in the database. The main advantage of this approach is reducing the computational complexity by two orders of magnitude by relying on the ridge lines. For the 2-D modality, we model the face by an attributed relational graph (ARG), where each node of the graph corresponds to a facial feature point. At each facial feature point, a set of attributes is extracted by applying Gabor wavelets to the 2-D image and assigned to the node of the graph. The edges of the graph are defined based on Delaunay triangulation and a set of geometrical features that defines the mutual relations between the edges is extracted from the Delaunay triangles and stored in the ARG model. The similarity measure between the ARG models that represent the probe and gallery images is used for 2-D face recognition. Finally, we fuse the matching results of the 3-D and the 2-D modalities at the score level to improve the overall performance of the system. Different techniques for fusion, such as the Dempster–Shafer theory of evidence and weighted sum of scores are employed and tested using the facial images in the third experiment dataset of the Face Recognition Grand Challenge version 2.0.      

On image matrix based feature extraction algorithms.

Principal component analysis (PCA) and linear discriminant analysis (LDA) are two important feature extraction methods and have been widely applied in a variety of areas. A limitation of PCA and LDA is that when dealing with image data, the image matrices must be first transformed into vectors, which are usually of very high dimensionality. This causes expensive computational cost and sometimes the singularity problem. Recently two methods called two-dimensional PCA (2DPCA) and two-dimensional LDA (2DLDA) were proposed to overcome this disadvantage by working directly on 2-D image matrices without a vectorization procedure. The 2DPCA and 2DLDA significantly reduce the computational effort and the possibility of singularity in feature extraction. In this paper, we show that these matrices based 2-D algorithms are equivalent to special cases of image block based feature extraction, i.e., partition each image into several blocks and perform standard PCA or LDA on the aggregate of all image blocks. These results thus provide a better understanding of the 2-D feature extraction approaches.     

Distributed H∞ containment control of multiagent systems over switching topologies with communication time delay

In this study, we consider the problem of distributed H_∞ containment control for multiagent systems over switching communication topologies. There exists a constant time‐delay and the energy‐bounded communication disturbances in the information transmission process, which are considered. Using the relative output, we develop an observer‐based containment control scheme such that the followers asymptotically converge to the convex hull formed by the leaders with a guaranteed H_∞ performance level. By constructing a Lyapunov functional and using the inequality technique, sufficient conditions for the existence of such dynamic controllers are obtained in terms of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed control protocol.     

Feedback stabilization of a class of distributed parameter systems with control constraints

The feedback stabilization problem for a class of infinite-dimensional linear systems with control constraints is investigated. The approach is developed using a state space system framework which is based on a semigroup formulation. In contrast to the previous works in this direction, it is not assumed that the C₀ semigroup we deal with is a semigroup of contractions. The main result links the problem with the positive invariance of appropriate polyhedral sets. Verifiable necessary and sufficient conditions for this latter property are stated.