Novel Techniques for Robust Voxelization and Visualization of Implicit Surfaces

Voxelization is the transformation of geometric surfaces into voxels. Up to date this process has been done essentially using incremental algorithms. Incremental algorithms have the reputation of being efficient but they lack an important property: robustness. The voxelized representation should envelop its continuous model. However, without robust methods this cannot be guaranteed. This article describes novel techniques of robust voxelization and visualization of implicit surfaces. First of all our recursive subdivision voxelization algorithm is reviewed. This algorithm was initially inspired by Duff's image space subdivision method. Then, we explain the algorithm to voxelize implicit surfaces defined in spherical or cylindrical coordinates. Next, we show a new technique to produce infinite replications of implicit objects and their voxelization method. Afterward, we comment on the parallelization of our voxelization procedure. Finally we present our voxel visualization algorithm based on point display. Our voxelization algorithms can be used with any data structure, thanks to the fact that a voxel is only stored once the last subdivision level is reached. We emphasize the use of the octree, though, because it is a convenient way to store the discrete model hierarchically. In a hierarchy the discrete model refinement is simple and possible from any previous voxelized scene thanks to the fact that the voxelization algorithms are robust.     

An Algebraic Approach to Camera Self-Calibration

This paper describes a new self-calibration method for a single camera undergoing general motions. It has the following main contributions. First, we establish new constraints which relate the intrinsic parameters of the camera to the rotational part of the motions. This derivation is purely algebraic. We propose an algorithm which simultaneously solves for camera calibration and the rotational part of motions. Second, we provide a comparison between the developed method and a Kruppa equation-based method. Extensive experiments on both synthetic and real image data show the reliability and outperformance of the proposed method. The practical contribution of the method is its interesting convergence property compared with that of the Kruppa equations method.     

Range Flow Estimation

We discuss the computation of the instantaneous 3D displacement vector fields of deformable surfaces from sequences of range data. We give a novel version of the basic motion constraint equation that can be evaluated directly on the sensor grid. The various forms of the aperture problem encountered are investigated and the derived constraint solutions are solved in a total least squares (TLS) framework. We propose a regularization scheme to compute dense full flow fields from the sparse TLS solutions. The performance of the algorithm is analyzed quantitatively for both synthetic and real data. Finally we apply the method to compute the 3D motion field of living plant leaves.     

Motion Analysis by Random Sampling and Voting Process

In computer vision, motion analysis is a fundamental problem. Applying the concepts of congruence checking in computational geometry and geometric hashing, which is a technique used for the recognition of partially occluded objects from noisy data, we present a new random sampling approach for the estimation of the motion parameters in two- and three-dimensional Euclidean spaces of both a completely measured rigid object and a partially occluded rigid object. We assume that the two- and three-dimensional positions of the vertices of the object in each image frame are determined using appropriate methods such as a range sensor or stereo techniques. We also analyze the relationships between the quantization errors and the errors in the estimation of the motion parameters by random sampling, and we show that the solutions obtained using our algorithm converge to the true solutions if the resolution of the digitalization is increased.     

Wavelet-Based Off-Line Handwritten Signature Verification

In this paper, a wavelet-based off-line handwritten signature verification system is proposed. The proposed system can automatically identify useful and common features which consistently exist within different signatures of the same person and, based on these features, verify whether a signature is a forgery or not. The system starts with a closed-contour tracing algorithm. The curvature data of the traced closed contours are decomposed into multiresolutional signals using wavelet transforms. Then the zero-crossings corresponding to the curvature data are extracted as features for matching. Moreover, a statistical measurement is devised to decide systematically which closed contours and their associated frequency data of a writer are most stable and discriminating. Based on these data, the optimal threshold value which controls the accuracy of the feature extraction process is calculated. The proposed approach can be applied to both on-line and off-line signature verification systems. Experimental results show that the average success rates for English signatures and Chinese signatures are 92.57% and 93.68%, respectively.     

3D Articulated Models and Multiview Tracking with Physical Forces

We present a method for automatically estimating the motion of an articulated object filmed by two or more fixed cameras. We focus our work on the case where the quality of the images is poor, and where only an approximation of a geometric model of the tracked object is available. Our technique uses physical forces applied to each rigid part of a kinematic 3D model of the object we are tracking. These forces guide the minimization of the differences between the pose of the 3D model and the pose of the real object in the video images. We use a fast recursive algorithm to solve the dynamical equations of motion of any 3D articulated model. We explain the key parts of our algorithms: how relevant information is extracted from the images, how the forces are created, and how the dynamical equations of motion are solved. A study of what kind of information should be extracted in the images and of when our algorithms fail is also presented. Finally we present some results about the tracking of a person. We also show the application of our method to the tracking of a hand in sequences of images, showing that the kind of information to extract from the images depends on their quality and of the configuration of the cameras.     

Perceptual Organization Based Computational Model for Robust Segmentation of Moving Objects

The role of perceptual organization in motion analysis has heretofore been minimal. In this work we present a simple but powerful computational model and associated algorithms based on the use of perceptual organizational principles, such as temporal coherence (or common fate) and spatial proximity, for motion segmentation. The computational model does not use the traditional frame by frame motion analysis; rather it treats an image sequence as a single 3D spatio-temporal volume. It endeavors to find organizations in this volume of data over three levels—signal, primitive, and structural. The signal level is concerned with detecting individual image pixels that are probably part of a moving object. The primitive level groups these individual pixels into planar patches, which we call the temporal envelopes. Compositions of these temporal envelopes describe the spatio-temporal surfaces that result from object motion. At the structural level, we detect these compositions of temporal envelopes by utilizing the structure and organization among them. The algorithms employed to realize the computational model include 3D edge detection, Hough transformation, and graph based methods to group the temporal envelopes based on Gestalt principles. The significance of the Gestalt relationships between any two temporal envelopes is expressed in probabilistic terms. One of the attractive features of the adopted algorithm is that it does not require the detection of special 2D features or the tracking of these features across frames. We demonstrate that even with simple grouping strategies, we can easily handle drastic illumination changes, occlusion events, and multiple moving objects, without the use of training and specific object or illumination models. We present results on a large variety of motion sequences to demonstrate this robustness.     

Optimal Robot Localization in Trees

The problem of localization, that is, of a robot finding its position on a map, is an important task for autonomous mobile robots. It has applications in numerous areas of robotics ranging from aerial photography to autonomous vehicle exploration. In this paper we present a new strategy LPS (Localize-by-Placement-Separation) for a robot to find its position on a map, where the map is represented as a geometric tree of bounded degree. Our strategy exploits to a high degree the self-similarities that may occur in the environment. We use the framework of competitive analysis to analyze the performance of our strategy. In particular, we show that the distance traveled by the robot is at most O( ) times longer than the shortest possible route to localize the robot, where n is the number of vertices of the tree. This is a significant improvement over the best known previous bound of O(n^2/3). Moreover, since there is a lower bound of Ω( ), our strategy is optimal up to a constant factor. Using the same approach we can also show that the problem of searching for a target in a geometric tree, where the robot is given a map of the tree and the location of the target but does not know its own position, can be solved by a strategy with a competitive ratio of O( ), which is again optimal up to a constant factor.     

Texture Classification and Segmentation by Cellular Neural Networks Using Genetic Learning

We present a new single-chip texture classifier based on the cellular neural network (CNN) architecture. Exploiting the dynamics of a locally interconnected 2D cell array of CNNs we have developed a theoretically new method for texture classification and segmentation. This technique differs from other convolution-based feature extraction methods since we utilize feedback convolution, and we use a genetic learning algorithm to determine the optimal kernel matrices of the network. The CNN operators we have found for texture recognition may combine different early vision effects. We show how the kernel matrices can be derived from the state equations of the network for convolution/deconvolution and nonlinear effects. The whole process includes histogram equalization of the textured images, filtering with the trained kernel matrices, and decision-making based on average gray-scale or texture energy of the filtered images. We present experimental results using digital CNN simulation with sensitivity analysis for noise, rotation, and scale. We also report a tested application performed on a programmable 22 × 20 CNN chip with optical inputs and an execution time of a few microseconds. We have found that this CNN chip with a simple 3 × 3 CNN kernel can reliably classify four textures. Using more templates for decision-making, we believe that more textures can be separated and adequate texture segmentation (< 1% error) can be achieved.     

Optimal Deterministic Protocols for Mobile Robots on a Grid

This paper studies a system of m robots operating in a set of n work locations connected by aisles in a × grid, where m≤n. From time to time the robots need to move along the aisles, in order to visit disjoint sets of locations. The movement of the robots must comply with the following constraints: (1) no two robots can collide at a grid node or traverse a grid edge at the same time; (2) a robot's sensory capability is limited to detecting the presence of another robot at a neighboring node. We present a deterministic protocol that, for any small constant ε>0, allows m≤(1-ε)n robots to visit their target locations in O( ) time, where each robot visits no more than d≤n targets and no target is visited by more than one robot. We also prove a lower bound showing that our protocol is optimal. Prior to this paper, no optimal protocols were known for d>1. For d=1, optimal protocols were known only for m≤ , while for general m≤n only a suboptimal randomized protocol was known.     

Test Sets for the Universal and Existential Closure of Regular Tree Languages

Finite test sets are a useful tool for deciding the membership problem for the universal closure of a given tree language, that is, for deciding whether a term has all its ground instances in the given language. A uniform test set for the universal closure must serve the following purpose: In order to decide membership of a term, it is sufficient to check whether all its test set instances belong to the underlying language. A possible application, and our main motivation, is ground reducibility, an essential concept for many approaches to inductive reasoning. Ground reducibility modulo some rewrite system is membership in the universal closure of the set of reducible ground terms. Here, test sets always exist, and several algorithmic approaches are known. The resulting sets, however, are often unnecessarily large. In this paper we consider regular languages and linear closure operators. We prove that universal as well as existential closure, defined analogously, preserve regularity. By relating test sets to tree automata and to appropriate congruence relations, we show how to characterize, how to compute, and how to minimize ground and non-ground test sets. In particular, optimal solutions now replace previous ad hoc approximations for the ground reducibility problem.     

Reconstruction and Simplification of Surfaces from Contours

In this paper we consider the problem of reconstructing triangular surfaces from given contours. An algorithm solving this problem must decide which contours of two successive slices should be connected by the surface (branching problem) and, given that, which vertices of the assigned contours should be connected for the triangular mesh (correspondence problem). We present a new approach that solves both tasks in an elegant way. The main idea is to employ discrete distance fields enhanced with correspondence information. This allows us not only to connect vertices from successive slices in a reasonable way but also to solve the branching problem by creating intermediate contours where adjacent contours differ too much. Last but not least we show how the 2D distance fields used in the reconstruction step can be converted to a 3D distance field that can be advantageously exploited for distance calculations during a subsequent simplification step.     

Fuzzy Distance Transform: Theory, Algorithms, and Applications

This paper describes the theory and algorithms of distance transform for fuzzy subsets, called fuzzy distance transform (FDT). The notion of fuzzy distance is formulated by first defining the length of a path on a fuzzy subset and then finding the infimum of the lengths of all paths between two points. The length of a path π in a fuzzy subset of the n-dimensional continuous space ⁿ is defined as the integral of fuzzy membership values along π. Generally, there are infinitely many paths between any two points in a fuzzy subset and it is shown that the shortest one may not exist. The fuzzy distance between two points is defined as the infimum of the lengths of all paths between them. It is demonstrated that, unlike in hard convex sets, the shortest path (when it exists) between two points in a fuzzy convex subset is not necessarily a straight line segment. For any positive number θ≤1, the θ-support of a fuzzy subset is the set of all points in ⁿ with membership values greater than or equal to θ. It is shown that, for any fuzzy subset, for any nonzero θ≤1, fuzzy distance is a metric for the interior of its θ-support. It is also shown that, for any smooth fuzzy subset, fuzzy distance is a metric for the interior of its 0-support (referred to as support). FDT is defined as a process on a fuzzy subset that assigns to a point its fuzzy distance from the complement of the support. The theoretical framework of FDT in continuous space is extended to digital cubic spaces and it is shown that for any fuzzy digital object, fuzzy distance is a metric for the support of the object. A dynamic programming-based algorithm is presented for computing FDT of a fuzzy digital object. It is shown that the algorithm terminates in a finite number of steps and when it does so, it correctly computes FDT. Several potential applications of fuzzy distance transform in medical imaging are presented. Among these are the quantification of blood vessels and trabecular bone thickness in the regime of limited special resolution where these objects become fuzzy.     

Using Human Perceptual Categories for Content-Based Retrieval from a Medical Image Database

It is often difficult to come up with a well-principled approach to the selection of low-level features for characterizing images for content-based retrieval. This is particularly true for medical imagery, where gross characterizations on the basis of color and other global properties do not work. An alternative for medical imagery consists of the “scattershot” approach that first extracts a large number of features from an image and then reduces the dimensionality of the feature space by applying a feature selection algorithm such as the Sequential Forward Selection method.This contribution presents a better alternative to initial feature extraction for medical imagery. The proposed new approach consists of (i) eliciting from the domain experts (physicians, in our case) the perceptual categories they use to recognize diseases in images; (ii) applying a suite of operators to the images to detect the presence or the absence of these perceptual categories; (iii) ascertaining the discriminatory power of the perceptual categories through statistical testing; and, finally, (iv) devising a retrieval algorithm using the perceptual categories. In this paper we will present our proposed approach for the domain of high-resolution computed tomography (HRCT) images of the lung. Our empirical evaluation shows that feature extraction based on physicians' perceptual categories achieves significantly higher retrieval precision than the traditional scattershot approach. Moreover, the use of perceptually based features gives the system the ability to provide an explanation for its retrieval decisions, thereby instilling more confidence in its users.     

Building Detection from Multiple Aerial Images in Dense Urban Areas

In this paper we present a novel approach for building detection from multiple aerial images in dense urban areas. The approach is based on accurate surface reconstruction, followed by extraction of building façades that are used as a main cue for building detection. For the façade detection, a simple but nevertheless flexible and robust algorithm is proposed. It is based on the observation that building façades correspond to the accumulation of 3D data, available from different views, in object space. Knowledge-driven thresholding of 3D data accumulators followed by Hough transform-based segment detection results in the extraction of façade positions. Three-dimensional planar regions resulting from surface reconstruction procedure and bounded by the extracted façades are detected as building hypotheses through testing a set of spatial criteria. Then, a set of verification criteria is proposed for the hypothesis confirmation.     

Optimal Series-Parallel Trade-offs for Reducing a Function to Its Own Graph

Consider the problem of computing a function given only an oracle for its graph. For this problem, we present optimal trade-offs between serial and parallel queries. In particular, we give a function for which parallel access to its own graph is exponentially more expensive than sequential access.     

Gray-Level Reduction Using Local Spatial Features

This paper proposes a new method for reduction of the number of gray-levels in an image. The proposed approach achieves gray-level reduction using both the image gray-levels and additional local spatial features. Both gray-level and local feature values feed a self-organized neural network classifier. After training, the neurons of the output competition layer of the SOFM define the gray-level classes. The final image has not only the dominant image gray-levels, but also has a texture approaching the image local characteristics used. To split the initial classes further, the proposed technique can be used in an adaptive mode. To speed up the entire multithresholding algorithm and reduce memory requirements, a fractal scanning subsampling technique is adopted. The method is applicable to any type of gray-level image and can be easily modified to accommodate any type of spatial characteristic. Several experimental and comparative results, exhibiting the performance of the proposed technique, are presented.     

Optimal Computing the Chessboard Distance Transform on Parallel Processing Systems

Thedistance transform(DT) is an image computation tool which can be used to extract the information about the shape and the position of the foreground pixels relative to each other. It converts a binary image into a grey-level image, where each pixel has a value corresponding to the distance to the nearest foreground pixel. The time complexity for computing the distance transform is fully dependent on the different distance metrics. Especially, the more exact the distance transform is, the worse execution time reached will be. Nowadays, quite often thousands of images are processed in a limited time. It seems quite impossible for a sequential computer to do such a computation for the distance transform in real time. In order to provide efficient distance transform computation, it is considerably desirable to develop a parallel algorithm for this operation. In this paper, based on the diagonal propagation approach, we first provide anO(N²) time sequential algorithm to compute thechessboard distance transform(CDT) of anN×Nimage, which is a DT using the chessboard distance metrics. Based on the proposed sequential algorithm, the CDT of a 2D binary image array of sizeN×Ncan be computed inO(logN) time on the EREW PRAM model usingO(N²/logN) processors,O(log logN) time on the CRCW PRAM model usingO(N²/log logN) processors, andO(logN) time on the hypercube computer usingO(N²/logN) processors. Following the mapping as proposed by Lee and Horng, the algorithm for the medial axis transform is also efficiently derived. The medial axis transform of a 2D binary image array of sizeN×Ncan be computed inO(logN) time on the EREW PRAM model usingO(N²/logN) processors,O(log logN) time on the CRCW PRAM model usingO(N²/log logN) processors, andO(logN) time on the hypercube computer usingO(N²/logN) processors. The proposed parallel algorithms are composed of a set of prefix operations. In each prefix operation phase, only increase (add-one) operation and minimum operation are employed. So, the algorithms are especially efficient in practical applications.     

Model-Based Object Recognition Using Geometric Invariants of Points and Lines

In this paper, we derive new geometric invariants for structured 3D points and lines from single image under projective transform, and we propose a novel model-based 3D object recognition algorithm using them. Based on the matrix representation of the transformation between space features (points and lines) and the corresponding projected image features, new geometric invariants are derived via the determinant ratio technique. First, an invariant for six points on two adjacent planes is derived, which is shown to be equivalent to Zhu's result [1], but in simpler formulation. Then, two new geometric invariants for structured lines are investigated: one for five lines on two adjacent planes and the other for six lines on four planes. By using the derived invariants, a novel 3D object recognition algorithm is developed, in which a hashing technique with thresholds and multiple invariants for a model are employed to overcome the over-invariant and false alarm problems. Simulation results on real images show that the derived invariants remain stable even in a noisy environment, and the proposed 3D object recognition algorithm is quite robust and accurate.