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.     

Lower Bounds for the Weak Pigeonhole Principle and Random Formulas beyond Resolution

We work with an extension of Resolution, called Res(2), that allows clauses with conjunctions of two literals. In this system there are rules to introduce and eliminate such conjunctions. We prove that the weak pigeonhole principle PHP^cn_n and random unsatisfiable CNF formulas require exponential-size proofs in this system. This is the strongest system beyond Resolution for which such lower bounds are known. As a consequence to the result about the weak pigeonhole principle, Res(log) is exponentially more powerful than Res(2). Also we prove that Resolution cannot polynomially simulate Res(2) and that Res(2) does not have feasible monotone interpolation solving an open problem posed by Krají ek.     

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.     

3-D Reconstruction of Urban Scenes from Aerial Stereo Imagery: A Focusing Strategy

A contribution to the automatic 3-D reconstruction of complex urban scenes from aerial stereo pairs is proposed. It consists of segmenting the scene into two different kinds of components: the ground and the above-ground objects. The above-ground objects are classified either as buildings or as vegetation. The idea is to define appropriate regions of interest in order to achieve a relevant 3-D reconstruction. For that purpose, a digital elevation model of the scene is first computed and segmented into above-ground regions using a Markov random field model. Then a radiometric analysis is used to classify above-ground regions as building or vegetation, leading to the determination of the final above-ground objects. The originality of the method is its ability to cope with extended above-ground areas, even in case of a sloping ground surface. This characteristic is necessary in a urban environment. Results are very robust to image and scene variability, and they enable the utilization of appropriate local 3-D reconstruction algorithms.     

An Algebra of Discrete Channels That Involve Combinations of Three Basic Error Types

Recently, the author introduced a nonprobabilistic mathematical model of discrete channels, the BEE channels, that involve the error-types substitution, insertion, and deletion. This paper defines an important class of BEE channels, the SID channels, which include channels that permit a bounded number of scattered errors and, possibly at the same time, a bounded burst of errors in any segment of predefined length of a message. A formal syntax is defined for generating channel expressions, and appropriate semantics is provided for interpreting a given channel expression as a communication channel (SID channel) that permits combinations of substitutions, insertions, and deletions of symbols. Our framework permits one to generalize notions such as error correction and unique decodability, and express statements of the form “The code K can correct all errors of type ξ” and “it is decidable whether the code K is uniquely decodable for the channel described by ξ”, where ξ is any SID channel expression.     

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.     

Specularities on Surfaces with Tangential Hairs or Grooves

Specularities on surfaces with tangential hairs or grooves are readily observable in nature. Examples of such phenomena are the arched or looped highlights observed on horses and human heads and the linear or curved specularities observed on parts of industrial machinery that have tangential grooves. We investigate the geometry of curvilinear specularities on surfaces of different curvature with tangential hairs or grooves of different orientations under controlled lighting and viewing conditions. First the nature of these specularities is investigated qualitatively. Then specularities on parametric surfaces and hair or groove orientations are calculated for some specific cases. Explicit calculations of specularities on some special surfaces, cylinders, cones, and spheres, are verified by photographs of the reflections. Aspects of the work are applicable to computer graphics and can be utilized for the image interpretation of surface specularities.     

Attentional Scene Segmentation: Integrating Depth and Motion

We present an approach to attention in active computer vision. The notion of attention plays an important role in biological vision. In recent years, and especially with the emerging interest in active vision, computer vision researchers have been increasingly concerned with attentional mechanisms as well. The basic principles behind these efforts are greatly influenced by psychophysical research. That is the case also in the work presented here, which adapts to the model of Treisman (1985, Comput. Vision Graphics Image Process. Image Understanding31, 156–177), with an early parallel stage with preattentive cues followed by a later serial stage where the cues are integrated. The contributions in our approach are (i) the incorporation of depth information from stereopsis, (ii) the simple implementation of low level modules such as disparity and flow by local phase, and (iii) the cue integration along pursuit and saccade mode that allows us a proper target selection based on nearness and motion. We demonstrate the technique by experiments in which a moving observer selectively masks out different moving objects in real scenes.     

Compact Recognizers of Episode Sequences

Given two strings X=a₁…a_n and P=b₁…b_m over an alphabet Σ, the problem of testing whether P occurs as a subsequence of X is trivially solved in linear time. It is also known that a simple O(n log |Σ|) time preprocessing of X makes it easy to decide subsequently, for any P and in at most |P| log |Σ| character comparisons, whether P is a subsequence of X. These problems become more complicated if one asks instead whether P occurs as a subsequence of some substring Y of X of bounded length. This paper presents an automaton built on the textstring X and capable of identifying all distinct minimal substrings Y of X having P as a subsequence. By a substring Y being minimal with respect to P, it is meant that P is not a subsequence of any proper substring of Y. For every minimal substring Y, the automaton recognizes the occurrence of P having the lexicographically smallest sequence of symbol positions in Y. It is not difficult to realize such an automaton in time and space O(n²) for a text of n characters. One result of this paper consists of bringing those bounds down to linear or O(n log n), respectively, depending on whether the alphabet is bounded or of arbitrary size, thereby matching the corresponding complexities of automata constructions for offline exact string searching. Having built the automaton, the search for all lexicographically earliest occurrences of P in X is carried out in time O(∑_i=1^mrocc_i·i) or O(n+∑_i=1^mrocc_i·i· log n), depending on whether the alphabet is fixed or arbitrary, where rocc_i is the number of distinct minimal substrings of X having b₁…b_i as a subsequence (note that each such substring may occur many times in X but is counted only once in the bound). All log factors appearing in the above bounds can be further reduced to log log by resorting to known integer-handling data structures.     

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.     

Efficient Computation of the Euclidean Distance Transform

We present a simple algorithm for the Euclidean distance transform of a binary image that runs more efficiently than other algorithms in the literature. We show that our algorithm runs in optimal time for many architectures and has optimal cost for the RAM and EREW PRAM.     

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.     

Convexity-Based Visual Camouflage Breaking

Camouflage is frequently used by animals and humans (usually for military purposes) in order to conceal objects from visual surveillance or inspection. Most camouflage methods are based on superposing multiple edges on the object that is supposed to be hidden, such that its familiar contours and texture are masked. In this work, we present an operator, D_arg, that is applied directly to the intensity image in order to detect 3D smooth convex (or equivalently: concave) objects. The operator maximally responds to a local intensity configuration that corresponds to curved 3D objects, and thus, is used to detect curved objects on a relatively flat background, regardless of image edges, contours, and texture. In that regard, we show that a typical camouflage found in some animal species seems to be a “counter measure” taken against detection that might be based on our method. Detection by D_arg is shown to be very robust, from both theoretic considerations and practical examples of real-life images. As a part of the camouflage breaking demonstration, D_arg, which is non-edge-based, is compared with a representative edge-based operator. Better performance is maintained by D_arg for both animal and military camouflage breaking.     

Vector-City Vector Distance Transform

This paper will examine the current chamfer and vector distance transforms for encoding objects as distance fields. A new vector distance transform is introduced which uses the city-block chamfer distance transform as a basis. Detailed error analysis using real CT data is presented, demonstrating the improved accuracy of the new approach over existing methods. The production of a subvoxel accurate distance field is also demonstrated by employing an improved classification. Distance fields are shown for skull and chess piece datasets.     

A Probabilistic Model for Recovering Camera Translation

This paper describes the mathematical basis and application of a probabilistic model for recovering the direction of camera translation (heading) from optical flow. According to the theorem that heading cannot lie between two converging points in a stationary environment, one can compute the posterior probability distribution of heading across the image and choose the heading with maximum a posteriori (MAP). The model requires very simple computation, provides confidence level of the judgments, applies to both linear and curved trajectories, functions in the presence of camera rotations, and exhibited high accuracy up to 0.1°–0.2° in random dot simulations.     

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.     

Improved Lower Bounds for Learning from Noisy Examples: An Information-Theoretic Approach

This paper presents a general information-theoretic approach for obtaining lower bounds on the number of examples required for Probably Approximately Correct (PAC) learning in the presence of noise. This approach deals directly with the fundamental information quantities, avoiding a Bayesian analysis. The technique is applied to several different models, illustrating its generality and power. The resulting bounds add logarithmic factors to (or improve the constants in) previously known lower bounds.     

Tracking of Human Limbs by Multiocular Vision

This article proposes a method for the tracking of human limbs from multiocular sequences of perspective images. These limbs and the associated articulations must first be modelled. During the learning stage, we model the texture linked to the limbs. The lack of characteristic points on the skin is compensated by the wearing of nonrepetitive texture tights. The principle of the method is based on the interpretation of image textured patterns as the 3D perspective projections of points of the textured articulated model. An iterative Levenberg–Marquardt process is used to compute the model pose in accordance with the analyzed image. The calculated attitude is filtered (Kalman filter) to predict the model pose in the following image of the sequence. The image patterns are extracted locally according to the textured articulated model in the predicted attitude. Tracking experiments, illustrated in this paper by cycling sequences, demonstrate the validity of the approach.     

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.     

The Undecidability of the First-Order Theories of One Step Rewriting in Linear Canonical Systems

By reduction from the halting problem for Minsky's two-register machines we prove that there is no algorithm capable of deciding the -theory of one step rewriting of an arbitrary finite linear confluent finitely terminating term rewriting system (weak undecidability). We also present a fixed such system with undecidable *-theory of one step rewriting (strong undecidability). This improves over all previously known results of the same kind.