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.     

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.     

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.     

一种新的完全欧氏距离变换算法

论文提出了一种基于边界剥离的二维完全欧氏距离变换算法。该算法从物体目标的最外层边界开始,自外向内、逐层对物体目标区域进行边界跟踪、剥离。在跟踪过程中,根据当前边界像素点的已获得距离变换结果或为背景的邻域像素信息,计算其与最近背景像素间的欧氏距离,从而实现距离变换。和已有算法相比,文中算法具有简单快速、容易实现,得到的是完全欧氏距离的优点,在分离粘连物体的应用中,取得了良好分离效果。     

Computing the Euclidean Distance Transform on a Linear Array of Processors

Given an n×n binary image of white and black pixels, we present an optimal parallel algorithm for computing the distance transform and the nearest feature transform using the Euclidean metric. The algorithm employs the systolic computation to achieve O(n) running time on a linear array of n processors.     

三维带符号的欧氏距离变换及其应用

距离变换是图像处理中历史悠久的研究课题．该文将二维带符号的欧氏距离变换推广到三维,对其进行了优化,分析了它的计算复杂度,并应用于解决计算机图形学中的两个重要问题：第一,将图形对象的三角网格表示转换为它的距离场表示．即首先将三角网格模型离散为体素表示,利用三维带符号的距离变换,将求空间一点到图形对象的最短距离的全局搜索过程,转化为求这一点到离它最近的特征体素所包含的图形对象部分的局部搜索过程;第二,利用类似的思想,求两张空间曲面之间的最短距离．     

Computation of Approximate Polynomial GCDs and an Extension

Computation of approximate polynomial greatest common divisors (GCDs) is important both theoretically and due to its applications to control linear systems, network theory, and computer-aided design. We study two approaches to the solution so far omitted by the researchers, despite intensive recent work in this area. Correlation to numerical Padé approximation enabled us to improve computations for both problems (GCDs and Padé). Reduction to the approximation of polynomial zeros enabled us to obtain a new insight into the GCD problem and to devise effective solution algorithms. In particular, unlike the known algorithms, we estimate the degree of approximate GCDs at a low computational cost, and this enables us to obtain certified correct solution for a large class of input polynomials. We also restate the problem in terms of the norm of the perturbation of the zeros (rather than the coefficients) of the input polynomials, which leads us to the fast certified solution for any pair of input polynomials via the computation of their roots and the maximum matchings or connected components in the associated bipartite graph.     

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.     

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.     

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.     

Formative Processes with Applications to the Decision Problem in Set Theory: I. Powerset and Singleton Operators

This paper introduces formative processes, composed by transitive partitions. Given a family of sets, a formative process ending in the Venn partition Σ of is shown to exist. Sufficient criteria are also singled out for a transitive partition to model (via a function from set variables to unions of sets in the partition) all set-literals modeled by Σ. On the basis of such criteria a procedure is designed that mimics a given formative process by another where sets have finite rank bounded by C(|Σ|), with C a specific computable function. As a by-product, one of the core results on decidability in computable set theory is rediscovered, namely the one that regards the satisfiability of unquantified set-theoretic formulae involving Boolean operators, the singleton-former, and the powerset operator. The method described (which is able to exhibit a set-solution when the answer is affirmative) can be extended to solve the satisfiability problem for broader fragments of set theory.     

On Computing the Exact Euclidean Distance Transform on Rectangular and Hexagonal Grids

In this paper we prove an equivalence relation between the distance transform of a binary image, where the underlying distance is based on a positive definite quadratic form, and the erosion of its characteristic function by an elliptic poweroid structuring element. The algorithms devised by Shih and Mitchell [18] and Huang and Mitchell [7], for calculating the exact Euclidean distance transform (EDT) of a binary digital image manifested on a square grid, are particular cases of this result. The former algorithm uses erosion by a circular cone to calculate the EDT whilst the latter uses erosion by an elliptic paraboloid (which allows for pixel aspect ratio correction) to calculate the square of the EDT. Huang and Mitchell's algorithm [7] is arguably the better of the two because: (i) the structuring element can be decomposed into a sequence of dilations by 3 × 3 structuring elements (a similar decomposition is not possible for the circular cone) thus reducing the complexity of the erosion, and (ii) the algorithm only requires integer arithmetic (it produces squared distance). The algorithm is amenable to both hardware implementation using a pipeline architecture and efficient implementation on serial machines. Unfortunately the algorithm does not directly transpose to, nor has a corresponding analogue on, the hexagonal grid (the same is also true for Shih and Mitchell's algorithm [7]). In this paper, however, we show that if the hexagonal grid image is embedded in a rectangular grid then Huang and Mitchell's algorithm [7] can be applied, with aspect ratio correction, to obtain the exact EDT on the hexagonal grid.     

On the Power of Las Vegas for One-Way Communication Complexity, OBDDs, and Finite Automata

The study of the computational power of randomized computations is one of the central tasks of complexity theory. The main goal of this paper is the comparison of the power of Las Vegas computation and deterministic respectively nondeterministic computation. We investigate the power of Las Vegas computation for the complexity measures of one-way communication, ordered binary decision diagrams, and finite automata.(i) For the one-way communication complexity of two-party protocols we show that Las Vegas communication can save at most one half of the deterministic one-way communication complexity. We also present a language for which this gap is tight.(ii) The result (i) is applied to show an at most polynomial gap between determinism and Las Vegas for ordered binary decision diagrams.(iii) For the size (i.e., the number of states) of finite automata we show that the size of Las Vegas finite automata recognizing a language L is at least the square root of the size of the minimal deterministic finite automaton recognizing L. Using a specific language we verify the optimality of this lower bound.     

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.     

A Monadic Second-Order Definition of the Structure of Convex Hypergraphs

We consider finite hypergraphs with hyperedges defined as sets of vertices of unbounded cardinality. Each such hypergraph has a unique modular decomposition, which is a tree, the nodes of which correspond to certain subhypergraphs (induced by certain sets of vertices called strong modules) of the considered hypergraph. One can define this decomposition by monadic second-order (MS) logical formulas. Such a hypergraph is convex if the vertices are linearly ordered in such a way that the hyperedges form intervals. Our main result says that the unique linear order witnessing the convexity of a prime hypergraph (i.e., of one, the modular decomposition of which is trivial) can be defined in MS logic. As a consequence, we obtain that if a set of bipartite graphs that correspond (in the usual way) to convex hypergraphs has a decidable monadic second-order theory (which means that one can decide whether a given MS formula is satisfied in some graph of the set) then it has bounded clique-width. This yields a new case of validity of a conjecture which is still open.     

Guaranteeing the Diversity of Number Generators

A major problem in using iterative number generators of the form x_i=f(x_i−1) is that they can enter unexpectedly short cycles. This is hard to analyze when the generator is designed, hard to detect in real time when the generator is used, and can have devastating cryptanalytic implications. In this paper we define a measure of security, called sequence diversity, which generalizes the notion of cycle-length for noniterative generators. We then introduce the class of counter-assisted generators and show how to turn any iterative generator (even a bad one designed or seeded by an adversary) into a counter-assisted generator with a provably high diversity, without reducing the quality of generators which are already cryptographically strong.     

The Complexity of the Exponential Output Size Problem for Top-Down and Bottom-Up Tree Transducers,

The exponential output size problem is to determine whether the size of output trees of a tree transducer grows exponentially in the size of input trees. In this paper the complexity of this problem is studied. It is shown to be NL-complete for total top-down tree transducers, DEXPTIME-complete for general top-down tree transducers, and P-complete for bottom-up tree transducers.