39603817
Optimal Terrain Construction Problems and Applications in Intensity-Modulated Radiation Therapy
In this paper, we study several rectilinear terrain construction problems, which model leaf sequencing problems in intensity-modulated radiation therapy (IMRT). We present a novel unified approach based on geometric techniques for solving these terrain construction problems. Our approach leads to the first algorithms for several leaf sequencing problems in IMRT that are practically fast and guarantee the optimal quality of the output solutions. Our implementation results show that our terrain construction algorithms run very fast on real medical data.
17909513
Determining an optimal penetration among weighted regions in two and three dimensions
We present efficient algorithms for solving the problem of computing an optimal penetration (a ray or a semi-ray) among weighted regions in 2-D and 3-D spaces. This problem finds applications in several areas, such as radiation therapy, geological exploration, and environmental engineering. Our algorithms are based on a combination of geometric techniques and optimization methods. Our geometric analysis shows that the d-D (d = 2, 3) optimal penetration problem can be reduced to solving O(n2(d−1)) instances of certain special types of non-linear optimization problems, where n is the total number of vertices of the regions. We also give implementation results of our 2-D algorithms.
27904596
Segmenting Doughnut-Shaped Objects in Medical Images
Image segmentation with specific constraints has found applications in several areas such as biomedical image analysis and data mining. In this paper, we study the problem of segmenting doughnut-shaped and smooth objects in 2-D medical images. Image objects of these shapes are often studied in medical applications. We present an O(IJU(U − L)logJlog(U − L)) time algorithm, where the size of the input 2-D image is I× J, M is the smoothness parameter with 1 ≤ M ≤ J, and L and U are the thickness parameters specifying the thickness between two border contours of a doughnut-shaped object. Previous approaches for solving this segmentation problem are computationally expensive and/or need a lot of user interference. Our algorithm improves the straightforward dynamic programming algorithm by a factor of \(O(\frac{J(U-L)M^2}{U{\rm log}J{\rm log}(U-L)})\). We explore some interesting observations, which make possible to apply the divide-and-conquer strategy combined with dynamic programming. Our algorithm is also based on computing optimal paths in an implicitly represented graph.
31548532
Efficient Algorithms and Implementations for Optimizing the Sum of Linear Fractional Functions, with Applications
This paper presents an improved algorithm for solving the sum of linear fractional functions (SOLF) problem in 1-D and 2-D. A key subproblem to our solution is the off-line ratio query (OLRQ) problem, which asks to find the optimal values of a sequence of m linear fractional functions (called ratios), each ratio subject to a feasible domain defined by O(n) linear constraints. Based on some geometric properties and the parametric linear programming technique, we develop an algorithm that solves the OLRQ problem in O((m+n)log (m+n)) time. The OLRQ algorithm can be used to speed up every iteration of a known iterative SOLF algorithm, from O(m(m+n)) time to O((m+n)log (m+n)), in 1-D and 2-D. Implementation results of our improved 1-D and 2-D SOLF algorithm have shown that in most cases it outperforms the commonly-used approaches for the SOLF problem. We also apply our techniques to some problems in computational geometry and other areas, improving the previous results.
2025275
Optimizing the sum of linear fractional functions and applications
1 I n t r o d u c t i o n . The problem of optimizing the sum of linear fractional functions (SOLF) is defined as follows: max f(Zl,...,Xd)= ~ [ : 7 : ~ d ) (Xl ..... z~t)ES i=1 ~ a r t m e n t of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN 46556, USA, {chen,odaescu,xwu,jxu}@cse.nd.edu. This research was supported in part by the National Science Foundation under Grant CCR-9623585. tThe work of this author was supported in part by a fellowship of the Center for Applied Mathematics, University of Notre Dame, Notre Dame, Indiana, USA. t Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152, Japan, daiOis.titech.ac.jp. Grant-in-Aid of Ministry of Science, Culture and Education of Japan, No. 10780274. §Department of Architecture and Architectural Systems, Graduate School of Engineering, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501 Japan, naoki~archi.kyoto-u.ac.jp. Partially supported by Grant-inAid for Scientific Research on Priority Areas (B) by the Ministry of Education, Science, Sports and Culture of Japan. IThe work of this author was supported in part by a fellowship of the Center for Applied Mathematics, and by a summer graduate research fellowship of the Graduate School, University of Notre Dame, Notre Dame, Indiana, USA. such that for each i = 1 , 2 , . . . , m , ni(Xl,...,Xd) and di(Xl,...,Xd) are linear functions in a fixed d-D space ]~d, di (x t , . . . ,Xd) ~ 0 for any (Xl , . . . ,Xd) E S, the feasible domain S is defined by n linear constraints (i.e., half-spaces in Rd), and S ~ ¢. Each linear fractional term ~(~1 ..... ~ ) ddxl,...,z~) is called a ratio. The SOLF problem appears in the algorithmic solutions for several geometric optimization problems. In layered manufacturing, Majhi et al. [17], [18] showed that the length-optimal supports for a simple nonconvex polygon can be obtained by solving the 1-D SOLF problem on an interval. Arkin et al. [1] formulated the problem of finding a minimum-area starshaped or monotone polygon that contains a simple polygon, which appears in material layout [1], [8], [21] (e.g., cloth manufacturing) and manufacturing [18], as one of optimizing the sum of 2-D fractional polynomials of degrees 3 and 2 under linear constraints. We will show that the objective function of this problem can actually be simplified to the 2-D SOLF form. For the problem of finding optimal penetrations among weighted regions [5], we shall show tha t the L1 and Leo versions of this problem have the 2-D SOLF problem as a key subproblem. The SOLF problem also arises in other areas. For example, in combinatorial optimization, a class of problems on finding a structure with the maximum (or minimum) mean-weight cost can be solved by optimizing some linear fractional functions [23]. In operations research, it was shown in [11], [24] that many economic applications (e.g., maximization of productivity, return on investment, and return/risk) can be reduced to solving the SOLF problem. Quite a few solutions have been given for the SOLF problem [1], [11], [15], [17], [18], [25]. It is interesting to note that the SOLF related problems were originally studied in economic applications, where the number of variables is usually much bigger than the number of fractional terms. As a result, many previous SOLF algorithms were designed to target problems with only a few fractional terms (less than 10) for a reasonable
47381475
Geometric algorithms for static leaf sequencing problems in radiation therapy
The static leaf sequencing (SLS) problem arises in radiation therapy for cancer treatments, aiming to accomplish the delivery of a radiation prescription to a target tumor in the minimum amount of delivery time. Geometrically, the SLS problem can be formulated as a 3-D partition problem for which the 2-D problem of partitioning a polygonal domain (possibly with holes) into a minimum set of monotone polygons is a special case. In this paper, we present new geometric algorithms for a basic case of the 3-D SLS problem (which is also of clinical value) and for the general 3-D SLS problem. Our basic 3-D SLS algorithm, based on new geometric observations, produces guaranteed optimal quality solutions using O(1) Steiner points in polynomial time; the previously best known basic 3-D SLS algorithm gives optimal outputs only for the case without considering any Steiner points, and its time bound involves a multiplicative factor of a factorial function of the input. Our general 3-D SLS algorithm is based on our basic 3-D SLS algorithm and a polynomial time algorithm for partitioning a polygonal domain (possibly with holes) into a minimum set of x-monotone polygons, and has a fast running time. Experiments of our SLS algorithms and software in clinical settings have shown substantial improvements over the current most popular commercial treatment planning system and the most well-known SLS algorithm in medical literature. The radiotherapy plans produced by our software not only take significantly shorter delivery times, but also have a much better treatment quality. This proves the feasibility of our software and has led to its clinical applications at the Department of Radiation Oncology at the University of Maryland Medical Center. Some of our techniques and geometric procedures (e.g., for partitioning a polygonal domain into a minimum set of x-monotone polygons) are interesting in their own right.
11584790
SLEVEs for planar spline curves
Given a planar spline curve and local tolerances, a matched pair of polygons is computed that encloses the curve and whose width (distance between corresponding break points) is below the tolerances. This is the simplest instance of a subdividable linear efficient variety enclosure, short sleve.We develop general criteria, that certify correctness of a global, polygonal enclosure built from a sequence of individual bounding boxes by extending and intersecting their edges. These criteria prove correctness of the sleve construction.
6198928
Energy-Efficient Deployment of Mobile Sensor Networks by PSO
Sensor deployment is an important issue in designing sensor networks. In this paper, particle swarm optimization (PSO) approach is applied to maximize the coverage based on a probabilistic sensor model in mobile sensor networks and to reduce cost by finding the optimal positions for the cluster-head nodes based on a well-known energy model. During the coverage optimization process, sensors move to form a uniformly distributed topology according to the execution of algorithm at base station. The simulation results show that PSO algorithm has faster convergence rate than genetic algorithm based method while achieving the goal of energy efficient sensor deployment.
29396765
Detection of LSB Steganography via Sample Pair Analysis
This paper introduces a new, principled approach to detecting LSB steganography in digital signals such as images and audio. It is shown that the length of hidden message embedded in the least significant bits of signal samples can be estimated with relatively high precision. The new steganalytic approach is based on some statistical measures of sample pairs that are highly sensitive to LSB embedding operations. The resulting detection algorithm is simple and fast. To evaluate the robustness of the proposed steganalytic approach, bounds on estimation errors are developed. Furthermore, the vulnerability of the new approach to possible attacks is also assessed, and counter measures are suggested.
35689582
A fast recurring two-dimensional entropic thresholding algorithm
Thresholding is an important form of image segmentation and is used in the processing of images for many applications. One of the criteria to select a suitable threshold is the maximization of the two-dimensional (2-D) entropies based on the 2-D (gray-level/local average gray-level) histogram. The rationale of this approach is introduced. In order to reduce the computation time of entropy function, a fast recurring algorithm for 2-D entropic thresholding method is presented. The experimental results show that the processing time to obtain the threshold vector from 2-D histogram is reduced from 30 to 0.15 s.
22986
Robust Routing in Malicious Environment for Ad Hoc Networks
Secure routing in ad hoc networks has been extensively studied in recent years. The vast majority of this work, however, has only focused on providing authenticity of the route. Availability of the network in a malicious environment has largely been ignored. 
 
In this paper, we divide the secure routing problem into two layers. The first layer provides authenticated routing and the second layer provides a route selection algorithm that selects a route with the highest probability of successful delivery rather than the shortest route. We provide a metric for evaluating this probability. We provide simulation results that demonstrate that our approach increases the throughput by at least ten percent in a network where fifty percent of the nodes are malicious when compared to an approach that selects the shortest route. Furthermore, our approach incurs only a small delay when compared to the delay along the shortest route.