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.
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.
123098354
Globally optimal segmentation of interacting surfaces with geometric constraints
Efficient detection of globally optimal surfaces representing object boundaries in volumetric datasets is important and remains challenging in many medical image analysis applications. We have developed an optimal surface detection method that is capable of simultaneously detecting multiple interacting surfaces, in which the optimality is controlled by the cost functions designed for individual surfaces and several geometric constraints defining the surface smoothness and interrelations. The method solves the surface detection problems by transforming them into computing minimum s-t cuts in the derived edge-weighted directed graphs. The proposed algorithm has low-order polynomial complexity and is computationally efficient. The method has been validated on over 100 computer generated volumetric images and 96 CT-scanned datasets of different-sized plexiglas tubes, yielding highly accurate results (mean signed error of the measured inner- and outer-diameters of the plexiglas tubes was 0.21 /spl plusmn/ 3.20%). Our approach can be readily extended to higher dimensional image segmentation.
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.
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
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.
26409739
Guest Editors' Introduction
11758071
Algorithms for optimal multi-resolution quantization
Multi-resolution quantization is a way of constructing a progressively refinable description of a discrete random variable. The underlying discrete optimization problem is to minimize an expected distortion over all refinement levels weighted by the probability or importance of the descriptions of different resolutions. This research is motivated by the application of multimedia communications via variable-rate channels. We propose an O(rN2) time and O(N2 log N) space algorithm to design a minimum-distortion quantizer of r levels for a random variable drawn from an alphabet of size N. It is shown that for a very large class of distortion measures the objective function of optimal multi-resolution quantization satisfies the convex Monge property. The efficiency of the proposed algorithm hinges on the convex Monge property. But our algorithm is simpler (even though of the same asymptotic complexity) than the well-known SMAWK fast matrix search technique, which is the best existing solution to the quantization problem. For exponential random variables our approach leads to a solution of even lower complexity: O(rN) time and O(N log N) space, which outperforms all the known algorithms for optimal quantization in both multi- and single-resolution cases. We also generalize the multi-resolution quantization problem to a graph problem, for which our algorithm offers an efficient solution.
33348508
Knowledge Discovery in Multiple Databases
36354030
Globally optimal uneven error-protected packetization of scalable code streams
In this paper, we present a family of new algorithms for rate-fidelity optimal packetization of scalable source bit streams with uneven error protection. In the most general setting where no assumption is made on the probability function of packet loss or on the rate-fidelity function of the scalable code stream, one of our algorithms can find the globally optimal solution to the problem in O(N/sup 2/L/sup 2/) time, compared to a previously obtained O(N/sup 3/L/sup 2/) complexity, where N is the number of packets and L is the packet payload size. If the rate-fidelity function of the input is convex, the time complexity can be reduced to O(NL/sup 2/) for a class of erasure channels, including channels for which the probability function of losing n packets is monotonically decreasing in n and independent erasure channels with packet erasure rate no larger than N/2(N + 1). Furthermore, our O(NL/sup 2/) algorithm for the convex case can be modified to rind an approximation solution for the general case. All of our algorithms do away with the expediency of fractional bit allocation, a limitation of some existing algorithms.
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.