Two Polynomial Time Graph Labeling Algorithms Optimizing Max-Norm-Based Objective Functions

Segmentation is one of the most important tasks in image processing that endeavors to classify pixels into two or more groups according to their intensity levels and a threshold value. Since traditional image processing techniques exhibit several difficulties when they are employed to segment images, the use of evolutionary algorithms has been extended to segmentation tasks in last years. The Harmony Search Algorithm (HSA) is an evolutionary method which is inspired in musicians improvising new harmonies while playing. Different to other evolutionary algorithms, HSA exhibits interesting search capabilities still keeping a low computational overhead. In this chapter, a multilevel thresholding (MT) algorithm based on the HSA is presented. The approach combines the good search capabilities of HSA with objective functions suggested by the popular MT methods of Otsu and Kapur. The presented algorithm takes random samples from a feasible search space inside the image histogram. Such samples build each harmony (candidate solution) in the HSA context whereas its quality is evaluated considering the objective function that is employed by the Otsu’s or Kapur’s method. Guided by these objective values, the set of candidate solutions are evolved through HSA operators until an optimal solution is found. The approach generates a multilevel segmentation algorithm which can effectively identify threshold values of a digital image in a reduced number of iterations. Experimental results show a high performance of the presented method for the segmentation of digital images.

A novel image thresholding method based on Parzen window estimate

Fuzzy c-means (FCM) is one of the most popular fuzzy clustering methods and it is used in various applications in computer science. Most clustering methods including FCM, suffer from bad initialization problem. If initial cluster centers (membership degree initialization in FCM) are not selected appropriately, it may yield poor results. In this paper we propose a method called MinMax FCM to overcome this problem. A new objective function is designed in MinMax FCM to this aim. We use maximum variance of clusters as objective function. In this regard, high-variance clusters are penalized. We compare MinMax FCM with FCM in terms of sum of clusters’ variances, maximum variance of clusters, and execution time using a number of UCI datasets.

We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing 2Ω(n2) n-vertex graphs as n→ ∞. This regime contains many classes of interest, for instance perfect graphs or comparability graphs, for which we obtain an adjacency labelling scheme with labels of n/4+o(n) bits per vertex. This implies the existence of a reachability labelling scheme for digraphs with labels of n/4+o(n) bits per vertex and comparability labelling scheme for posets with labels of n/4+o(n) bits per element. All these results are best possible, up to the lower order term.

Optimization problems are used to model many real-life problems. Therefore, solving these problems is one of the most important goals of algorithm design. A general optimization problem can be defined by specifying a set of constraints that defines a subset in some underlying space (like the Euclidean space) called the feasible subset and an objective function that we are trying to maximize or minimize, as the case may be, over the feasible set. The difficulty of solving such problems typically depends on how ‘complex’ the feasible set and the objective function are. For example, a very important class of optimization problems is linear programming. Here the feasible subset is specified by a set of linear inequalities (in the Euclidean space); the objective function is also linear. A more general class of optimization problems is convex programming, where the feasible set is a convex subset of a Euclidean space and the objective function is also convex. Convex programs (and hence, linear programs) have a nice property that any local optimum is also a global optimum for the objective function. There are a variety of techniques for solving such problems – all of them try to approach a local optimum (which we know would be a global optimum as well). These notions are discussed in greater detail in a later section in this chapter. The more general problem, the so-called non-convex programs, where the objective function and the feasible subset could be arbitrary can be very challenging to solve. In particular, discrete optimization problems, where the feasible subset could be a (large) discrete subset of points falls under this category.

Online kernel selection is a more complex problem compared with offline kernel selection, which intermixes training and selection at each round and requires a sublinear regret and low computational complexities. But existing online kernel selection approaches have at least linear time and space complexities at each round with respect to the number of rounds, or lack sublinear regret guarantees for an uncountably infinite number of candidate kernels. To address these issues, we propose a novel online kernel selection approach using tensor sketching, which has constant computational complexities at each round and enjoys a sublinear regret bound for an uncountably infinite number of candidate kernels. We represent the data using the tensor products and construct data sketches using the Taylor series and the Count Sketch matrices, which yields a sketched reproducing kernel Hilbert space (SRKHS). Then we update the optimal kernels and the hypotheses using online gradient descent in SRKHS. We prove that the kernel corresponding to SRKHS satisfies the reproducing property, the hypotheses in SRKHS are convex with respect to the kernel parameter, and the proposed online kernel selection approach in SRKHS enjoys a regret bound of order $O(\sqrtT )$ for an uncountably infinite number of candidate kernels, which is optimal for a convex loss function, where T is the number of rounds. By the fast Fourier transform, the hypotheses in SRKHS can be computed in a quasilinear time complexity and a logarithmic space complexity with respect to the sketch size at each round, where the sketch size is a constant. Experimental results demonstrate that our online kernel selection approach is more accurate and efficient for online kernel learning on high dimension data.

Synthesis of optimal controllers for a class of maximization problems

Minimizing a linear multiplicative-type function under network flow constraints

With the growing need of big-data processing in diverse application domains, MapReduce (e.g., Hadoop) has become one of the standard computing paradigms for large-scale computing on a cluster system. Despite its popularity, the current MapReduce framework suffers from inflexibility and inefficiency inherent to its programming model and system architecture. In order to address these problems, we propose Vispark, a novel extension of Spark for GPU-accelerated MapReduce processing on array-based scientific computing and image processing tasks. Vispark provides an easy-to-use, Python-like high-level language syntax and a novel data abstraction for MapReduce programming on a GPU cluster system. Vispark introduces a programming abstraction for accessing neighbor data in the mapper function, which greatly simplifies many image processing tasks using MapReduce by reducing memory footprints and bypassing the reduce stage. Vispark provides socket-based halo communication that synchronizes between data partitions transparently from the users, which is necessary for many scientific computing problems in distributed systems. Vispark also provides domain-specific functions and language supports specifically designed for high-performance computing and image processing applications. We demonstrate the performance of our prototype system on several visual computing tasks, such as image processing, volume rendering, K-means clustering, and heat transfer simulation.

An initial study of time complexity in infinite-domain constraint satisfaction

Finite state automata are of fundamental importance in theory as well as applications in computer science. This dissertation presents several parallel algorithms for processing of them, including minimizing the number of states for a given finite state automaton, finding homing sequences for a given finite state automaton, converting a regular expression into an $\\varepsilon$-free nondeterministic finite automaton. Implementation of some algorithms, as well as their experimental results, are also given. For minimization of deterministic finite automata, we present a parallel algorithm and discuss its implementation on a practical parallel machine using data parallel and message passing paradigms. Its time complexity on a p processor EREW PRAM $(p\\le n)$ for inputs of size n is $O({n\\log\\sp2n\\over p}+\\log n\\ \\log p)$ uniformly on almost all instances. The work done by our algorithm is thus within a factor of $O(\\log n)$ of the best known sequential algorithm. The space used by our algorithm is linear in the input size. The actual resource requirements of our implementations are consistent with these estimates. For finding homing sequences of deterministic finite automata, we present both deterministic and randomized algorithms. Two randomized algorithms of time complexity $O(log\\sp2n)$ are given. One of the randomized algorithms produces a homing sequence of length $O(n\\ log\\sp2n)$ for almost all DFA's with n states. The deterministic algorithm is based on a well known sequential algorithm. The implementation of the deterministic parallel algorithm shows that it can handle randomly generated instances with millions of states, and all DFA's with thousands of states using currently available parallel machines. Finally, we present an efficient parallel algorithm for converting a regular expression into an $\\varepsilon$-free nondeterministic finite automaton. The resulting automaton is natural representation of a regular expression and has some important and practical uses in the document processing area. The time complexity of the algorithm is $O(\\log n)$ and the number of processors used is $O(n\\sp2).$

This research explores the design and implementa- tion of an image and video processing platform using recon- figurable FPGA technology. With the increasing demand for real- time, high-performance multimedia processing in modern applications, a flexible and efficient solution is essential. The project utilizes an FPGA-based platform to handle image and video data processing through edge detection, image scaling, and real-time motion detection techniques. By integrating custom- built hardware components, this study aims to provide a low- power, high-throughput solution that meets the requirements of both image and video processing tasks. Materials and methods involved using Xilinx Virtex-II Pro FPGA for hardware imple- mentation, alongside embedded memory processors and multi- functional design elements such as zoom-in/out utilities. The system was evaluated through several experimental procedures, using various image and video datasets, to measure performance in terms of speed, resource utilization, and power consumption. Results indicated that the system achieved notable improvements in terms of efficiency, utilizing 25% of logic resources, 55% of on- chip memory, and 180mW of power. The discussion focuses on the adaptability of FPGA-based systems in meeting the diverse needs of multimedia processing, emphasizing scalability and low power consumption. In conclusion, the reconfigurable FPGA platform provides a versatile and efficient environment for image and video processing. The system’s resource efficiency, scalability, and potential for further enhancements make it a promising candidate for future multimedia applications. Keywords-Reconfigurable FPGA Architecture, Video and Image Processing, Edge Detection, Image Rescaling.

This paper considers the combination of the general sum-of-processing-time effect and position-dependent effect on a single machine. The actual processing time of a job is defined by functions of the sum of the normal processing times of the jobs processed and its position and control parameter in the sequence. We consider two monotonic effect functions: the nondecreasing function and the nonincreasing function. Our focus is the following objective functions, including the makespan, the sum of the completion time, the sum of the weighted completion time, and the maximum lateness. For the nonincreasing effect function, polynomial algorithm is presented for the makespan problem and the sum of completion time problem, respectively. The latter two objective functions can also be solved in polynomial time if the weight or due date and the normal processing time satisfy some agreeable relations. For the nondecreasing effect function, assume that the given parameter is zero. We also show that the makespan problem can remain polynomially solvable. For the sum of the total completion time problem and a 1 is the deteriorating rate of the jobs, there exists an optimal solution for a 1 ≥ M ; a V-shaped property with respect to the normal processing times is obtained for 0 < a 1 ≤ 1 . Finally, we show that the sum of the weighted completion problem and the maximum lateness problem have polynomial-time solutions for a 1 > M under some agreeable conditions, respectively.

Adaptive Neighbors Graph Learning for Large-Scale Data Clustering using Vector Quantization and Self-Regularization

We introduce a notion of inexact model of a convex objective function, which allows for errors both in the function and in its gradient. For this situation, a gradient method with an adaptive adjustment of some parameters of the model is proposed and an estimate for the convergence rate is found. This estimate is optimal on a class of sufficiently smooth problems in the presence of errors. We consider a special class of convex nonsmooth optimization problems. In order to apply the proposed technique to this class, an artificial error should be introduced. We show that the method can be modified for such problems to guarantee a convergence in the function with a nearly optimal rate on the class of convex nonsmooth optimization problems. An adaptive gradient method is proposed for objective functions with some relaxation of the Lipschitz condition for the gradient that satisfy the Polyak--Lojasievicz gradient dominance condition. Here, the objective function and its gradient can be given inexactly. The adaptive choice of the parameters is performed during the operation of the method with respect to both the Lipschitz constant of the gradient and a value corresponding to the error of the gradient and the objective function. The linear convergence of the method is justified up to a value associated with the errors.