Computational geometry

여성사 쓰기의 가능성과 현주소, 한국여성연구소 여성사연구실 지음, 청년사 1999

This new textbook by R. E. Blahut, which deals with the theory and design of efficient algorithms for digital signal processing, contains perhaps the most comprehensive coverage of fast algorithms todate.Alargecollectionofalgorithmsistreated,withanemphasis on implementing the two canonical signal processing operations of convolution and discrete Fourier transformation. In recent years, there has been much work done on fast algorithms,andBlahutdoesafinejobofblendingmaterialfromdiverse sources to form a coherent and self-contained approach to his subject.The mathematical level of this book is high, reflecting the rather abstract nature of the theoretical underpinnings of fast computational techniques. Although electrical engineers are forthe most part mathematically sophisticated, they tend to lack training in abstract algebra and number theory, both of which are essential to any thorough discussion of fast algorithms. Thus this audience should find the tutorial chapters which the text provides on these topics to be quite helpful. An additional feature of the text, which the nonspecialist should find useful, is that each new algorithm is described through three different formats: a simple example, a flowchart, and a set of matrix equations. This use of repetition assists the reader in grasping subject matter which for the most part is nonintuitive. Operation counts (as measured by the number of multiplications and the number of additions) for each algorithm are tabulated for avarietyof blocklengths (i.e., lengths of data segments), making performance comparisons easy. As the author points out, run-time comparisons may be quite different. Each chapter concludes with a set of problems of varying difficulty. These problems are well integrated with the text and serve to supplement the many examples worked out in the text. The book is devoted to how one rapidly computes various mathematical operators such as transform and convolutions. For a deeper understanding of the meaning of theseoperators,one must consult other sources in which their use is discussed. The text emphasizes algorithms which employ a reduced, or minimum, numberof muItiplications,althoughadditioncountsarealsotaken into consideration. However, an algorithm which is the “fastest” as measured in arithmetic operation counts may not be the fastest in execution time, particularly if dedicated hardware is employed. Indeed in practiceother considerationsfrequentlypropel oneaway from the computationally ”optimal” algorithm. Much work has been done on the theory and application of signal processing algorithms which are “efficient” in terms other than rnultiply/add counts, such as roundoff noise, limit cycles, coefficient quantization, memory access, hardware costs, etc. It is clearly necessary to limit the scope of any treatise, and the exclusion of differing performance measures is certainly appropriate. A description of the contents of the book will now be given, followed by some concluding remarks of a more general nature. Chapter 2 i s a tutorial on abstract algebra. It i s quite readable and is liberally laced with examples. In addition to the standard modern algebra fare (groups, rings, fields, vector spaces, matrices), the ubiquitous Chinese remainder theorem is discussed in detail. Chapters 3 and 4, and their extensions in Chapters 7and 8, form the core of the text. The third chapter addresses fast algorithms for short convolutions. The Cook-Toom convolution algorithm is discussed, followed by the Winograd convolution algorithm. A proof of the optimality of the Winograd algorithm, with respect to multiplications, for performing cyclic convolutions, is presented at the close of the chapter. The fourth chapter addresses fast algorithms for computing the discrete Fourier transform. The CooleyTukey algorithm is considered first. The approach taken is to view this algorithm as a means of mapping a onedimensional Fourier transform into a multidimensional transform. Variations of the algorithm are discussed, including the Rader-Brenner transform. Next, the Good-Thomas algorithm is discussed. This algorithm is again presented as a means of mapping a onedimensional transform into a higher dimensional transform, this time based on the Chinese remainder theorem. Rader’s algorithm for computing primelength Fouriertransforms by useofconvolution ispresented next. Extensions of the algorithm to blocklengths which are the power of an odd prime are considered. The chapter closes with the Winograd-Fourier transform which builds upon the Rader prime algorithm. Certain short blocklengthsareconsidered in detail,and the corresponding algorithms are compiled into an Appendix. Chapter 5 i s a mathematical interlude, tutorially covering items from number theory and algebraic field theory which are needed in later chapters. Topics include the totient function, Euler’s theorem,  Fermat’s theorem, minimal polynomials, and cyclotomic polynomials. Chapter 6 is devoted to number theoretic transforms. These transforms proceed by representing the data values themselves in the field of integers modulo a prime. Convolution in integer fields is also covered. Chapters 7and 8 extend the convolution and transform methods of Chapters 3 and 4 to higher dimensions. Multidimensional transforms (convolutions) are used both to efficiently compute onedimensional transforms (convolutions) and to process data which are inherently higher dimensional. Both applications are treated in these chapters. Topics include the Agarwal-Cooley convolution algorithm, polynomial transforms, the family of Johnson-Burrus transforms and the Nussbaumer-Quandalle FFT. Chapter 9 discusses architectures for transforms and digital filters and includes treatmentsof FFT butterfly networks and overlapadd convolution. The remaining three chapters are mostly independent from the rest of the book. Chapter 10 covers fast algorithms based on doubling strategies. Computational tasks for which such fast algorithms are derived include sorting, matrix transposition, matrix multiplication, polynomial division, computation of trigonometric functions, and coordinate rotation. Many of theseoperations arise as steps in the solution of oneor more signal processing problems. Fast algorithms for solving Toeplitz systems is the theme of Chapter 11. There is a variety of fast algorithms discussed, the proper choice of which depends on the specific structure of the Toeplitz system at hand (such as whether or not the system is symmetric and whether or not the right-hand vector is arbitrary). The final chapter addresses fast algorithms for Trellis and tree search and includes the Viterbi, Stack, and Fano algorithms. These

Fast algorithms for digital signal processing

We have developed a pixellated high energy X-ray detector instrument to be used in a variety of imaging applications. The instrument consists of either a Cadmium Zinc Telluride or Cadmium Telluride (Cd(Zn)Te) detector bump-bonded to a large area ASIC and packaged with a high performance data acquisition system. The 80 by 80 pixels each of 250 μm by 250 μm give better than 1 keV FWHM energy resolution at 59.5 keV and 1.5 keV FWHM at 141 keV, at the same time providing a high speed imaging performance. This system uses a relatively simple wire-bonded interconnection scheme but this is being upgraded to allow multiple modules to be used with very small dead space. The readout system and the novel interconnect technology is described and how the system is performing in several target applications.

/pdf/pixellated-cd-zn-te-high-energy-x-ray-instrument-f21vyffig5.pdf

Pixellated Cd(Zn)Te high-energy X-ray instrument

Was 1974 a watershed? It saw an increase in the rate of technological change in the production of new equipment. It was the start of a sharp rise in income inequality. It signaled the beginning of the productivity slowdown. Were these phenomena related? Could they have been the result of an Industrial Revolution associated with the introduction of information technologies?

李觀雨著 新調査方法論, 螢雪出版社 1974

Large-volume CdZnTe (CZT) single crystals with electron lifetime exceeding 10 mus have recently become commercially available. This opened the opportunity for making room temperature CZT gamma-ray detectors with extended thicknesses and larger effective areas. However, the extended defects that are present even in the highest-quality material remain a major drawback which affects the availability and cost of large CZT detectors. In contrast to the point defects that control electron lifetime and whose effects on the charge collection can be electronically corrected, the extended defects introduce significant fluctuations in the collected charge, which increase with a crystal's thickness. The extended defects limit the uniformity in the electrons' drift distance in CZT crystals, above which electron trapping cannot effectively be corrected. In this paper, we illustrate the roles of the extended defects in CZT detectors with different geometries. We emphasize that the crystallinity of commercial CZT materials remains a major obstacle on the path to developing thick, large-volume CZT detectors for gamma-ray imaging and spectroscopy.

Extended Defects in CdZnTe Radiation Detectors

Given a (multi) set S of n positive integers and a target integer u, the subset sum problem is to decide if there is a subset of S that sums up to u. We present a series of new algorithms that compute and return all the realizable subset sums up to the integer u in O(min l Snu,u5/4,σ r), where σ is the sum of all elements of S and O hides polylogarithmic factors. We also present a modified algorithm for integers modulo m, which computes all the realizable subset sums modulo m in O(min l Snm,m5/4r) time.Our contributions improve upon the standard dynamic programming algorithm that runs in O(nu) time. To the best of our knowledge, the new algorithms are the fastest deterministic algorithms for this problem. The new results can be employed in various algorithmic problems, from graph bipartition to computational social choice. Finally, we also improve a result on covering Zm, which might be of independent interest.

Faster Pseudopolynomial Time Algorithms for Subset Sum

A closed curve in the plane is weakly simple if it is the limit (in the Frechet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.

Detecting Weakly Simple Polygons

Given a multiset S of n positive integers and a target integer t, the subset sum problem is to decide if there is a subset of S that sums up to t. We present a new divide-and-conquer algorithm that computes all the realizable subset sums up to an integer u in [EQUATION], where σ is the sum of all elements in S and O hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in O(nu) time. To the best of our knowledge, the new algorithm is the fastest general deterministic algorithm for this problem. We also present a modified algorithm for finite cyclic groups, which computes all the realizable subset sums within the group in [EQUATION] time, where m is the order of the group.

A faster pseudopolynomial time algorithm for subset sum

A closed curve in the plane is weakly simple if it is the limit (in the Frechet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.

/pdf/detecting-weakly-simple-polygons-378xnsn3fw.pdf

Chao Xu

Papers

Extended Defects in CdZnTe Radiation Detectors

Faster Pseudopolynomial Time Algorithms for Subset Sum

Detecting Weakly Simple Polygons

A faster pseudopolynomial time algorithm for subset sum

Detecting weakly simple polygons