In this thesis, we primarily focus on one common type of symmetry, the translational symmetry. We first review the current state-of-the-art methods for translational symmetry detection, and discuss their benefits and drawbacks. Towards an efficient, automatic and widely applicable translational symmetry detector, we develop a novel method for automatically detecting translational symmetry patterns, and extracting the corresponding lattices from images without pre-segmentation or reconstructing the underlying 3D geometry. In particular, we employ a region-based feature and fully utilize its regional properties (shape, orientation and well-defined boundary) to propose the repeated candidates. Compared with traditional treatments, which usually rely on point-based features and group them to propose repeated candidates, our treatment is more efficient and stable to perspective projection, distortion or noise. By clustering the candidate regions and indexing the major clusters using a GPU KD-tree, the parallel lattice formation processes turn out to be very efficient and achieve a real-time rate. By using a set of spatially varying vectors with a loose neighboring constraint to represent the underlying lattice, we successfully detect most of translational symmetry patterns over arbitrary surfaces, which can be planar or curve, without or with perspective projection, and even when suffered from global and local deformations. Moreover, the parallel searching and saving scheme enables us to simultaneously detect multiple disjoint symmetry patterns from an input images. / Symmetry has been an important concept in the nature, science and art. There is an abundant of biological, chemical, and artificial structures captured in many real-world images, exhibiting various forms of symmetry. The symmetry patterns and the repetitive elements reinforce the visual importance and usually make an image more attractive. Although our humans have an excellent innate ability in recognizing symmetry and perceiving its beauty, efficient and automatic symmetry detection from images remains a unsolved challenging problem in computer vision and graphics. Without understanding the high-level semantics of symmetry, editing such images while preserving the repetitions and their relations turns out to be difficult to perform, such as image resizing, image inpainting and image replacement. / The significant improvements of our method in both efficiency and accuracy make it a useful tool from which many applications can benefit. One of them is image resizing. We demonstrate that image resizing can be achieved more effectively if we have a better understanding of the image semantics. By analyzing the translational symmetry patterns, and detecting the underlying lattices in an image, we can summarize, instead of only distorting or cropping, the image content. This opens a new space for image resizing that allows us to manipulate, not only image pixels, but also the semantic cells in the lattice. As a general image contains both symmetry & non-symmetry regions and their natures are different, we propose to resize symmetry regions by summarization and non-symmetry region by optimized warping. In addition, by smoothing the intensity of cells across the lattice, we can further maintain the seamlessness of illumination during the summarization. As the difference in resizing strategy between symmetry regions and non-symmetry region leads to discontinuity at their shared boundary, we propose a framework to minimize the artifact. Experimental results show that, with the high-level knowledge of symmetry, our method outperforms the state-of-the-art resizing techniques. / Wu, Huisi. / Advisers: Tien-Tsin Wong; Pheng-Ann Heng. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 92-100). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

在約束滿足問題中，破壞指數量級數量的所有對稱通常過於昂貴。在實踐中，我們通常只有效地破壞對稱的一個子集。我們稱之為目標對稱。在靜態對稱破壞中，我們的目標是發佈一套約束去破壞這些目標對稱，以達到減少解集以及搜索空間的效果。一個問題中的所有對稱之間是互相交織的。一個旨在特定對稱的破壞對稱約束几乎總會產生副作用，而不僅僅破壞了預期的對稱。破壞相同目標對稱的不同約束可以有不同的副作用。傳統智慧告訴我們應該選擇一個破壞更多對稱從而有更多副作用的破壞對稱約束。雖然這樣的說法在許多方面上都是有效的，我們應該更加注意副作用發生的地方。 / 給與一個約束滿足問題，一個對稱被一個約束保留當且僅當該對稱仍然是新的約束滿足問題的對稱。這個新的約束滿足問題是有原問題加上該約束組成的。我們給出定律和例子，以表明發佈儘量保留目標對稱以及限制它的副作用發生在非目標對稱上的破壞約束是有利的。這些好處來自于被破壞的對稱數目以及一個對稱被破壞（或消除）的程度，并導致一個較小的解集和搜索空間。但是，對稱不一定會被保留。我們顯示，旨在一個已經被消除的目標對稱的破壞對稱約束仍然可以被發佈。我們建議根據問題的約束以及其他破壞對稱約束來選擇破壞對稱約束，以繼續消除更多的對稱。我們進行了廣泛的實驗來確認我們的建議的可行性與效率。 / Breaking the exponential number of all symmetries of a constraint satisfaction problem is often too costly. In practice, we often aim at breaking a subset of the symmetries efficiently, which we call target symmetries. In static sym-metry breaking, the goal is to post a set of constraints to break these target symmetries in order to reduce the solution set and thus also the search space. Symmetries of a problem are all intertwined. A symmetry breaking constraint intended for a particular symmetry almost always breaks more than just the intended symmetry as a side-effect. Different constraints for breaking the same target symmetry can have different side-effects. Conventional wisdom suggests that we should select a symmetry breaking constraint that has more side-effects by breaking more symmetries. While this wisdom is valid in many ways, we should be careful where the side-effects take place. / A symmetry σ of a CSP P =(V, D, C) is preserved by a set of symmetry breaking constraints C{U+02E2}{U+1D47} i σ is a symmetry of P¹ =(V, D, CU C{U+02E2}{U+1D47}). We give theorems and examples to demonstrate that it is beneficial to post symmetry breaking constraints that preserve the target symmetries and restrict the side-effects to only non-target symmetries as much as possible. The benefits are in terms of the number of symmetries broken and the extent to which a symmetry is broken (or eliminated), resulting in a smaller solution set and search space. However, symmetry preservation may not always hold. We illustrate that symmetry breaking constraints, which aim at a target symmetry that is already eliminated, can still be posted. To continue eliminating more symmetries, we suggest to select symmetry breaking constraints based on problem constraints and other symmetry breaking constraints. Extensive experiments are also conducted to confirm the feasibility and efficiency of our proposal empirically. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Li, Jingying. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 101-112). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Constraint Satisfaction Problems --- p.1 / Chapter 1.2 --- Motivation and Goals --- p.3 / Chapter 1.3 --- Outline of Thesis --- p.5 / Chapter 2 --- Background --- p.8 / Chapter 2.1 --- Constraint Satisfaction Problems --- p.8 / Chapter 2.1.1 --- Backtracking Search --- p.9 / Chapter 2.1.2 --- Consistency Techniques --- p.12 / Chapter 2.1.3 --- Local Consistencies with Backtracking Search --- p.15 / Chapter 2.2 --- Symmetry Breaking in CSPs --- p.16 / Chapter 2.2.1 --- Symmetry Classes --- p.18 / Chapter 2.2.2 --- Breaking Symmetries --- p.22 / Chapter 2.2.3 --- Variable and Value Symmetries --- p.23 / Chapter 2.2.4 --- Symmetry Breaking Constraints --- p.26 / Chapter 3 --- Effects of Symmetry Breaking Constraints --- p.29 / Chapter 3.1 --- Removing Symmetric Search Space --- p.29 / Chapter 3.1.1 --- Properties --- p.30 / Chapter 3.1.2 --- Canonical Variable Orderings --- p.31 / Chapter 3.1.3 --- Regenerating All Solutions --- p.33 / Chapter 3.1.4 --- Remaining Solution Set Sizes --- p.36 / Chapter 3.2 --- Constraint Interactions in Propagation --- p.43 / Chapter 4 --- Choices of Symmetry Breaking Constraints --- p.45 / Chapter 4.1 --- Side-Effects --- p.45 / Chapter 4.2 --- Symmetry Preservation --- p.50 / Chapter 4.2.1 --- De nition and Properties --- p.50 / Chapter 4.2.2 --- Solution Reduction --- p.54 / Chapter 4.2.3 --- Preservation Examples --- p.55 / Chapter 4.2.4 --- Preserving Order --- p.64 / Chapter 4.3 --- Eliminating Eliminated Symmetries --- p.65 / Chapter 4.3.1 --- Further Elimination --- p.65 / Chapter 4.3.2 --- Aggressive Elimination --- p.71 / Chapter 4.4 --- Interactions with Problem Constraints --- p.72 / Chapter 4.4.1 --- Further Simplification --- p.72 / Chapter 4.4.2 --- Increasing Constraint Propagation --- p.73 / Chapter 5 --- Experiments --- p.75 / Chapter 5.1 --- Symmetry Preservation --- p.75 / Chapter 5.1.1 --- Diagonal Latin Square Problem --- p.76 / Chapter 5.1.2 --- NN-Queen Problem --- p.77 / Chapter 5.1.3 --- Error Correcting Code - Lee Distance (ECCLD) --- p.78 / Chapter 5.2 --- Eliminating Eliminated Symmetries --- p.80 / Chapter 5.2.1 --- Equidistance Frequency Permutation Array Problem --- p.80 / Chapter 5.2.2 --- Cover Array Problem --- p.82 / Chapter 5.2.3 --- Sports League Scheduling Problem --- p.83 / Chapter 6 --- Related Work --- p.86 / Chapter 6.1 --- Symmetry Breaking Approaches --- p.86 / Chapter 6.2 --- Reducing Overhead and Increasing Propagation --- p.90 / Chapter 6.3 --- Selecting and Generating Choices --- p.91 / Chapter 6.3.1 --- Reducing Conflict with Search Heuristic --- p.92 / Chapter 6.3.2 --- Choosing the Subset of Symmetries --- p.93 / Chapter 6.4 --- Detecting Symmetries --- p.93 / Chapter 7 --- Conclusion and Remarks --- p.95 / Chapter 7.1 --- Conclusion --- p.95 / Chapter 7.2 --- Discussions --- p.97 / Chapter 7.3 --- Future Work --- p.99 / Bibliography --- p.101

Mode of access: World Wide Web. / Thesis (Ph. D.)--University of Hawaii at Manoa, 2004. / Includes bibliographical references (leaves 159-164). / Electronic reproduction. / Also available by subscription via World Wide Web / x, 164 leaves, bound ill. 29 cm

Thesis (Ph. D.)--Oregon State University, 2010. / Printout. Includes bibliographical references (leaves 96-102). Also available on the World Wide Web.

Uses the techniques of symmetric presentations to manipulate elements of large sporadic groups and to represent elements of these groups in much shorter forms than their corresponding permutation or matrix representation. Undertakes to develop a nested algorithm and a computer program to manipulate elements of large sporadic groups.

Variable symmetries in constraint satisfaction problems can be broken by adding lexicographic ordering constraints. Existing general methods of generating such sets of ordering constraints can produce a huge number of additional constraints. This adds an unacceptable overhead to the solving process. Methods exist by which this large set of constraints can be reduced to a much smaller set automatically, but their application is also prohibitively costly. In contrast, this thesis takes a bottom up approach to generating symmetry breaking constraints. This will involve examining some commonly-occurring families of mathematical groups and deriving a general formula to produce a minimal set of ordering constraints which are sufficient to break all of the symmetry that each group describes. In some cases it is known that there exists no manageable sized sets of constraints to break all symmetries. One example of this occurs with matrix row and column symmetries. In such cases, incomplete symmetry breaking has been used to great effect. Double lex is a commonly used incomplete symmetry breaking technique for row and column symmetries. This thesis also describes another similar method which compares favourably to double lex. The general formulae investigated are used as building blocks to generate small sets of ordering constraints for more complex groups, constructed by combining smaller groups. Through the utilisation of graph automorphism tools and the groups and permutations software GAP we provide a method of defining variable symmetries in a problem as a group. Where this group can be described as the product of smaller groups, with known general formulae, we can construct a minimal set of ordering constraints for that problem automatically. In summary, this thesis provides the theoretical background necessary to apply efficient static symmetry breaking to constraint satisfaction problems. It also goes further, describing how this process can be automated to remove the necessity of having an expert CP practitioner, thus opening the field to a larger number of potential users.

This dissertation has two areas of interest with regard to the four dimensional Kerr black hole; the rst being its conformal nature in its near region and second it characteristic frequencies. With it now known that the scalar solution space of the four dimensional Kerr black hole has a two dimensional conformal symmetry in its near region, it was the rst focus of this dissertation to see if this conformal symmetry is unique to the near region scalar solution space or if it is also present in the spin-half solution space. The second focus of this dissertation was to explore techniques which can be used to calculate these quasinormal mode (characteristic) frequencies, such as the WKB(J) approximation which has been improved from third order to sixth order recently and applied to the perturbations of a Schwarzschild black hole. The additional correction terms show a signi cant increase of accuracy when comparing to numerical methods. This dissertation shall use the sixth order WKB(J) method to calculate the quasinormal mode frequencies for both the scalar and spin-half perturbations of a four dimensional Kerr black hole. An additional method used was the asymptotic iteration method, a relatively new technique being used to calculate the quasinormal mode frequencies of black holes that have been perturbed. Prior to this dissertation it had only been used on a variety of Schwarzschild black holes and their possible perturbations. For this dissertation the asymptotic iteration method has been used to calculate the quasinormal frequencies for both the scalar and spin-half perturbations of the four dimensional Kerr black hole. The quasinormal mode frequencies calculated using both the sixth order WKB(J) method and the asymptotic iteration method were compared to previously published values and each other. For the most part, they both compare favourably with the numerical values, with di erences that are near negligible. The di erences did become more apparent when the mode number (or angular momentum per unit mass increased), but less so when the angular number increased. The only factor that separates these two methods signi cantly, was that the computational time for the sixth order WKB(J) method is less than than that of the asymptotic iteration method.

There is a well established connection between one parameter Lie groups of transformations and conservation laws for differential equations. In this thesis, we construct conservation laws via the invariance and multiplier approach based on the wellknown result that the Euler-Lagrange operator annihilates total divergences. This technique will be applied to some plasma physics models. We show that the recently developed notion of the association between Lie point symmetry generators and conservation laws lead to double reductions of the underlying equation and ultimately to exact/invariant solutions for higher-order nonlinear partial di erential equations viz., some classes of Schr odinger and KdV equations.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, in requirement for the degree Doctor of Philosophy, Johannesburg, 2013. / The (1+3) dimensional wave and Klein-Gordon equations are constructed using the covariant d'Alembertian operator on several spacetimes of interest. Equations on curved geometry inherit the nonlinearities of the geometry. These equations display interesting properties in a number of ways. In particular, the number of symmetries and therefore, the conservation laws reduce depending on how curved the manifold is. We study the symmetry properties and conservation laws of wave equations on Freidmann-Robertson-Walker, Milne, Bianchi, and de Sitter universes. Symmetry structures are used to reduce the number of unknown functions, and hence contribute to nding exact solutions of the equations. As expected, properties of reduction procedures using symmetries, variational structures and conservation laws are more involved than on the well known at (Minkowski) manifold.

Thesis (MSc)--University of Stellenbosch, 2008. / Bibliography. Also available via the Internet.