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Compatible Embedding for 2D Shape Animation
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 2008
"... Abstract—We present new algorithms for the compatible embedding of 2D shapes. Such embeddings offer a convenient way to interpolate shapes having complex, detailed features. Compared to existing techniques, our approach requires less user input, is faster, more robust, and simpler to implement, maki ..."
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Abstract—We present new algorithms for the compatible embedding of 2D shapes. Such embeddings offer a convenient way to interpolate shapes having complex, detailed features. Compared to existing techniques, our approach requires less user input, is faster, more robust, and simpler to implement, making it ideal for interactive use in practical applications. Our new approach consists of three parts. First, our boundary matching algorithm locates salient features using the perceptuallymotivated principles of scalespace and uses these as automatic correspondences to guide an elastic curve matching algorithm. Second, we simplify boundaries while maintaining their parametric correspondence and the embedding of the original shapes. Finally, we extend the mapping to shapes ’ interiors via a new compatible triangulation algorithm. The combination of our algorithms allows us to demonstrate 2D shape interpolation with instant feedback. The proposed algorithms exhibit a combination of simplicity, speed, and accuracy that has not been achieved in previous work.
Approximate convex decomposition and its applications
, 2006
"... Geometric computations are essential in many realworld problems. One important issue in geometric computations is that the geometric models in these problems can be so large that computations on them have infeasible storage or computation time requirements. Decomposition is a technique commonly us ..."
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Geometric computations are essential in many realworld problems. One important issue in geometric computations is that the geometric models in these problems can be so large that computations on them have infeasible storage or computation time requirements. Decomposition is a technique commonly used to partition complex models into simpler components. Whereas decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this work, we have developed an approximate technique, called Approximate Convex Decomposition (ACD), which decomposes a given polygon or polyhedron into “approximately convex ” pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficiently. Indeed, for many applications, an ACD can represent the important structural features of the model more accurately by providing a mechanism for ignoring less significant features, such as wrinkles and surface texture. Our study of a wide range of applications shows that in addition to providing computational efficiency, ACD also provides natural multiresolution or hierarchical representations. In this dissertation, we provide some examples of ACD’s many potential applications, such as particle simulation, mesh generation, motion planning, and skeleton extraction.
Chapter 1 Fundamentals
"... Real RAM Model. A memory cell stores a real number. Any single arithmetic operation or comparison can be computed in constant time. In addition, sometimes also roots, logarithms, other analytic functions, indirect addressing (integral), or floor and ceiling are used. This is a quite powerful (and so ..."
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Real RAM Model. A memory cell stores a real number. Any single arithmetic operation or comparison can be computed in constant time. In addition, sometimes also roots, logarithms, other analytic functions, indirect addressing (integral), or floor and ceiling are used. This is a quite powerful (and somewhat unrealistic) model of computation, as a single real number in principle can encode an arbitrary amount of information. Therefore we have to ensure that we do not abuse the power of this model. Algebraic Computation Trees (BenOr [1]). A computation is regarded as a binary tree. The leaves contain the (possible) results of the computation. Every node v with one child has an operation of the form +,−,,/,,... associated to it. The operands of this operation are constant input values, or among the ancestors of v in the tree. Every node v with two children has associated to it a branching of the form> 0,�0, or = 0. The branch is with respect to the result of v’s parent node. If the expression yields true, the computation continues with the left child of v; otherwise, it continues with the right child of v. a − b ≤ 0 a − c b − c
Chapter 1 Fundamentals
"... When designing algorithms, one has to agree on a model of computation according to which these algorithms can be executed. There are various such models, but when it comes to geometry some are more convenient to work with than others. Even using very elementary geometric operations—such as taking th ..."
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When designing algorithms, one has to agree on a model of computation according to which these algorithms can be executed. There are various such models, but when it comes to geometry some are more convenient to work with than others. Even using very elementary geometric operations—such as taking the center of a circle defined by three points or computing the length of a given circular arc—the realms of rational and even algebraic numbers are quickly left behind. Representing the resulting real numbers/coordinates would be a rather painful task in, for instance, a Turing machine type model of computation. Therefore, other models of computation are more prominent in the area of geometric algorithms and data structures. In this course we will be mostly concerned with two models: the Real RAM and the algebraic computation/decision tree model. The former is rather convenient when designing algorithms, because it sort of abstracts from the aforementioned representation issues by simply assuming that it can be done. The latter model typically appears in the context of lower bounds, that is, proofs that certain problems cannot be solved more efficiently than some function depending on the problem size (and possibly some other parameters). So let us see what these models are in more detail. Real RAM Model. A memory cell stores a real number (that is what the “Real ” stands for) 1. Any single arithmetic operation (addition, subtraction, multiplication, division, and kth root, for small constant k) or comparison can be computed in constant time. 2 This is a quite powerful (and somewhat unrealistic) model of computation, as a single real number in principle can encode an arbitrary amount of information. Therefore we 1 RAM stands for random access machine, meaning that every memory cell can be accessed in constant time. Not like, say, a list where one always has to start from the first element. 2 In addition, sometimes also logarithms, other analytic functions, indirect addressing (integral), or floor and ceiling are used. As adding some of these operations makes the model more powerful, it is usually specified and emphasized explicitly when an algorithm uses them.