Browsing by Author "Goldman, Ron"
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Item A computational framework for evaluating outcomes in infant craniosynostosis reconstruction(2016-02-11) Yuan, Binhang; Goldman, RonHistorically, surgical outcomes in craniosynostosis have been evaluated by qualitative analysis, direct and indirect anthropometry, cephalometrics, and CT craniometric analysis. Three-dimensional meshes constructed from 3dMD images acquired on patients with synostosis at multiple times across the course of surgical treatment provide ideal raw data for a novel approach to 3D geometric shape analysis of surgical results. We design a automatic computational framework for evaluating and visualizing the results of infant cranial surgeries based on 3dMD images. The goal of this framework is to assist surgeons in evaluating the efficacy of their surgical techniques. Feedback from surgeons in Texas Children's Hospital confirms that this framework is a robust computational system within which surgical outcomes in synostosis can be accurately and meaningfully evaluated. We also propose an algorithm to generate normative infant cranial models from the input of 3D meshes, which are extracted from CT scans of normal infant skulls. Comparing of the head shape of an affected subject with a normal control will more clearly illustrate in what aspect the subject's head deviates from the norm. Comparing of a post-treatment subject's head shape and an age-matched control would allow assessing of a specific treatment approach or surgical technique.Item A computational method for constructing Sylvester-style sparse resultants(2005) Song, Ning; Goldman, RonWe present a computational approach for constructing Sylvester style resultants for sparse systems of bivariate polynomial equations. Necessary and sufficient conditions are derived which guarantee that a multiplying set of monomials generates an exact Sylvester style resultant for three bivariate polynomials with a given planar Newton polygon. These conditions include a set of Diophantine equations that can be solved to generate multiplying sets of monomials and therefore the corresponding Sylvester resultants. We have implemented this method in Mathematica, and the results show that such Sylvester style sparse resultants often exist, and they appear in certain specific patterns. This method of Diophantine equations can also be used together with moving planes and moving quadrics [16, 17] to find the implicit equation of a rational surface. Moving planes and moving quadrics were originally introduced for tensor product surfaces---that is, bivariate polynomial systems whose Newton polygons are rectangles. Now by a method similar to our technique for generating Sylvester style sparse resultants, we can use moving quadrics to generate implicit equations for certain rational parametric surfaces whose Newton polygons are not rectangles.Item A Polynomial Blossom for the Askey–Wilson Operator(Springer, 2018) Simeonov, Plamen; Goldman, RonWe introduce a blossoming procedure for polynomials related to the Askey–Wilson operator. This new blossom is symmetric, multiaffine, and reduces to the complex representation of the polynomial on a certain diagonal. This Askey–Wilson blossom can be used to find the Askey–Wilson derivative of a polynomial of any order. We also introduce a corresponding Askey–Wilson Bernstein basis for which this new blossom provides the dual functionals. We derive a partition of unity property and a Marsden identity for this Askey–Wilson Bernstein basis, which turn out to be the terminating versions of Rogers’ 6ϕ5 summation formula and a very-well-poised 8ϕ7 summation formula. Recurrence and symmetry relations and differentiation and degree elevation formulas for the Askey–Wilson Bernstein bases, as well as degree elevation formulas for Askey–Wilson Bézier curves, are also given.Item Analytic functions in computer-aided geometric design(2002) Morin, Geraldine; Goldman, RonThis thesis presents a new paradigm for geometric modeling based on analytic functions. This model includes not only a representation of analytic curves and surfaces, but also tools and algorithms to manipulate this representation. Analytic functions on a given domain represent a very large class of infinitely smooth functions, including trigonometric functions and functions with poles outside the domain. Thus, the model is very rich; in particular the model is able to represent an object of optimal smoothness as well as functions as close as desired to singularities. The Bezier representation for polynomials generalizes to the Poisson representation for analytic curves and surfaces. The coefficients in the Poisson basis not only characterize an analytic function, but also are geometrically meaningful and intuitive control parameters for the curve or surface the function defines---as the Bezier control points are for polynomial shapes. Based on this Poisson representation, we derive standard geometric modeling algorithms for analytic curves and surfaces, including subdivision, trimming, evaluation and change of basis procedures. We also develop a notion of blossoming, as well as a de Boor-Fix formula and a Marsden identity, for analytic curves. These algorithms and tools provide an efficient and complete framework for using analytic functions in geometric modeling.Item Birational Quadratic Planar Maps with Generalized Complex Rational Representations(MDPI, 2023) Wang, Xuhui; Han, Yuhao; Ni, Qian; Li, Rui; Goldman, RonComplex rational maps have been used to construct birational quadratic maps based on two special syzygies of degree one. Similar to complex rational curves, rational curves over generalized complex numbers have also been constructed by substituting the imaginary unit with a new independent quantity. We first establish the relationship between degree one, generalized, complex rational Bézier curves and quadratic rational Bézier curves. Then we provide conditions to determine when a quadratic rational planar map has a generalized complex rational representation. Thus, a rational quadratic planar map can be made birational by suitably choosing the middle Bézier control points and their corresponding weights. In contrast to the edges of complex rational maps of degree one, which are circular arcs, the edges of the planar maps can be generalized to hyperbolic and parabolic arcs by invoking the hyperbolic and parabolic numbers.Item de Boor-suitable (DS) T-splines of Bidegree (3, 3)(2018-11-30) Zhang, Yang; Goldman, RonNecessary and sufficient conditions under which the de Boor algorithm can be applied to evaluate points on a T-spline surface of bidegree (3, 3) are investigated. These de Boor-suitable (DS) T-splines are compared to the standard analysis-suitable (AS) T-splines. An algorithm is also developed to search in a T-mesh for the appropriate control points and blend them using the de Boor algorithm to compute the corresponding point on the T-spline surface.Item Functional representation and manipulation of shapes with applications in surface and solid modeling(2013-09-16) Feng, Powei; Warren, Joe; Goldman, Ron; O'Malley, Marcia K.Real-valued functions have wide applications in various areas within computer graphics. In this work, we examine three representation of shapes using functions. In particular, we study the classical B-spline representation of piece-wise polynomials in the univariate domain. We provide a generalization of B-spline to the bivariate domain using intuition gained from the univariate construction. We also study the popular scheme of representing 3D density distribution using a uniform, rectilinear grid, where we provide a novel contouring scheme that culls occluded inner geometries. Lastly, we examine a ray-based representation for 3D indicator functions called ray-rep, for which we present a novel meshing scheme with multi-material extensions.Item Kodaira dimensions of some moduli spaces of special hyperkähler fourfolds(2020-08-10) Petok, Jack; Várilly-Alvarado, Anthony; Goldman, RonWe study the Noether-Lefschetz locus of the moduli space $\mathcal{M}$ of $K3^{[2]}$-fourfolds with a polarization of degree $2$. Following Hassett's work on cubic fourfolds, Debarre, Iliev, and Manivel have shown that the Noether-Lefschetz locus in $\mathcal{M}$ is a countable union of special divisors $\mathcal{M}_d$, where the discriminant $d$ is a positive integer congruent to $0,2,$ or $4$ modulo 8. In this thesis, we compute the Kodaira dimensions of these special divisors for all but finitely many discriminants; in particular, we show that for $d>176$ and for many other small values of $d$, the space $\mathcal{M}_d$ is a variety of general type. The main idea of the proof is to study the Kodaira dimension of the moduli spaces using the "quasi-pullback" trick of Gritsenko-Hulek-Sankaran: by explicitly constructing certain modular forms on the period domain, we can show the plurigenera of a smooth compactification of $\mathcal{M}_d$ grow fast enough to conclude that $\mathcal{M}_d$ is of general type for all but $40$ values of $d$. We also give information about the Kodaira dimension of $\mathcal{M}_d$ for 6 additional values of $d$, leaving only 34 values of $d$ for which we cannot yet say anything about the Kodaira dimension.Item Lower order solvability of links(2013-09-16) Martin, Taylor; Harvey, Shelly; Cochran, Tim D.; Goldman, RonThe n-solvable filtration of the link concordance group, defined by Cochran, Orr, and Teichner in 2003, is a tool for studying smooth knot and link concordance that yields important results in low-dimensional topology. We focus on the first two stages of the n-solvable filtration, which are the class of 0-solvable links and the class of 0.5-solvable links. We introduce a new equivalence relation on links called 0-solve equivalence and establish both an algebraic and a geometric characterization 0-solve equivalent links. As a result, we completely characterize 0-solvable links and we give a classification of links up to 0-solve equivalence. We relate 0-solvable links to known results about links bounding gropes and Whitney towers in the 4-ball. We then establish a sufficient condition for a link to be 0.5-solvable and show that 0.5-solvable links must have vanishing Sato-Levine invariants.Item On geometry along grafting rays in Teichmuller space(2012-09-05) Laverdiere, Renee; Wolf, Michael; Hardt, Robert M.; Goldman, RonIn this work, we investigate the mid-range behavior of geometry along a grafting ray in Teichm\"{u}ller space. The main technique is to describe the hyperbolic metric $$\sigma_{t}$$ at a point along the grafting ray in terms of a conformal factor $$g_{t}$$ times the Thurston (grafted) metric and study solutions to the linearized Liouville equation. We give a formula that describes, at any point on a grafting ray, the change in length of a sum of distinguished curves in terms of the hyperbolic geometry at the point. We then make precise the idea that once the length of the grafting locus is small, local behavior of the geometry for grafting on a general manifold is like that of grafting on a cylinder. Finally, we prove that the sum of lengths of is eventually monotone decreasing along grafting rays.Item q-Blossoming for analytic functions(Springer, 2018) Goldman, Ron; Simeonov, PlamenWe construct aᅠq-analog of the blossom for analytic functions, the analyticᅠq-blossom. Thisᅠq-analog also extends the notion ofᅠq-blossoming from polynomials to analytic functions. We then apply this analyticᅠq-blossom to derive identities for analytic functions represented in terms of theᅠq-Poisson basis, includingᅠq-versions of the Marsden identity and the de Boor-Fix formula for analytic functions.Item Rational curves over generalized complex numbers(Elsevier, 2019) Du, Juan; Goldman, Ron; Wang, XuhuiComplex rational curves have been used to represent circular splines as well as many classical curves including epicycloids, cardioids, Joukowski profiles, and the lemniscate of Bernoulli. Complex rational curves are known to have low degree (typically half the degree of the corresponding rational planar curve), circular precision, invariance with respect to Möbius transformations, special implicit forms, an easy detection procedure, and a fast algorithm for computing their μ-bases. But only certain very special rational planar curves are also complex rational curves. To construct a wider collection of curves with similar appealing properties, we generalize complex rational curves to hyperbolic and parabolic rational curves by invoking the hyperbolic and parabolic numbers. We show that the special properties of complex rational curves extend to these hyperbolic and parabolic rational curves. We also provide examples to flesh out the theory.Item Strong $\mu$-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity(Society for Industrial and Applied Mathematics, 2017) Shen, Li-Yong; Goldman, RonWe investigate conditions under which the resultant of a $\mu$-basis for a rational tensor product surface is the implicit equation of the surface without any extraneous factors. In this case, we also derive a formula for the implicit degree of the rational surface based only on the bidegree of the rational parametrization and the bidegrees of the elements of the $\mu$-basis without any knowledge of the number or multiplicities of the base points, assuming only that all the base points are local complete intersections. We conclude that in this case the implicit degree of a rational surface of bidegree $(m,n)$ is at most $mn$, so the rational surface must have at least $mn$ base points counting multiplicity. When the resultant of a $\mu$-basis generates extraneous factors, we show how to predict and compute these extraneous factors from either the existence of bad base points or anomalies occurring in the parametrization at infinity. Examples are provided to flesh out the theory.Item Syzygies for translational surfaces(Elsevier, 2018) Wang, Haohao; Goldman, RonA translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces are invariant under rigid motions: translating and rotating the two generating curves translates and rotates the translational surface by the same amount. We construct three special syzygies for a translational surface from a μ-basis of one of the generating space curves, and we show how to compute the implicit equation of a translational surface from these three special syzygies. Examples are provided to illustrate our theorems and flesh out our algorithms.Item Three approaches to building curves and surfaces in computer-aided geometric design(1997) Habib, Ayman Wadie; Goldman, RonModeling free-form curves and surfaces is one of the fundamental problems in computer aided geometric design. To solve this problem, several modeling techniques have been proposed. Three of these techniques, are investigated. The unifying theme of these three techniques is the use and the control of geometric continuity. The first technique deals with constructing parametric spline curves with controlled continuity between the spline segments at the knots. An axiomatic approach to geometric continuity for parametric representations is proposed. Based on this totally algebraic approach, many new flexible notions of continuity are developed. Corresponding to these notions, new spline curves are constructed in a way that gives the designer more control over the curve shape. Many examples are given. When derivative information is available Hermite interpolation can be used to build high continuity surfaces. A dynamic programming algorithm that solves the problem of interpolating bivariate Hermite data where the interpolation positions are aligned on a triangular grid is developed and analyzed. The third geometric continuity problem arises when modeling with subdivision surfaces, in reducing the continuity of these surfaces to allow for the insertion of sharp edges/vertices on these surfaces. A new approach to solving this problem is introduced and analyzed with illustrative examples.Item Topics in resultants and implicitization(2000) Zhang, Ming; Goldman, RonResultants are computational tools for determining whether or not a system of polynomials has a common root without actually solving for the roots of these equations. Resultants can also be used to solve for the common roots of polynomial systems. Classical resultants are typically represented as determinants whose entries are polynomials in the coefficients of the original polynomials in the system. The work in this dissertation on classical resultants focuses on bivariate polynomials. It is shown that bivariate resultants can be represented as determinants in a variety of innovative ways and that these various formulations are interrelated. Remarkable internal structures in these resultant matrices are exposed. Based on these structures, efficient computational algorithms for calculating the entries of these resultant matrices are developed. Sparse resultants are used for solving systems of sparse polynomials, where classical resultants vanish identically and hence fail to give any useful information about the common roots of the sparse polynomials. Nevertheless, sparse polynomial systems frequently appear in surface design. Sparse resultants are usually represented as GCDs of a collection of determinants. These GCDs are extremely awkward for symbolic computation. Here a new way is presented to construct sparse resultants as single determinants for a large collection of sparse systems of bivariate polynomials. An important application of both classical and sparse resultants in geometric modeling is implicitization. Implicitization is the process of converting surfaces from parametric form into algebraic form. Classical resultant methods fail when a rational surface has base points. The method of moving quadrics, first introduced by Professor Tom Sederberg at Brigham Young University, is known empirically to successfully implicitize rational surfaces with base points. But till now nobody has ever been able to give a rigorous proof of the validity of this technique. The first proof of the validity of this method when the surfaces have no base points is provided in this dissertation.