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35
Cortical surfacebased analysis II: Inflation, flattening, and a surfacebased coordinate system
 NEUROIMAGE
, 1999
"... The surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds. As such, it is a difficult domain for computational as well as visualization purposes. We have therefore designed a set of procedures for modifying the representation of the c ..."
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Cited by 738 (57 self)
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The surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds. As such, it is a difficult domain for computational as well as visualization purposes. We have therefore designed a set of procedures for modifying the representation of the cortical surface to (i) inflate it so that activity buried inside sulci may be visualized, (ii) cut and flatten an entire hemisphere, and (iii) transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surfacebased coordinate system.
ThreeDimensional Face Recognition
, 2005
"... An expressioninvariant 3D face recognition approach is presented. Our basic assumption is that facial expressions can be modelled as isometries of the facial surface. This allows to construct expressioninvariant representations of faces using the bendinginvariant canonical forms approach. The re ..."
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Cited by 150 (24 self)
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An expressioninvariant 3D face recognition approach is presented. Our basic assumption is that facial expressions can be modelled as isometries of the facial surface. This allows to construct expressioninvariant representations of faces using the bendinginvariant canonical forms approach. The result is an efficient and accurate face recognition algorithm, robust to facial expressions, that can distinguish between identical twins (the first two authors). We demonstrate a prototype system based on the proposed algorithm and compare its performance to classical face recognition methods. The numerical methods employed by our approach do not require the facial surface explicitly. The surface gradients field, or the surface metric, are sufficient for constructing the expressioninvariant representation of any given face. It allows us to perform the 3D face recognition task while avoiding the surface reconstruction stage.
Hyperbolic geometry: the first 150 years
 Bull. Amer. Math. Soc., New Ser
, 1982
"... This will be a description of a few highlights in the early history of noneuclidean geometry, and a few miscellaneous recent developments. An Appendix describes some explicit formulas concerning volume in hyperbolic 3space. The mathematical literature on noneuclidean geometry begins in 1829 with ..."
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Cited by 82 (0 self)
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This will be a description of a few highlights in the early history of noneuclidean geometry, and a few miscellaneous recent developments. An Appendix describes some explicit formulas concerning volume in hyperbolic 3space. The mathematical literature on noneuclidean geometry begins in 1829 with publications by N. Lobachevsky in an obscure Russian journal. The infant subject grew very rapidly. Lobachevsky was a fanatically hard worker, who progressed quickly from student to professor to rector at his university of Kazan, on the Volga. Already in 1829, Lobachevsky showed that there is a natural unit of distance in noneuclidean geometry, which can be characterized as follows. In the right triangle of Figure 1 with fixed edge a, as the opposite vertex A moves infinitely far away, the angle 9 will increase to a limit 90 which is assumed to be strictly less than 7r/2. He showed that a =log tan(0o/2) if the unit of distance is suitably chosen. In particular, a « (TT/2) 0O if a is very small. (In the interpretation introduced by Beltrami forty years later, this unit of distance is chosen so that curvature =1.) FIGURE 1. A right triangle in hyperbolic space By early 1830, Lobachevsky was testing his "imaginary geometry " as a possible model for the real world. If the universe is noneuclidean in Lobachevsky's sense, then he showed that our solar system must be extremely small, in terms of this natural unit of distance. More precisely, taking the vertex A in Figure 1 to be the star Sirius and taking the edge a to be a suitably chosen radius of the Earth's orbit, he used the (unfortunately incorrect) estimate 7T — 20 ss 1.24 seconds of arc s 6 x 10~6 radians
Robotic Origami Folding
 IEEE INTERNATIONAL CONFERENCE ON ROBOTICS AND AUTOMATION
, 2004
"... Origami, the human art of paper sculpture, is a fresh challenge for the field of robotic manipulation, and provides a concrete example for many difficult and general manipulation problems. This thesis will present some initial results, including the world’s first origamifolding robot, some new theo ..."
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Cited by 36 (0 self)
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Origami, the human art of paper sculpture, is a fresh challenge for the field of robotic manipulation, and provides a concrete example for many difficult and general manipulation problems. This thesis will present some initial results, including the world’s first origamifolding robot, some new theorems about foldability, definition of a simple class of origami for which I have designed a complete automatic planner, analysis of the kinematics of more complicated folds, and some observations about the configuration spaces of compound spherical closed chains.
Modified Gravity and Cosmology
, 2012
"... In this review we present a thoroughly comprehensive survey of recent work on modified theories of gravity and their cosmological consequences. Amongst other things, we cover General Relativity, ScalarTensor, EinsteinAether, and Bimetric theories, as well as TeVeS, f(R), general higherorder theo ..."
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Cited by 28 (0 self)
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In this review we present a thoroughly comprehensive survey of recent work on modified theories of gravity and their cosmological consequences. Amongst other things, we cover General Relativity, ScalarTensor, EinsteinAether, and Bimetric theories, as well as TeVeS, f(R), general higherorder theories, HořavaLifschitz gravity, Galileons, Ghost Condensates, and models of extra dimensions including KaluzaKlein, RandallSundrum, DGP, and higher codimension braneworlds. We also review attempts to construct a Parameterised PostFriedmannian formalism, that can be used to constrain deviations from General Relativity in cosmology, and that is suitable for comparison with data on the largest scales. These subjects have been intensively studied over the past decade, largely motivated by rapid progress in the field of observational cosmology that now allows, for the first time, precision tests of fundamental physics on the scale of the observable Universe. The purpose of this review is to provide a reference tool for researchers and students in cosmology and gravitational physics, as well as a selfcontained, comprehensive and uptodate introduction to the subject as a whole.
Canonical quantum gravity
"... Abstract. This is a review of the aspirations and disappointments of the canonical quantization of geometry. I compare the two chief ways of looking at canonical gravity, geometrodynamics and connection dynamics. I capture as much of the classical theory as I can by pictorial visualization. Algebrai ..."
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Cited by 22 (0 self)
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Abstract. This is a review of the aspirations and disappointments of the canonical quantization of geometry. I compare the two chief ways of looking at canonical gravity, geometrodynamics and connection dynamics. I capture as much of the classical theory as I can by pictorial visualization. Algebraic aspects dominate my description of the quantization program. I address the problem of observables. The reader is encouraged to follow the broad outlines and not worry about the technical details. CLASSICAL CANONICAL GRAVITY Dynamical laws and instantaneous laws One of the main preoccupations of classical physics has been finding the laws governing physical data. One of the oldest schemes of quantization has been to subject such data to canonical commutation relations. I am going to review where this program leads us when it is applied to geometry. Classical physics deals with two kinds of laws: dynamical laws, and instantaneous laws. The discovery of dynamical laws started the Newtonian revolution. The first instantaneous law was found by Gauss: at any instant of time, the divergence of the electric field is determined by the distribution of charges. In empty space, the instantaneous electric field is divergencefree. Theorema egregium Without knowing it, and without most of us viewing it this way, Gauss also came across the fundamental instantaneous law of general relativity: the Hamiltonian constraint. This constraint is a simple reinterpretation of the famous result Gauss obtained when studying curved surfaces embedded in a flat Euclidean space [1]. 1 Figure 1. Intrinsic
The Campbell–Magaard Theorem is inadequate and inappropriate as a protective theorem for relativistic field equations, grqc/0409122
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Existence and uniqueness of complete constant mean curvature surfaces at infinity of H 3
 J. Geom. Anal
, 1999
"... A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in H3. It is shown that analogs of these equivalences exist for surfaces in S2∞, the bounding ideal sphere of H3, leading to a notion of constant mean curvature at infinity of H3. A parametrization of all ..."
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Cited by 3 (1 self)
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A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in H3. It is shown that analogs of these equivalences exist for surfaces in S2∞, the bounding ideal sphere of H3, leading to a notion of constant mean curvature at infinity of H3. A parametrization of all complete constant mean curvature surfaces at infinity of H3 is given by holomorphic quadratic differentials on Ĉ, C, and D. 1
1 DIFFERENTIAL INVARIANTS FOR HIGHER–RANK TENSORS. A PROGRESS REPORT
, 2008
"... We outline the construction of differential invariants for higher–rank tensors. 1 ..."
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We outline the construction of differential invariants for higher–rank tensors. 1