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Purpose Prospective motion correction for MRI and other imaging modalities are commonly based on the assumption of affine motion, i.e., rotations, shearing, scaling and translations. In addition it often involves transformations between different reference frames, especially for applications with an external tracking device. The goal of this work is to develop a computational framework for motion correction based on homogeneous transforms. Theory and Methods The homogeneous representation of affine transformations uses 4 × 4 transformation matrices applied to four-dimensional augmented vectors. It is demonstrated how homogenous transforms can be used to describe the motion of slice objects during an MRI scan. Furthermore, we extend the concept of homogeneous transforms to gradient and k-space vectors, and show that the fourth dimension of an augmented k-space vector encodes the complex phase of the corresponding signal sample due to translations. Results The validity of describing motion tracking in real space and k-space using homogeneous transformations only is demonstrated on phantom experiments. Conclusion Homogeneous transformations allows for a conceptually simple, consistent and computationally efficient theoretical framework for motion correction applications. Magn Reson Med 75:274–279, 2016. © 2015 Wiley Periodicals, Inc.

We present a novel framework for polyhedral mesh editing with face-based projective maps that preserves planarity by definition. Such meshes are essential in the field of architectural design and rationalization. By using homogeneous coordinates to describe vertices, we can parametrize the entire shape space of planar-preserving deformations with bilinear equations. The generality of this space allows for polyhedral geometric processing methods to be conducted with ease. We demonstrate its usefulness in planar-quadrilateral mesh subdivision, a resulting multi-resolution editing algorithm, and novel shape-space exploration with prescribed transformations. Furthermore, we show that our shape space is a discretization of a continuous space of conjugate-preserving projective transformation fields on surfaces. Our shape space directly addresses planar-quad meshes, on which we put a focus, and we further show that our framework naturally extends to meshes with faces of more than four vertices as well.

The consolidation of the link among four fields in computational mathematics and mechanics is the main objective of this work. Surfaces based on NURBS (non-uniform rational B-spline), mathematical optimization, the finite element method (FEM) in structural analysis, and automatic differentiation (AD) are applied in shape optimization of shells. This problem is performed taking into account the fact that material and mechanical characteristics influence both, the structural shape and the thickness variation, in order to obtain the best performance with respect to a specific criterium. Some techniques were implemented to modify the shell geometry conserving the same parameterization without a new finite element mesh generation. The shape modification is carried out using an optimization code based on the data obtained by a finite element analysis and gradients evaluation. In this work the optimization procedure is performed using an SQP (Sequential Quadratic Programming) algorithm, where the variables are the control points in homogeneous coordinates, the knot vectors, and the thickness. The functional value is determined by the FEM and gradients are evaluated using AD. Some examples are analyzed and discussed. As a consequence of the shape optimization, shells with high structural performance and aesthetically beautiful shapes can be obtained. Copyright © 2011 John Wiley & Sons, Ltd.

The natural homogeneous coordinate system is the analog of the Cartesian coordinate system for projective geometry. Roughly speaking a projective geometry adds an axiom that parallel lines meet at a point at infinity. This removes the impediment to line-point duality that is found in traditional Euclidean geometry. The natural homogeneous coordinate system is surprisingly useful in a number of applications including computer graphics and statistical data visualization. In this article, we describe the axioms of projective geometry, introduce the formalism of natural homogeneous coordinates, and illustrate their use with four applications. WIREs Comp Stat 2010 2 678–685 DOI: 10.1002/wics.122 For further resources related to this article, please visit the WIREs website.

Optical methods of contouring that utilise image recordings by cameras are based on a fundamental discipline, projective geometry. The 3-D world is projected in 2-D utilising a camera modelled in the technical literature by the pinhole camera. To get back 3-D information, the fundamental property measured is parallax. Parallax is a vector resulting from the difference of the projective coordinates of a point in space when projected onto a plane from two different points. The oldest method used for measuring parallax is photogrammetry. It is assumed to be the most precise technique, with the capability of obtaining 10−5 of the largest dimension of the measured object. This study summarises the state-of-the-art methods based on projecting a spatial carrier. Starting with the concept of moire as a form of photogrammetry, the different optical techniques for parallax determination are discussed. Although the moire method has reached 1 μm accuracy in laboratory work, a question remains: can moire become a standardised contouring technique yielding 10−6 m accuracy? This study is devoted to the analysis of high accuracy contour measurements, through both theoretical derivations and experimental verifications.

This paper presents the design and implementation of a hyperelliptic curve cryptography (HECC) coprocessor over affine and projective coordinates, along with measurements of its performance, hardware complexity, and power consumption. We applied several design techniques, including parallelism, pipelining, and loop unrolling, in designing field arithmetic units, group operation units, and scalar multiplication units to improve the performance and power consumption. Our affine and projective coordinate-based HECC processors execute in 0.436 ms and 0.531 ms, respectively, based on the underlying field GF(2 89 ). These results are about five times faster than those for previous hardware implementations and at least 13 times better in terms of area-time products. Further results suggest that neither case is superior to the other when considering the hardware complexity and performance. The characteristics of our proposed HECC coprocessor show that it is applicable to high-speed network applications as well as resource-constrained environments, such as PDAs, smart cards, and so on.

The sphericality of tangential varieties of rational homogeneous varieties is determined. The homogeneous coordinate rings and rings of covariants of the tangential varieties of homogenously embedded compact Hermitian symmetric spaces (CHSS) are determined. Bounds on the degrees of generators of the ideals of tangential varieties of CHSS are given, and more explicit information is obtained about the ideals in certain cases.

Direct solutions are very attractive because they obviate the need for initial approximations associated with non-linear solutions. The direct linear transformation (DLT) establishes itself as a method of choice for direct solutions in photogrammetry and other fields. The use of the DLT with coplanar object space points leads to a rank deficient model. This rank deficient model leaves the DLT defined up to a 2D projective transformation, which makes the direct retrieval of the exterior orientation parameters (EOPs) a non-trivial task. This paper presents a novel direct algorithm to retrieve the EOPs from the 2D projective transformation. It is based on a direct relationship between the 2D projective transformation and the collinearity model using homogenous coordinates. This representation offers a direct matrix correspondence between the 2D projective transformation parameters and the collinearity model parameters. This correspondence lends itself to a direct matrix factorisation to retrieve the EOPs. An important step in the proposed algorithm is a normalisation process that provides the actual link between the 2D projective transformation and the collinearity model. This paper explains the theoretical basis of the proposed algorithm as well as the necessary steps for its practical implementation. In addition, numerical examples are provided to demonstrate its validity.

Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas ℚ–Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.

A general method for visualizing high-dimensional phase diagrams of systems containing a mixture of molecular and ionic species is presented. Based on geometric modeling with homogeneous coordinates, canonical coordinates are developed to represent cuts and projections, which, being reaction-invariant, do not depend on the degree of ionization. Examples are provided to illustrate the application of this method for identifying useful transforms with potential applications in separation system synthesis.

The dual intersection test is described. It is a new method for deciding whether a line intersects a window. The concept of semi-homogeneous coordinates is introduced. It allows us to define line segments in projective space, and to derive a generalized Cohen-Sutherland end-point t test for such segments. When this is combined with the dual intersection test, we obtain new clipping algorithms for 2D and 3D projective space.

A comparison is made between geometry optimization in Cartesian coordinates, using an appropriate initial Hessian, and natural internal coordinates. Results on 33 different molecules covering a wide range of symmetries and structural types demonstrate that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internals to converge to global minima whereas Cartesian optimizations converge to the local minimum closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internals are to be preferred over Z-matrix coordinates. General optimization strategies using internal coordinates and/or Cartesians are discussed for both unconstrained and constrained optimization. © John Wiley & Sons, Inc.

We generalize the usual complex virial theorem using a procedure of complex scaling for only some of the coordinates of the system. This result can also be considered as an extension of the virial theorem obtained in a previous work using real scaling for a subset of coordinates.

Purpose: Single-isocenter, multiple target VMAT SRS for multiple brain metastases typically features field isocenter positional displacement from the individual lesion centroids. This study examines the planning margin implications of this isocenter displacement for VMAT SRS owing strictly to verification tolerances of the image guidance system. Methods: SRS image guidance at this institution is provided by Brainlab ExacTrac in combination with 6DOF couch. The system provides patient position correction through a combination of couch rotations and translations. It is well-known that affine transformations can be compactly represented in matrix form using homogeneous coordinates. The maximum geometric error directly attributable to the x-ray verification tolerances specified in ExacTrac was determined in homogenous coordinates for a range of practical radial distances from isocenter to lesion centroid. These tolerances are selectable and are based on typical uncertainties achieved by the method of patient immobilization. The calculated maximum geometric error is dependent on the understanding that the translational and rotational tolerances selected for x-ray verification represent the ability of the image guidance system to correct errors up to, and including, the specified tolerance. The additional planning margin in the worst-case is assumed to be the magnitude of the maximum geometric error. Results: The maximum error is a non-linear function of the radial distance from field isocenter to lesion centroid. It is dependent on the angular and translational tolerances specified based on setup uncertainties associated with typical immobilization options for SRS. The typical institutional SRS tolerance of 0.5 mm and 0.5° consistent with Brainlab mask immobilization yields maximum error of 0.867 – 1.23 mm over the range 0 – 7 cm radial distance. Conclusion: An expression for the maximum error attributable to the image guidance system for multiple-target, single-isocenter VMAT SRS was developed, enabling proper specification of planning margins to account for the associated geometric uncertainty.

SUMMARY An explanation of Feller's result enabling the contours of mean viability at a triallelic locus to be rendered circular is offered, and a proof given which does not involve the direct use of homogeneous coordinates. In a subsequent paper we propose to use the fact that the same transformation also maps the contours of equal average effect for each gene into circles to establish further results.