Leo Dorst

University of Amsterdam | UVA · Institute of Informatics

This paper provides a definition of back-propagation through geometric correspondences for morphological neural networks. In addition, dilation layers are shown to learn probe geometry by erosion of layer inputs and outputs. A proof-of-principle is provided, in which predictions and convergence of morphological networks significantly outperform con...

Oriented elements are part of geometry, and they come in two complementary types: intrinsic and extrinsic. Those different orientation types manifest themselves by behaving differently under reflection. Dualization in geometric algebras can be used to encode them; or vice versa, orientation types inform the interpretation of dualization. We employ...

Pooling is essentially an operation from the field of Mathematical Morphology, with max pooling as a limited special case. The more general setting of MorphPooling greatly extends the tool set for building neural networks. In addition to pooling operations, encoder-decoder networks used for pixel-level predictions also require unpooling. It is comm...

Pooling is essentially an operation from the field of Mathematical Morphology, with max pooling as a limited special case. The more general setting of MorphPooling greatly extends the tool set for building neural networks. In addition to pooling operations, encoder-decoder networks used for pixel-level predictions also require unpooling. It is comm...

We design a computational method to align pairs of counter-fitting fracture surfaces of digitized archaeological artefacts. The challenge is to achieve an accurate fit, even though the data is inherently lacking material through abrasion, missing geometry of the counterparts, and may have been acquired by different scanning practices. We propose to...

p>This paper provides a definition of back-propagation through geometric correspondences for morphological neural networks. In addition, dilation layers are shown to learn probe geometry by erosion of layer inputs and outputs. A proof-of-principle is provided, in which predictions and convergence of morphological networks significantly outperform c...

p>This paper provides a definition of back-propagation through geometric correspondences for morphological neural networks. In addition, dilation layers are shown to learn probe geometry by erosion of layer inputs and outputs. A proof-of-principle is provided, in which predictions and convergence of morphological networks significantly outperform c...

Orientation measurements of attitudes estimate relative rotations of objects. The non-commutative algebra of rotations makes transference of techniques inspired by the usual vector-based approaches for translations non-trivial. We treat three different metrics that may be used to compare orientations, compute the corresponding optimal averages, and...

We show that if Projective Geometric Algebra (PGA), i.e. the geometric algebra with degenerate signature (n, 0, 1), is understood as a subalgebra of Conformal Geometric Algebra (CGA) in a mathematically correct sense, then flat primitives share the same representation in PGA and CGA. Particularly, we treat duality in PGA in the framework of CGA. Th...

The direct construction of geometric elements in an N dimensional geometric algebra by taking the outer product between \(N-1\) primitive points is one of the cornerstone tools. It is used to construct a variety of objects, from spheres in CGA [6], up to quadric [2] and even cubic surfaces [9] in much higher dimensional algebras. Initial implementa...

When fitting archaeological artifacts, one would like to have a representation that simplifies fragments while preserving their complementarity. In this paper, we propose to employ the scale-spaces of mathematical morphology to hierarchically simplify potentially fitting fracture surfaces. We study the masking effect when morphological operations a...

This paper introduces a novel and matrix-free implementation of the widely used Levenberg-Marquardt algorithm, in the language of Geometric Algebra. The resulting algorithm is shown to be compact, geometrically intuitive, numerically stable and well suited for efficient GPU implementation. An implementation of the algorithm and the examples in this...

We propose to employ scale spaces of mathematical morphology to hierarchically simplify fracture surfaces of complementarily fitting archaeological fragments. This representation preserves complementarity and is insensitive to different kinds of abrasion affecting the exact fitting of the original fragments. We present a pipeline for morphologicall...

We consider Villarceau circles as the orbits of specific composite rotors in 3D conformal geometric algebra that generate knots on nested tori. We compute the conformal parametrization of these circular orbits by giving an equivalent, position-dependent simple rotor that generates the same parametric track for a given point. This allows compact der...

An orthogonal sphere representation of arcs on spatial circles can be used to compactly perform Boolean combinations of such arcs. We formulate this using conformal geometric algebra, of which the oriented nature allows both minor and major arcs to be treated. Easily computable quantities discriminate the cases of relative positions. An application...

We propose to employ scale spaces of mathematical morphology to hierarchically simplify fracture surfaces of complementarily fitting archaeological fragments. This representation preserves contact and is insensitive to different kinds of abrasion affecting the exact complementarity of the original fragments. We present a pipeline for morphologicall...

In GRAVITATE, two disparate specialities will come together in one working platform for the archaeologist: the fields of shape analysis, and of metadata search. These fields are relatively disjoint at the moment, and the research and development challenge of GRAVITATE is precisely to merge them for our chosen tasks. As shown in chapter 7 the small...

This paper exposes a very geometrical yet directly computational way of working with conformal motions in 3D. With the increased relevance of conformal structures in architectural geometry, and their traditional use in CAD, its results should be useful to designers and programmers. In brief, we exploit the fact that any 3D conformal motion is gover...

The GRAVITATE project is developing techniques that bring together geometric and semantic data analysis to provide a new and more effective method of re-associating, reassembling or reunifying cultural objects that have been broken or dispersed overtime. The project is driven by the needs of archaeological institutes, and the techniques are exempli...

It is possible to set up a correspondence between 3D space and \({\mathbb{R}^{3,3}}\), interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become represented as rotors in the geometric algebra of \({\mathbb{R}^{3,3}}\). We show explicitly how various primitive projective transformati...

We fit k-spheres optimally to n-D point data, in a geometrically total least squares sense. A specific practical instance is the optimal fitting of 2D-circles to a 3D point set. Among the optimal fitting methods for 2D-circles based on 2D (!) point data compared in Al-Sharadqah and Chernov (Electron. J. Stat. 3:886–911, 2009), there is one with an...

Riemannian geometry allows for the generalization of statistics designed for Euclidean vector spaces to Riemannian manifolds. It has recently gained popularity within computer vision as many relevant parameter spaces have such a Riemannian manifold structure. Approaches which exploit this have been shown to exhibit improved efficiency and accuracy....

In this paper, we derive a method to determine a conformal transformation in n-dimensional Euclidean space in closed form given exact correspondences between data. We show that a minimal data set needed for correspondence is a localized vector frame and an additional point. In order to determine the conformal transformation, we use the representati...

The motion rotors, or motors, are used to model Euclidean motion in 3D conformal geometric algebra. In this chapter we present a technique for estimating the motor which best transforms one set of noisy geometric objects onto another. The technique reduces to an eigenrotator problem and has some advantages over matrix formulations. It allows motors...

Conformal transformations are described by rotors in the conformal model of geometric algebra (CGA). In applications there is a need for interpolation of such transformations, especially for the subclass of 3D rigid body motions. This chapter gives explicit formulas for the square root and the logarithm of rotors in 3D CGA. It also classifies the t...

We describe a new algorithm to reconstruct a rigid body motion from point correspondences. The algorithm works by constructing a series of reflections which align the points with their correspondences one by one. This is naturally and efficiently implemented in the conformal model of geometric algebra, where the resulting transformation is represen...

Using conformal geometric algebra, Euclidean motions in n-D are represented as orthogonal transformations of a representational space of two extra dimensions, and a well-chosen metric. Orthogonal transformations are representable as multiple reflections, and by means of the geometric product this takes an efficient and structure preserving form as...

This paper presents a fast monocular visual odometry algorithm. We propose a closed form solution for the computation of the unknown scale ratio between two consecutive image pairs. Our method requires only 1 2D-3D correspondence. A least square solution can also be found in closed form when more correspondences are available. Additionally we provi...

Subspaces are powerful tools for modeling geometry. In geometric algebra, they are represented using blades and constructed using the outer product. Producing the actual geometrical intersection (meet) and union (join) of subspaces, rather than the simplified linearizations often used in Grassmann-Cayley algebra, requires efficient algorithms when...

A new and useful set of homogeneous coordinates has been discovered for the treatment of Euclidean geometry. They render Euclidean motions not merely linear (as the classical homogeneous coordinates do), but even turn them into orthogonal transformations, through a clever choice of metric in two (not one) additional dimensions. To take full advant...

We derive a method to determine a conformal transformation in nD in closed form given exact correspondences between data. We show that a minimal dataset needed for correspondence is a localized vector frame and an additional point. In order to determine the conformal transformation we use the representation of the conformal model of geometric algeb...

There are n independent 1-D directions in an n-dimensional physical space, and they can conveniently be drawn as vectors at the origin. Mathematically, they form a vector space Rn that can be added and scaled with real numbers to produce other legitimate 1-D directions. The metric of the directions in the physical space (typically Euclidean) can be...

The homogeneous model is well suited to applications in which incidences of offset flat subspaces are central, but less so when metric properties are also important. It plays a role similar to Grassmann–Cayley algebra. They require 6-D vectors and corresponding matrices, which appear extraneous to the usual 4-D data structures in homogeneous coordi...

The basic constructions in geometric algebra are spanning and intersection. The meet is greatly extended in its capabilities to intersect arbitrary flats and rounds, or to compute other incidences. The results can be real or imaginary, or even the infinitesimal tangents. The dual of the meet provides a novel operation: the plunge, which constructs...

This chapter presents two ways to implement the linear products and operations of geometric algebra. Both implementation approaches are based on the linearity and distributivity of the products and operations. The first approach uses linear algebra to encode the multiplying element as a square matrix acting on the multiplied element, which is encod...

This chapter discusses the implementation of nonlinear geometric algebra operations. The exponentials of general bivectors generate the continuous motions in geometry, and only in fewer than four dimensions are bivectors always 2-blades. In the important conformal model, rigid body motions are exponentials of bivectors that are not 2-blades. At tim...

This chapter presents the development of the conformal model of Euclidean geometry. The blades of the conformal model can represent many more elements that are useful in Euclidean geometry. They provide spheres, circles, point pairs, and tangents as direct elements of computation. The sharp distinction between size and weight occurs for the first t...

All geometric algebra implementations that represent multivectors as a weighted sum of basis blades will have to compute products of basis blades. Among the various products, the capability to compute the geometric product (also for non-Euclidean metrics) is clearly the minimum requirement, as the outer product and inner products can be derived fro...

Differentiation is the process of computing with changes in quantities. When the changes are small, those computations can be linear to a good approximation, and it is not difficult to develop a calculus for geometry by analogy to classical analysis. When formulated with geometric algebra, it becomes possible to differentiate not only with respect...

This chapter focuses on the algebra of subspaces highlighting the familiar lines and plans. The properties of homogeneous lines includes that a line through the origin is a 1-D homogeneous subspace of the vector space Rn. It is characterized by any nonzero vector a, such that any vector x denoting a point on the line is a multiple of a. A plane thr...

The homogeneous model differs from the vector space model in representation method: it embeds Rn in a space Rn+1 with one more dimension, and then uses the algebra of Rn+1 to represent those elements of Rn in a structured manner. A metric of a homogeneous representation space is completely determined by the inner product, so given the value of x ·...

The conformal model is especially designed for Euclidean geometry, which is the geometry of transformations preserving the Euclidean distances of and within objects. These Euclidean transformations are sometimes called isometries, and they include translations, rotations, reflections, and their compositions. The Euclidean points are represented by...

A geometry (affine, Euclidean, conformal, projective, or any kind) is characterized by certain operators that act on the objects of the geometry. These objects can range from geometrical entities such as a triangle to properties such as length, so they may have various dimensionalities that are called grades. The operators in the geometry change th...

The efficient implementation of geometric algebra requires addressing three issues: multivectors, metrics, and operations. The metric is a very fundamental feature of a geometric algebra, affecting the most basic products. Many different metrics are useful and need to be allowed; yet looking up the metric at run-time is too costly. The number of ba...

The geometric algebra approach considerably extends the classical techniques of homogeneous coordinates. The homogeneous model permits representing offset subspaces as blades and transformations on them as linear transformations and their outermorphisms. The geometric algebra approach exposes some weaknesses in the homogeneous model. It turns out t...

All conformal transformations are generated by versor products using the elementary vectors of the conformal model. The most elementary conformal transformation is the reflection in a unit sphere, called (spherical) inversion. The inversion in the sphere does not preserve the class of the element it acts on: the consequence of essentially swapping...

A ray tracer is a program that takes as input the description of a scene, including light sources, models, and camera, and produces an image of that scene as seen from the camera. The representation of rays and their intersections with objects is central to a ray tracer. To render an image, the ray tracer spawns rays through the optical center of t...

Geometric algebra contains operations to determine the union and intersection of subspaces, the join and meet products. When applied to the subspaces at the origin, meet and join generalize some specific formulas from 3-D linear algebra into a more unified framework, and extend them to subspaces intersecting in n-dimensional space. The join is ofte...

This chapter provides an introduction to geometric algebra, a powerful computational system used to describe and solve geometrical problems. The main features of geometric algebra includes that vectors can be used to represent aspects of geometry, but the precise correspondence is a modeling choice. Geometric algebra offers three increasingly power...

Linear transformations of a vector space Rn change its vectors. When this happens, the blades spanned by those vectors change quite naturally to become the spans of the transformed vectors. That defines the extension of a linear transformation to the full subspace algebra. An extension of a map of vectors to the whole of the Grassmann algebra is ca...

The scalar product is a mapping from a pair of k-blades to the real numbers and is denoted by an asterisk. The inner product of vectors is a special case of the scalar product, as applied to vectors. When applied to k-blades, it should at least be backwards compatible with that vector inner product in the case of 1-blades. The scalar product of two...

The concept of a versor (a product of vectors to be used as an operator in a sandwiching product) combines all the representations of orthogonal transformations. The versors preserve the structure of geometric constructions and can be universally applied to any geometrical element. This is a unique feature of geometric algebra, and it can simplify...

Suppose we only know of some elements in a geometric algebra how a versor has transformed them, can we then reconstruct the unknown versor V? We present an O(2^n) method that works in n-D geometric algebra for n exact vector correspondences. This makes it usable for determining, for instance, a Euclidean rigid body motion in n-D from a frame corres...

The classical Vahlen matrix representation of conformal transformations on R(n) is directly related to the versor representation of conformal geometric algebra (CGA) using R(n+1;1). This paper spells out the relationship, which enriches both fields with insights and techniques. We extend the Vahlen matrices to include the representation of blades i...

Until recently, all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex-often a lot of effort is required to bring about even modest performance enhance...

This paper introduces a new algorithm for computing multi-resolution optical flow, and compares this new hierarchical method with the traditional combination of the Lucas-Kanade method with a pyramid transform. The paper shows that the new method promises convergent optical flow computation. Aiming at accurate and stable computation of optical flow...

In geometric algebra, the conformal model provides a much more powerful framework to represent Euclidean motions than the customary “homogeneous coordinate” methods. It permits elementary universal operators that can act to displace not only points, but also lines and planes, and even circles and spheres. We briefly explain this model, focusing on...

The first two parts of this book have given an abstract description of geometric algebra that is mostly free of coordinates and other low-level implementation details. Even though many programming examples have been provided, this may still have left a somewhat unreal feeling about geometric algebra, and an impression that any implementation of geo...

This chapter starts afresh and introduces the basics of Clifford algebra to develop a powerful geometric algebra. This geometric algebra incorporates operators on subspaces into the framework and permits to displace the constructions of the subspace algebra in a structure-preserving manner. The crucial construction is to unify the qualitative and q...

Reflection in a line is represented by a sandwiching construction involving the geometric product. Though that may have seemed like a curiosity in the previous chapter, it is crucial to the representation of operators in geometric algebra. Geometrically, all orthogonal transformations can be considered as multiple reflections. Algebraically, this l...

When geometric algebra was developed in the first part of this book, it illustrated the principles with pictures in which vectors are represented as arrows at the origin, bi-vectors as area elements at the origin, and so on. This is the purest way to show the geometric properties corresponding to the algebra. The examples show this algebra of the m...

The homogeneous model of Euclidean geometry is reasonably effective since it linearizes Euclidean transformations, and geometric algebra extends the classical homogeneous coordinate techniques nicely through its outermorphisms. In the next few chapters, the book presents the new conformal model for Euclidean geometry, which can represent Euclidean...

This is the first book on geometric algebra that has been written especially for the computer science audience. When reading it, you should remember that geometric algebra is fundamentally simple, and fundamentally simplifying. That simplicity will not always be clear; precisely because it is so fundamental, it does basic things in a slightly diffe...

The well-known Procrustes method determines the optimal rigid body motion that registers two point clouds by minimizing the square distances of the residuals. In this paper, we perform the first order error analysis of this method for the 3D case, fully specifying how directional noise in the point clouds affects the estimated parameters of the rig...

The language of geometric algebra can be used in the development of computer graphics applications. This paper proposes a method to describe a 3D polygonal mesh model using a representation technique based on geometric algebra and the conformal model of the 3D Euclidean space. It describes also the stages necessary to develop an application that us...

This article compares five models of 3D Euclidean geometry-not theoretically, but by demonstrating how to implement a simple recursive ray tracer in each of them. It's meant as a tangible case study of the profitability of choosing an appropriate model, discussing the trade-offs between elegance and performance for this particular application. The...

This report describes a method to implement the main abstract operations characteristic to Cli#ord algebra. The design goal was to maintain the algorithms simple, their specifications correspond (mostly) to the basic definitions. The report is addressed to those who want to understand the low-level processing mechanisms underlying the abstract conc...

The problem of finding collision-free placements for an object amid obstacles has two well-known solutions: the task space approach and the configuration space approach. In this correspondence, we study the mathematical structure of the placement problem, and show that Minkowski decomposition of the object produces a hierarchy of intermediate refor...

1 Abstract Computations of 3D Euclidean geometry can be performed using various com- putational models of different effectiveness. In this paper we compare five al- ternatives: 3D linear algebra, 3D geometric algebra, a mix of 4D homogeneous coordinates and Plucker coordinates, a 4D homogeneous model using geomet- ric algebra, and the 5D conformal...

A* graph search effectively computes the optimal solution path from start nodes to goal nodes in a graph, using a heuristic function. In some applications, the graph may change slightly in the course of its use and the solution path then needs to be updated. Very often, the new solution will differ only slightly from the old. Rather than perform th...

Every vector space with an inner product has a geometric algebra, whether or not you choose to use it. This article shows how to call on this structure to define common geometrical constructs, ensuring a consistent computational framework. The goal is to show you that this can be done and that it is compact, directly computational, and transcends t...

A method for estimating ego-motion with vehicle mounted stereo cameras is presented. This approach is based on finding corresponding features in stereo images and tracking them between succeeding stereo frames. Our approach estimates stereo ego-motion with geometric algebra techniques. Starting with a simple linear estimate of the ego-motion, estim...

Making derived products out of the geometric product requires care in consistency. We show how a split based on outer product and scalar product necessitates a slightly different inner product than usual. We demonstrate the use of...

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Geometric algebra has established itself as a powerful and valuable mathematical tool for solving problems in computer science, engineering, physics, and mathematics. The articles in this volume, written by experts in various fields, reflect an interdisciplinary approach to the subject, and highlight a range of techniques and applications. Relevant...

This article deals with the problem of planning for robotic re surveillance in an oce building. It is shown that the planning search space, although it cannot be handled with traditional MDP solving methods, can be reduced through abstracting the environment and the expected cost. In this method the expected costs are computed per path and - given...

The set theory relations \in, \backslash, \Delta, \cap, and \cup have corollaries in subspace relations. Geometric Algebra is introduced as the ideal framework to explore these subspace operations. The relations \in, \backslash, and \Delta are easily subsumed by Geometric Algebra for Euclidean metrics. A short computation shows that the meet (\cap)...

This paper reports on some issues encountered when preparing the wealth of geometric algebra for its application in the computer sciences. They involve simply making the internal structure explicit (section 2.2); redesigning the operators (even the rather basic inner product can be improved, in section 2.3); the development of new techniques to ena...

This article deals with the problem of planningfor robotic re surveillance in an oce building.It is shown that the planning search space,although it cannot be handled with traditionalMDP solving methods, can be reduced throughabstracting the environment and the expectedcost. In this method the expected costs arecomputed per path and - given an appr...

Geometric algebra extends Clifford algebra with geometrically meaningful operators with the purpose of facilitating geometrical computations. Present textbooks and implementation do not always convey this geometrical flavor or the computational and representational convenience of geometric algebra, so we felt a need for a computer tutorial in which...

The motivation behind this work is to make the computation of collision-free motions of robots efficiently computable. For translational motions, the boundary of permissible translations of a reference point is obtained from the obstacles and the robot by a kind of dilation, ‘thickening’ the obstacle (see below for details) to produce the forbidden...

The sense of touch and the capability to analyze potential contacts is important to many interactions of robots, such as planning exploration, handling objects, or avoiding collisions based on sensing of the environment. It is a pleasant surprise that the mathematics of touching and contact can be developed along the same algebraic lines as that of...

We present a directional boundary representation which deals locally and consistently with the boundary's “inside”. We show that collision and wave propagation are reduced to addition on the spectrum of directions, and we derive transformation laws for differential geometrical properties such as directed curvature

this paper d- d- d- d- d- 2* The signifies our confidence that our representation can be extended to m-D and yield algorithms; we are planning do this using geometric algebra

Driving a car involves simultaneous consideration of events at different spatio-temporal scales. Proper interpretation and planning then leads to behaviour such as the parallel parking manoeuvre, the three-point turn, free Euclidean driving in a desert, following a road, and translationally passing other vehicles at high speed. In the study of auto...

The Dutch government is considering placing Automatic Debiting Systems (ADS) for electronic fee collection (EFC) on the highways. These systems would interact via a transponder in each passing car, and subtract a fee from the driver's credit card. Nonpayers would be photographed and fined. The ultimate goal is to use these systems to influence road...

An ADS (Automatic Debiting System) is an electronic fee collection (EFC) system on a freeway, which interacts with a transponder in each car, and subtracts a fee from a credit card. Non-payers are to be photographed and fined. In requirements on such systems in The Netherlands, privacy laws demand separation between the financial transaction and th...