Affine Transformations

2008 ◽  
pp. 35-50 ◽  
Keyword(s):  
Affine Transformations

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

scholarly journals Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

The dimension of self-affine fractals II

1992 ◽  
Vol 111 (1) ◽  
pp. 169-179 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Exact Expression ◽  
Singular Value ◽  
Compact Set ◽  
Value Functions ◽  
Affine Transformations ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Almost All

AbstractA family {S1, ,Sk} of contracting affine transformations on Rn defines a unique non-empty compact set F satisfying . We obtain estimates for the Hausdorff and box-counting dimensions of such sets, and in particular derive an exact expression for the box-counting dimension in certain cases. These estimates are given in terms of the singular value functions of affine transformations associated with the Si. This paper is a sequel to 4, which presented a formula for the dimensions that was valid in almost all cases.

scholarly journals A Remark on the Principle of Zero Utility

1982 ◽  
Vol 13 (2) ◽  
pp. 133-134 ◽  
Author(s):  
Hans U. Gerber
Keyword(s):  
Risk Aversion ◽  
Utility Function ◽  
Affine Transformation ◽  
Random Variable ◽  
Affine Transformations ◽  
Premium Calculation ◽  
Image Position

Let u(x) be a utility function, i.e., a function with u′(x)>0, u″(x)<0 for all x. If S is a risk to be insured (a random variable), the premium P = P(x) is obtained as the solution of the equationwhich is the condition that the premium is fair in terms of utility. It is clear that an affine transformation of u generates the same principle of premium calculation. To avoid this ambiguity, one can standardize the utility function in the sense thatfor an arbitrarily chosen point y. Alternatively, one can consider the risk aversionwhich is the same for all affine transformations of a utility function.Given the risk aversion r(x), the standardized utility function can be retrieved from the formulaIt is easily verified that this expression satisfies (2) and (3).The following lemma states that the greater the risk aversion the greater the premium, a result that does not surprise.

Non-Lebesgue measurability of finite unions of Vitali selectors related to different groups

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Venuste Nyagahakwa ◽  
Gratien Haguma
Keyword(s):  
Topological Group ◽  
Finite Union ◽  
Affine Transformations ◽  
Real Numbers ◽  
Group Isomorphism ◽  
Lebesgue Measurability ◽  
Dense Subgroups

In this paper, we prove that each topological group isomorphism of the additive topological group $(\mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $\mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(\mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $\mathbb{R}$ onto itself.

scholarly journals An affine invariant approach for dense wide baseline image matching

2016 ◽  
Vol 12 (12) ◽  
pp. 155014771668082
Author(s):  
Fanhuai Shi ◽  
Jian Gao ◽  
Xixia Huang
Keyword(s):  
Image Matching ◽  
Intelligent Systems ◽  
Region Growing ◽  
Low Rank ◽  
Affine Invariant ◽  
Affine Transformations ◽  
Visual Sensor ◽  
Dense Matching ◽  
Rank Matrix ◽  
Distributed Intelligent Systems

Visual sensor networks have emerged as an important class of sensor-based distributed intelligent systems, where image matching is one of the key technologies. This article presents an affine invariant method to produce dense correspondences between uncalibrated wide baseline images. Under affine transformations, both point location and its neighborhood texture are changed between views, so dense matching becomes a tough task. The proposed approach tends to solve this problem within a sparse-to-dense framework. The contribution of this article is in threefolds. First, a strategy of reliable sparse matching is proposed, which starts from affine invariant features extraction and matching and then these initial matches are utilized as spatial prior to produce more sparse matches. Second, match propagation from sparse feature points to its neighboring pixels is conducted in the way of region growing in an affine invariant framework. Third, the unmatched points are handled by low-rank matrix recovery technique. Comparison experiments of the proposed method versus existing ones show a significant improvement in the presence of large affine deformations.

scholarly journals Projective Connections and Projective Transformations

1957 ◽  
Vol 12 ◽  
pp. 1-24 ◽  
Author(s):  
Noboru Tanaka
Keyword(s):  
Affine Transformations ◽  
Connected Manifold ◽  
Projective Transformations

The main purpose of the present paper is to establish a theorem concerning the relation between the group of all projective transformations on an affinely connected manifold and the group of all affine transformations.

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