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Affine invariant points

Mathieu Meyer, Carsten Schütt, Elisabeth M. Werner
2015 Israel Journal of Mathematics  
We answer in the negative a question by Grünbaum who asked if there exists a finite basis of affine invariant points.  ...  We give a positive answer to another question by Grünbaum about the "size" of the set of all affine invariant points.  ...  Remark (i) provides examples of non-proper affine invariant points: once there are two different affine invariant points, there are affine invariant points p(K) / ∈ K, i.e. nonproper affine invariant points  ... 
doi:10.1007/s11856-015-1196-2 fatcat:i4qocivdb5grnld4qyfgzuk7ly

Affine invariant points [article]

Mathieu Meyer, Carsten Schuett, Elisabeth M. Werner
2013 arXiv   pre-print
We answer in the negative a question by Gruenbaum who asked if there exists a finite basis of affine invariant points.  ...  Crucial to establish these results, are new affine invariant points, not previously considered in the literature.  ...  Remark (i) provides examples of non-proper affine invariant points: once there are two different affine invariant points, there are affine invariant points p(K) / ∈ K, i.e. nonproper affine invariant points  ... 
arXiv:1301.2606v1 fatcat:7v3vrm2s5bgahnzzor63hk5fpe

Affine invariant triangulations [article]

Prosenjit Bose, Pilar Cano, Rodrigo I. Silveira
2020 arXiv   pre-print
Springer, 1993] that uses the inverse of the covariance matrix of S to define an affine invariant norm, denoted A_S, and an affine invariant triangulation, denoted DT_A_S[S].  ...  In addition, we provide different affine invariant sorting methods of a point set S and of the vertices of a polygon P that can be combined with known algorithms to obtain other affine invariant triangulation  ...  affine invariant sorting methods.  ... 
arXiv:2011.02197v1 fatcat:e3itntpbirbetdfxygpuevnvmy

Extended affinization of Invariant Affine Reflection Algebras [article]

Saeid Azam, S. Reza Hosseini, Malihe Yousofzadeh
2011 arXiv   pre-print
Invariant affine reflection algebras are the last and the most general known extension of affine Kac-Moody Lie algebras, introduced in recent years.  ...  , and applying the so called "loop construction", we obtain again an invariant affine reflection algebra.  ...  To be more precise, we note that the main difference of the class of invariant affine reflection algebras with extended affine Lie algebras or locally extended affine Lie algebras, is that in the latter  ... 
arXiv:1105.1679v2 fatcat:hpu5twly3zeirch46a5t6imw3m

Some Affine Invariants Revisited [chapter]

Alina Stancu
2013 Fields Institute Communications  
We present several sharp inequalities for the SL(n) invariant Ω 2,n (K) introduced in our earlier work on centro-affine invariants for smooth convex bodies containing the origin.  ...  The technique employed prompts us to conjecture that any SL(n) invariant of convex bodies with continuous and positive centro-affine curvature function can be obtained as a limit of normalized p-affine  ...  Inequalities for a second order centro-affine invariant In [34] , we introduced a class of SL(n) invariants for smooth convex bodies in R n .  ... 
doi:10.1007/978-1-4614-6406-8_16 fatcat:do7s37hkbnazrn5dlmzxc3fna4

Cutting Affine Moment Invariants

Jianwei Yang, Ming Li, Zirun Chen, Yunjie Chen
2012 Mathematical Problems in Engineering  
The traditional affine moment invariants (AMIs) method is applied to the new image. Consequently, cutting affine moment invariants (CAMIs) are derived.  ...  The extraction of affine invariant features plays an important role in many fields of image processing.  ...  More affine invariant features, cutting affine moment invariants CAMIs , are extracted. Furthermore, we combine CAMIs with the original AMIs we call the obtained affine invariants as CCAMIs .  ... 
doi:10.1155/2012/928161 fatcat:fuu2yyoa4jfytk34tmirtori4i

Dual Affine invariant points [article]

Mathieu Meyer, Carsten Schuett, Elisabeth M. Werner
2013 arXiv   pre-print
An affine invariant point on the class of convex bodies in R^n, endowed with the Hausdorff metric, is a continuous map p which is invariant under one-to-one affine transformations A on R^n, that is, p(  ...  We define a product on the set of affine invariant points, in relation with duality. Finally, examples are given which exhibit the rich structure of the set of affine invariant points.  ...  Now we recall the definitions of affine invariant points and of affine invariant sets [3, 13] . Definition 1.  ... 
arXiv:1310.0128v1 fatcat:kv7fcebpobfgva7rq4wr7iy554

Robust Affine Invariant Descriptors

Jianwei Yang, Zirun Chen, Wen-Sheng Chen, Yunjie Chen
2011 Mathematical Problems in Engineering  
An approach is developed for the extraction of affine invariant descriptors by cutting object into slices.  ...  Gray values associated with every pixel in each slice are summed up to construct affine invariant descriptors. As a result, these descriptors are very robust to additive noise.  ...  Affine Invariant Affine maps parallel lines onto parallel lines, intersecting lines into intersecting lines.  ... 
doi:10.1155/2011/185303 fatcat:tfibvnnjf5dqravf5xsdebarae

Dual Affine invariant points

Elisabeth Werner, Mathieu Meyer, Carsten Schuett
2015 Indiana University Mathematics Journal  
An affine invariant point on the class of convex bodies Kn in R n , endowed with the Hausdorff metric, is a continuous map from Kn to R n which is invariant under one-to-one affine transformations A on  ...  Now we recall the definitions of affine invariant points and of affine invariant sets [3, 13] . Definition 1.  ...  Let p be a proper affine invariant point.  ... 
doi:10.1512/iumj.2015.64.5514 fatcat:5i5fegbgubfmzox7kaj7rpkp5u

Some Affine Invariants Revisited [article]

Alina Stancu
2012 arXiv   pre-print
We present several sharp inequalities for the SL(n) invariant Ω_2,n(K) introduced in our earlier work on centro-affine invariants for smooth convex bodies containing the origin.  ...  The technique employed prompts us to conjecture that any SL(n) invariant of convex bodies with continuous and positive centro-affine curvature function can be obtained as a limit of normalized p-affine  ...  Inequalities for a second order centro-affine invariant In [32] , we introduced a class of SL(n) invariants for smooth convex bodies in R n .  ... 
arXiv:1208.0783v1 fatcat:4uabanb7qfck3exb7jsbxn6sti

Affine-Invariant Midrange Statistics [chapter]

Cyrus Mostajeran, Christian Grussler, Rodolphe Sepulchre
2019 Lecture Notes in Computer Science  
We formulate and discuss the affine-invariant matrix midrange problem on the cone of n× n positive definite Hermitian matrices ℙ(n), which is based on the Thompson metric.  ...  Affine-invariant Finsler metrics on P(n) Consider the family of affine-invariant metric distances d Φ on P(n) defined as d Φ (A, B) = log A −1/2 BA −1/2 Φ , (1) where • Φ is any unitarily invariant norm  ...  In Subsection 1.2, we define the affine-invariant midrange problem for a collection of N points.  ... 
doi:10.1007/978-3-030-26980-7_51 fatcat:u2gn5tfh2jb6zcg73kxs3h325u

Affine invariants of annuli

Waldemar Cieślak, Elżbieta Szczygielska
2012 Annales Universitatis Mariae Curie-Sklodowska. Sectio A. Mathematica  
Affine invariants of annuli are introduced. 2000 Mathematics Subject Classification. 53A04.  ...  Invariants of annuli. We note that Theorem 2.1. Let an annulus CD belongs to CΛ.  ...  The number c o (CD) given by the formula (2.1) c o (CD) = exp   − 2π 0 |ż(t)| λ (t) dt   = exp   − 2π 0 R (t) λ (t) dt   does not depend on parametrizations of C, D and affine transformations.  ... 
doi:10.2478/v10062-012-0002-4 fatcat:3qresiipfjbb7az27bbm3br6f4

Affine Invariant Image Segmentation

A. Bhalerao, R. Wilson
2004 Procedings of the British Machine Vision Conference 2004  
with a set of affine transformation coefficients.  ...  An appropriate set of prototype blocks is determined for a given an image by clustering in a feature subspace of discrete samples from an affine symmetry group.  ...  Results based on 1 or more chosen prototypes using affine invariant block based texture model.  ... 
doi:10.5244/c.18.18 dblp:conf/bmvc/BhaleraoW04 fatcat:7po5zmqo2rh7xasdc7ckhjfw7u

Dual affine moment invariants [article]

You Hao, Hanlin Mo, Qi Li, He Zhang, Hua Li
2019 arXiv   pre-print
In this paper, we propose a general framework to derive moment invariants under DAT for objects in M-dimensional space with N channels, which can be called dual-affine moment invariants (DAMI).  ...  And affine transformations often occur in both shape and color space simultaneously, which can be termed as Dual-Affine Transformation (DAT).  ...  Invariant(M, N, P, Q, O S , O C ) is invariant under DAT in the M-D space with N channels, which can be called Dual-Affine Moment Invariants (DAMI(M, N)).  ... 
arXiv:1911.08233v1 fatcat:f3juqt4pqbdqhorg6oery3xwqu

Affine-invariant scene categorization

Xue Wei, Son Lam Phung, Abdesselam Bouzerdoum
2014 2014 IEEE International Conference on Image Processing (ICIP)  
This paper presents a scene categorization method that is invariant to affine transformations.  ...  ABSTRACT This paper presents a scene categorization method that is invariant to affine transformations.  ...  The existing approaches to affine invariance can be divided into three categories: invariance by training, invariance by image normalization, and invariance by feature extraction.  ... 
doi:10.1109/icip.2014.7025205 dblp:conf/icip/WeiPB14 fatcat:oupimxe42ncjxhk5rja4igphei
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