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[[Meta:Meta:Meta]]
 
[[User:IntegralPython/sandbox/Fractal measure| Fractal Measure]]
<big>Commence the Stuff</big>
 
==Open set condition==
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 |doi=10.1090/S0002-9939-05-08300-0 |jstor=4097989 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ<sub>''i''</sub>, the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.
 
Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15–23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>
 
An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>
 
===Computing Hausdorff measure===
 
When the open set condition holds and each ψ<sub>''i''</sub> is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>
 
:<math> \sum_{i=1}^m r_i^s = 1. </math>
 
where r<sub>i</sub> is the magnitude of the dilation of the similitude.
 
With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub> in the plane '''R'''<sup>2</sup> and let ψ<sub>''i''</sub> be the dilation of ratio 1/2 around ''a<sub>i</sub>''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>
 
Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.
 
==Hand-eye calibration problem==
{{collapse top| Hand-eye calibration problem}}
In robotics, the '''hand-eye calibration problem''', or the '''robot-sensor calibration problem''', is the problem of determining the transformation between a robot [[end-effector]] and a camera or the transformation between a robot base and the world coordinate system.<ref> Amy Tabb, Khalil M. Ahmad Yousef. [https://arxiv.org/abs/1907.12425 "Solving the Robot-World Hand-Eye(s) Calibration Problem with Iterative Methods."] 29 Jul 2019.</ref> It takes the form of {{math|AX{{=}}ZB}}, where ''A'' and ''B'' are two systems, usually a robot base and a camera, and {{math|X}} and {{math|Z}} are unknown transformation matrices. A highly studied special case of the problem occurs where {{math|X{{=}}Z}}, taking the form of the problem {{math|AX{{=}}XB}}. Solutions to the problem take the forms of several types of methods, including "separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions".<ref>Mili I. Shah, Roger D. Eastman, Tsai Hong Hong. [https://www.nist.gov/publications/overview-robot-sensor-calibration-methods-evaluation-perception-systems?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."] 22 March, 2012</ref> The covariance of {{math|X}} in the equation can be calculated for any randomly perturbed matrices {{math|A}} and {{math|B}}.<ref>Huy Nguyen, Quang-Cuong Pham. [https://arxiv.org/pdf/1706.03498.pdf "On the covariance of X in AX = XB."] 12 June, 2017.</ref>
 
===Methods===
Many different methods and solutions developed to solve the problem, broadly defined as either Separable, simultaneous solutions. Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem. A common theme throughout all of the methods is the common use of [[quaternion]]s to represent rotation matrices.
 
====Separable solutions====
Given the equation {{math|AX{{=}}ZB}}, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where {{math|'''R'''<sub>A</sub>}} represents a 3×3 rotation matrix and {{math|'''t'''<sub>A</sub>}} a 3×1 translation vector, the equation can be broken into two parts:<ref>[{{cite journal | url=https://arxiv.org/pdf/1907.12425.pdf] | arxiv=1907.12425 | doi=10.1007/s00138-017-0841-7 | title=Solving the robot-world hand-eye(s) calibration problem with iterative methods | year=2017 | last1=Tabb | first1=Amy | last2=Ahmad Yousef | first2=Khalil M. | journal=Machine Vision and Applications | volume=28 | issue=5–6 | pages=569–590 | s2cid=20150713 }}</ref>
:{{math|'''R'''<sub>A</sub>'''R'''<sub>X</sub>{{=}}'''R'''<sub>Z</sub>'''R'''<sub>B</sub>}}
:{{math|'''R'''<sub>A</sub>'''t'''<sub>X</sub>+'''t'''<sub>A</sub>{{=}}'''R'''<sub>Z</sub>'''t'''<sub>B</sub>+'''t'''<sub>Z</sub>}}
Equation 2 becomes linear if {{math|'''R'''<sub>Z</sub>}} is known. As such, the most frequent approach is to {{math|'''R'''<sub>x</sub>}} and {{math|'''R'''<sub>z</sub>}} using the first equation then using it to solve for the second two variables in the second equation. Rotation is represented using [[quaternion]]s, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the rotation matrices is compounded when being applied to the translation vector.<ref name="tsapps">Mili Shah, et al. [https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."]</ref> Other solutions avoid this problem.
 
====Simultaneous solutions====
Simultaneous solutions are based on solving for both {{math|X}} and {{math|Z}} at the same time (rather than basing the solution of one part off of the other as in seperable solutions), propogation of error is significantly reduced.<ref name="dual-quaternions"> Algo Li, et al. [https://pdfs.semanticscholar.org/225d/e4ea2d3f18b7743bfeabf925fa603fc47bcb.pdf "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product."] International Journal of the Physical Sciences Vol. 5(10), pp. 1530-1536, 4 September, 2010. </ref> By formulating the matrices as [[dual quaternion]]s, it is possible to get a linear equation by which {{math|X}} is solvable in a linear format.<ref name="tsapps"/> An alternative way applies the [[least squares| least squares method]] to the [[Kronecker product]] of the matrices {{math|A⊗B}}. As confirmed by experimental results, simultaneous solutions have less error than seperable quaternion solutions.<ref name="dual-quaternions"/>
By formulating the matrices as [[dual quaternion]]s, it is possible to get a linear equation by which {{math|X}} is solvable in a linear format.<ref name="tsapps"/>
 
====Iterative solutions====
Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing {{math|{{!}}{{!}}AX−XB{{!}}{{!}}}}. As the program iterates, it will converge on a solution to {{math|X}} independent to the initial robot orientation of {{math|'''R'''<sub>B</sub>}}. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into [[dual quaternion]]s.<ref>Zhiqiang Zhang, et al. [https://link.springer.com/article/10.1007/s11548-017-1646-x "A computationally efficient method for hand–eye calibration."] 19 July, 2017.</ref> However, while iterative solutions to the problem are generally simultaneous and accurate, they can be computationally taxing to carry out and may not always converge on the optimal solution.<ref name="tsapps"/>
 
*[http://math.loyola.edu/~mili/Calibration/]
*[https://ieeexplore.ieee.org/abstract/document/8788685/keywords#keywords] - Octonion solution
*[https://arxiv.org/abs/1706.03498]
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{{collapse top | Ugandan knuckles}}
 
Ugandan Knuckles is an [[internet meme]] from January 2018 depicting a deformed version of [[Knuckles the Echidna]]. Players would go in hords to the virtual reality video game ''[[VRChat]]'' to troll other players. The people would say quotes such as "Do you know the way?", which originate from the 2010 Ugandan action film ''[[Who Killed Captain Alex?]]'', as well as "spitting" on other users whom they felt did not know "de way".<ref name="dailydot">{{Cite web|url=https://www.dailydot.com/unclick/ugandan-knuckles-vrchat-meme/|title=How Ugandan Knuckles turned VRChat into a total trollfest|last=Hathaway|first=Jay|date=11 January 2018|website=The Daily Dot|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref><ref>{{Cite web|url=https://heavy.com/games/2018/01/controversial-ugandan-knuckles-meme/|title=Controversial ‘Ugandan'Ugandan Knuckles’Knuckles' Meme Has Infested VRChat|last=MacGregor|first=Collin|date=9 January 2018|website=Heavy.com|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref> The meme was a significant trend followed by several news organisations, including ''[[USA Today]]''.<ref>https://www.usatoday.com/story/tech/news/2018/02/09/ugandan-knuckles-do-you-know-de-wey-meme-explained/307575002/ Retrieved October 9 2018</ref>
 
===History===
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===Controversy===
Because of its use of a fake Ugandan accent as well as the quotations from ''Who Killed Captain Alex?'', the meme was widely criticized for being racially insensitive;<ref name="dailydot"/><ref name=Polygon2/> ''[[Polygon (website)|Polygon]]'' described it as problematic.<ref name=Polygon2>{{cite web|url=https://www.polygon.com/2018/1/8/16863932/ugandan-knuckles-meme-vrchat|title=‘Ugandan'Ugandan Knuckles’Knuckles' is overtaking VRChat|last=Alexander|first=Julia|publisher=[[Vox Media, Inc.]]|work=Polygon|date=October 9, 2018|accessdate=January 9, 2018}}</ref> On January 27 2018, the company [[Razer Inc.|Razer]] was brought under fire for posting a Ugandan Knuckles meme that was widely criticised as a racist misstep.<ref>https://gizmodo.com/does-razer-know-it-posted-a-racist-meme-1822485212 Retrieved October 9 2018</ref>
 
The original creator of the 3D avatar, [[DeviantArt]] user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of ''VRChat'' and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar.<ref>{{Cite web|url=http://www.gamerevolution.com/news/362289-creator-vrchats-ugandan-knuckles-meme-regrets-decision|title=Creator of VRChat’sVRChat's ‘Ugandan'Ugandan Knuckles’Knuckles' Meme Regrets His Decision|last=Tamburro|first=Paul|date=8 January 2018|website=GameRevolution|archive-url=|archive-date=|dead-url=|access-date=9 October 2018}}</ref> In response to the trolling in the game, the developers of ''VRChat'' published an open letter on ''[[Medium (website)|Medium]]'', stating that they were developing "new systems to allow the community to better self moderate" and asking users to use the built-in muting features.<ref>{{Cite web |url=https://www.polygon.com/2018/1/10/16875716/vrchat-safety-concerns-open-letter-players-behavior |title=VRChat team speaks up on player harassment in open letter |last=Alexander |first=Julia |date=January 10, 2018 |website=Polygon |access-date=October 9, 2018}}</ref>
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