Curve cuspless reconstruction via sub-Riemannian geometry∗∗∗. (25th June 2014)

• Record Type: Journal Article
• Authors: Boscain, Ugo; Duits, Remco; Rossi, Francesco; Sachkov, Yuri
• Is Part Of: ESAIM. Volume 20:Number 3(2014:Jul.)
• Page Start: 748
• Page End: 770
• Publication Date: 2014-06-25
• Subjects: System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2013082
• Languages: English
• ISSNs: 1292-8119

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>We consider the problem of minimizing <inline-formula><alternatives><tex-math id="tex_eq1"><![CDATA[\hbox{$\int_{0}^\ell \sqrt{\xi^2 +K^2(s)}\, {\rm d}s $}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2jt01x6r" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq1" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:msup><mml:mi /><mml:mi mathvariant="normal">∫</mml:mi></mml:msup><mml:mn mathvariant="normal">0</mml:mn><mml:mi mathvariant="italic">ℓ</mml:mi></mml:msubsup><mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:msup><mml:mi mathvariant="italic">ξ</mml:mi><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo mathvariant="normal">+</mml:mo><mml:msup><mml:mi mathvariant="italic">K</mml:mi><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo mathvariant="normal">(</mml:mo><mml:mi mathvariant="italic">s</mml:mi><mml:mo mathvariant="normal">)</mml:mo></mml:mrow></mml:msqrt></mml:mrow></mml:mrow><mml:mo> </mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="italic">s</mml:mi></mml:math></alternatives></inline-formula> for a planar curve having fixed initial and final positions and directions. The total length <italic>ℓ</italic> is free. Here <italic>s</italic> is the arclength parameter, <italic>K</italic>(<italic>s</italic>) is the curvature of the curve and <italic>ξ</italic> > 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.</p> </abstract>