v2010.10.26 - Convex Optimization

ˇx 4ˇx 3ˇx 2Figure 2: Application of trilateration (5.4.2.3.5) is localization (determiningposition) of a radio signal source in 2 dimensions; more commonly known byradio engineers as the process “triangulation”. In this scenario, anchorsˇx 2 , ˇx 3 , ˇx 4 are illustrated as fixed antennae. [209] The radio signal source(a sensor • x 1 ) anywhere in affine hull of three antenna bases can beuniquely localized by measuring distance to each (dashed white arrowed linesegments). Ambiguity of lone distance measurement to sensor is representedby circle about each antenna. Trilateration is expressible as a semidefiniteprogram; hence, a convex optimization problem. [320]a triangle and its interior, for example, four points a tetrahedron), thenwe can ascribe shape of a set of points to their convex hull. It should beapparent: from distance, these shapes can be determined only to within arigid transformation (rotation, reflection, translation).Absolute position information is generally lost, given only distanceinformation, but we can determine the smallest possible dimension in whichan unknown list of points can exist; that attribute is their affine dimension(a triangle in any ambient space has affine dimension 2, for example). Incircumstances where stationary reference points are also provided, it becomespossible to determine absolute position or location; e.g., Figure 2.23