By Webber R.E.

During this thesis, numerous features of quadtree representations are analyzed. The quadtree is a hierarchical variable-resolution info constitution appropriate for representing the geometric gadgets of special effects, the polygonal maps of laptop cartography, and the digitized photos of desktop vision.The research of quadtrees is gifted in 3 parts:A) a proper semantics for quadtree algorithms,B) better algorithms for manipulating the traditional quarter quadtree, andC) variations of the quadtree technique to the duty of representing polygonal maps.

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P], Ym(t, p) x(O) = xo(p), = h[x(t), pl, where f and h are assumed to be infinitely continuously differentiable. i. is therefore (Pohjanpalo, 1978) 0, I, ... ,k max • ~ J\ '" p=p, where k max is some positive integer, small enough for the computation to remain tractable. 4 state~space Consider the matrix of all impulse responses of a LI Y mU, p) model (with D =0) =C(p)exp[A(p)/]B(p). ;0+ t the Taylor series approach amounts to testing identifiability from identity of the ltlarkol' parameters (Fisher, 1966; Grewal and Glover, 1976) a method that usually turns out to be more complicated than the LapJace transfOlm and similarity transformation approaches.

This corresponds to an AutoRegressive with eXogenolls variable structure (or ARK). Once n a , Ilb and Ilr have been chosen, the unknown parameters to be estimated are For models obtained by discretization of an Ilth-order continuous-time LI model, the inputs of which are maintained constant between sampling times (by a zero-order hold), lIa and lib are equal to 12, which simplifies characterization. This very simple structure may turn out not be l1exiblc enough to describe the properties of the perturbation.

R3, * a == 0 It must hold true for any x* around 0 and almost any p *,so P2 l\. * Processmg . the f 'In the same manner proves th * so M IS . an d P4 = P4. i. 5 Use of elimination theory Computer algebra software (Davenport, Siret and Tournier, 1987, 1993) such as AXIOM, MACSYMA, MAPLE or REDUCE can be used to obtain equations expressing M(p) = j\1(p*) and solve them. , 1985; Lecourtier and Raksanyi, 1987). These sets of triangular equations can then be solved by considering a sequence of single-variable polynomials.

### Analysis of quadtree algorithms by Webber R.E.

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