(404) 868-9078

visual .net ANSI/AIM Code 39 for .NET

x B x B use none none creator togenerate none for none7029882618 for all A, B H(X).. printing barcode c# 1.12 The Hausdorff metric data matrix c# Figure 1.35 Level none none sets for the shortest-distance function D L (x ) (in red) and the furthest-distance function F L (x ) (in black) for the set L R2 . The set L is the line segment, shown in green, at the centre.

The underlying metric is the euclidean distance. Contours are labelled with corresponding distances. This array of level sets can be used to estimate the Hausdorff distance from a set B that is overlayed on the contours.

Imagine that the ant is very small, relative to the spacing of the contours, and located as indicated by the arrow. Then you can estimate d H (ant, L ) very accurately! Can you nd an upper bound for d H (leaf, L ) . 7088458349 Proof creating code 39 asp.net This follows from none for none dH (A, B) = max{D A (B), D B (A)},. printing ean 13 vb.net where D A (B) = max D A (x) and create data matrix vb.net D B (A) = max min none for none d(a, b) min max d(a, b) = min F A (x).. beclout a A b B b B a A x B Upc Barcodes printer for .net c# Thus, the Hausdorf none none f distance from A to B is bounded by the larger of the maximum value (on B) of the shortest-distance function D A (x) and the minimum value (on B) of the furthest-distance function F A (x). E x e r c i s e 1.12.

26 Use Figure 1.35 to obtain an upper bound for dH (L , leaf ); see the gure caption. E x e r c i s e 1.

12.27 In Figure 1.36 trace the paths of shortest descent for both ants, with respect to D L (x) and F L (x).

. Codes, metrics and topologies Figure 1.36 This none for none gure is similar to Figure 1.35, but here the underlying metric is d max .

Sketch some paths of steepest descent. What route will each ant follow to decrease d H (ant, L ) as rapidly as possible Where will each ant end up . E x e r c i s e 1. 12.28 Find the minimum value of f (x, y) in Equation (1.

12.6) and a value of (x, y) at which it occurs, when A = {(x, y) R2 : x 2 + y 2 = 1, x 0} and B = {(x, y) R2 : x 2 + y 2 = 1 } {(0, y) R2 : 1 y 1 }. 2 2 4 Hausdorff distances on code spaces Here we consider distance functions on H( ), the set of nonempty compact subsets of a code space.

The code space may be the metric space ( , d ), where d is de ned in Equation (1.6.1), or ( , d.

A. ), where d A. is de ned in Equa tion (1.6.6), or more generally ( , d ), where : R2 is an embedding and d ( 1 , 2 ) = .

( 1 ) ( 2 ). for all 1 , 2 ,. as in Theorem 1.5.5. When the underlying metric is obtained by embedding, as in the case of d A. and more generall none none y d , it is possible to make pictures of the associated shortestdistance functions and to think quite geometrically and optically about the metric, as illustrated in Figures 1.37 and 1.38.

. 1.12 The Hausdorff metric In Figure 1.37 we consider the code space ( , d ), where the embedding function : {0,1,2,3} R2 is de ned by ( ) = lim f 1 f 2 f n (x0 , y0 ).

75 =. {0,1,2,3}. for all for some xed (x0 , y0 ) R2 . Here f (x, y) = 2 y (1 ( 1) ) x + , + 3 3 2 3 3 for all (x, y) R none none 2 , {0, 1, 2, 3}, where [x] denotes the greatest integer less than or equal to the real number x. We defer until 4 a proof that is indeed an embedding function and a more precise discussion of such embeddings. What matters here is that the embedded set ( ) looks like the set of green points in the bottom left panel of Figure 1.

37. What you cannot see is that each small green rectangle represents many more green rectangles organized in the same sort of way as those green rectangles that you can see, and so on. ( ) is in fact of the form C C R2 , where C R is a classical Cantor set.

The top right panel of Figure 1.37 represents the embedded set (S), where S . The top left panel shows level sets of D (S) (x) for x R2 .

Of course, this top left panel is not a picture of the level sets of D S ( ) for , because most points x on level sets of D (S) (x) do not correspond to points in . But the level sets of D (S) (x) accurately describe the distances from points in to points in S for all x R , the range of , because D (S) ( ( )) = D S ( ) for all ..

The function D (S ) : R2 [0, ) is a continuous extension of D (S) : R R2 [0, ) to all R2 . The bottom right panel shows the level sets of D (S) (x), for x , superimposed on ( ). Since this panel contains in principle the points of both (S) and ( ), we can use it to estimate D (S) ( ) and hence, since D ( (S)) = 0, dH ( , S).

But our purpose, of course, is not so much to do this as it is to enable us to think geometrically and visually about code-space metrics. In Figure 1.38 the level sets of D (S) (x), in the right-hand panel, may be compared with the level sets of D ( ) (x), in the left-hand panel, for x R2 .

In this example = {0,1} {0,1} , where {0,1} and {0,1} are the code spaces de ned in Section 1.4, and a different embedding function : R2 is used, similar to the one used in Figure 1.15.

The points of ( {0,1} ) are situated at the branch points, also called nodes, of a tree-like structure in R2 much the same as that seen in Figure 1.15 and the points of ( {0,1} ) are located on the canopy of the tree-like structure. Although it does not appear to be so, the canopy ( {0,1} ).

Copyright © usebarcode.com . All rights reserved.