s d(x, y) x, y R2

Read UCC - 12 In Visual Studio .NETUsing Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications.

(101)

Creating GS1 - 12 In Visual Studio .NETUsing Barcode printer for VS .NET Control to generate, create UPC A image in .NET framework applications.

s d(x, y) x, y R2

Recognizing UPC-A Supplement 2 In .NETUsing Barcode reader for .NET Control to read, scan read, scan image in .NET applications.

(102)

Print Bar Code In .NETUsing Barcode encoder for .NET framework Control to generate, create barcode image in .NET framework applications.

lim on (x) = xf

Bar Code Decoder In .NETUsing Barcode decoder for .NET Control to read, scan read, scan image in .NET applications.

x R2

UPC-A Supplement 2 Printer In C#Using Barcode generation for .NET Control to generate, create GS1 - 12 image in .NET applications.

(103)

Universal Product Code Version A Generation In VS .NETUsing Barcode printer for ASP.NET Control to generate, create UPC Code image in ASP.NET applications.

10214 Hausdorff distance Let us consider the metric space (R2 , d) The symbol H(R2 ) indicates a space whose elements are the non-empty compact subsets of R2 The distance [TRI 93] from the point x element of R2 to the set B element of H(R2 ), noted d(x, B), is de ned by: d(x, B) = min{d(x, y) : y B} The distance from the set A element of H(R2 ) to the set B element of H(R2 ), noted d(A, B), is de ned by: d(A, B) = max{d(x, B) : x A}

Create Universal Product Code Version A In Visual Basic .NETUsing Barcode creator for VS .NET Control to generate, create UPCA image in Visual Studio .NET applications.

Iterated Function Systems and Applications in Image Processing

Code-39 Printer In .NET FrameworkUsing Barcode maker for .NET Control to generate, create Code 3 of 9 image in .NET applications.

The Hausdorff distance between two sets A and B elements of H(R2 ), noted hd (A, B), is de ned by: hd (A, B) = max{d(A, B), d(B, A)} (104)

Barcode Creator In .NET FrameworkUsing Barcode drawer for VS .NET Control to generate, create barcode image in VS .NET applications.

Only when applied to closed and bounded sets also referred to as compacts does the Hausdorff distance verify all the properties of a distance (in particular, commutativity) Evidently, it is not to be confused with the concept of the Hausdorff dimension presented in this volume in 1 and 3 10215 Contracting transformation on the space H(R2 ) Let : R2 R2 be a contracting transformation de ned on the metric space (R , d) with the real s as contraction factor The transformation : H(R2 ) H(R2 ) de ned by:

Data Matrix ECC200 Printer In .NET FrameworkUsing Barcode creator for Visual Studio .NET Control to generate, create Data Matrix ECC200 image in Visual Studio .NET applications.

(B) = { (x) : x B},

I-2/5 Drawer In Visual Studio .NETUsing Barcode generation for .NET Control to generate, create ANSI/AIM ITF 25 image in Visual Studio .NET applications.

B H(R2 )

Printing Bar Code In .NETUsing Barcode generation for ASP.NET Control to generate, create barcode image in ASP.NET applications.

(105)

Bar Code Recognizer In JavaUsing Barcode scanner for Java Control to read, scan read, scan image in Java applications.

is contracting on (H(R2 ), hd ), with contraction factor s The symbol hd indicates the Hausdorff distance 10216 Iterated transformation system An IFS de ned on the complete metric space (R2 , d) is composed of a set of N transformations i : R2 R2 (i = 1, , N ), each of them associated with a Lipschitz factor si From now on, in this section, it will be considered that the N transformations are contracting: the transformation system is in this case called hyperbolic IFS The contraction factor of the hyperbolic IFS, noted s, is equal to max{si : i = 1, , N } 1022 Attractor of an iterated transformation system Let us consider an IFS {R2 ; i , i = 1, , N } It has been demonstrated [BARN 93] that the operator W : H(R2 ) H(R2 ) de ned by:

Code 3/9 Reader In .NETUsing Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications.

W (B) =

Reading USS Code 128 In Visual Studio .NETUsing Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications.

i (B),

Printing DataMatrix In C#.NETUsing Barcode printer for .NET framework Control to generate, create Data Matrix 2d barcode image in .NET framework applications.

B H(R2 )

Barcode Recognizer In .NETUsing Barcode decoder for VS .NET Control to read, scan read, scan image in VS .NET applications.

(106)

Create Barcode In .NET FrameworkUsing Barcode creation for ASP.NET Control to generate, create bar code image in ASP.NET applications.

is contracting and that its contraction factor corresponds with that of the IFS The operator W possesses a single xed point At H(R2 ) given by: At = W (At ) = lim W on (X),

GS1 - 13 Generation In VB.NETUsing Barcode creator for VS .NET Control to generate, create EAN-13 Supplement 5 image in VS .NET applications.

X H(R2 )

(107)

The object At is also called an IFS attractor It is invariant under the transformation W and is equal to the union of N copies of itself transformed by 1 , , N This invariant object is called self-similar or self-af ne when the elementary transformations i are af ne

Scaling, Fractals and Wavelets

EXAMPLE 101 Let us consider the IFS {R2 ; i , i = 1, , 3} composed of the following af ne transformations: x y 1 2 = 0 1 2 x y = 2 0 1 3 x y = 2 0 0 x + y0 1 y 2 2 x0 0 x 2 y + y 0 1 2 2 x 0 0 x 2 y + y0 1 2 2 0

(108)

Its contraction factor is equal to 025 The attractor coded by the IFS, called the Sierpinski triangle, is represented in Figure 101

x y ( 20 , y 0 ) w1 (0,0) w2 (-x 0 , -y 0 ) w3

Figure 101 Attractor of the iterated transformation system of Example 101 The initial square, originally centered and with sides of length 2x0 , 2y0 , is transformed into three homothetic squares by contracting transformations 1 , 2 and 3 This process is then iterated

1023 Collage theorem This theorem, shown in [BARN 93], provides an upper bound to the Hausdorff distance hd between a point A included in H(R2 ) and the attractor At of an IFS THEOREM 101 We consider the complete metric space (R2 , d) Given a point A belonging to H(R2 ) and an IFS {R2 ; 1 , 2 , , n } with a real contraction factor