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4. Affine invariance

As we mentioned before,  by  s(D,M)we denote both the IFS and the  interpolation scheme (depending on M = {Mi, i = 1,.., n})  that associates to  D Ì R3 the  function  f whose graph FD,M is the unique attractor of s(D,M).

 Definition 2. An  interpolation scheme s(D,M)  is invariant upon  transformation w: R3® R iff for any DÌ R3,

                                                    w( FD,M) = Fw(D),M ,                                               (12)

where FD,M  denotes the attractor of the IFS s(D,M).

Definition 3. An  interpolation scheme s(D,M)  is  weakly  invariant upon  w  iff  there exists a set of matrices M' = {Mi', i = 1,.., n} such that, for any DÌ R3,
 

                                                    w( FD,M) = Fw(D),M' ,                                              (13)
where is the attractor of the IFS s(D,M').

Note that invariance upon w is weak invariance with M' =M.

In this section we will examine the invariance of the interpolation scheme  introduced according to Theorem 1  with respect  to affine mappings w: R3® R3 having the same structure as transformations wi, more precisely
 

                                    (14)

 with a, a1,  b, b and  g, g real parameters, and  a1 > 0. We will give conditions for the strict invariance of the general scheme, and will also discuss an interesting and very useful case of  weak  invariance.

Let  w transform the data set D into the set of points  w(D) = D*= { Pi* = [xi*  yi* z i*], i = 0, 1,..., n}. Notice that the special form of the matrix  W introduced in (14) guarantees that w transform pairs of points having strictly increasing abscissas into pairs of points having the same property, therefore it is possible to build  the IFS s(D*,M ) = {R3; w1*, ..., wn*} and consider the  interpolation function corresponding to the transformed data set. At this point we need a general lemma,  first. Also let (f° g)(x) = g(f(x)).

Lemma 2. Given the IFS s(D) = {R3; w1, ..., wn} and  a  transformation  w having equation (14),  if wi° w° wi (i = 1,..., n) then

  w(FD) = Fw(D) .

Proof. Let G denote the piecewise linear interpolant of D. Then the   Hutchinson orbit of G is a sequence of polygonal lines  converging (in Hausdorff  metric) to FD, let this be {Lk = W k(G), k = 0, 1,...}. Also denote by  L0* = w(L0)  the piecewise linear interpolant of  D* = w(D), and let Lk* = W*(Lk-1*), k = 1, 2,... where W*( . ) = Èi wi*( . ). The sequence {Lk*} is the Hutchinson orbit of w(L0) and, therefore, Fw(D) = limk ®¥ Lk* . If the diagram
 

                                                  (15)

commutes for all i, namely  if  wi° w = w° wi* , then at  every new iteration k it holds  Lk* = w(Lk), which means that Fw(D) = limk®¥ Lk*  = limk ®¥ w(Lk) = w(FD) , being w a continuous function in R3 à

Lemma 3 (Translation case) Let  the affine mapping  w be the  translation  w(P) = P + v,  or,  equivalently, let W = I  in (14). Then the interpolation scheme is strictly invariant under  w, namely (12) holds.

Proof. According to (5)  wi (P) = Pi +Ai (P -Pn) and, analogously wi*(P) = Pi*+Ai *(P -Pn*),  with
 

                                                     (16)


Then  w(wi (P)) = Pi + Ai (P-Pn) + v  and  wi* (w(P)) = Pi +  v + Ai*(P -Pn). Therefore, for any arbitrary point P Î R3 we have di(P)= wi*(w(P)) - w(wi (P)) = (w°wi*-wi° w)(P) = (Ai* - Ai)(P -Pn), and therefore  di(P)= if and only if  the matrix   D  = Ai*- Ai = [djk] is the zero matrix. Direct computation shows that for M=M', dj2 = dj3 = 0,  j = 1,2,3,  while  d11ai*-ai = 0,  d21ci*-ci = 0,   d31 = ki*-ki = 0 . So,  di(P) = 0, i.e., the diagram (15) commutes.  à

Lemma 4 (Linear case). Let  the affine mapping  have zero translation part, v = 0, so that it reduces to the linear transformation  w(P) =WPLet  W have lower-triangular form with g2= 0. Then,  (12) is valid if and only if  b2 = g3.

Proof.  Under the assumptions of the Lemma, the difference di  will have the value di = (w°wi*- wi° w)(P) = (Ai*W - WAi)(P -Pn), where Ai* is given by (16). For M=M', the matrix D = Ai*W WAi has the following form

So, is the zero matrix for all feasible M if and only if  b2 = g3. Then, di= 0, the diagram (15) commute, and w(FD,M) = Fw(D),M. à

The Lemmas 2, 3 and 4 can be combined into

Theorem 3.  Let the affine mapping  w be defined by (14). Let the matrix W have  lower-triangular form with g2= 0. Then, the interpolation scheme s(D,M)is strictly invariant under  w, namely (12) holds if and only if  b2 = g3.

The next example uses a specific visual approach to illustrate Theorem 3.

 Example 3.  Let D = {(0, 0, 0), (1/4, 0, 1/2), (1/2, 1/2, 1/2), (3/4, 1, 1/2), (1, 1, 0)}be the data,  and let the choice d1= 0,  d2= 1/4,  d3= 1/4, d4= 0,  h1= 0, h2= 0, h3= 0, h4= -1/4, l1= 1/4 , l2= 0, l3=l4=-1/4, m1= 0, m2= -1/4, m3= 1/4, m4= 0 be made. By this, the parameter matrix set M is fully determined as well as the hyperbolic IFS s(D) ={R3; w1, w2, w3, w4}with the attractor FD. Further, consider two different linear maps w(P) =WP and q(P) =W1P, where and W1 are defined by

The Figure 4 (above) shows three diagrams.  The leftmost one is the graph w(FD), the middle one is Fw(D) and the rightmost one represents  both graphs simultaneously. Let the vector valued function g(x), 0 £ x £ 1, correspond to the graph w(FD) and let g*(x), 0 £ x £ 1, correspond to Fw(D). To compare two graphs better, the function e(x) = |g(x) -g*(x)|, 0 £ x £ 1, is calculated (Fig. 4, below).

Figure 4. Affine invariance.

 Let q(FD) be the graph of the function h(x),  0 £ x £ 1, and let Fq(D) be the graph of  h*(x),   0 £ x £ 1. The Figure 5 shows the similar comparison as above. This time the difference between the  graphs q(FD) and Fq(D) (or h and h*) is visible (left). This is proved by the graph of e (x) = |h(x) - h*(x) |, 0 £ x £ 1, being different from zero.

Figure 5. Failure of affine invariance .

The following theorem gives the conditions of weak invariance of an interpolating IFS.

Theorem 4. Let s(D,M)be a hyperbolic block diagonal IFS. Let  w be defined by (14) with W being a regular block diagonal matrix (  b1 = g1= 0 ).Then,  the scheme  s(D,M) is weakly invariant under  w, namely (13)  holds.

Proof. Since s(D,M) is a block diagonal, there exists a hyperbolic IFS t(Dyz,M) so that s(D,M) is its lifting (Definition 1), i.e., the 2D data set  Dyz = {Qi = [ y zi]T , i = 0, 1,..., n} obey conditions (11). In the matrix form, these conditions are
 

 Qi-1 = Mi Q0 + riQi = Mi Qn+ ri,                                          (17)

where ri = [ f  gi]T .  Since the scheme s(D,M) is invariant upon translation (Lemma 3), one can restrict transformation w given by (14) to the linear case w(P) = WP with W being a regular block diagonal matrix

Now, w(Pi) = [a1xi b2yi+ b3zi g2yi+ g3zi]T or, shortly w(Pi) = [a1xi (NQi)T ]T. Multiplying (17) with N from the left, and using I = N-1N  (since W is regular, N  is invertable) one gets

NQi-1 = NMi N-1NQ0 + Nri, NQi = NMiN-1NQn+ Nri,

which, by denoting

Mi* = NMiN -1, ri* = Nri,                                                      (18)
takes the form
Qi-1* = Mi*Q0*+ ri*, Qi*= Mi*Qn*+ ri*.                                          (19)

Equations (19) are conditions (11) for the data set Dyz* with vi*(y, z= Mi*[y   z]T + ri*. Since Mi*and Mi are similar matrices, they share eigenvalues, so t(Dyz*,M') is also a hyperbolic IFS and M' = {Mi*, i = 1,.., n}. Let D* be a lifting of the data Dyz* on the mesh  x0< x1< ...< xn and  let s(D*,M') be the lifting of t(Dyz*,M'). By Theorem 2, s(D*,M') = {R3; w1*, ..., wn*} is also a hyperbolic block diagonal IFS.  Let Mi* be the lower-right 2x2 block of the matrix Ai* that corresponds to the mapping wi*. Then, the first equality in (18) implies  Ai*= WAiW-1 or Ai*W = WAi, for i = 1,..., n, i.e., by Lemma 2w( FD,M) = Fw(D),M' .

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