lauwer@cs.hw.ac.uk (Jean-Marc de Lauwereyns) (05/07/90)
I have posted, more than 10 days ago, an article which was rising, I
think, important questions about the credibility of certain formula given in
the book edited by O. Peitgen and D. Saupe. I know that I am only a student,
but the studies I have done in France gave me the taste for the exactitude in
the mathematical proof of the validity of a formula. The previous posting said
that that problem was urgent and it was : I had to give back the report with
a terrible dilemna : were I right or not ?
Well, 10 days are a long time enough I think for this posting to
reach the main important parts of the net, and I had no answer. So there
are two alternatives :
1) Nobody on this net has read the posting, and I cannot believe that,
2) There are people who have read the posting but didn't want to answer
for maybe four different reasons :
a) because, I am a student and students are no worth to be
answered : they are asking so trivial questions. Well, that is an opinion, but
in that case the people who think that way could have, at least told me that.
Nobody has done that : I had no answer at all.
b) it is for a project and so, don't help the student : he just
has to figure out a way ( but I gave the prooves I made : one had just to
to explain me why I was wrong or what kind of errors there were in the book.
I didn't want an algorithm like some people dared to ask : I proved that I
worked seriously on the subject I think).
c) nobody cares because the mathematical part is not
interesting : "hey why would you bother about the prooves : you have the
formula, use them and don't ask any question." To those people, I will
answer :"It is not on the sand that you are building a solid house". It seems
that it is the way most of the people work : picking a mathematical formula
without even checking if the hypothesis are verified or not, and thus without
knowing if the formula can be applied or not.
d) nobody on the net cares because nobody knows enough
mathematics to answer me, which I cannot believe.
e) nobody cares because fractals are "out". It is not my
opinion : I think on the contrary that fractal are to be used more and more
intensively in simulations of natural phenomenon. A typic field in which
it can be applied is fluid mechanics, for example.
I don't know what to think, so I post the problem again, just for the
sake of the truth now (I will go back to France soon and I don't know if I will
have access to the net in my future job or not). Please send me your answers
at lauwer@cs.hw.ac.uk . I will write a summury that I will post if I have
enough answers. Or you can post your answer to this group. Flame me if you
want, I don't care : I just want people to answer me why the reasonment I
have written is false or if it is the book which is non consistent.
Jean-Marc
------------------------ Here are the questions ----------------------------
I wanted to verify some of the mathematical parts in The Science
of Fractal Images (O. Peitgen, D. Saupe, SPRINGER VERLAG 1988) concerning
three different methods to generate fractal mountains : the displacement of
interpolated points, the midpoint algorithm, the spectral synthesis algo.
Ther seem to be some inconsistencies or errors.
First in the formula for the discrete Fourier transform (formula 1.14
page 49) :
N/2-1
----- 2 pi i fm tn
Vn = \ v e
/____ m
m=0
there seems to be something forgotten because n is running from 0 to N-1 and
because Vn should be real : thus the real formula should be
N/2-1 2 pi i f t 2 pi i f t
----- m n _ N-m n
Vn = v0 + \ (v e + v e )
/____ m m
m=1
This formula is a classic one which every student in science knows
perfectly well. Am I right to think that that formula is the one that should
figure as Eqn. 1.14, and not the other one ?
* * *
There also seems to be contradictions as regard the variances of the
Gaussian random variable. Please, could you answer me by E-mail. I will
post a summary of the answers on the different interested newsgroup.
First p. 51 to 53, the formulas given seem to be correct (I even
repeat the calculus in the general case : that means T replaces 1 and
r replaces 1/2) which gives the formula :
2H 2-2H 2
var(Delta n) = (rT) [ 1 - r ] sigma
But with the calculus made page 89, there seems to be something wrong :
we know that
X(t) = X(0) + t/T [X(T) - X(0)] + D
2
where D is a random variable of variance Delta , so if we replace
X(t) by its expression in
2H 2
var( X(t) - X(0) ) = t sigma
X(0) disappears and we have
2H 2
var( t/T [X(T) - X(0)] + D ) = t sigma
but there is the properties : var ( sum Xi ) = sum ( var (Xi)) for uncorrelated
variables, and var(rX) = r var(X), such that we have :
2H 2 2 2 2H 2
var(D) = t sigma - (t / T ) T sigma
or
2 2H 2 2-2H
Delta = t sigma [ 1 - (t/T) ]
2
and not the half of that quantity. Why is there a 2 Delta in the last
formula of page 89 ? ( I may have forgotten an important property of the
Gaussian function such normalisation of a gaussian function ... !)
* * *
My second question is about the generalisation of this algorithm to
higher functions. Once again I have been very courageous and I tried to
calculate the variance of the random function between two levels granted
the following facts : I considered that
2H 2
var(X(T,T) - X(0,0)) = T sigma , which implies that
H 2H 2
var(X(T,0) - X(0,0)) = var(X(0,T) - X(0,0)) = (1/2) T sigma
In that cases, by using interpolation between 4 points it comes
X(t,t) = (1 - t/T)(1 - t/T)X(0,0) + (1 - t/T)(t/T)X(0,T) +
(t/T)(1 - t/T)X(T,0) + (t/T)(t/T)X(T,T) + D
2
where D is a gaussian variable of variance Delta , with the same reasonment
as in 1-D and using r = t/T, we have first :
2H 2H 2 2H 2
var(X(t,t) - X(0,0)) = r T sigma = t sigma
we have also,
2 2H 2 1-H 2 2 2H 2 4 2H 2
Delta = t sigma - 2 (1-r )r T sigma - r T sigma
which can be simplified into
2 2 2H _ 2-2H
Delta = sigma t [1 - (\/2 r) ..... ]
the other terms can be neglected as infinitly small values of order
higher than (2 - 2H).
In that case why have Saupe used the same function dor Delta in the
1-D algorithm page 88 and in the 2-D algorithm page 104 ?
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Thanks on advance for the answers.
Jean-Marc
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Jean-Marc de Lauwereyns | ____ | e-mail addresses :
Heriot-Watt University | |\ /| | \ | JANET: lauwer@uk.ac.hw.cs
Computer Science Department | | \/ | |___/ | ARPA.: lauwer@cs.hw.ac.uk
Edinburgh | \___/ |___| \ |
Jean-Marc de Lauwereyns | ____ | e-mail addresses :
Heriot-Watt University | |\ /| | \ | JANET: lauwer@uk.ac.hw.cs
Computer Science Department | | \/ | |___/ | ARPA.: lauwer@cs.hw.ac.uk
Edinburgh | \___/ |___| \ |