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In-Depth Information
In summary one can say, in the language of signal processing, that nodal bases
localize well in time and multi-scales bases localize well in frequency. Now we
want to combine the two approaches - nodal basis and multi-scales basis. (We will
come to this later.) The Heisenberg uncertainty principle applies here: the product
of time and frequency resolutions is always greater than a natural constant. Using
Δ
t
as the time interval and
Δ
f
as the frequency interval, we have:
>
4
:
Δ
t Δ
f
ð
6
:
2
Þ
If the time resolution increases, of necessity the frequency resolution decreases,
and vice versa. This is a fundamental relationship inmulti-scale approximation theory.
In practice, nodal bases are mostly used to formulate problems: for instance, in
signal processing, sounds (1D) or images (2D) can immediately be recorded
electronically in the nodal basis. Nodal bases are also preferred for solution of
differential equations in the field of FEM, because of their flexibility for modeling.
In practice however most of the cases we are dealing with are for the most part
smooth functions (speech in signal processing, images in image processing, defor-
mations or flows in differential equations, models in data mining, etc.). In most
cases these can be better approximated in a multi-scale basis.
The result of this is to use the most efficient method as shown in Fig.
6.1
.
The problem is formulated in a nodal basis
Φ
n
, then a basis transformation is
applied to convert the function (or the error) into a multi-scale basis
Ψ
n
; the
problem is solved there and by means of the inverse basis transformation converted
back into the nodal basis
Φ
n
.
Examples 6.1
We now give a few examples of the use of basis transformations into
multi-scale bases:
•
Data compression
: The signals (sounds, images, etc.) are converted by basis
transformations such as Fourier or wavelet transformation (called encoding) into
the multi-scale basis. There they can be represented efficiently, that is, with few
coefficients
d
i
and thus efficiently stored and transmitted. As soon as they are
required again, the inverse basis transformation (decoding) is performed, and the
signals are once again available in the practical nodal basis.
•
Signal processing
: Process as described in data compression. In addition the
signals can be better analyzed and smoothed in the multi-scale basis. For
instance, for smoothing, the coefficients
d
i
of the high-frequency basis functions
can simply be set to 0.
•
Solution of differential and integral equations
: The operator equation is formu-
lated in the nodal basis, after which a basis transformation is applied to convert it
to the multi-scale basis. For most important differential and integral equations,
it can be shown that the solution by means of iteration methods in the space
of good multi-scale bases is asymptotically optimal. After the efficient solution
in the space of the multi-scale basis, the solution is transformed back to the
nodal basis.
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