√
φ(t)= h(n)2φ(2t . n) (A.8)
n
√
ψ(t)= h1(n)2φ(2t . n) (A.9)
n
This forms a recursive equation set where the initial approximate vector is the transform input aj0 = f(t) (Eq. A.10
and
A.11).
The
input
to
any
scale aj is the approximate output from the previous one.
dj,k = h1(n . 2k)aj+1,n (A.10) n
aj,k = h(n . 2k)aj+1,n (A.11) n
Reconstruction
(Eq.
A.12)
starts
with
the
lowest
level
detail
and
approx-
imate vectors, which are used for reconstructing the second lowest approxi-mate. The algorithm the recurses back until the original aj0 coe.cients are obtained. Consequently, only the lowest level approximate vector, in addi-tion to the entire detail matrix, is necessary for reconstruction of the original input.
aj+1,k = h(k . 2n)aj,n + h1(k . 2n)dj,n (A.12) nn
The 2k translation factor of the .lters stems from the factor-of-two rela-tion
between
each
scale
(Eq.
A.4).
This
ensures
that
each
scale
has
half
the
number of coe.cients of the previous one. Consequently, the total number of coe.cients for any number of iterations will be equal to the number of in-put samples. By guarding only high-value coe.cients, a reasonably accurate reconstruction can be made using only a fraction of the original coe.cients. By disposing of the coe.cients representing noise, reconstruction gives a de-noised version of the original input.
A.3.3 Stationary Wavelet Transform
For de-noising, better results are often obtained using the Stationary Wavelet Transform
(SWT)
[8].
SWT
di.ers
from
DWT
by
upsampling
the
.lters
at
each scale instead of downsampling the coe.cients. This creates a redun-dancy as each layer (j) produces the same number of coe.cients as the input signal. The tradeo. is better time (k) precision for coarse scales.
This study deals only with .nite signals. In order to keep .lter output length the same as input length, while reducing transients, signals / coe.-cients are padded at start and end. Padding consist of samples having the mean value of the K .rst / last samples. This provides better results than periodic padding (circular convolution), as the start and end of the signals used in this study can have very di.erent amplitude.
A.4 Nonlinear Optimization
Optimization problems can normally be transformed into minimization prob-lems, which consist in .nding the x which minimizes an object function f(x). The variable x can be a vector, so that f(x) is a function of one or more vari-ables. For linear functions, such as polynomials, the solution can be found using symbolic derivation. For non-linear problems, the solution must be found using other methods.
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