Digital Signal Processing Reference
In-Depth Information
manner. For each coefficient
β
j
,,
k
, we test the following hypothesis:
H
0
:
α
j
,,
k
=
0
against
H
1
:
α
j
,,
k
=
0
,
(6.3)
where
H
0
is the null hypothesis and
H
1
is the alternative. For wavelets and curvelets,
H
0
can also be interpreted as saying that the data
X
are locally constant at scale
j
,
orientation
0 (indeed, wavelet or curvelet
coefficients computed from locally homogeneous parts of the data are zero-valued
for sufficiently regular wavelets and curvelets).
Rejection of hypothesis
H
0
depends on the double-sided
p
-value of each coeffi-
cient:
, and around position
k
, that is,
α
j
,,
k
=
>
β
j
,,
k
H
0
)
<
−
β
j
,,
k
H
0
)
p
=
Pr(
U
+
Pr(
U
.
(6.4)
α
>
α
Given a type 1 error level,
, the null hypothesis is not excluded. Although
nonzero, the value of the coefficient could be due to noise. On the other hand, if
p
,if
p
, the coefficient value is likely not to be due to the noise alone, and so the null
hypothesis is rejected. In this case, a significant coefficient has been detected.
A key step to putting equation (6.4) into practice is to specify the distribution
of the coefficient
≤ α
β
j
,,
k
under the null hypothesis
H
0
. This is the challenging part
of thresholding by hypothesis testing as it requires study of the distribution of
η
j
,,
k
under the null hypothesis, which also amounts to investigating how the distribution
of the noise
in the original domain translates into the transform domain. We first
consider, in the following section, the additive Gaussian case.
ε
6.2.1.1 Additive Gaussian Noise
Assume that
η
k
is additive zero-mean Gaussian, with subband-dependent stan-
j
,,
dard deviation
σ
j
,
, or equivalently,
2
j
β
j
,,
k
∼N
(
α
j
,,
k
,σ
)
.
(6.5)
,
This is typically the case when a zero-mean white Gaussian noise in the original do-
main,
, is transformed into a zero-mean colored Gaussian process whose covariance
is dictated by the Gram matrix of the dictionary
ε
.
Thus the double-symmetric
p
-value in equation (6.4) is given by
+∞
(
β
j
,,
k
/σ
j
,
))
1
e
−
t
2
j
/
2
σ
p
=
2
√
2
,
dt
=
2(1
−Φ
,
(6.6)
πσ
j
,
|
β
j
,,
k
|
where
) is the standard normal cumulative distribution function.
Equivalently, for a given type 1 error level
Φ
(
.
α
, one can compute the critical thresh-
old level as
τ
=Φ
−
1
(1
−α
/
2)
(6.7)
and then threshold the coefficients
β
j
,,
k
with threshold
τσ
j
,
by declaring
if
β
j
,,
k
≥
τσ
j
,
then
β
j
,,
k
is significant;
if
β
j
,,
k
<τσ
j
,
(6.8)
then
β
j
,,
k
is not significant
.
Often
τ
is chosen as 3, which corresponds to
α =
0
.
0027.
