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Global and local spectral analysis of oscillating signals and images

Global'nyj i lokal'nyj spektral'nyj analiz ostsilirujushchich signalov i izobrazhenij

Title: Global and local spectral analysis of oscillating signals and images

Textbook , 2014 , 88 Pages

Autor:in: Dr. Eugene Postnikov (Author)

Mathematics - Applied Mathematics
Excerpt & Details   Look inside the ebook

This textbook presents the basics of the Fourier, the Gabor and the continuous wavelet analysis with the Morlet wavelet. The course started from the theoretical derivation based on the averaging approach, which is close connected with corresponding methods in the theory of dynamical systems. Thus, the introductory Chapter 1 is supplied with the examples illustrating analogies between the considered signal processing and the analytical approximations for ODE solutions. Chapter 2 explores the representation of the continuous wavelet transform with the Morlet wavelet as a solution of partial differential equations. Chapter 3 contains the practical recipes for MATLAB’s code realizing the described algorithms and the examples of their implementations.

Учебное пособие представляет основы преобразований Фурье, Габора и непрерывного вейвлет-преобразования с вейвлетом Морле. Курс начинается с теоретического вывода, основанного на подходе усреднения, который тесно связан с соответствующими методами в теории динамических систем. Таким образом, вводная Глава 1 сопровождается примерами, иллюстрирующими аналогии обработки сигналов и аналитических аппроксимаций для решений ОДУ. Глава 2 исследует представление непрерывного вейвлет-преобразования с вейвлетом Морле как решения дифференциальных уравнений в частных производных. Глава 3 содержит практические рецепты программного кода для MATLAB, реализующего описанные алгоритмы, и примеры их применения.

Excerpt


1
2
1.1 . . . . . . . . . . . . .
2
1.2 ;
. . . . . . . . . . . . . . . . . .
9
1.3 - -
. . . . . . . . . . . . . . . . . .
17
1.4 -
. . . . . . . . . . . . . . . . .
24
1.5
. . . . . . . . . . . . . . . . . . . . . . . .
32
2 -
-
38
2.1 . . . . . . . . . . . . . . . . . . .
38
2.2 42
2.3 :
. . . . . . . . . .
45
3
48
3.1 . . . . . . . .
48
3.2
. . . . . . . . . . . . . . . . .
57
1
3.3 . . . . . . . .
63
3.4
-
. . . . . . . . . . . . . . . .
64
3.5 -
71
3.6 - -
: . . . . . .
76
3.7
. . . . . . . . . . . . .
81
2
1
1.1
-
, sin(t) cos(t) 2,
.. sin(t + 2n) = sin(t), cos(t + 2n) = cos(t), n Z
sin
2
(t) + cos
2
(t) = 1, , -
-
.
-
,
exp(it) = cos(t) + i sin(t),
3
, f(t)
f (t) = f (t
0
) +
df
dt
t=t
0
(t
- t
0
) +
d
2
f
dt
2
t=t
0
(t
- t
0
)
2
2
+
+
d
3
f
dt
3
t=t
0
(t
- t
0
)
3
6
+ ...
d
n
f
dt
n
t=t
0
(t
- t
0
)
n
n!
+ ... (1.1)
­ (
), ­ ( ­ ),
t
0
= 0
d
n
dt
n
sin(t)
|
t=0
=
-(-1)
n
, n = 2m - 1;
0, n = 2m,
m N
sin(t) = t
-
x
3
6
+
t
5
120
- ... - (-1)
n
x
2n-1
(2n
- 1)!
, n N. (1.2)
,
d
n
dt
n
cos(t)
|
t=0
=
0, n = 2m
- 1;
(
-1)
n-1
, n = 2m,
m N
cos(t) = 1
-
x
2
2
+
t
4
24
- ... + (-1)
n
x
2n
(2n)!
, n N.
(1.3)
, ..
d
n
dt
n
e
it
t=0
= i
n
, n N.
e
it
= 1 + it
-
x
2
2
- i
x
3
3
+ ... + i(
-1)
n-1
x
n
n!
, n N.
(1.4)
, (1.4) ­ (1.2) (1.3).
4
0
5
10
15
20
-1
0
1
-1
0
1
t
exp(it)
cos(t)
i
sin(t)
0
5
10
15
20
-1
0
1
-1
0
1
t
exp(-it)
cos(t)
-i
sin(t)
. 1.1.
, exp(±it) = cos(t) ± isin(t) ( ).
( ) (- ) ­
- , -
­ ,
.
-
-
­ :
cos(t) =
e
it
+ e
-it
2
,
sin(t) =
e
it
- e
-it
2i
.
(1.5)
(1.5)
, . . 1.1: , -
"-
" () "-
" () . -
.
5
x = sin(t), a y = cos(t).
dx
dt
= y,
dy
dt
=
-x,
, -
.
, -
d
2
x/dt
2
+ x = 0, x -
.
sin(t), cos(t) exp(it) ­
2, sin(nt), cos(nt) exp(int) ­ -
( 2/n). n ­ -
( ), 2 ­
( 2 -
n 2/n). ,
f (t) =
+
n=-
c
n
e
int
(1.6)
2.
6
, () -
c 2 :
f (t) = f
0
+
+
n=1
a
n
sin(nt) + b
n
cos(nt).
(1.7)
, (1.5) -
(1.6), c
0
= f
0
, c
n
=
(b
n
- ia
n
) /2, c
-n
= (b
n
+ ia
n
) /2.
, (1.6), (1.7) -
: c
n
= c
-n
, ..
, -
, a
n
= 0 ­ -
(
); c
n
=
-c
-n
, ,
b
n
= 0, f
0
= 0
.
-
. 1.2, 1.3. , -
-.
(1.6), (1.7) -
. -
-
1
.
(1.6), (1.7)
.
, -
f(t), -
.
1
, (),
() . : Jerri A. The Gibbs Phenomenon in Fourier
Analysis, Splines and Wavelet Approximations. Springer, 1998.
7
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
f(t)
t
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
(-2/
)sin(2
t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
n=1
2
(-2/
n)sin(2
nt)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
n=1
5
(-2/
n)sin(2
nt)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
n=1
10
(-2/
n)sin(2
nt)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1
0
1
t
n=1
100
(-2/
n)sin(2
nt)
. 1.2. -
.
8
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
f(t)
t
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
1+(4/
)cos(2
t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
1+
n=1
2
(-(-1)
n
4/
(2n-1))cos(2
(2n-1)t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
1+
n=1
5
(-(-1)
n
4/
(2n-1))cos(2
(2n-1)t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
1+
n=1
100
(-(-1)
n
4/
(2n-1))cos(2
(2n-1)t)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0
1
2
t
1+
n=1
100
(-(-1)
n
4/
(2n-1))cos(2
(2n-1)t)
. 1.3. - -
.
9
1.2 ;
-
, -
, -
,
-
.
, -
exp(i
0
t) -
exp(-it), ,
, t,
K(, t) =
1,
=
0
;
exp [i(
0
- )t] , =
0
.
(1.8)
f = exp(2t) -
. 1.4.
, K(, t) -
=
0
, ­ -
,
. , (1.8), ­ -
( ) (
), .. ­ -
, , , .. -
, .
, -
, ,
0
,
. , (. . 1.4)
0
= 2m
( . 1.4 m = 1), = 2n, n Z, -
10
0
5
10
0
5
10
15
-1
0
1
/
t
Re[K(
,t)]
0
5
10
0
5
10
15
-1
0
1
/
t
Im[K(
,t)]
. 1.4.
(1.8)
0
= 2. -
( ) .
, T = 2/
0
,
T
0
K(, t)dt =
T
0
e
2i(m-n)
dt
T, n = m;
0. n = m.
(1.9)
, -
-
, ,
f (x) =
+
n=-
c
n
e
2n
T
x
,
,
11
:
c
n
=
1
T
T
0
f (x)e
-
2n
T
x
dx.
,
,
f (x) = f
0
+
n=1
a
n
sin
2n
T
x + b
n
cos
2n
T
x
,
:
f
0
=
1
T
T
0
f (x)dx,
a
n
=
2
T
T
0
f (x) sin
2n
T
x dx, b
n
=
2
T
T
0
f (x) cos
2n
T
x dx.
-
, , -
2
. :
,
dx
dt
= y + µf (x, y, ; µ),
(1.10)
dy
dt
=
-
2
0
x + µg(x, y, ; µ),
(1.11)
2
Andronov A. Les cycles limites de Poicar ´e et la th ´eorie des oscillations auto-entrenues.
Comptes Rendus de l'Acad ´emie des Sciences de Paris. 189 (1929) 559-561.
12
µ ­ .
, -
, ­
. ,
, .. µ ,
.
(1.10)-(1.11) µ = 0, .. -
­ , -
x = R cos(
0
t), y = -
0
R sin(
0
t).
, -
, 2/. -
(1.10)-(1.11) ,
, -
-
2
0
0
[f (R cos(
0
), -
0
R sin(
0
); 0) cos
0
-
- g(R cos(
0
), -
0
R sin(
0
); 0) sin(
0
)] d = 0, (1.12)
R.
, , -
, .
z = x + i
-1
0
y. -
:
z = R cos(
0
t) - i
-1
0
0
R sin(
0
t) = R exp(-i
0
t).
(1.11) i
-1
0
,
dz
dt
=
-i
0
z + µZ(x, y; µ),
(1.13)
Z(x, y; µ) = f (x, y; µ)+i
-1
0
g(x, y; µ).
13
(1.13) exp(i
0
t)
T = 2/
0
:
2
0
0
dz
dt
e
i
0
t
dt = -
0
2
0
0
ze
i
0
t
dt +
2
0
0
Ze
i
0
t
dt.
,
2
0
0
e
i
0
t
dz
dt
dt = ze
i
0
t
2
0
0
-
2
0
0
z
de
i
0
t
dt
dt = -
0
2
0
0
ze
i
0
t
dt,
( ­ , z(0) = z(2/
0
))
.
2
0
0
Ze
i
0
t
dt = 0,
(1.14)
, ,
(1.10)-(1.11)
, -
0
, ..
c -
(1.10)-(1.11).
(1.12) (1.14) -
µ = 0:
Ze
i
0
t
= (f + i
-1
0
g)(cos(
0
t) + i sin(
0
t)) =
f cos(
0
t) - g
-1
0
sin(
0
t) + i f sin(
0
t) + g
-1
0
cos(
0
t) .
14
, -
, -
(1.12)
(1.14)
3
.
. 1.5.
(1.8)
0
= 2. -
,
0
,
, .
, -
f(t)
F () =
1
2
+
-
f (t)e
-it
dt.
3
, -
,
/2, -
.
15
, , -
(1.8) (. c. 1.5)
(
0
- ) =
+
-
K(, t)dt =
=
+
-
e
i(-
0
)t
dt =
, =
0
;
0.
=
0
.
(1.15)
=
0
-
, ­ , -
, ..
-
.
,
f (t) =
+
-
F ()e
it
d
e
i
0
t
=
+
-
(
0
- )e
it
d.
-
(1.15),
-
,
f (
0
) =
+
-
f ()(
0
- )d,
(1.16)
16
-
f() -
, .
(1.16) : (1.15)
, - , -
, , f() , -
0
, -
.
(1.16) f 1, , -
, ,
, - :
+
-
(
0
- )d = 1.
(1.17)
-
( -
) , -
­ ­ -
, .
,
F () =
1
2
+
-
f (t)e
-it
dt, f (t) =
+
-
F ()e
it
d
(1.18)
-
(.. -
). F () -
- -
( -
, .. (1.6)).
17
0
5
10
15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Re[f(t)]
t
-20
0
20
0
1
2
3
4
5
6
7
8
|F(
)|
. 1.6. (
) () -
().
1.3 -
, , -
, -
: f(t) = exp(i2t), f(t) =
exp(i4t), . . 1.6(). -
, . . 1.6(),
. , -
­ , -
, -
-
.
,
18
exp(-it), , , -
K(, t) =
1,2
t
, . . 1.7.
, , -
-
,
, .. t (-,+),
t ­
(t
- ,t + ).
-
ab initio, -
. , c , -
, ,
K(, t, ) ´o (-
) K(, t, )
,
K(, t, + ) =
1
2
[K(, t
- t,) + K(,t + t,)].
t -
t, ,
(master equation) (1.1),
,
K(, t, ) +
K
=
1
2
2K(, t, ) + 2
2
K
t
2
(t)
2
2
,
(
- , ±t).
, (t)
2
/t =
2
= const -
K
=
2
2
2
K
t
2
,
(1.19)
19
0
5
10
0
5
10
15
-1
0
1
/
t
Re[K(
,t)]
0
5
10
0
5
10
15
-1
0
1
/
t
Im[K(
,t)]
. 1.7. :
. 1.6 exp(-it). : -
, =
1,2
.
20
.
(1.18). -
K(, t, ) =
+
-
^
K(, , )e
it
d,
(1.20)
.
2
K
t
2
=
+
-
^
K(, , )
2
t
2
e
it
d =
+
-
-
2
^
K(, , ) e
it
d.
, -
(1.19) -
^
K
=
-
2
2
2
^
K
^
K = ^
K
0
exp
-
2
2
/2 ,
(1.21)
^
K
0
­ ­ - -
:
^
K
0
(, ) =
1
2
+
-
K(, t
, 0)e
-it
dt
.
(1.22)
(1.22), (1.21) (1.20),
K(, t, ) =
1
2
+
-
+
-
K(, t
, 0)e
-
2
2
/2-i(t
-t)
ddt
.
21
,
,
K(, t, ) =
1
2
+
-
K(, t
, 0)e
-
(t
-t)2
22
dt
+
-
e
-
2
+i
(t
-t)
2
2
d,
4
+
-
e
-
2
d = ,
K(, t, ) =
+
-
K(, t
, 0)e
-
(t
-t)2
22
dt
2
2
.
(1.23)
(1.23) -
, , K, , K(, t
, 0)
t
,
(t-3,t+3) (.
. 1.8).
4
: ,
+
-
e
-
2
d
2
=
+
-
e
-
2
d
+
-
e
-
2
d
=
+
-
e
-
(
2
+
2
)
dd,
,
2
+
2
= r
2
( )
dd = d(r
2
), ( )
( 0 , )
+
0
e
-
r
2
d(r
2
) = - e
-
r
2
0
= -(0 - 1) = ,
.
22
0
t
exp[-t
2
/2
2
]/(2
2
)
1/2
-
1/2
1/2
-2
1/2
-3
1/2
2
1/2
3
1/2
. 1.8. .
(1.23) (1.8). -
, K(, t
, 0) -
(1.23)
( , K
, -
).
+
-
e
i(
0
-)t
e
-
(t
-t)2
22
dt
2
2
=
= e
i(
0
-)t
+
-
e
i(
0
-)(t
-t)-
(t
-t)2
22
dt
2
2
,
-
.
23
,
+
-
e
i(
0
-)t
e
-
(t
-t)2
22
dt
2
2
= e
i(
0
-)t
e
-2
2
(
0
-)
2
,
(1.24)
(1.9) "-
" / -
0
. -
(1.23), , -
, , .
, . ,
-
2
,
.
= 1.
, , -
, -
-
. , . 1.9 -
, -
. , -
,
, -
(1.24)
,
, .
24
0
5
10
15
-1
0
1
-1
0
1
t
Re[f(t)]
Im
[f
(t
)]
t
|K( ,t,1)|
0
5
10
15
2
4
6
8
10
12
14
0.2
0.4
0.6
0.8
. 1.9. ( -
) (. . 1.6) , -
-
(1.23).
1.4 -
-
(1.24) , -
exp(
0
t) , -
0
= . , (1.24)
. (1.24)
exp(it),
e
it
K(, t, ) = e
i
0
t
e
-2
2
(
0
-)
2
-
.
25
0
5
10
15
-1
0
1
t
f(t)
=0.25
0
5
10
15
5
10
15
=1
0
5
10
15
5
10
15
t
=2
0
5
10
15
5
10
15
. 1.10.
( ­ ) , -
exp(i2t) t 5, 0.5exp(i2t) + 0.5exp(i4t) 5 < t 10
exp(i4t) t > 10 ( ­ -
, ­ ). ,
, .
f(t), -
G(, t) =
+
-
f (t
)e
i(t
-t)
e
-
(t
-t)2
22
dt
2
2
,
(1.25)
(
).
,
, -
( ) ( -
, ) -
, . . 1.10.
26
0
2
4
6
8
10
12
14
16
18
0
0.5
1
G(
,2.5)
0
2
4
6
8
10
12
14
16
18
0
0.5
1
G(
,7.5)
0
2
4
6
8
10
12
14
16
18
0
0.5
1
G(
,12.5)
. 1.11. ,
. 1.10, (
G) -
.
, ,
,
,
, -
, . -
, , . 1.10,
. 1.11 -
(1.25) = 1 ( -
­
. 1.10). ,
2 = 2.
27
(
-
) -
.
, -
,
, (1.13), ­
-
(
-
)
dz
dt
= i
0
z + µ 1 - |z|
2
z,
(1.26)
(1.26) z = Z(t) exp(i
0
t+ i
0
),
dz
dt
=
dZ
dt
e
i
0
t+i
0
+ i
0
Ze
i
0
t+i
0
=
dZ
dt
e
i
0
t+i
0
+ i
0
z.
(1.26) -
,
dZ
dt
= µ 1
- Z
2
Z,
(1.27)
Z(0) = Z
0
Z(t) = Z
0
e
µt
Z
2
0
(e
2µt
- 1) + 1
.
(1.28)
: µ ,
,
5
.
5
-
- : Garcia-Morales V., Krischer K. The
complex Ginzburg-Landau equation: an introduction. Contemporary Physics. 53 (2012)
79-95.
28
0
20
40
60
80
100
-1
0
1
-1.5
-1
-0.5
0
0.5
1
1.5
t
Re(z)
Im(z)
. 1.12. ­ -
(1.26)
0
= 1, µ = 0.1, = 1 z = Z
0
= 0.01 ( ),
(1.28) ( ,
- ).
. 1.12.
, . . 1.13. , -
(
),
-
,
- (. (1.28),
Z = 1/ t; , , dZ/dt = 0
, (1.27)).
-
(1.28), -
, ,
, .
29
10
20
30
40
50
60
70
80
90
100
0.5
1
1.5
2
t
|G(
,t)|
10
20
30
40
50
60
70
80
90
100
0
0.5
1
t
|G(1,t)|
10
20
30
40
50
60
70
80
90
100
-2
-1
0
1
2
t
Re[G(1,t)]
10
20
30
40
50
60
70
80
90
100
-2
-1
0
1
2
t
-Im[G(1,t)]
. 1.13. ­
- (. . 1.12): , -
, ( -
) .
- ().
30
,
, .
-
, -
. -
,
,
6
,
dx
dt
= f (x, t, )
-
( ) -
X , -
, -
T :
dX
dt
= ¯
f (X, t, ),
¯
f =
1
T
T
0
f (x, t
, )dt
.
, -
-.
,
-
(, , -
, ­ )
. -
(, , ).
6
. , Jan A.
Sanders (2006) Averaging. Scholarpedia, 1(11):1760, -
: ..
-
( ).
31
-
-
, -
,
-
, -
.
, -
-
, -
, a
,
:
w(t, a, b) =
1
2a
2
e
-
(t-b)
2
2a2
,
´ b
.
, exp [i(t - b)]
. -
, -
, =
0
/a,
0
­ -
,
-
.
(1.25) -
w(a, b) =
+
-
f (t)e
-i
0
(t-b)
a
e
-
(t-b)
2
2a2
dt
2a
2
,
(1.29)
- -
.
,
, ( -
) , . 1.14.
32
-5
0
5
-1
-0.5
0
0.5
1
-5
0
5
-1
-0.5
0
0.5
1
Gabor
-5
0
5
-1
-0.5
0
0.5
1
-5
0
5
-1
-0.5
0
0.5
1
-5
0
5
-1
-0.5
0
0.5
1
Morlet
-5
0
5
-1
-0.5
0
0.5
1
. 1.14. ( ) -
( ) : = 0.5, , 2
( ). , -
0
= (
,
a =
0
/). , -
­ .
1.5 -
,
7
(Morlet wavelet)
() = Ce
i
0
e
-
2
2
,
(1.30)
C ­ ( -
).
7
,
, -
, (1.30) , -
: = C
0
[exp(i
0
) - exp(-
2
0
/2)] exp(-
2
/2)
33
, , -
­
,
+
-
||d = 1.
(1.30) -
,
+
-
||
2
d = C
+
-
e
-
2
2
d = C2 = 1,
C = 1/2. -
(1.29)
= (t - b)/a, d = dt/a, (1.29) -
a
2
.
--
f(t)
.
, , (1.29)
w(a, b) =
+
-
f (t)
()d,
c = (t - b)/a
() = exp[
-i
0
-
2
/2]/2,
- (1.30) .
:
-
2
2
- i
0
= -
1
2
2
+ 2i
0
+ (i
0
)
2
- (i
0
)
2
=
=
-
1
2
( + i
0
)
2
-
2
0
2
.
34
() =
1
2e
-
2
0
2
e
-
1
2
(+i
0
)
2
.
, -
:
w(a, b) = e
-
2
0
2
+
-
f (t)
e
-
[(t-b)+i0a]
2
2a2
2a
2
dt.
(1.31)
(1.31) -
a = 0. ,
8
, -
. -
exp[-
2
/2a
2
]/2a
2
.
,
+
-
e
-
2
2a2
d
2a
2
= 1;
= 0 a 0
, a , = 0
( -
), ­ ,
. :
() = lim
a0
e
-
2
2a2
2a
2
=
, = 0;
0.
= 0,
- .
8
. §V.7 : .., .. -
. .: , 1973.
35
, (1.16)
lim
a0
+
-
f (t)
e
-
((t-b)+i0a)
2
2a2
2a
2
dt =
+
-
f (t)(t - b)dt = f(b).
,
- a = 0
w(0, b) = f (b)e
-
2
0
2
,
(1.32)
-
, -
.
(1.32) ,
f (b) 1, -
+
-
()d =
+
-
()d = e
-
2
0
2
,
(0.0072
0
= , 2.7
· 10
-9
0
= 2), -
9
.
-
.
9
- -
, -
a
b (. §2.4 : . . : "
", 2001.). , ,
(. . 32) ( -
). , ,
"-" -
,
-.
36
,
f (t) = exp(it), -
(1.29) , -
-
:
w(a, b) =
+
-
e
it
e
-i
0
(t-b)
a
e
-
(t-b)
2
2a2
dt
2a
2
=
= e
ib
+
-
e
i
(
-
0
a
)
(t-b)
e
-
(t-b)
2
2a2
dt
2a
2
=
= e
ib
e
-
(a-0)
2
2
+
-
e
-
[(t-b)+ia(a-0)]
2
2a2
dt
2a
2
=
= e
ib
e
-
(a-0)
2
2
. (1.33)
, -
´ ( b) -
, -
( a). -
a
max
, -
a
max
-
0
= 0. , -
-
T = 2/T , ,
-
:
a
max
=
0
2
T.
(1.34)
(1.34) ,
­ ,
37
. 1.15. -
, ( -
. . 1.10) ,
0
=
( )
0
= 2 ( ). -
.
, ­ .
0
= 2 -
.
. 1.15
,
1 0.5 -
. , , -
-
( ).
, , , ´ -
,
..
.
38
2
-
2.1
- -
-
, -
, -
-
39
-
, -
1
.
,
,
(, , -
)
2
.
-
() = exp(
-i
0
-
2
/2)
(- , -
(1.29);
):
() = e
i
0
-
2
2
(
-i
0
- ) = -
()(i
0
+ ),
() =
-
()(i
0
+ )
-
().
,
-, a and b,
, = (t - b)/a,
a
=
a
=
t - b
-a
2
=
-
a
,
b
=
b
=
-
1
a
,
1
: .. -
- -
. . . . . . 2006 46 77-78
( , : ..
"" 2005
128 752-759.)
2
: Haase M.A. Family of Complex Wavelets
for the Characterization of Singularities. In: Paradigms of Complexity, ed. M.M Novak.
World Scientific, 2000, pp.287-288. , -
-,
40
2
b
2
=
-
1
a
b
=
1
a
2
=
-
1
a
2
()(i
0
+ ) +
() . (2.1)
, -
. -
, , -
a
-1
,
w
a
=
a
1
a
+
-
f (t)
()
dt
2=
-
1
a
2
+
-
f (t)
()
dt
2 +
1
a
+
-
f (t)
()
-
a
dt
2.
(2.1) -
,
2
w
b
2
=
-
1
a
3
+
-
f (t)
()(i
0
+ )
dt
2 +
+
-
f (t)
()
dt
2,
a ,
a
2
w
b
2
=
-
i
0
a
2
+
-
f (t)
()
dt
2
i
0
w
b
-
1
a
2
+
-
f (t)
()dt +
+
-
f (t)
()
dt
2
w
a
.
41
,
:
w
a
= a
2
w
b
2
- i
0
w
b
.
(2.2)
(2.2) -
, ´y
, b, "
", a ­ " -
". (2.2) -
( ,
),
exp(
-i
0
) .
, (2.2) -
, ­
(1.32). , -
" ", -
(2.2) -
w(0, b) = f(b), -
exp(
-
2
0
/2)
3
- -
a b. -
(2.2).
3
(1.32), , -
, , (
0
= 2) . (2.2) -
, (1.32),
´ .
42
2.2 -
a = 0 (2.2) -
w
a
=
-i
0
w
b
,
(2.3)
, ,
- .
,
w(a, b) = u(a, b) + iv(a, b) (2.3) -
:
u
(a/
0
)
=
v
b
,
v
(a/
0
)
=
-i
u
b
.
,
(1.33) -
:
a
e
ib
e
-
(a-0)
2
2
=
-(a -
0
) w;
b
e
ib
e
-
(a-0)
2
2
= iw,
a 0 (2.3).
, u = cos(t) u = cos(t) -
exp(it),
f = exp(it) (2.2) , f = exp(iz),
z = b + ia/
0
­ ,
- (2.3), (2.2) -
.
43
, (-
) , , -
u(t), , -
, -
, a = 0 (2.3).
­ , -
­ ,
, , , -
­ , ..
. -
(1.5),
cos(t) =
e
it
2
+
e
-it
2
,
sin(t) =
-i
e
it
2
+ i
e
-it
2
,
, -
() -i, -
c (-) i,
, -
.
, - ^u()
^v() -
^
v() = -isgn()^u(),
sgn() =
1,
0;
-1, 0.
44
, -
,
: -
-
- :
^
F
+
-
f (t
)g(t
- t
)dt
= 2 ^
f ()^
g().
(2.4)
, , -
sgn():
1
2
+
-
h(t)e
-it
dt = -isgn().
, -
, § 1.3
, ,
-
, ,
-it
1
2
+
-
h(t)e
-it
dt = -i2(),
:
t
+
-
1
2
+
-
h(t)e
-it
dt
e
it
d = 2
+
-
()e
it
d,
h(t) = 2/t , , -
,
:
v(t) = H [u(t)] =
1
+
-
u(t
)
1
t
- t
dt
.
(2.5)
45
, - -
f(t) -
(2.2) , -
a = 0, .. (2.2)
c -
: w(0, b) = f(t) + H [f(t)].
-
A(b) =
f (b)
2
+ (H [f (b)])
2
,
(b) = arctg
H [f (b)]
f (b)
w(0, b) =
A(b) exp(i(b)) - -
,
4
2.3 :
(2.2),
w
a
= a
2
w
b
2
- i
0
w
b
,
, , , ­
, -
. -
­ ,
4
-
,
( ) -
: Vela-Arevalo, L.V. (2002) Time-frequency analysis based on wavelets
for Hamiltonian systems. Dissertation (Ph.D.), California Institute of Technology.
http://resolver.caltech.edu/CaltechETD:etd-03302004-115559
46
w ,
. -
, -
, -
. , , -
,
0
, ,
-
. -
w
a
= a
2
w - i
0
· w,
(2.6)
, ,
0
= (
0x
,
0y
).
( ) (2.6)
w
a
= a
2
W
b
2
x
+
2
W
b
2
y
- i
0x
W
b
x
+
0y
W
b
y
.
(2.7)
b
x
, b
y
­ ,
- b = (b
x
, b
y
),
, - a ­ .
(2.7)
w(a, b
x
, b
y
) =
+
+
f (x, y)
(
x
,
y
)dxdy,
(
x
,
y
) =
1
2a
2
e
i(
0x
x
+
0y
y
)
e
-
1
2
(
2
x
+
2
y
)
.
(2.8)
, (2.7) (1.32):
W (0, b
x
, b
y
) =) = f (b
x
, b
y
)e
-
1
2
(
2
0x
+
2
0y
)
= f (b
x
, b
y
)e
-
2
0
2
,
47
0x
=
0
cos(),
0y
=
0
sin(), -
( -
, -
).
,
§ 2.1
5
, , (2.8)
(1.30)
.
5
. : Postnikov E.B., Singh V.K. Local spectral
analysis of images via the wavelet transform based on partial differential equations.
Multidim. Syst. Sign. Proces, 2012 (doi:10.1007/s11045-012-0196-1)
48
3
3.1
, -
MATLAB
, -
(Fast Fourier Transform, FFT), -
fft .
F () =
1
2
+
-
f (t)e
-it
dt
F
n
=
N
j=1
f
j
e
-
2i(j-1)(n-1)
N
,
(3.1)
f
j
­ ,
. , f(t)
49
[t
min
t
max
, f
j
= f (t
j
) t
j
, -
N - 1 (t
1
= t
min
, t
N
= t
max
).
fft -
(Cooley-Tukey), N -
(.., , 2
6
= 64, 2
10
= 1024
..). N (.. n = 1..N), -
.
. 3.1 -
( t
max
N):
N=64;
tmax =5;
t=l i n s p a c e ( 0 , tmax ,N) ;
nu =1;
f =exp ( i 2pinu t ) ;
F= f f t ( f ) ;
, . 3.1 (3.1),
: fft ,
, , -
, , ,
, ­
, , -
, ;
2(n - 1)/N, n = 1..N
(-
­ - -
), -
, "", .. -
, -
/ ­ -
, ­ -
50
0
5
10
-1
0
1
t
f(t)
0
50
100
0
50
100
150
n
|F
n
|
0
5
10
-1
0
1
t
f(t)
0
50
100
0
50
100
150
n
|F
n
|
0
5
10
-1
0
1
t
f(t)
0
50
100
0
50
100
150
n
|F
n
|
. 3.1. ­ ( -
-
, ; -
), .
, (.
)
-
, -
.
,
,
, .
, , -
.
, , ,
, ; ,
.
51
0
5
10
-1
0
1
t
f(t)
0
100
200
-200
0
200
n
Re(F
n
)
0
100
200
-200
0
200
n
Im(F
n
)
0
100
200
-200
0
200
n
|F
n
|
0
5
10
-1
0
1
t
f(t)
0
100
200
-200
0
200
n
Re(F
n
)
0
100
200
-200
0
200
n
Im(F
n
)
0
100
200
-200
0
200
n
|F
n
|
. 3.2. , ( ) -
fft : ,
.
( ,
,
­ -
(Nyquist));
+
, -
,
-
=
-
-
, (3.1)
fft , -
(, ,
(1.5)), ,
n (, , F
n
) -
.
-
, . 3.2.
52
- (
(1.5))
F [cos(
0
t)] = 0.5( -
0
) + 0.5( +
0
)
F [sin(
0
t)] = -0.5i( -
0
) + 0.5i( +
0
)
.
- -
,
­ , -
, ­
. , . 3.2, fft
, (
, ),
­ ­ -
. , ,
( -
,
0 2), -
-
.
-
MATLAB
fftshift , . . 3.2
:
N=256;
tmax =20;
t=l i n s p a c e ( 0 , tmax ,N) ;
nu =1;
f =exp ( i 2pinu t ) ;
F0= f f t ( f ) ;
F= f f t s h i f t ( F0 ) ;
53
0
5
10
-1
0
1
t
f(t)
0
100
200
-200
0
200
n
Re(F
n
)
0
100
200
-200
0
200
n
Im(F
n
)
0
100
200
-200
0
200
n
|F
n
|
0
5
10
-1
0
1
t
f(t)
0
100
200
-200
0
200
n
Re(F
n
)
0
100
200
-200
0
200
n
Im(F
n
)
0
100
200
-200
0
200
n
|F
n
|
. 3.3. , ( . 3.2)
, fftshift .
, - -
, -
, ­ -
, ­ .
-
.
, , , -
-N/2 N/2-1
( ,
N/2 -
-N/2)
max
= t
max
- t
min
, 2/T
max
-
-
54
.
, -
. -
+
-
|f(t)|
2
dt =
1
2
+
-
|f()|
2
d
(3.2)
-
´ .
, ,
,
( -
MATLAB
-
trapz).
,
, -
-
, ­
, .
function [ omega , F]= F o u r i e r ( t , f )
n=length ( t ) ; t v ( : , 1 ) = t ; f v ( : , 1 ) = f ;
N=2^( c e i l ( log2 ( n ) ) ) ;
t i ( : , 1 ) = l i n s p a c e ( t v ( 1 ) , t v ( end ) ,N) ;
f i =interp1q ( tv , f v , t i ) ;
Fk= f f t ( f i ) ;
Fks= f f t s h i f t ( Fk ) ;
omega=([-N/2:N/2-1])2 pi /( t i (end)- t i ( 1 ) ) ;
fNorm=trapz ( t i , abs ( f i ) . ^ 2 ) ;
FNorm=trapz ( omega , abs ( Fks ) . ^ 2 ) ;
F=Fkssqrt (2 pifNorm/FNorm) ;
55
, -
.
,
MATLAB
-
-, -
, tv (:,1)= t ;
fv (:,1)= f ;
­ .
-
-
( fft
),
( fi =interp1q(tv,fv, ti ); )
, -
( linspace) -
- ( -
N=2^(ceil(log2(n)));, ceil ­ -
´ ).
Fourier , -
, .3.4. , -
, -
-
. ,
( -
),
( ). -
, ­
;
-
­ .. - ).
56
0
5
10
-1
0
1
t
f(t)
-5
0
5
0
5
/2
|F(
)|
-2
0
2
0
5
/2
|F(
)|
0
5
10
-1
0
1
t
f(t)
-10
0
10
0
5
/2
|F(
)|
-2
0
2
0
5
/2
|F(
)|
0
5
10
-1
0
1
t
f(t)
-5
0
5
0
5
10
/2
|F(
)|
-2
0
2
0
5
10
/2
|F(
)|
0
5
10
-1
0
1
t
f(t)
-2
0
2
-10
0
10
/2
Re[F(
)]
-2
0
2
-10
0
10
/2
Im[F(
)]
-2
0
2
-10
0
10
/2
|F(
)|
0
5
10
-1
0
1
t
f(t)
-2
0
2
-10
0
10
/2
Re[F(
)]
-2
0
2
-10
0
10
/2
Im[F(
)]
-2
0
2
-10
0
10
/2
|F(
)|
. 3.4. , . (3.1)
(3.2) Fourier.
±2, -
, 2.
57
. 3.5. -
.
3.2
, , -
,
, -
, ( -
) . , -
,
f (x, y) = sin(10x) + sin(5y) + sin(7(x + y)),
(3.3)
. 3.5 .
58
F (
x
,
y
) =
1
4
2
+
-
+
-
f (x, y)e
-i
(
x
x+
y
y
)dxdy.
(3.4)
-
MATLAB
fft2 , -
, , fft , , -
-
.
function [ omegax , omegay , F2]= F o u r i e r 2 ( x , y , f )
[ X, Y]= meshgrid ( x , y ) ;
nx=length ( x ) ;
ny=length ( y ) ;
Nx=2^( c e i l ( log2 ( nx ) ) ) ;
Ny=2^( c e i l ( log2 ( ny ) ) ) ;
x i ( : , 1 ) = l i n s p a c e ( x ( 1 ) , x ( end ) , Nx ) ;
y i ( : , 1 ) = l i n s p a c e ( y ( 1 ) , y ( end ) , Ny ) ;
[ Xi , Y i ]= meshgrid ( x i , y i ) ;
f i =interp2 (X, Y , f , Xi , Yi , ' l i n e a r ' , 0 ) ;
F=f f t 2 ( f i ) ;
F2= f f t s h i f t (F ) ;
omegax=([-Nx/2:Nx/2-1])2 pi /( x (end)-x ( 1 ) ) ;
omegay=([-Ny/2:Ny/2-1])2 pi /( y (end)-y ( 1 ) ) ;
-
x y ,
,
- -, .
- -
-
59
. 3.6.
( ,
­ ). , -
x
= 0,
y
= 0
x
=
y
, ..
.
, , , -
.
, Fourier2
. , -
,
, trapz
­ ­ -
. ,
-
, -
.
(3.3)
. 3.6.
60
- 6 -
.
-
, . 3.6. -
, -
, -
­ (. (3.3))
, , .
-
, . 3.6
-
, ( ,
-
, .. , -
- ). -
, -
, -
-
, . -
.
[ ox , oy , F]= F o u r i e r 2 ( x , y , f ) ;
[OX,OY]= meshgrid ( ox , oy ) ;
rho_max=sqrt ( ox ( end)^2+ oy ( end ) ^ 2 ) ;
rho=l i n s p a c e ( 0 , rho_max , 1 2 8 ) ;
p h i=l i n s p a c e ( 0 , 2 pi ,101);
[RHO, PHI]= meshgrid ( rho , p h i ) ;
OXR=RHO. cos (PHI ) ;
OYR=RHO. sin (PHI ) ;
absF=abs (F ) ;
pF=interp2 (OX, OY, absF ,OXR,OYR, ' n e a r e s t ' , 0 ) ;
61
0
1
2
0
5
10
15
0.5
1
30
210
60
240
90
270
120
300
150
330
180
0
. 3.7. - (. . 3.6) -
,
.
(
x
,
y
)
(, ),
=
2
x
+
2
y
, = arctg
y
x
;
x
= cos(),
y
= sin(),
0 2 (.. 360
0
, . . 3.7 ).
-
( 128 × 101), -
interp2 (
' nearest ' -
; , -
: ' linear ', 'cubic', ' spline ').
. 3.7 .
,
62
(
)
, ().
,
( = 0) c
0
= 10, -
( = /2 = 90
0
) ­
/2
= 5
( = /4 = 45
0
) ­
/2
= 72
9.9 ( -
, , ).
-
, . . 3.7 . -
-, -
, |F(,)|
2
, ..
|F|
2
0
|F(,)|
2
d.
. 3.7
IpF=trapz ( pF . ^ 2 ' ) ;
polar ( phi , IpF /max( IpF ) ) ;
, -
, trapz -
-
; ,
, -
-
.
. 3.7 -
, -
(3.3), .
, , ,
63
-
,
, .
-
(3.2), -
,
. , -
(, -
) .
, .
3.3
MATLAB
ifft ifft2 -
.
-
, .. ( -
) -, .
F=fft(f ) f=fft2(F),
f= ifft (F) f=ifft2 (F) -
.
, -
F , -
(.. -
). -
, ,
(2.4),
.
64
3.4 -
-
-
(1.25),
G(, t) =
+
-
f (t
)e
i(t
-t)
e
-
(t
-t)2
22
dt
2
2
,
, -
, -
(2.4),
( -
^
F
-1
) (-
^
F )
( 2):
G(, t) = ^
F
-1
^
F [f (t)] ^
F e
-it
e
t2
22
2
2
2
.
^
f (
) = ^
F [f (t)] -
, § 1.3, -
:
G(, t) = 2 ^
F
-1
^
f (
)e
-2
2
(
-)
2
.
(3.5)
, -
-
, -
MATLAB
-
, .
65
F= f f t ( f ) ;
nrm=2pi /( t (end)-t ( 1 ) ) ;
omega_ = ( [ ( 0 : (N/ 2 ) ) ((( -N/2)+1): -1)])nrm ;
for k=1: length ( omega ) ;
window=exp(-2sigma ^2(omega_+omega(k ) ) . ^ 2 ) ;
c o n v o l ( : , k)=window . F;
G( : , k)= i f f t ( c o n v o l ( : , k ) ) ;
end
,
ifft 1/2,
^
F
-1
, 2 ^
F
-1
. ,
, omega_, -
­
(
§ 3.3) -
fft .
t [0, 1] -
-
f (t) = e
-4t
cos(20t) + [
1
3
,
2
3
](t) sin(40t),
(3.6)
[t
b
,t
e
]
(t) =
1, t
[t
b
, t
e
];
0, t /
[t
b
, t
e
].
(3.6), . 3.8 , -
-
: -
(
)
(
).
66
. 3.8. -
= 0.05,
. ­ -
, .
-
,
, . 3.8 .
,
, , .
,
­ . -
,
fft
, ,
20, , -
,
.
-
67
-0.5
0
0.5
1
1.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
t
f(t)
. 3.9. -
.
­ ,
-
, . . 3.9.
, fft ,
,
, .
-
, -
.
fftGabor, , Fourier, -
, -
,
. -
. 3.10. .
68
. 3.10. ,
.
function [ t i ,G]= f f t G a b o r ( t , f , omega , sigma ) ;
n=length ( t ) ; t v ( : , 1 ) = t ; f v ( : , 1 ) = f ;
N=2^( c e i l ( log2 ( n ) ) ) ;
t i ( : , 1 ) = l i n s p a c e ( t v ( 1 ) , t v ( end ) ,N) ;
f i =interp1q ( tv , f v , t i ) ;
f p ( 1 :N/2)= f l i p u d ( f i ( 2 :N/ 2 + 1 ) ) ;
f p (N/ 2 + 1 : 1 . 5 N)= f i ;
f p ( 1 . 5 N+1:2N)=flipud ( f i (N/2:N-1));
F= f f t ( f p ) ; nrm=pi / ( t ( end)-t ( 1 ) ) ;N=2N;
omega_ = ( [ ( 0 : (N/ 2 ) ) ((( -N/2)+1): -1)])nrm ;
for k=1: length ( omega ) ;
window=exp(-2sigma ^2(omega_+omega(k ) ) . ^ 2 ) ;
cnv ( k , : ) = window . F;G(k ,:)= i f f t ( cnv (k , : ) ) ;
end
G=G( : ,N/4+1:3N/4);
69
- -
, ,
:
function [ t i ,G]= f f t M o r l e t ( t , f , a , omega0 ) ;
n=length ( t ) ; t v ( : , 1 ) = t ; f v ( : , 1 ) = f ;
N=2^( c e i l ( log2 ( n ) ) ) ;
t i ( : , 1 ) = l i n s p a c e ( t v ( 1 ) , t v ( end ) ,N) ;
f i =interp1q ( tv , f v , t i ) ;
f p ( 1 :N/2)= f l i p u d ( f i ( 2 :N/ 2 + 1 ) ) ;
f p (N/ 2 + 1 : 1 . 5 N)= f i ;
f p ( 1 . 5 N+1:2N)=flipud ( f i (N/2:N-1));
F= f f t ( f p ) ;N=2N;
nrm=pi / ( t ( end)-t (1))
omega_ = ( [ ( 0 : (N/ 2 ) ) ((( -N/2)+1): -1)])nrm ;
i f a (1)==0
w( 1 , : ) = f p exp(-omega0 ^2/2);
k1 =2;
else
k1 =1;
end
for k=k1 : length ( a ) ;
omega_s=a ( k ) omega_ ;
window=exp( -(omega_s-omega0 ) . ^ 2 / 2 ) ;
cnv ( k , : ) = window . F;
w( k , : ) = i f f t ( cnv ( k , : ) ) ;
end
w=w ( : ,N/4+1:3N/4);
­ -
,
a = 0 -
, (1.32).
70
. 3.11. -
0
= 2: , -
(
).
71
3.5 - -
-
-
-
(2.2),
, u(a, b)
v(a, b) -
w(a, b) = u(a, b) + iv(a, b) (2.2) -
a
u
v
=
b
a
u
b
+
0
v
a
v
b
-
0
u
,
(3.7)
-
MATLAB
pdepe.
, ,
, -
. , -
-
. ,
. -
, -
. , -
,
;
. -
,
-
, , -
72
,
, -
..
,
-
, , -
-
.
, -
-
/ , -
. -
,
a = 0, :
u(a, ·)
v(a, ·)
+
a
10
-n
+ a
n
b
1
0
0 b
2
a
u
b
+
0
v
a
v
b
-
0
u
=
u(0, ·)
v(0, ·)
.
n ­ -
4 8,
[0, 1] (
). b
1
b
2
{b
1
= 1, b
2
= 0
} {b
1
= 0, b
2
= 1
}, -
.
(3.7) (3.5)
, , -
MATLAB
' -
pdepe, -
(..
,
),
.
73
- :
function w=pdeMorlet ( b , u , omega0 , a , lB , rB , n )
global xu0 uu0 xmax omega u l e f t
global u r i g h t LeftB RightB pow
omega=omega0 ; pow=n ; bmax=length ( b ) ;
xu0=b/b ( bmax ) ; au=a /b ( bmax ) ;
uu0 ( : , 1 ) = r e a l ( u ) ; uu0 ( : , 2 ) = imag ( u ) ;
LeftB=lB ; RightB=rB ;
u l e f t =uu0 ( 1 , : ) ; u r i g h t=uu0 ( bmax , : ) ;
m = 0 ;
s o l = pdepe (m, @pde , @ic , @bc , xu0 , au ) ;
w=( s o l ( : , : , 1 ) + i sol ( : , : , 2 ) ) ;
w=wexp(-omega0 ^2/2);
function [ c , f , s ]= pde ( x , t , u , DuDx)
global omega
c = [ 1 ; 1 ] ;
f = [ t DuDx(1)+omegau (2 );
t DuDx(2)-omegau (1)] ;
s = [ 0 ; 0 ] ;
function u0=i c ( x )
global xu0 uu0
u0 =[ interp1q ( xu0 , uu0 ( : , 1 ) , x ) ;
interp1q ( xu0 , uu0 ( : , 2 ) , x ) ] ;
function [ pl , ql , pr , qr ]= bc ( xl , ul , xr , ur , t )
global u l e f t u r i g h t LeftB RightB pow
e p s l n =10^(-pow ) ;
p l = [ u l (1) - u l e f t (1 ) ; ul (2)- u l e f t ( 2 ) ] ;
q l = [ LeftB ( 1 ) ; LeftB ( 2 ) ] t /( t ^pow+epsln ) ;
p r = [ ur (1) - uright (1 ); ur(2)- uright ( 2 ) ] ;
qr = [ RightB ( 1 ) ; RightB ( 2 ) ] t /( t ^pow+epsln ) ;
74
, , -
b, u,
­ \omega_0
a ( , -
,
­ -
),
lB rB, -
b
1
, b
2
[b1, b2], n
(3.5) .
, -
(3.7) (pde), (ic) (bc) -
. (pdeMorlet)
-
(global), -
. ­ (-
) -
w=(sol (:,:,1)+ isol (:,:,2)); , -
(w=wexp(-omega0^2/2);) -
( -
-
, exp(-
2
0
/2).
-, -
b a -
.
" ", .. ,
, , -
, -
, ( -
, pdepe, -
-
75
0
0.2
0.4
0.6
0.8
1
-1
0
1
t
f(t)
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
a
0
0.01
0.02
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
a
0.1
0.2
0.3
. 3.12. ( ) ,
0.3 cos(20t) sin(40t) ­ , -
-, -
() ().
1
).
, , -
, , -
, ,
(
0
= ). -
. . 3.12, , -
- -
-
.
1
: .., ..
- , -
.
XIX . . 2. - .: , 2007, C. 7-9;
http://spravka.akin.ru/Docs/Rao/Ses19/AI2.PDF
76
3.6 - -
: -
-
, -
.
, , , -
, -
-
. ­ -
§ 2.3, ­ , -
-
.
, , -
w(b
x
, b
y
, a, ) ­ -
,
(b
x
, b
y
), a . , -
.
, - -
( ) -
. -
w(b
x
, b
y
, a, ), -
,
a
-
w
a,
(b
x
, b
y
).
,
. 3.13, -
( -
77
. 3.13. ,
:
,
-, ­ -
.
sin(10x), sin(5y) sin(2(x + y))),
.
.
, -
-
(. . 3.13). -
, -
- -
( = 0, /4, /2), -
­ , -
( [mx,imx]=max(pF');
­ -
78
, ,
plot(phi/pi,rho(imx)/pi) -
; § 3.1). ,
,
.
,
- -
(??),
w(b
x
, b
y
, a, ) =
=
+
-
f (x, y)e
-i
0
a
[(x-b
x
) cos +(y-b
y
) sin ]
e
-
(x-bx)
2+(y-by)2
2a2
dxdy
2a
2
,
=
0
/a .
,
w
,
(b
x
, b
y
) = e
i(b
x
cos +b
y
sin )
×
×
+
-
f (x, y)e
-i(xcos+y sin)
e
-
(x-bx)
2+(y-by)2
2(0/)2
dxdy
2a
2
. (3.8)
(3.8) , -
§§ 1.4-1.5: -
, .
-
, -
, -
.
,
, , ,
­ -
.
79
MATLAB
fspecial imfilter .
. 'gaussian'),
-
.
( , fspecial
,
, -
, ­ nrmx, nrmy).
imfilter (
conv ­ "convolution") -
. -
: 'symmetric'
-
, ' circular ' -
, ' replicate ' -
, ­ (
).
= /4, = 52 -
0
= 2 :
v a r p h i=pi / 4 ; omega=5sqrt (2) pi ;
omega0=2pi ; amv=omega0/omega ;
r d o t p h i=XXcos ( varphi )+YYsin ( varphi ) ;
f v=exp(- i omega( rdotphi ) ) ; fvz=fv . z ;
nrmx=length ( xx ) cos ( varphi )/( xx (end)-xx ( 1 ) ) ;
nrmy=length ( yy ) sin ( varphi )/( yy (end)-yy ( 1 ) ) ;
sigma=amv (nrmx+nrmy ) ;
g f= f s p e c i a l ( ' g a u s s i a n ' , round ( sigma ) , sigma ) ;
vw= i m f i l t e r ( f v z , gf , ' symmetric ' , ' conv ' ) ;
80
. 3.14. ,
-
. (
) .
­ , .. abs(vw), -
. 3.14.
, -
, -
.
.
-
, -
( = 10), -
a =
0
/ = 0.2 -
( -
). -
,
0
= 4 -
.
81
3.7
. 3.11,
. 3.14, ,
0.5, -
-
( . 3.11 ­ -
). ,
§ 1.4 -
, (-
) ,
- .
, , -
-
-
( -,
, -
), -
-
. , -
.
, § 2.2 ,
-
( ) -
,
. -
, , -
.
82
MATLAB
hilbert , -
. ,
,
-
,
( ).
,
,
- (-
,
).
:
function [ t i , f a ]= s A n a l y t i c ( t , f ) ;
n=length ( t ) ; t v ( : , 1 ) = t ; f v ( : , 1 ) = f ;
N=2^( c e i l ( log2 ( n ) ) ) ;
t i ( : , 1 ) = l i n s p a c e ( t v ( 1 ) , t v ( end ) ,N) ;
f i =interp1q ( tv , f v , t i ) ;
f p ( 1 :N/2)= f l i p u d ( f i ( 2 :N/ 2 + 1 ) ) ;
f p (N/ 2 + 1 : 1 . 5 N)= f i ;
f p ( 1 . 5 N+1:2N)=flipud ( f i (N/2:N-1));
N=2N;
f a= h i l b e r t ( f p ) ;
f a=f a ( : ,N/4+1:3N/4);
fa
ta, -
, -
-.
-
, , -
. 3.15.
83
0
0.5
1
-2
-1
0
1
2
t
f(t)
-500
0
500
0
0.05
0.1
0.15
0.2
|F(omega)|
0
0.5
1
-2
0
2
-2
0
2
t
f(t)
H[f(t)]
-500
0
500
0
0.1
0.2
0.3
0.4
|F
analyt
(omega)|
. 3.15. : -
() -
, , ().
: ( ) .
, -
,
, -
-
. , -
, , -
. -
, -
- ( -
´ , -
, -
) .
, -
84
. 3.16. , (
) - -
0
= 2: ( )
( ).
, [ ta , fa]=sAnalytic(t , f );
[b,w]=fftMorlet(ta , fa ,a,omega0); ,
. 3.16.
. 3.16 . 3.11
,
,
/ -
-1 1, -
, ( ,
0
= 2 ) a = 0.05, = 40,
-
, .
--
, ,
85
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1
0
1
t
f(t)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1
0
1
b
Re[w(0.1,b)]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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87
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Details

Title
Global and local spectral analysis of oscillating signals and images
Subtitle
Global'nyj i lokal'nyj spektral'nyj analiz ostsilirujushchich signalov i izobrazhenij
College
Kursk State University
Author
Publication Year
2014
Pages
88
Catalog Number
V281539
ISBN (eBook)
9783656841876
ISBN (Book)
9783656841883
Language
Russian
Tags
wavelets Fourier transform Morlet wavelet spectral analysis MATLAB