Transcript of ウェーブレット理論と工学への応用 - AIMaP...The original chromatic scale All the...
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Wavelet theory and its applications to engineering
OKU & AIMaP 2019 Workshop
2019 12 5 – 12 6
1 (A)
2019 12 5 13:20 – 18:00
13:20–13:30
()
()
DWT
DES
10:00 – 11:00
Mellin
()
543-0054 4-88 (06)6775-6611
JR 10
JR 5
https://aimap.imi.kyushu-u.ac.jp/wp/
Contents 1. Preliminaries.
2. We introduce our proposed Hilbert transform pairs of orthonormal
bases of chromatic-scale wavelets.
3. Comparing between Gabor and our wavelets, we appeal why we want
to use the Gabor wavelet in the discrete wavelet transform.
4. For this aim, we propose “Tight wavelet frame using complex
wavelet designed in free shape of frequency domain”.
5. We construct the tight wavelet frame using an approximate Gabor
wavelet.
6. The dual wavelet frame using an approximate Gabor wavelet. The
contents are like these.
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fff ,
Our proposed “Hilbert transform pairs of orthonormal bases of
chromatic-scale wavelets”
The original chromatic scale
R
j
j
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applications to engineering2
0 500 1 103 1
0
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n
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50 0 50 0.4
0,1ˆ
1
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Imag.
Real
t0,1
t
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applications to engineering 3
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0.2
0
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Gabor wavelet
Trapezium=
Gaussian function
Imag.
Real
tG
Gˆ
0.2
0
0.2
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←→
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Gabor wavelet
Trapezium=
Gaussian function
0.5
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1
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1.5
Complex wavelet based on Meyer wavelet Trapezium= Considering the
uncertainty principle …
Time domain
Frequency domain
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For this aim, we propose the following method:
1. We design the original mother wavelet having free shape on the
frequency domain.
2. Using the original mother wavelet, we construct an approximate
tight wavelet frame.
3. Based on it, we construct a tight wavelet frame with minor
modification.
We want to use the Gabor wavelet for digital transform!
The tight wavelet frame using an approximate Gabor wavelet
tf tf Transform Inv. transform
tf N nj
P nj dd ,, ,
Orig nj
N nj
j n
Orig nj
P njOrig
tdtd A
tf ,,,, 1
Rem nj
Rem nj
j n
Rem nj
Rem njRem
tftf A
Rem nj
N nj
j n
Rem nj
P njRem
tdtd A
tf ,,,, 1
N nj
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Theorem 1 for tight wavelet frames The Fourier transform of has a
compact support of length as follows:
,,,ˆsupp 1010
.20 p is a constant real number.
We define the following transform of . 0p
R2Ltf
RR 21 LLt
,ˆ0 sup R
fWF
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
Origˆ
OrigˆThe original mother wavelet must be normalized and have a
compact support on the frequency domain, but its shape is
free!
We recommend the positive real function value, easy to treat
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Time domain
Time domain
Time domain
Frequency domain
Frequency domain
Frequency domain
An example of the original mother wavelet based on Gabor wavelet
tOrig
,
'
'
1
2
3
Gˆ
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The distance p>0 between wavelets in level 0 and the dilation
a>1
,20 p
.1 0
a
Example of p>0 and a>1 based on Gabor wavelet can be set
based on .2,3, 10
,1p
4 2 0 2 4 0
2
4
6
8
., ,,
domain
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., ,,
,, nptaat jOrigjOrig nj
.ˆ fOrigfWF Orig
2
4
6
8
Orig nj
Orig nj
j n
Orig nj
Orig njOrig
tftf A
2
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8
Orig
4 2 0 2 4 0
2
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Rem nj
Rem nj
j n
Rem nj
Rem njRem
tftf A
constructs a tight wavelet frame as follows: Znjtt Rem nj
Rem nj ,:, ,,
The third step is the construction of a tight wavelet frame using a
remaking mother wavelet
0 100 200 300 400 0
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Gˆ
1
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Gˆ
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0.1
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The dual orthogonal wavelet “B spline wavelet of order 4”
0 5 10 0.3
Orthogonal relation nnjjnjnj ,,,, ~,
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0.2
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The dual orthogonal wavelet “B spline wavelet of order 4”
0 5 10 0.3
tf tf Transform Inv. transform
tf N nj
P nj dd ,, ,
Orig nj
N nj
j n
Orig nj
P njOrig
tdtd A
tf ,,,, 1
Rem nj
Rem nj
j n
Rem nj
Rem njRem
tftf A
Rem nj
N nj
j n
Rem nj
P njRem
tdtd A
tf ,,,, 1
N nj
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tf tf Transform Inv. transform
Orig nj
M nj
Orig nj
Orig nj
N nj
j n
Orig nj
P njOrig
tdtd A
tf ,,,, 1
Orig nj
Orig nj
j n
Orig nj
Orig njOrig
tftf A
Orig nj
N nj
j n
Orig nj
P njOrig
tdtd A
tf ,,,, 1
N nj
,,ˆ~ 10
Orig Orig
Orig Orig
A Remaking
Theorem 1 for tight wavelet frames The Fourier transform of has a
compact support of length as follows:
,,,ˆsupp 1010
.20 p is a constant real number.
We define the following transform of . 0p
R2Ltf
RR 21 LLt
,ˆ0 sup R
fWF
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Theorem 2 for dual wavelet frames The Fourier transform of and have
a compact support of length as follows:
RR 21~ LLt
,~0 sup R
,:~~ Z nnpttn
We define the following transform of . 0p
R2Ltf
.ˆˆ~1,~ f
0.1
0.2
0.3
0.4
0.5
1
2
3
1
2
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tOrig~ tOrig
t t
OrigOrig
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0.2
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tOrig~ tOrig
t t
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tOrig~ tOrig
t t
OrigOrig
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0.5
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Mˆ
OrigOrig
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Orig~
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The analysis by the dual wavelet frame based on Meyer
0 500 1 103 1
0
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n
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n
Conclusion 1. We introduced our proposed Hilbert transform pairs
of
orthonormal bases of chromatic-scale wavelets.
2. We introduced the Gabor wavelets.
3. We introduced the tight wavelet frame using complex wavelet
designed in free shape on frequency domain.
4. We constructed the tight wavelet frame using an approximate
Gabor wavelet.
5. We constructed the dual wavelet frame using an approximate Gabor
wavelet.
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20dB
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• (Auditory Brainstem Response :
ABR)
• auditory steadystate response : ASSR
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40-Hz ASSR 80-Hz ASSR
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40Hz ASSR
Kawase 2015, Audiology Japan 581 p.47
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– MB11
– MASTER
–
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An example of MASTER output window:
- 70 dB : 512 epochs --> 32 sweeps - 60 dB : 474 epochs -->
29 sweeps - 80 dB : 184 epochs --> 11 sweeps (1 sweep = 16
epochs)
(Aoyagi[1])
80Hz-ASSR
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In the case of 80-Hz ASSR, MASTER system uses both relationship
between the carrier frequency of input stimuli and the modulation
frequency of cochlea responses. 2019/12/5
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Diatec HP https://www.diatec-
diagnostics.jp/solutions/products/abr/interacoustics-eclipse-complete-solution#
•CEChirp® •ASSR8
=>
=>
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applications to engineering 29
Diatec HP https://www.diatec-
diagnostics.jp/solutions/products/abr/interacoustics-eclipse-complete-solution#
2019/12/5
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CF= 125 8000 Hz, MF0100H
Sound intensity: 0 90 dB (nHL)
CF= 1000 Hz, MF40H Intensity:
80 dB (nHL)
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Averaged EEG data:
•β1430Hz
βγ28Hz
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D1
D2
D3
D4
D5
D6
D7
A7
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Here, we put , 2,
Using MATLAB function cmorwavf(Lb, Ub, U, N, fb, fc),
where
N = 1000; Lb = 5; Ub = 5; fb = 1.5; fc = 1;
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40-Hz ASSR(70dB, 50dB, 30dB) 40-Hz ASSR
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[1] , , , 2004, .
[2] N. Ikawa, T. Yahagi and H. Jiang, "Waveform analysis based on latency
frequency characteristics of auditory brainstem response using wavelet transform,"
Journal of Signal Processing, Vol. 9, No.6, pp. 505518, 2005.11.
[3] M. S. John, et al., MASTER: aWindow
program for recording multiple auditory
steadystate
responses, Comput. Methods Programs Biomed., 61, 125–150, 1998.
[4] , PXI4461 , CFME, 2009.
[5] , , , “40Hz
”, JSIAM, 47–48, 2011 [6] , ,
, “
”,, Vol.27 No.2, pp.216238,
2017.6.25.
[7] , , ,“ ” ,
http://www.osakakyoiku.ac.jp/~morimoto/WSPRO/wavelet2013proceedings.pdf,
25 ,2013.11.23.
[8], , ,“
”, JSIAM, 2501_2018080520080097313.pdf, 2018.9..
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• 18 6
DWT DES
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applications to engineering42
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•
• DWT
• • • •
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applications to engineering 43
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DESAES
• DES DES AES
• AESAdvanced Encryption StandardAES- 128AES-192AES-2563
AES
• AES AES
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applications to engineering44
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RSA
• RSA
RSA VS AES
• RSA
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applications to engineering 45
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/ • ”SSL”
• ””SSL
•
•
•
•
•
•
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applications to engineering46
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SSL
• Web
• SSL
• SSL
WordPress (PHP) • Web
•
• WordPress
https://techacademy.jp/briefing-lp-wordpress-s?utm_source=yahoo&utm_medium=cpc&utm_campaign
=03_briefing_wordpress&yclid
=YSS.1000002331.EAIaIQobChMIkeD5076d5gIVQ7aWCh1vbQuaEAAYASAAEgItfPD_BwE
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applications to engineering 47
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• • 1DES() • 1643
•
• DES
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applications to engineering48
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• DESDES3 DES
• UNIX DES
DES
• DES 3 8 DES 3 DATA 8
• • 2 • 3 2 • DES
• 3 • 2 • 2
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applications to engineering 49
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• 3DWT RGB 4
• 4R GBHH
• 5IDWT
.docx.xlsx.pdf .txtK1K2 K3
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applications to engineering50
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• 2HH
• 43DES
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• (a) Airplane.png • (b) Arctichare.png • (c) Baboon.png • (d)
Boat.png • (e) Boy.bmp • (f) Cat.png • (g) Fruits.png • (h)
Frymire.png • (i) Lena.png • (j) Peppers.png
TYPES OF SUPPORTED DOCUMENTS
• 12 KB docx 9.67 KBxlsx22.3 KBpdf161 txt
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applications to engineering52
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PC: Let’s Note CF-R9
• CPU: Intel Core i7 (1.07GHz) • RAM: 4GB • 32 bit OS • X64 Base
Processor
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applications to engineering 53
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• MSE PSNR
Airplane 512x512
MSE 18 14 - 2
MSE 18 16 - 2
512x512 MSE 18 15 - 2 PSNR 23 25 - 43
Boy 768x512
MSE 14 11 22 1 PSNR 26 27 21 46
Cat 490x733
MSE 14 12 23 2 PSNR 25 27 21 45
Fruits 512x512
Frymire 1118x1106
PSNR 34 35 31 52 Lena
512x512 MSE 17 15 - 2 PSNR 23 25 - 43
Peppers 512x512
MSE 17 15 - 2
PSNR 23 25 - 43
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Airplane.png HH 50 airplane.encripsi.nilai.50.png Original Document
43 dB
Airplane.png HH 60 airplane.encripsi.nilai.60.png Original Document
41 dB
Airplane.png HH 70 airplane.encripsi.nilai.70.png Original Document
40 dB
Airplane.png HH 80 airplane.encripsi.nilai.80.png Original Document
39 dB
Airplane.png HH 90 airplane.encripsi.nilai.90.png Original Document
38 dB
Airplane.png HH 100 airplane.encripsi.nilai.100.png Original
Document 37 dB
Airplane.png HH 150 airplane.encripsi.nilai.150.png Original
Document 34 dB
Airplane.png HH 200 airplane.encripsi.nilai.200.png Original
Document 32 dB
Airplane.png HH 255 airplane.encripsi.nilai.255.png Original
Document 31 dB
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Airplane.txt.png Brightness 30% Brightness.30%.png Original
Document
Airplane.txt.png Brightness 50% Brightness.50%.png Original
Document
Airplane.txt.png Brightness 100% Brightness.100%.png Original
Document
Airplane.txt.png Brightness 150% Brightness.150%.png Original
Document
Airplane.txt.png Brightness -150% Brightness.minus150%.png Original
Document
Airplane.txt.png Contrast 100% Contrast.100%.png Original
Document
Airplane.txt.png Contrast -50% Contrast.minus50%.png Original
Document
Airplane.txt.png Crop Up Crop.atas.png Original Document
Airplane.txt.png Crop Down Crop.bawah.png Blank Document
Airplane.txt.png Crop Right Crop.kanan.png “Can’t be
Extracted”
Airplane.txt.png Crop Left Crop.kiri.png "The Key That You Entered
Is Incorrect"
Airplane.txt.png Resize 400x400 Resize.400x400.png “Can’t be
Extracted”
Airplane.txt.png Resize 500x500 Resize.500x500.png “Can’t be
Extracted”
Airplane.txt.png Resize 510x510 Resize.510x510.png “Can’t be
Extracted”
Airplane.txt.png Resize 514x514 Resize.514x514.png “Can’t be
Extracted” Airplane.txt.png Rotate 90o CW Rotate.kanan.png Blank
Document Airplane.txt.png Rotate 90o CCW Rotate.kiri.png “Can’t be
Extracted” Airplane.txt.png Rotate 180o Rotate.penuh.png “Can’t be
Extracted”
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Original Encryption Result
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applications to engineering 57
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Original Encryption Result
Original Encryption Result
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applications to engineering58
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Original Encryption Result
• HLHHMSEPSNR
• MSEPSNR Insertion Value50
•
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References
• Kohei Arai, Kaname Seto, Data Hiding Based on Wavelet
Multi-resolution Analysis, Journal of Visualization Society of
Japan, Vol.22, Suppl.No.1, 229-232, 2002.
• Kohei Arai, Kaname Seto, Data Hiding Based on Multi- resolution
Analysis Utilizing Information Content Concentrations by Means of
Eigen Value decomposition, Journal of Visualization Society of
Japan, Vol.23, No.8, pp.72- 79,2003.
• Kohei Arai, Kaname Seto, Information Hiding Method Based on
Coordinate Conversion, Journal of Visualization Society of Japan,
25, Suppl.No.1, 55-58,(2005)
• Kohei Arai, Kaname Seto, Data hiding based on Multi-Resolution
Analysis taking into account scanning of the embedded image for
improvement of invisibility, Journal of Visualization Society of
Japan, 29, Suppl.1, 167-170, 2009.
• Kohei Arai and Yuji Yamada, Improvement of secret image
invisibility in circulation image with Dyadic wavelet based data
hiding with run-length coding, International Journal of Advanced
Computer Science and Applications, 2, 7, 33-40, 2011
• Kohei Arai, Method for data hiding based on Legal 5/2 (Cohen-
Daubechies-Feauveau: CDF 5/3) wavelet with data compression and
random scanning of secret imagery data, International Journal of
Wavelets Multi Solution and Information Processing, 11, 4, 1-18,
B60006 World Scientific Publishing Company, DOI:
I01142/SO219691313600060, 1360006-1, 2013.
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applications to engineering60
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• Kohei Arai, Data Hiding Method Replacing LSB of Hidden Portion
for Secrete Image with Run-Length Coded Image, International
Journal of Advanced Research on Artificial Intelligence, 5, 12,
8-16, 2016.
• Cahya Rahmed Kohei Arai, Arief Prasetyo, Noriza Arigki, Noble
Method for Data Hiding using Steganography Discrete Wavelet
Transformation and Cryptography Triple Data Encryption Standard:
DES, International Journal of Advanced Computer Science and
Applications: IJACSA, 9, 11, 261-266, 2018.
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Mf(ω) =
f(t) tωdt (1)
ω ∈ R+ ω n Mellin n
Mf(ω) C
f(t) = 1
1.1 Mellin
Mellin (1) Roger Godement Springer “Analysis III” [1] Wolfram
MathWorld [2] Wikipedia [3] Mellin
Mf(s) = φ(s) =
1
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Mellin
2019 12 5
1 Mellin R+ := {t ∈ R | t ≥ 0} f(t) Mellin
c φ(s) φ(s) φ(s) t−s Mellin
B[f(t)](s) = ∫ ∞
B[f(e−t)](s) =
=
f(η) ηs−1 dη
= M[f ](s− 1)
η = e−t t : −∞ → ∞ η : ∞ → 0 dη = −e−tdt Wolfram B[f(e−t)](s) = M[f
](s) M[e−t](s)
f(t) = ∑ n
an en(t)
generating function z-
2.1
{an}n=0,1,... n tn
f(t) = ∞∑ n=0
2
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n t = 0
an = f (n)(0)
{e2πint}n∈Z {an}n∈Z
f(t) = ∞∑
n=−∞
an e 2πint (4)
f(t) 1 [0, 1) {e2πint}n∈Z an
an =
∫ 1
0
f(t) e−2πint dt (5)
Fourier 1807 1 f(t) (5) (4) 1904 L2([0, 1)) 1845 “Uber die
Darstellbarkeit einer Function durch eine trigonometrische
Reihe”
2.3
L[f ](s) = ∫ ∞
e−st = ∞∑ n=0
3
OKU & AIMaP 2019 Workshop on Wavelet theory and its
applications to engineering64
∫ ∞
3 Mellin fα(t) = f(αt) Mellin
Mfα(ω) =
.
1 sin(t) sin(αt), α = 0.7 0.2 Mellin α ω
0 2 4 6 8 10t -1
-0.5
0
0.5
1
0.7
0.702
0.704
0.706
0.708
a
w
1: sin(t) sin(αt), α = 0.7 0.2α
t = 0 2
4
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0 5 10 15 20t -1
-0.5
0
0.5
1
0.64
0.66
0.68
0.7
0.72
0.74
-0.5
0
0.5
1
0.7
0.705
0.71
0.715
0.72
0.725
1 Mellin f(t) fα(t) = f(αt)
f(ξ) =
fα(ξ) =
α =
1
5
OKU & AIMaP 2019 Workshop on Wavelet theory and its
applications to engineering66
Mellin p > 0
Mellin
α
M f (p, ω) =⇒ α =
M fα (p, ω)
M f (p, ω)
4.1.1
3 2 [0, 2π] sin(2πt) 2 1 = 0.01 α = 1.3 sin(2παt) 2 [0, 3π] 2 =
0.007 fft 0
fft Matlab fftshift
• −π/1
• 2π/1/
6
OKU & AIMaP 2019 Workshop on Wavelet theory and its
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0 2 4 6 8 10 -1
-0.5
0
0.5
0.2
0.4
0.6
0.8
1
[rad]
3: α = 1.3 [−100, 100] [rad]
3 p α 4 5
p = 3 α = 1.3 ω = p− 1
-100 -50 0 50 100 0
0.2
0.4
0.6
0.8
1
[rad] 0 2 4 6 1.18
1.2
1.22
1.24
1.26
1.28
w
a
0.2
0.4
0.6
0.8
1
[rad] 0 2 4 6 1.05
1.1
1.15
1.2
1.25
1.3
w
a
4: p = 1, 2 α = 1.3 ω
7
OKU & AIMaP 2019 Workshop on Wavelet theory and its
applications to engineering68
-100 -50 0 50 100 0
0.2
0.4
0.6
0.8
1
[rad] 0 2 4 6 1.05
1.1
1.15
1.2
1.25
1.3
w
a
0.2
0.4
0.6
0.8
1
[rad] 0 2 4 6 1.297
1.298
1.299
1.3
1.301
w
a
5: p = 3, 4 α = 1.3 ω
5 2 Mellin Mellin 1 [0,∞)
2 (r, θ) θ r Mellin Mellin t ω f(t)
2 Mellin
5.1 Mellin
6 (r, θ) θ 0 ≤ θ ≤ 90 7 α = 0.73 θ 0
90 0.5 ω 1 10 0.1 α 8 θ 0 90 ω ω = 1 ω = 10 α 0.71
0.74 8 α
8
OKU & AIMaP 2019 Workshop on Wavelet theory and its
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r
q
r
O
6: (r, θ) ×
7: 0.73 0.73 0.91
7 α1 = 0.73 α2 = 0.91 α 9
5.2 2 Mellin
R2 f(x1, x2) Mellin
Mf(ω) =
∫ R2
2) ω/2dx1dx2
9
OKU & AIMaP 2019 Workshop on Wavelet theory and its
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a = 0.73
1 10w
0
45
90
0.72
0.725
0.73
0.735
0.74
q
5
8: 0.73 Mellin α θω
α
a = 0.73, a = 0.91
1 10w5
1 2
9: 0.73 0.91 Mellin α
θω α α1 = 0.73
α2 = 0.91
Mfα(ω) =
∫ R2
2) ω/2dx1dx2
10
OKU & AIMaP 2019 Workshop on Wavelet theory and its
applications to engineering 71
α
ω + 2
7 0.73 α 10 1 15 ω α = 0.73
0 5 10 15
w
10: 0.73 Mellin α ω α = 0.73
6 2 Mellin 2 f(x1, x2)
f(ξ1, ξ2) =
fα(x1, x2) = f(αx1, αx2)y1 = αx1,
y2 = αx1 dy1dy2 = α2dx1dx2
fα(ξ1, ξ2) =
=
dy1dy2 α2
11
OKU & AIMaP 2019 Workshop on Wavelet theory and its
applications to engineering72
f(ξ1, ξ2) f(ξ1, ξ2) p > 0 Mellin
f(ξ1, ξ2)p (ξ21 + ξ22) ω/2dξ1dξ2
M|fα| (p, ω) = ∫ R2
fα(ξ1, ξ2)p (ξ21 + ξ22) ω/2dξ1dξ2
=
)ω/2 α2dη1dη2
= αω+2−2pM|f | (p, ω)
η1 = ξ1/α, η2 = ξ2/α, dη1dη2 = dξ1dξ2/α 2 α
M|f | (p, ω) M|fα| (p, ω)
αω+2−2p = M|fα| (p, ω) M|f | (p, ω)
=⇒ α =
) 1
AIMaP (C) 17K05363
[1] R. Godement, Analysis III, Analytic and Differential Functions,
Manifolds
and Riemann Surfaces, Springer, 2015.
[2] Wolfram MathWorld Mellin Transform http://mathworld.
wolfram.com/MellinTransform.html 2019 12 5
wiki/Mellin_transform 2019 12 5
12
OKU & AIMaP 2019 Workshop on Wavelet theory and its
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wavelet2019
wavelet2019programchair2
1_toda
2_AIMap2019aki_v006ikawa
3_Arai_New
4_MorimotoTR