1 丁建均 (jian-jiun ding) national taiwan university 辦公室:明達館 723 室,...
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丁建均 (Jian-Jiun Ding)
National Taiwan University
辦公室:明達館 723 室, 實驗室:明達館 531 室
聯絡電話: (02)33669652
Major : Digital Signal Processing
Digital Image Processing
2Research Fields
[A. Time-Frequency Analysis]
(1) Time-Frequency Analysis (page 4)
(2) Music Signal Analysis (page 17)
(3) Fractional Fourier Transform (page 20)
(4) Wavelet Transform (page 34)
[B. Image Processing]
(5) Image Compression (page 37)
(6) Edge and Corner Detection (page 45)
(7) Segmentation (page 49)
(8) Pattern Recognition (Face, Character) (page 54)
: main topics that I researched in recent years
3[C. Fast Algorithms]
(9) Fast Algorithms
(10) Integer Transforms (page 56)
(11) Number Theory, Haar Transform, Walsh Transform
[D. Applications of Signal Processing]
(12) Optical Signal Processing (page 62)
(13) Acoustics
(14) Bioinformatics (page 64)
[E. Theories for Signal Processing]
(15) Quaternions (page 68)
(16) Eigenfunctions, Eigenvectors, and Prolate Spheroidal Wave Function
(17) Signal Analysis (Cepstrum, Hilbert, CDMA)
41. Time-Frequency Analysis
http://djj.ee.ntu.edu.tw/TFW.htm
Fourier transform (FT)
Time-Domain Frequency Domain
Some things make the FT not practical:
(1) Only the case where t0 t t1 is interested.
(2) Not all the signals are suitable for analyzing in the frequency domain.
It is hard to analyze the signal whose instantaneous frequency varies with time.
2j f tX f x t e dt
5Example: x(t) = cos( t) when t < 10,
x(t) = cos(3 t) when 10 t < 20,
x(t) = cos(2 t) when t 20 (FM signal)
0 5 10 15 20 25 30-1
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Fouriertransform
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x(t) = cos( t) when t < 10, x(t) = cos(3 t) when 10 t < 20,
x(t) = cos(2 t) when t 20 (FM signal)
Left : using Gray level to represent the amplitude of X(t, f)
Right : slicing along t = 15
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Using Time-Frequency analysis
7Several Time-Frequency Distribution
Short-Time Fourier Transform (STFT) with Rectangular Mask
2,t B j f
t BX t f x e d
Gabor Transform
2 2 ( )( ) 2,t
j ftxG t f e e x d
Wigner Distribution Function
* 2, / 2 / 2 j fxW t f x t x t e d
Gabor-Wigner Transform (Proposed)
, ( , ) ( , )x x xD t G t W t
avoid cross-term
less clarity
with cross-term
high clarity
avoid cross-term
high clarity
8Cohen’s Class Distribution
S Transform
* 2, / 2 / 2 j txA x t x t e dt
where
, , , exp 2 ( )x xC t f A j t f d d
2 2( , ) exp exp 2xS t f x f t f j f d
Hilbert-Huang Transform
9
1
( ) exp( ( ))N
k kk
x t a j t
31 2 '( ) '( )'( ) '( ), , , ,
2 2 2 2Nt tt t
Instantaneous Frequency 瞬時頻率
If
then the instantaneous frequency of x(t) are
自然界瞬時頻率會隨時間而改變的例子
音樂,語音信號 , Doppler effect, seismic waves, optics, radar system,
rectangular function, ………………………
In fact, in addition to sinusoid-like functions, the instantaneous frequencies
of other functions will inevitably vary with time.
10
(1) Finding Instantaneous Frequency
(2) Music Signal Analysis
(3) Sampling Theory
(4) Modulation and Multiplexing
(5) Filter Design
(6) Random Process Analysis
(7) Signal Decomposition
(8) Electromagnetic Wave Propagation
(9) Optics
(10) Radar System Analysis
Applications of Time-Frequency Analysis
(11) Signal Identification
(12) Acoustics
(13) Biomedical Engineering
(14) Spread Spectrum Analysis
(15) System Modeling
(16) Image Processing
(17) Economic Data Analysis
(18) Signal Representation
(19) Data Compression
(20) Seismology
(21) Geology
11
Conventional Sampling Theory
Nyquist Criterion1
2t B
New Sampling Theory
(1) t can vary with time
(2) Number of sampling points == Area of time frequency distribution
12假設有一個信號,
The supporting of x(t) is t1 t t1 + T, x(t) 0 otherwise
The supporting of X( f ) 0 is f1 f f1 + F, X( f ) 0 otherwise
根據取樣定理, t 1/F , F=2B, B: 頻寬
所以,取樣點數 N 的範圍是
N = T/t TF
重要定理:一個信號所需要的取樣點數的下限,等於它時頻分佈的面績
13
Modulation and Multiplexing
not overlapped
spectrum of signal 1
spectrum of signal 2
B1-B1
B2-B2
14Improvement of Time-Frequency Analysis
(1) Computation Time
(2) Tradeoff of the cross term problem and clarification
15
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left: x1(t) = 1 for |t| 6, x1(t) = 0 otherwise, right: x2(t) = cos(6t 0.05t2)
WDF
Gabor
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Gabor-Wigner Transform
avoiding the cross-term problem and high clarity
1.5 0.25, ( , ) ( , )f f fC t G t W t
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172. Music Signal Analysis
time (sec)
freq
uenc
y H
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Fs=44100Hz window size=0.2sec
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Using the time-frequency analysis
聲音檔: http://djj.ee.ntu.edu.tw/Chord.wav
SoMiDo
LaMiDo
LaFaRe
18聲音檔: http://djj.ee.ntu.edu.tw/air.mp3
time (sec)
freq
uenc
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time-frequency analysis
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目標: 音樂信號搜尋
( 運用音的高低和拍子 )
音樂信號壓縮
203. Fractional Fourier Transform Performing the Fourier transform a times (a can be non-integer)
Fourier Transform (FT)
generalization
Fractional Fourier Transform (FRFT)
, = a/2
When = 0.5, the FRFT becomes the FT.
dttfeF tj
2
1
dttfj
uFt
juju
j
eee t
22 cot2csccot
2
2
cot1
21
Fractional Fourier Transform (FRFT)
, = a/2.
When = 0: (identity)
When = 0.5:
When is not equal to a multiple of 0.5, the FRFT is equivalent to doing /(0.5 ) times of the Fourier transform.
when = 0.1 doing the FT 0.2 times;
when = 0.25 doing the FT 0.5 times;
when = /6 doing the FT 1/3 times;
dttfj
uFt
juju
j
eee t
22 cot2csccot
2
2
cot1
22
Physical Meaning: Transform a Signal into the Fractional domain, which is the intermediate of the time domain and the frequency domain.
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f(t): rectangle
F(w): sinc function
23Time domain Frequency domain fractional domain
Modulation Shifting Modulation + Shifting
Shifting Modulation Modulation + Shifting
Differentiation j2f Differentiation and j2f
−j2f Differentiation Differentiation and −j2f
0 0 0exp 2 ( ) exp 2 (cos s )injFRFT j f t f t e j f u F u f t
is some constant phase
( ) 2 ( )s (n os)i cFRFT f t j u F u F u
24
Conventional filter design:
x(t): input x(t) = s(t) (signal) + n(t) (noise) y(t): output (We want that y(t) s(t)) H(): the transfer function of the filter.
Filter design by the fractional Fourier transform (FRFT):
(replace the FT and the IFT by the FRFTs with parameters and )
txFTHIFTty
txFRFTuHFRFTty
Example: Filter Design
Why do we use the fractional Fourier transform?
To solve the problems that cannot be solved by the Fourier transform
25When x(t) = triangular signal + chirp noise exp[j 0.25(t 4.12)2]
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Fourier transform of x(t) x(t) = signal + noise
fractional Fourier transform of x(t)
(separable)
(non-separable)
recovered signal
26
The Fourier transform is suitable to filter out the noise that is a combination of sinusoid functions exp(j0t).
The fractional Fourier transform (FRFT) is suitable to filter out the noise that is a combination of higher order exponential functions exp[j(nk tk + nk-1 tk-1 + nk-2 tk-2 + ……. + n2 t2 + n1 t)]
For example: chirp function exp(jn2 t2)
With the FRFT, many noises that cannot be removed by the FT will be filtered out successfully.
27
horizon: t-axis, vertical: f-axis
FRFT = with angle
The Gabor Transform for the FRFT of the rectangular function.
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[Theorem] The FRFT with parameter is equivalent to the clockwise rotation operation with angle for Wigner distribution functions (or for Gabor transforms)
= 0 (identity), /6 2/6 /2 (FT) 4/6 5/6
From the view points of Time-Frequency Analysis:
[Ref 1] S. C. Pei and J. J. Ding, “Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing,” IEEE Trans. Signal Processing, vol. 55, no. 10, pp. 4839-4850, Oct. 2007.
28 Filter designed by the fractional Fourier transform
o F F ix t O O x t H u
f-axis
Signal noise
t-axis
FRFT FRFT
noise Signal
cutoff line
Signal
cutoff line
noise
( ) ( ( )) ( )o ix t IFT FT x t H f比較: Filter Designed by the Fourier transform
29以時頻分析的觀點,傳統濾波器是垂直於 f-axis 做切割的
t-axis
f0
f-axis
cutoff linepass band
stop band
而用 fractional Fourier transform 設計的濾波器是,是由斜的方向作切割
u0
f-axis
cutoff line
pass band
stop band
cutoff line 和 f-axis 在逆時針方向的夾角為
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Signal
noise
t-axis
fractional axis
Gabor Transform for signal + 0.3exp[j0.06(t1)3 j7t]
Advantage: Easy to estimate the character of a signal in the fractional domain Proposed an efficient way to find the optimal parameter
31
In fact, all the applications of the Fourier transform (FT) are also the applications of the fractional Fourier transform (FRFT), and using the FRFT instead of the FT for these applications may improve the performance.
Filter Design : developed by us improved the previous works
Signal synthesis (compression, random process, fractional wavelet transform)
Correlation (space variant pattern recognition)
Communication (modulation, multiplexing, multiple-path problem)
Sampling
Solving differential equation
Image processing (asymmetry edge detection, directional corner detection)
Optical system analysis (system model, self-imaging phenomena)
Wave propagation analysis (radar system, GRIN-medium system)
32 Invention:
[Ref 2] N. Wiener, “Hermitian polynomials and Fourier analysis,” Journal of Mathematics Physics MIT, vol. 18, pp. 70-73, 1929.
Re-invention
[Ref 3] V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Maths. Applics., vol. 25, pp. 241-265, 1980.
Introduction for signal processing
[Ref 4] L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Processing, vol. 42, no. 11, pp. 3084- 3091, Nov. 1994.
Recent development
Pei, Ding (after 1995), Ozaktas, Mendlovic, Kutay, Zalevsky, etc.
33 My works related to the fractional Fourier transform (FRFT)
Extensions: Discrete fractional Fourier transform
Fractional cosine, sine, and Hartley transform,
Two-dimensional form, N-D form,
Simplified fractional Fourier transform
Fractional Hilbert transform,
Solving the problem for implementation
Foundation theory: relations between the FRFT and the well-known time-frequency analysis tools (e.g., the Wigner distribution function and the Gabor transform)
Applications: sampling, encryption, corner and edge detection, self-imaging phenomena, bandwidth saving, multiple-path problem analysis, random process analysis, filter design
344. Wavelet Transform
只將頻譜分為「低頻」和「高頻」兩個部分,大幅簡化了 Fourier transform
( 對 2-D 的影像,則分為四個部分 )
x[n]
g[n]
2
x1,L[n]
x1,H[n]
2
h[n]
「低頻」部分
「高頻」部分
Example: g[n] = [1, 1], h[n] = [1, -1]
or
0.0106 0.0329 0.0308 0.1870 0.0280 0.6309 0.7148 0.2304g n
0.2304 0.7148 0.6309 0.0280 0.1870 0.0308 0.0329 0.0106h n
35
The result of the wavelet transform for a 2-D image
lowpass for x
lowpass for y
lowpass for x
highpass for y
highpass for x
lowpass for y
highpass for x
highpass for y
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-- JPEG 2000 (image compression)
-- filter design
-- edge and corner detection
-- pattern recognition
-- biomedical engineering
Applications for Wavelets
375. Image Compression
Conventional JPEG method:
Separate the original image into many 8*8 blocks, then using the DCT to code each blocks.
DCT: discrete cosine transform
PS: 感謝 2008 年畢業的黃俊德同學
38
壓縮的基本原理:
影像在經過 discrete cosine transform (DCT) 之後,大部分的能量都集中在低頻
DCT
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JPEG 是當前最普及的影像壓縮格式。
問題:壓縮率高的時候,會產生 blocking effect
Compression ratio = 53.4333RMSE = 10.9662
40New Method: Edge-Based Segmentation and Compression
和小時候畫圖的方法類似
41
Image Segment Compression
Bit stream
ImageSegmentation
Boundary Compression
Image Segment
Boundary
An image
• Segmentation-based image compression
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Original Image By JPEG
An 100x100 image Bytes: 1295, RMSE: 2.39
By Proposed Method
Bytes: 456, RMSE: 2.54
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原圖 (10000 bytes)
使用 JPEG (233 bytes)
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使用 JPEG (692 bytes)
使用新方法 (165 bytes)
44
技術上的問題:
(1) 如何找到物體的邊緣並切割? ( 努力中 )
(2) 如何針對不規則的區域,找到 orthogonal transform ( 已解決 )
(3) 如何避免讓邊緣區域的高頻成分影響到壓縮的結果 ( 已解決 )
(4) 如何用最小的資料量,對邊界的部分做紀錄 ( 已解決 )
(5) 如何用最小的資料量,對內部的部分做紀錄 ( 努力中 )
(6) 減少壓縮和解壓縮的運算時間 ( 努力中 )
456. Edge and Corner Detection
Why should we perform edge and corner detection?
--Segmentation
--Compression
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Simplest way for edge detection: differentiation
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Other ways for edge detection: convolution with a longer odd function
Doing difference x[n] x[n1] = x[n] (convolution) with h[n].
h[n] = 1 for n = 0,
h[n] = -1 for n = 1,
h[n] = 0 otherwise.
x[n]
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by Harris’ algorithm by proposed algorithm
Corner Detection
497. Segmentation
Important for compression
biomedical engineering
object identification
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Conventional method:
97.87 sec
New method:
1.02 sec
51
未受過傷的老鼠肌肉纖維 受過傷的老鼠肌肉纖維
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未受過傷的老鼠肌肉纖維「分區」的結果
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受過傷的老鼠肌肉纖維「分區」的結果
548. Pattern Recognition
應用很廣: security,
identification,
computer vision …………
including face recognition
character recognition
55
最簡單的方法: matched filter
但技術上的問題頗多……… .
scaling
shadow
rotation
partially distortion
其他的方法: 特徵拮取
臉有哪些特徵?
, , , , ,y m n x m n h m n x m n h
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xAy
),()3,()2,()1,(
),3()3,3()2,3()1,3(
),2()3,2()2,2()1,2(
),1()3,1()2,1()1,1(
NNANANANA
NAAAA
NAAAA
NAAAA
A
10. Integer Transform Conversion Integer Transform: The discrete linear operation whose entries are summations of 2k.
, ak = 0 or 1 or , C is an integer. k
kkanmA 2,
b
CnmA
2,
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310.0520.0210.0
320.0273.0593.0
113.0584.0297.0
A
16/516/816/3
16/516/416/9
16/216/916/5
B
713.1110.1006.1
649.0274.0006.1
625.0961.0006.11A
16/2716/1816/16
16/1016/416/16
16/1016/1516/16~B
IBB ~
Problem: Most of the discrete transforms are non-integer ones.
DFT, DCT, Karhunen-Loeve transform, RGB to YIQ color transform
--- To implement them exactly, we should use floating-point processor
--- To implement them by fixed-point processor, we should approximate it by an integer transform.
However, after approximation, the reversibility property is always lost.
58Integer RGB to YCbCr Transform
310.0520.0210.0
320.0273.0593.0
113.0584.0297.0
A0.25 0.5 0.25
1 1 0
0 1 1
B
1 0.75 0.25
1 0.25 0.25
1 0.25 0.75
B
713.1110.1006.1
649.0274.0006.1
625.0961.0006.11A
B B IThis is used in JPEG 2000.
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[Integer Transform Conversion]:
Converting all the non-integer transform into an integer transform that achieve the following 6 Goals:
A, A-1: original non-integer transform pair, B, B ̃: integer transform pair
(Goal 1) Integerization , , bk and b- k are integers.
(Goal 2) Reversibility .
(Goal 3) Bit Constraint The denominator 2k should not be too large.
(Goal 4) Accuracy B A, BA A-1 (or B A, BA -1A-1)
(Goal 5): Less Complexity
(Goal 6) Easy to Design
kkb
nmB2
, kkb
nmB2
~,
~
IBB ~
60
Development of Integer Transforms:
(A) Prototype Matrix Method (Partially my work)(suitable for 2, 4, 8 and 16-point DCT, DST, DFT)
(B) Lifting Scheme
(suitable for 2k-point DCT, DST, DFT)
(C) Triangular Matrix Scheme
(suitable for any matrices, satisfies Goals 1 and 2)
(D) Improved Triangular Matrix Scheme (My works)
(suitable for any matrices, satisfies Goals 1 ~ 6)
61
References Related to the Integer Transform
[Ref. 1] W. K. Cham, “Development of integer cosine transform by the principles of dynamic symmetry,” Proc. Inst. Elect. Eng., pt. 1, vol. 136, no. 4, pp. 276-282, Aug. 1989.
[Ref. 2] S. C. Pei and J. J. Ding, “The integer Transforms analogous to discrete trigonometric transforms,” IEEE Trans. Signal Processing, vol. 48, no. 12, pp. 3345-3364, Dec. 2000.
[Ref. 3] T. D. Tran, “The binDCT: fast multiplierless approximation of the DCT,” IEEE Signal Proc. Lett., vol. 7, no. 6, pp. 141-144, June 2000.
[Ref. 4] P. Hao and Q. Shi., “Matrix factorizations for reversible integer mapping,” IEEE Trans. Signal Processing, vol. 49, no. 10, pp. 2314-2324, Oct. 2001.
[Ref. 5] S. C. Pei and J. J. Ding, “Reversible Integer Color Transform with Bit-Constraint,” accepted by ICIP 2005.
[Ref. 6] S. C. Pei and J. J. Ding, “Improved Integer Color Transform,” in preparation
62
12. Optical Signal Processing and Fractional Fourier Transform
lens, (focal length = f)
free space, (length = z1) free space, (length = z2)
f = z1 = z2 Fourier Transform
f z1, z2 but z1 = z2 Fractional Fourier Transform (see page 20)
f z1 z2 Fractional Fourier Transform multiplied by a chirp
63Depth recovery:
如何由照片由影像的模糊程度,來判斷物體的距離
註:感謝 2008 年畢業的的林于哲同學
64
There are four types of nucleotide in a DNA sequence: adenine (A), guanine (G), thymine (T), cytosine (C)
Unitary Mapping
bx[] = 1 if x[] = ‘A’, bx[] = 1 if x[] = ‘T’, bx[] = j if x[] = ‘G’, bx[] = j if x[] = ‘C’.
y = ‘AACTGAA’, by = [1, 1, j, 1, j, 1, 1].
14. Discrete Correlation Algorithm for DNA Sequence Comparison
[Reference] S. C. Pei, J. J. Ding, and K. H. Hsu, “DNA sequence comparison and alignment by the discrete correlation algorithm,” submitted.
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Discrete Correlation Algorithm for DNA Sequence Comparison
For two DNA sequences x and y, if
where
Then there are s[n] nucleotides of x[n+] that satisfies x[n+] = y[].
Example: x = ‘GTAGCTGAACTGAAC’, y = ‘AACTGAA’,
.
x = ‘GTAGCTGAACTGAAC’, y (shifted 7 entries rightward) = ‘AACTGAA’.
1 22Re
4nz n z n L
s n
1 x yz n b n b n
]0,1,3,1,0,0,2,7,2,0,0,1,6,2,0,1,1,1,2,0,0[141312721016 n
s n
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Example: x = ‘GTAGCTGAACTGAAC’, y = ‘AACTGAA’,
s[n] = .
Checking:
x = ‘GTAGCTGAACTGAAC’, y = ‘AACTGAA’. (no entry match)
x = ‘GTAGCTGAACTGAAC’, y = (shifted 2 entries rightward) ‘AACTGAA’. (6 entries match)
x = ‘GTAGCTGAACTGAAC’, y (shifted 7 entries rightward) = ‘AACTGAA’. (7 entries match)
]0,1,3,1,0,0,2,7,2,0,0,1,6,2,0,1,1,1,2,0,0[141312721016 n
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Advantage of the Discrete Correlation Algorithm:
---The complexity of the conventional sequence alignments is O(N2)
---For the discrete correlation algorithm, the complexity is reduced to O(N log2N) or O(N log2N + b2) b: the length of the matched subsequences
Experiment: Local alignment for two 3000-entry DNA sequences
Using conventional dynamic programming Computation time: 87 sec. Using the proposed discrete correlation algorithm: Computation time: 4.13 sec.
6815. Quaternion 翻譯成“四元素”, Generalization of complex number
Complex number: a + ib i2 = 1
real part imaginary part
Quaternion: a + ib + jc + kd i2 = j2 = k2 = 1
real part 3 imaginary parts
[Ref 18] S. C. Pei, J. J. Ding, and J. H. Chang, “Efficient implementation of quaternion Fourier transform,” IEEE Trans. Signal Processing, vol. 49, no. 11, pp. 2783-2797, Nov. 2001.
[Ref 19] S. C. Pei, J. H. Chang, and J. J. Ding, “Commutative reduced biquaternions for signal and image processing,” IEEE Trans. Signal Processing, vol. 52, pp. 2012-2031, July 2004.
69Application of quaternion a + ib + jc + kd:
--Color image processing
a + iR + jG + kB represent an RGB image
--Multiple-Channel Analysis
4 real channels or 2 complex channels
a
b
c
d
a+jb
c+jd=
70實驗室研究的規定(1) 原則上,一週 meeting 一次
(a) 碩二上學期的其中二週 ( 腦力激盪 ) 和碩二下學期 4 月 5 月 ( 準備碩士論文口試 ) ,將一週 meeting 二次
例外:
(b) 碩一上下學期可以選三週不必 meeting ,碩二上學期每個學期可以選二週不必 meeting ,以準備學校的考試(c) 碩二下學期碩士論文口試 (5 月底 ) 結束之後,只需再 meeting 一次即可。
(2) 碩一升碩二的暑假,要參加國內的研討會 CVGIP
Take it easy ,雖然是學術研討會,就當作是旅行就可以了。
(3) 畢業之前,都要有自己創新的新點子
創新,是研究所教育和大學教育之間最大的不同
(d) 農曆新年休息二週,預官考試休息一週。
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(5) 畢業之前,希望大家至少能曾經幫忙寫過一篇研討會論文
(6) 每週 meeting 所規定的工作,儘可能達成。
但如果已經盡了力仍然難以達成目標,我是可以接受的。
(7) 只要有事情,不管是什麼原因,一律都可以請假,或延後 meeting 時間。
但如果請假一週,將來要選一週補回來 (也就是那一週要 meeting 二次 )
(4) 碩一上學期和下學期四月以前,同學們可以自由選擇有興趣的題目來研究,每三個月可以換一次題目。
到了碩一下學期四月,則要從我所列出的 10 個研究領域,選擇一個領域 ( 自由選擇 ) ,來當成將來碩士論文的研究主題。
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(10) 這一屆的同學,第一次 meeting 的時間是 2010 年 7 月底。
(9) 每學期會有二至三次的導生會,歡迎學生多多參加。
(8) 每三個月將請同學針對自己所研究的領域,做一次口頭報告。
一方面,讓其他同學了解你的研究 (你也同時了解其他同學的研究 ) ,一方面,也訓練演講和報告的能力。
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研究所的生活,和大學比起來,更有彈性 ,但是也離近入社會更近。
希望各位同學能妥善運用時間,好好充實自已,
並且多訓練自己「創造發明」以及「思考」的能力