# 丁建均 (Jian-Jiun Ding) National Taiwan University

## Presentation on theme: "丁建均 (Jian-Jiun Ding) National Taiwan University"— Presentation transcript:

Research 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

[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)

1. Time-Frequency Analysis
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.

Example: x(t) = cos( t) when t < 10,
x(t) = cos(2 t) when t  (FM signal)

Using Time-Frequency analysis
x(t) = cos( t) when t < 10, x(t) = cos(3 t) when 10  t < 20, x(t) = cos(2 t) when t  (FM signal) Left：using Gray level to represent the amplitude of X(t, f) Right：slicing along t = 15 f -axis t -axis t -axis

Several Time-Frequency Distribution
Short-Time Fourier Transform (STFT) with Rectangular Mask Gabor Transform avoid cross-term less clarity Wigner Distribution Function with cross-term high clarity Gabor-Wigner Transform (Proposed) avoid cross-term high clarity

Cohen’s Class Distribution
where S Transform Hilbert-Huang Transform

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.

Applications of Time-Frequency Analysis
(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 (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

Conventional Sampling Theory
Nyquist Criterion New Sampling Theory (1) t can vary with time (2) Number of sampling points == Area of time frequency distribution

Modulation and Multiplexing
spectrum of signal 1 -B1 B1 spectrum of signal 2 not overlapped B2 -B2

Improvement of Time-Frequency Analysis
(1) Computation Time (2) Tradeoff of the cross term problem and clarification

left: x1(t) = 1 for |t|  6, x1(t) = 0 otherwise, right: x2(t) = cos(6t  0.05t2)
Gabor  -axis t -axis WDF  -axis t -axis

Gabor-Wigner Transform
avoiding the cross-term problem and high clarity  -axis t -axis

2. Music Signal Analysis Using the time-frequency analysis

3. 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.

 Fractional Fourier Transform (FRFT)
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;

 Physical Meaning: Transform a Signal into the Fractional domain, which is
the intermediate of the time domain and the frequency domain.

Time domain Frequency domain fractional domain
Modulation Shifting Modulation + Shifting Shifting Modulation Modulation + Shifting Differentiation  j2f Differentiation and  j2f  −j2f Differentiation Differentiation and  −j2f  is some constant phase

Why do we use the fractional Fourier transform?
To solve the problems that cannot be solved by the Fourier transform Example: Filter Design 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 )

When x(t) = triangular signal + chirp noise exp[j 0.25(t 4.12)2]

 The Fourier transform is suitable to filter out the noise that is a combination of sinusoid functions exp(j0t).  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.

 From the view points of Time-Frequency Analysis:
[Theorem] The FRFT with parameter  is equivalent to the clockwise rotation operation with angle  for Wigner distribution functions (or for Gabor transforms) FRFT = with angle  The Gabor Transform for the FRFT of the rectangular function.  = 0 (identity), / / /2 (FT) / /6 horizon: t-axis, vertical: f-axis [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 , Oct

 Filter designed by the fractional Fourier transform

stop band f0 cutoff line pass band t-axis 而用 fractional Fourier transform 設計的濾波器是，是由斜的方向作切割 f-axis stop band cutoff line 和 f-axis 在逆時針方向的夾角為  u0 pass band cutoff line

 Gabor Transform for signal + 0.3exp[j0.06(t1)3  j7t]
fractional axis t-axis Advantage:  Easy to estimate the character of a signal in the fractional domain  Proposed an efficient way to find the optimal parameter 

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)

 Invention: [Ref 2] N. Wiener, “Hermitian polynomials and Fourier analysis,” Journal of Mathematics Physics MIT, vol. 18, pp , 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 , 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 3091, Nov Recent development Pei, Ding (after 1995), Ozaktas, Mendlovic, Kutay, Zalevsky, etc.

 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,  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  Solving the problem for implementation

4. 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) g[n]  2 x1,L[n] 「低頻」部分 x[n] h[n]  2 x1,H[n] 「高頻」部分 Example: g[n] = [1, 1], h[n] = [1, -1] or

The result of the wavelet transform for a 2-D image
lowpass for x highpass for y lowpass for x lowpass for y highpass for x lowpass for y highpass for x highpass for y

Applications for Wavelets
-- JPEG 2000 (image compression) -- filter design -- edge and corner detection -- pattern recognition -- biomedical engineering

5. 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

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 = RMSE =

New Method: Edge-Based Segmentation and Compression

Image Segment Compression
Segmentation-based image compression Boundary Boundary Compression Image Segmentation Bit stream An image Image Segment Compression Image Segment

Original Image By JPEG By Proposed Method An 100x100 image Bytes: 1295, RMSE: 2.39 Bytes: 456, RMSE: 2.54

6. Edge and Corner Detection
Why should we perform edge and corner detection? --Segmentation --Compression

Simplest way for edge detection: differentiation

Doing difference x[n]  x[n1] = x[n]  (convolution) with h[n].
h[n] = 1 for n = 0, h[n] = -1 for n = 1, h[n] = 0 otherwise. Other ways for edge detection: convolution with a longer odd function x[n] 

 Corner Detection by Harris’ algorithm by proposed algorithm

7. Segmentation Important for compression biomedical engineering
object identification

Conventional method: 97.87 sec New method: 1.02 sec

8. Pattern Recognition including face recognition
character recognition 應用很廣： security, identification, computer vision …………

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.

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.

Integer RGB to YCbCr Transform
This is used in JPEG 2000.

[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, B̃: integer transform pair (Goal 1) Integerization , , bk and b̃k are integers. (Goal 2) Reversibility (Goal 3) Bit Constraint The denominator 2k should not be too large. (Goal 4) Accuracy B  A, B̃  A-1 (or B  A, B̃  -1A-1) (Goal 5): Less Complexity (Goal 6) Easy to Design

 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)

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 , Aug [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 , Dec [Ref. 3] T. D. Tran, “The binDCT: fast multiplierless approximation of the DCT,” IEEE Signal Proc. Lett., vol. 7, no. 6, pp , June 2000. [Ref. 4] P. Hao and Q. Shi., “Matrix factorizations for reversible integer mapping,” IEEE Trans. Signal Processing, vol. 49, no. 10, pp , Oct [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

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

Depth 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

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.  There are four types of nucleotide in a DNA sequence: adenine (A), guanine (G), thymine (T), cytosine (C)  Unitary Mapping bx[] = if x[] = ‘A’, bx[] =  if x[] = ‘T’, bx[] = j if x[] = ‘G’, bx[] = j if x[] = ‘C’. y = ‘AACTGAA’,  by = [1, 1, j, 1, j, 1, 1].

 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’.

 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)

 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.

15. 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 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 , Nov [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 , July 2004.

Application 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 =

(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 (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。 (d) 農曆新年休息二週，預官考試休息一週。 (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

(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。

(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。