1. 1 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

2. 2 Research Fields [A. Signal Analysis] (1) Time-Frequency Analysis (2) Fractional Fourier Transform (3) Wavelet Transform (4) Eigenfunctions, Eigenvectors, and Prolate Spheroidal Wave Function (5) Signal Analysis (Cepstrum, Hilbert, CDMA) [B. Fast Algorithm] (6) Integer Transforms (7) Fast Algorithms (8) Number Theory, Haar Transform, Walsh Transform : the main topics I researched in recent years : the main topics I research before

3. 3 [C. Applications of Signal Processing] (9) Optical Signal Processing (10) Acoustics (11) Bioinformatics [D. Image Processing] (12) Image Compression (13) Edge and Corner Detection (14) Pattern Recognition [E. Theories for Signal Processing] (15) Quaternion : the main topics I research before : the main topics I researched in recent years

4. 4 1. 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 t 0  t  t 1 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.

5. 5 Example: 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)

6. 6 Instantaneous Frequency 瞬時頻率 If then the instantaneous frequency of f (t) are 其他瞬時頻率會隨時間而改變的例子 音樂，語音信號 Chirp Signal

7. 7 Several Time-Frequency Distribution Short-Time Fourier Transform (STFT) with Rectangular Mask Gabor Transform Wigner Distribution Function Gabor-Wigner Transform (Proposed) avoid cross-term less clarity with cross-term high clarity avoid cross-term high clarity

8. 8 Cohen’s Class Distribution S Transform where Hilbert-Huang Transform

9. 9 Example: 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 f -axis t -axis

10. 10 (1) Finding Instantaneous Frequency (2) Sampling Theory (3) Filter Design (4) Signal Decomposition (5) Modulation and Multiplexing (6) Electromagnetic Wave Propagation (7) Optics (8) Radar System Analysis (9) Random Process Analysis Applications of Time-Frequency Analysis (10) Signal Identification (11) Acoustics (12) Biomedical Engineering (13) Spread Spectrum Analysis (14) System Modeling (15) Image Processing (16) Economic Data Analysis (17) Signal Representation (18) Data Compression

11. 11 Conventional Sampling Theory Nyquist Criterion New Sampling Theory (1)  t can vary with time (2) Number of sampling points == Area of time frequency distribution

12. 12 假設有一個信號，  The supporting of x(t) is t 1  t  t 1 + T, x(t)  0 otherwise  The supporting of X( f )  0 is f 1  f  f 1 + 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. 13 Modulation and Multiplexing not overlapped spectrum of signal 1 spectrum of signal 2 B1B1 -B 1 B2B2 -B 2

14. 14 Improvement of Time-Frequency Analysis (1) Computation Time (2) Tradeoff of the cross term problem and clarification

15. 15  -axis t -axis left: x 1 (t) = 1 for |t|  6, x 1 (t) = 0 otherwise, right: x 2 (t) = cos(6t  0.05t 2 ) WDF Gabor  -axis t -axis

16. 16 Gabor-Wigner Transform avoiding the cross-term problem and high clarity  -axis t -axis

17. 17 2. 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.

18. 18  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;

19. 19  Physical Meaning: Transform a Signal into the Fractional domain, which is the intermediate of the time domain and the frequency domain.

20. 20 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

21. 21 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  )  Why do we use the fractional Fourier transform? To solve the problems that cannot be solved by the Fourier transform Example: Filter Design

22. 22 When x(t) = triangular signal + chirp noise exp[j 0.25(t  4.12) 2 ]

23. 23  The Fourier transform is suitable to filter out the noise that is a combination of sinusoid functions exp(j  0 t).  The fractional Fourier transform (FRFT) is suitable to filter out the noise that is a combination of higher order exponential functions exp[j(n k t k + n k-1 t k-1 + n k-2 t k-2 + ……. + n 2 t 2 + n 1 t)] For example: chirp function exp(jn 2 t 2 )  With the FRFT, many noises that cannot be removed by the FT will be filtered out successfully.

24. 24 (2) Gabor transform [Ref 10] 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. (1) Wigner distribution function (WDF) [Ref 9] S. C. Pei and J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” IEEE Trans. Signal Processing, v. 49, pp 1638-1655, (2001).  From the view points of Time-Frequency Analysis:

25. 25 horizon: t-axis, vertical: f-axis FRFT  = with angle  The Gabor Transform for the FRFT of the rectangular function. [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

26. 26  Filter designed by the fractional Fourier transform f-axis Signal noise t-axis FRFT  FRFT  noiseSignal cutoff line Signal cutoff line noise 比較： Filter Designed by the Fourier transform

27. 27 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 f0f0 f-axis cutoff line pass band stop band 而用 fractional Fourier transform 設計的濾波器是，是由斜的方向作切割 u0u0 f-axis cutoff line pass band stop band  cutoff line 和 f-axis 在逆時針方向的夾 角為 

28. 28 t-axis fractional axis  Gabor Transform for signal + 0.3exp[j0.06(t  1) 3  j7t] Advantage:  Easy to estimate the character of a signal in the fractional domain  Proposed an efficient way to find the optimal parameter 

29. 29  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)

30. 30  Invention: [Ref 1] N. Wiener, “Hermitian polynomials and Fourier analysis,” Journal of Mathematics Physics MIT, vol. 18, pp. 70-73, 1929.  Re-invention [Ref 2] 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 3] 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.

31. 31 [Ref 5] S. C. Pei, W. L. Hsue, and J. J. Ding, “Discrete fractional Fourier transform based on new nearly tridiagonal commuting matrices,” accepted by IEEE Trans. Signal Processing. Type 1: Sampling Form Complexity: 2N + Nlog 2 N [Ref 4] S. C. Pei and J. J. Ding, “Closed form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Processing, vol. 48, no. 5, pp. 1338-1353, May 2000. Type 2: Eigenfunction Decomposition Form E: eigenvectors of the DFT (many choices), D: eigenvalues  Extension 1: Discrete Fractional Fourier Transform

32. 32  Extension 2: Fractional Cosine Transform [Ref 6] S. C. Pei and J. J. Ding, “Fractional, canonical, and simplified fractional cosine, sine and Hartley transforms,” IEEE Trans. Signal Processing, vol. 50, no. 7, pp. 1611-1680, Jul. 2002. [Ref 7] S. C. Pei and J. J. Ding, “Two-dimensional affine generalized fractional Fourier transform,” IEEE Trans. Signal Processing, vol. 49, no. 4, pp. 878-897, Apr. 2001. Extension 3: N-D Affine Generalized Fractional Fourier Transform

33. 33 [Ref 8] S. C. Pei and J. J. Ding, “Simplified fractional Fourier transforms,” J. Opt. Soc. Am. A, vol. 17, no. 12, pp. 2355-2367, Dec. 2000. (easier for digital implementation) (easier for optical implementation)  Extension 4: Simplified Fractional Fourier Transform

34. 34  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

35. 35 3 Wavelet Transform New Research field Useful for JPEG 2000 (image compression), filter design, edge and corner detection 只將頻譜分為「低頻」和「高頻」兩個部分 ( 對 2-D 的影像，則分為四個部分 ) x[n]x[n] h[n]h[n]  2 x 1,L [n] x 1,H [n]  2 g[n]g[n] 「低頻」部分 「高頻」部分

36. 36 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

37. 37 6. Integer Transform Conversion  Integer Transform: The discrete linear operation whose entries are summations of 2 k., a k = 0 or 1 or, C is an integer.

38. 38 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.

39. 39 [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,, b k and b̃ k are integers. (Goal 2) Reversibility. (Goal 3) Bit Constraint The denominator 2 k should not be too large. (Goal 4) Accuracy B  A, B̃  A -1 (or B   A, B̃   -1 A -1 ) (Goal 5): Less Complexity (Goal 6) Easy to Design

40. 40  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 2 k -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)

41. 41 Problem: The number of bits is increased (due to 3 triangular matrices) Number of bit tradeoff Accuracy The number of time cycles is increased (due to 3 triangular matrices) How to find the optimal one  Basic idea of the triangular matrix scheme: Any matrix can be decomposed as A = PDLUSQ P, Q: permuting matrices, D: diagonal matrix L: lower triangular matrix, U: upper triangular matrix, S: One row lower triangular matrix

42. 42 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.

43. 43 9. Optical Signal Processing and Fractional Fourier Transform lens, (focal length = f) free space, (length = z 1 )free space, (length = z 2 ) f = z 1 = z 2  Fourier Transform f  z 1, z 2 but z 1 = z 2  Fractional Fourier Transform f  z 1  z 2  Fractional Fourier Transform multiplied by a chirp

44. 44 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

45. 45  There are four types of nucleotide in a DNA sequence: adenine (A), guanine (G), thymine (T), cytosine (C)  Unitary Mapping b x [  ] = 1 if x[  ] = ‘A’, b x [  ] =  1 if x[  ] = ‘T’, b x [  ] = j if x[  ] = ‘G’, b x [  ] =  j if x[  ] = ‘C’. y = ‘AACTGAA’,  b y = [1, 1,  j,  1, j, 1, 1]. 11. 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.

46. 46  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’.

47. 47  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)

48. 48  Advantage of the Discrete Correlation Algorithm: ---The complexity of the conventional sequence alignments is O(N 2 ) ---For the discrete correlation algorithm, the complexity is reduced to O(N log 2 N) or O(N log 2 N + b 2 ) 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.

49. 49 12. 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

50. 50 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 = 53.4333 RMSE = 10.9662

51. 51 New Method: Edge-Based Segmentation and Compression

52. 52 Image Segment Compression Bit stream Image Segmentation Boundary Compression Image Segment Boundary An image Segmentation-based image compression

53. 53 Original ImageBy JPEG An 100x100 imageBytes: 1295, RMSE: 2.39 By Proposed Method Bytes: 456, RMSE: 2.54

54. 54 13. Edge and Corner Detection Why should we perform edge and corner detection? Segmentation Compression

55. 55 Simplest way for edge detection: differentiation

56. 56 by differentiation

57. 57 Other ways for edge detection: convolution with a longer odd function 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. x[n] 

58. 58 ++  (  +  /2)  Corner Detection Conventional Algorithm: Observing the variation along x-axis and y-axis, Proposed Algorithm: Observing the variation along +  axis,  -axis, +(  +  /2)-axis and  (  +  /2)-axis, -- +(  +  /2) Corner: the edge of an edge

59. 59 by Harris’ algorithm by proposed algorithm

60. 60 14. Pattern Recognition 應用很廣： security, identification ………… 但技術上的問題頗多 ………. scaling shadow rotation partially distortion 最簡單的方法： matched filter 其他的方法： 特徵拮取

61. 61 15. Quaternion 翻譯成 “ 四元素 ” ， Generalization of complex number  Complex number: a + ib i 2 =  1 real part imaginary part  Quaternion: a + ib + jc + kd i 2 = j 2 = k 2 =  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.

62. 62 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 abcdabcd a+jb c+jd =

63. 63 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。

64. 64 (4) 畢業之前，最低限度要曾經幫忙寫過一篇研討會論文 (5) 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。 唯獨，有的時候 ( 不是經常 ) ，有些工作若被特別強調一定要做到 ( 比如說， 下週一定要把某個資料看完 ) ，那就一定要完成。若沒有做到，則要補 meeting 。 (6) 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 )

65. 65 (10) 這一屆的同學，第一次 meeting 的時間是 2009 年 8 月中旬。 (9) 每學期會有二至三次的導生會，歡迎學生多多參加。 (7) 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 15 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。 (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。

66. 66 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閒瑕的時間，好好充實自已。