Presentation on theme: "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."— Presentation transcript:
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 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 [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 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 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 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 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 Cohen’s Class Distribution S Transform where Hilbert-Huang Transform
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 (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 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 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 Modulation and Multiplexing not overlapped spectrum of signal 1 spectrum of signal 2 B1B1 -B 1 B2B2 -B 2
14 Improvement of Time-Frequency Analysis (1) Computation Time (2) Tradeoff of the cross term problem and clarification
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 Gabor-Wigner Transform avoiding the cross-term problem and high clarity -axis t -axis
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 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 Physical Meaning: Transform a Signal into the Fractional domain, which is the intermediate of the time domain and the frequency domain.
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 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 When x(t) = triangular signal + chirp noise exp[j 0.25(t 4.12) 2 ]
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 (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 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 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 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 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 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 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 [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 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 [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 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 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 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 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 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 [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 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 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 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 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
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 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 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 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 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
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 ++ ( + /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
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 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 =