Presentation on theme: "丁建均 (Jian-Jiun Ding) National Taiwan University"— 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)Major：Digital Signal ProcessingDigital Image Processing
2 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
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)
4 1. Time-Frequency Analysis Fourier transform (FT)Time-Domain Frequency DomainSome 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.
5 Example: x(t) = cos( t) when t < 10, x(t) = cos(2 t) when t (FM signal)
6 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 = 15f -axist -axist -axis
7 Several Time-Frequency Distribution Short-Time Fourier Transform (STFT) with Rectangular MaskGabor Transformavoid cross-termless clarityWigner Distribution Functionwith cross-termhigh clarityGabor-Wigner Transform (Proposed)avoid cross-termhigh clarity
8 Cohen’s Class Distribution whereS TransformHilbert-Huang Transform
9 Instantaneous Frequency 瞬時頻率 Ifthen 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 frequenciesof other functions will inevitably vary with time.
10 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
11 Conventional Sampling Theory Nyquist CriterionNew 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 spectrum of signal 1-B1B1spectrum of signal 2not overlappedB2-B2
14 Improvement of Time-Frequency Analysis (1) Computation Time(2) Tradeoff of the cross term problem and clarification
20 3. Fractional Fourier Transform Performing the Fourier transform a times (a can be non-integer) Fourier Transform (FT)generalization Fractional Fourier Transform (FRFT), = a/2When = 0.5, the FRFT becomes the FT.
21 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;
22 Physical Meaning: Transform a Signal into the Fractional domain, which is the intermediate of the time domain and the frequency domain.
23 Time domain Frequency domain fractional domain Modulation Shifting Modulation + ShiftingShifting Modulation Modulation + ShiftingDifferentiation j2f Differentiation and j2f −j2f Differentiation Differentiation and −j2f is some constant phase
24 Why do we use the fractional Fourier transform? To solve the problems that cannot be solved by the Fourier transformExample: Filter DesignConventional 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 )
25 When x(t) = triangular signal + chirp noise exp[j 0.25(t 4.12)2]
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 From the view points of Time-Frequency Analysis: [Theorem] The FRFT with parameter is equivalent to the clockwise rotationoperation 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) / /6horizon: 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
28 Filter designed by the fractional Fourier transform 比較：Filter Designed by the Fourier transformf-axisSignalnoiset-axisFRFTFRFTcutoff line
29 以時頻分析的觀點，傳統濾波器是垂直於 f-axis 做切割的 stop bandf0cutoff linepass bandt-axis而用 fractional Fourier transform 設計的濾波器是，是由斜的方向作切割f-axisstop bandcutoff line 和 f-axis 在逆時針方向的夾角為 u0pass bandcutoff line
30 Gabor Transform for signal + 0.3exp[j0.06(t1)3 j7t] fractional axist-axisAdvantage: 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 , 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-frequencyrepresentations,” IEEE Trans. Signal Processing, vol. 42, no. 11, pp3091, NovRecent developmentPei, 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, 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
45 6. Edge and Corner Detection Why should we perform edge and corner detection?--Segmentation--Compression
46 Simplest way for edge detection: differentiation
47 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.Other ways for edge detection: convolution with a longer odd functionx[n]
48 Corner Detectionby Harris’ algorithm by proposed algorithm
49 7. Segmentation Important for compression biomedical engineering object identification
56 10. Integer Transform Conversion Integer Transform: The discrete linear operation whose entries aresummations of 2k., ak = 0 or 1 or , C is an integer.
57 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.
58 Integer RGB to YCbCr Transform This is used in JPEG 2000.
59 [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
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 , 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
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 Transformf z1, z2 but z1 = z2 Fractional Fourier Transform (see page 20)f z1 z2 Fractional Fourier Transform multiplied by a chirp
64 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 Mappingbx = 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].
65 Discrete Correlation Algorithm for DNA Sequence Comparison For two DNA sequences x and y, ifwhereThen 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’.
66 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)
67 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 subsequencesExperiment: Local alignment for two 3000-entry DNA sequencesUsing conventional dynamic programming Computation time: 87 sec Using the proposed discrete correlation algorithm: Computation time: 4.13 sec.
68 15. Quaternion 翻譯成“四元素”，Generalization of complex number Complex number: a + ib i2 = 1real part imaginary part Quaternion: a + ib + jc + kd i2 = j2 = k2 = 1real 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.
69 Application of quaternion a + ib + jc + kd: --Color image processinga + iR + jG + kB represent an RGB image--Multiple-Channel Analysis4 real channels or 2 complex channelsabcda+jbc+jd=