Wavelet and multiresolution based signal-image processing


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  • - The first time I heard about wavelets was at the end of the eighties while reading a paper by Stephane Mallat. I have been immediately hit by the simplicity, even the beauty of the multiresolution concept exposed in this exceptional article. It was the beginning for me and I am still studying, working, thinking about wavelet today. But, the story was already an old one, the first paper talking of wavelets appeared in 1982, we will come back to this point later. Since the nineties, the wavelet analysis has been extensively studied and many applications have been proposed. In this talk I would like to overview the basics about wavelet and to sample some applications in industrial processes. In the reference section, you will find more than 180 papers covering nearly all the fields involving signal or image processing in industrial applications. Therefore I chose to give more an introduction to the topic - the wavelet transform and its industrial application - than a true exhaustive review. I selected only few examples, in signal processing field as well as in image processing. I chose to focus more on the diversity of the application domains than on the wavelet properties involved, even if we will scan the most important one along the examples.
  • We will consider here the information contained in signal or image, this information is made of quantitative data and processing them can be seen the same way a chemist deals with the matter: analyze, transform and, sometimes, synthesize. Analyzing for a chemist is to find the constituting atoms; in signal processing, the atoms can be chosen and the result of the processing depends heavily on this choice. It has been shown during those last twenty years that often Wavelets are a fine choice as signal or image processing “atoms”.
  • If we begin with an simple example and without using math. What can be a signal processing work in music?
  • In this musical sequence one can find two parts, the first one with separate notes and in the second a bunch of notes are played at the same time in a chord.
  • During the “chord section” all the notes are played together and the problem is to distinguish their pitches or frequencies. No clear periodicity can be seen in the temporal representation, but in a spectral representation it is easy to see that 3 simple notes are used in the chord.
  • In the other section of the music sequence, all the notes are mixed in the spectral representation and the temporal representation only allows a time segmentation of the music and therefore gives the sequence of notes used in the melody. The two representations are therefore needed at the same time for a complete analysis of the musical sequence.
  • The same conclusion is reached for other sounds as, for instance, as for a simple word emitted by a human being. In the sound, it can be seen that the time representation give hints about the syllabic segmentation but, once again, the pitch is given by the spectral representation.
  • In the time-frequency representation both the information are presented at the same time. The question is: how to do that?, How to measure the time and frequency components at the same time and with the best resolution?
  • The first simple idea could be to look for some sinusoïdal wave in the signal. But to estimate the frequency, this wave must contain a sufficient number of periods. And its duration is constant, the number of period depends on the frequency. And for short duration, the frequency precision can fall critically.
  • – The new idea proposed at the beginning of the eighties by J. Morlet, is to adapt automatically the duration of the wave to its frequency. This lead to the concept of wavelet. Basically, the atom-wavelet is simply an oscillating function well localized as well in time as well in frequency. A family is constructed from a mother.
  • To conclude with the musical example, a question: a note is a wave or a wavelet? It is not really a wavelet it is more a windowed wave, the size of the window being independent of the pitch.
  • We, now, come through the same question but with some mathematics.
  • Hint: work in the Fourier domain
  • Wavelet and multiresolution based signal-image processing

    1. 1. Wavelet and multiresolution based signal-image processing By Fred Truchetet Le2i, UMR 5158 CNRS-Université de Bourgogne, France [email_address]
    2. 2. Overview
    3. 3. Wavelet play field: signal and image processing <ul><li>Signal or image: quantitative information </li></ul><ul><li>Process: Analyze </li></ul><ul><li>Transform </li></ul><ul><li>Synthesize </li></ul>32 45 3 4 5 67 7 8 3 7
    4. 4. Wavelets, why? <ul><li>Signal processing: analysis, transformation, characterization, synthesis </li></ul><ul><li>Example : </li></ul><ul><li>Analysis of a musical sequence </li></ul><ul><li>For automatic creation of score (music sheet) </li></ul><ul><li>Synthesis of music from score </li></ul><ul><li>For automatic reading and playing score </li></ul>
    5. 5. A musical sound: a function of time, a signal Separate notes and chord
    6. 6. Analysis and synthesis ? <ul><li>In a score the music stream is segmented into « atoms » or notes defined by their </li></ul><ul><li>Pitch: C, D, E, etc… </li></ul><ul><li>Duration (whole note, half note, quarter note, etc…) </li></ul><ul><li>Position in time (measure bars) </li></ul><ul><li>It provides an analysis of the musical signal </li></ul><ul><li>With the score the musician can play the music as it has been originally created </li></ul><ul><li>It is the synthesis stage </li></ul>
    7. 7. Distinguish the frequencies In a chord
    8. 8. Distinguish the times and the frequencies for series of notes
    9. 9. A sound: a wave
    10. 10. A sound: a function of time and frequency
    11. 11. Wave and impulse
    12. 12. A wavelet ? <ul><li>Oscillating mother function, well localized both in time and frequency : </li></ul><ul><li> (t) </li></ul>
    13. 13. Wavelet ? <ul><li>A family built by dilation </li></ul><ul><li>  (t)  (t/2)  (t/4) </li></ul>
    14. 14. Wavelets ? <ul><li>and translation : </li></ul><ul><li>  (t)  (t-20)  (t-40) </li></ul>
    15. 15. Waves or wavelets ? <ul><li>Wave </li></ul><ul><li>frequency </li></ul><ul><li>Infinite duration </li></ul><ul><li>No temporal localization </li></ul><ul><li>Wavelet </li></ul><ul><li>scale </li></ul><ul><li>Duration (window size) </li></ul><ul><li>Temporal localization </li></ul><ul><li>then </li></ul><ul><li>Wavelet = Note ? </li></ul>
    16. 16. Why the wavelets?
    17. 17. The wavelets, why? (again but with mathematical arguments) <ul><li>In the real world, a signal is not stationary . </li></ul><ul><li>The information is in the statistical, frequential, temporal, spatial varying features </li></ul><ul><li>Examples: vocal signal, music, images… </li></ul><ul><li>Joseph Fourier, in 1822, proposed a global analysis: </li></ul><ul><ul><li>Integrals are from -  to +  </li></ul></ul><ul><ul><li>Spatial or temporal localization is lost </li></ul></ul><ul><li>Fourier Transform : </li></ul>
    18. 18. The wavelets, why? <ul><li>A straightforward idea: cut the integration domain into sliding windows </li></ul><ul><li>Window Fourier transform or Short Time Fourier Transform (STFT) : </li></ul><ul><li>We denote the “window function” as: </li></ul><ul><li>When t and  vary It constitutes a family which can be considered as a kind of « basis » </li></ul>
    19. 19. The wavelets, why? <ul><li>This transform can be seen as the projection over the sliding window functions : </li></ul><ul><li>With the inner product: </li></ul>
    20. 20. The wavelets, why? <ul><li>Many window functions are used: Hanning, Hamming, and Gauss : </li></ul><ul><li>For the Gauss window, the transform is called “Gabor transform”. The basis function is called “gaboret». These functions are normalized with </li></ul><ul><li>Gabor Transform : </li></ul>
    21. 21. The wavelets, why? Example of “gaboret” for two frequencies  (real part) The window size does not depend on the frequency
    22. 22. The wavelets, why? <ul><li>The resolution in the frequency-time plane can be estimated by the variance of the window function: </li></ul><ul><li>With x=t or x=f for time and frequency * resolution respectively </li></ul><ul><li>For a “gaboret”: </li></ul><ul><li>As </li></ul><ul><li>Then whatever the frequency: </li></ul>Show that *
    23. 23. The wavelets, why? Time-frequency plane tiling provided by the Gabor Transform Not optimal As some periods are necessary for frequency measurement a low temporal resolution comes naturally for low frequencies, for high frequencies a finer temporal resolution is possible. Question : how to find an automatic trade-off between time and frequency resolution for all the frequencies?
    24. 24. The wavelets, why? <ul><li>Answer : the Wavelet Transform </li></ul>a is the scale factor and b the translation parameter and  is the wavelet function (basis window function). The scale factor a is as 1/  the greater a the larger the wavelet. If a is small, the frequency is high and the window is small allowing a high temporal resolution for the analysis.  is called the mother of a family of functions built by dilation and translation following:
    25. 25. A wavelet, what is it? <ul><li>A mother function oscillating, localized: </li></ul><ul><li> (t) </li></ul>
    26. 26. A wavelet, what is it? <ul><li>A family built by dilation </li></ul><ul><li>  (t)  (t/2)  (t/4) </li></ul>
    27. 27. A wavelet, what is it? <ul><li>and translation: </li></ul><ul><li> (t)  (t-20)  (t-40) </li></ul>
    28. 28. The wavelets, why? The norm does not depend on a: The wavelet transform (WT) can be denoted as: If the temporal resolution of the mother wavelet is taken as unit, then
    29. 29. The wavelets, why? Then And for the frequency resolution, taking in the same way the frequency variance of the mother wavelet as unit And finally Show that  0 ~ 1/a then Q=constant
    30. 30. The wavelets, why? <ul><li>Time-frequency plane tiling </li></ul>The wavelet transform produces a constant Q analysis Uncertainty principle:  f.  t = constant
    31. 31. Continuous wavelet transform
    32. 32. Wavelet Transform <ul><li>Analysis </li></ul><ul><li>Searching for the weight of each wavelet (atom of signal) in a function f(t) </li></ul>
    33. 33. Continuous wavelet transform: CWT <ul><li>Continuous wavelet transform: </li></ul><ul><li>In the Fourier space: </li></ul><ul><li>Inverse transform : </li></ul>
    34. 34. A wavelet has to be admissible <ul><li>Admissibility condition: </li></ul><ul><li>For ordinary localized functions: </li></ul><ul><li>Or, more generally: </li></ul>
    35. 35. Wavelet Transform <ul><li>Synthesis </li></ul><ul><li>Add the wavelets weighted by their respective weights </li></ul>
    36. 36. Wavelets for CWT <ul><li>Some examples of admissible wavelets </li></ul><ul><ul><li>Haar (this example is presented further) </li></ul></ul><ul><ul><li>Mexican hat </li></ul></ul><ul><ul><li>Morlet </li></ul></ul>Show that the Morlet wavelet is only close to admissible
    37. 37. Wavelets for CWT Wavelets in the Fourier domain Morlet for a=1 and a=2 Mexican hat As a is increasing, the frequency size shrinks while the temporal window enlarges. The original trade-off is maintained whatever the scale factor.
    38. 38. Wavelet Transform as time-frequency analysis
    39. 39. Sampling for discrete wavelet transform The time-scale plane can be sampled to avoid or limit the redundancy of the CWT. To respect the Q-constant analysis principle, the sampling must be such that: i is the discrete scale factor and n the discrete translation parameter, both are integer.
    40. 40. Discrete wavelet transform: DWT <ul><li>Discrete analysis with continuous wavelet </li></ul><ul><li>Isomorphism between L 2 (R) and l 2 (R) (continuous functions ↔ discrete sequences) </li></ul><ul><li>a=a 0 i with i integer b=nb 0 a 0 i with n integer </li></ul><ul><li>Dyadic analysis: a 0 =2 b 0 =1 </li></ul><ul><li>Discrete tiling of the scale-time space </li></ul>
    41. 41. Which Wavelet Transform? <ul><li>Continuous, CWT, for signal analysis, without synthesis: redundant </li></ul><ul><li>Discrete, DWT, (dyadic or not, Mallat or lifting scheme), for signal or image analysis if synthesis is required </li></ul><ul><ul><li>Non redundant : </li></ul></ul><ul><ul><ul><li>Orthogonal basis </li></ul></ul></ul><ul><ul><ul><li>Non orthogonal basis (biorthogonal) </li></ul></ul></ul><ul><ul><li>Redundant : non decimated DWT, Frame </li></ul></ul><ul><ul><li>Wavelet packets (redundant or not) </li></ul></ul>
    42. 42. Who invented wavelets? <ul><li>From Joseph Fourier to Jean Morlet </li></ul><ul><li>and after ... </li></ul><ul><li>almost a French story </li></ul><ul><li>The ancestor </li></ul><ul><li>Joseph FOURIER born in Auxerre </li></ul><ul><li>(Burgundy, France) in 1768, </li></ul><ul><li>amateur mathematician, provost of Isère </li></ul><ul><li>published in 1822 a theory of heat… </li></ul><ul><li>Every « physical » function can be </li></ul><ul><li>written as a sum of sine-waves: </li></ul><ul><li>Fourier Transform </li></ul>
    43. 43. Who invented wavelets? <ul><li>The grandfather </li></ul><ul><li>Dennis GABOR electrical engineer </li></ul><ul><li>and physicist, Hungarian born English, </li></ul><ul><li>Nobel price of physics in 1971 </li></ul><ul><li>for inventing holography </li></ul><ul><li>Decomposition into constant duration </li></ul><ul><li>« wave pulses »: </li></ul><ul><li>Short Time Fourier Transform (1946) </li></ul>
    44. 44. <ul><li>The father </li></ul><ul><li>Jean MORLET French engineer from Ecole Polytechnique, geologist for petrol company </li></ul><ul><li>Elf Aquitaine </li></ul><ul><li>Decomposition into wavelets with duration in inverse proportion to frequency (1982) </li></ul><ul><li>The children </li></ul><ul><li>A.Grossmann (1983), Y.Meyer (1986), </li></ul><ul><li>S.Mallat (1987), I.Daubechies (1988), J.C.Fauveau (1990), W. Sweldens (1995)... </li></ul>Who invented wavelets?
    45. 45. references <ul><li>Daubechies, «Ten Lectures on Wavelets», SIAM, Philadelphia, PA, 1992. </li></ul><ul><li>S. Mallat, «A theory for multiresolution signal decomposition : the wavelet representation», IEEE, PAMI, vol. 11, N° 7, pp. 674-693, july 1989. </li></ul><ul><li>S. Mallat, “Wavelet Tour of Signal Processing”, Academic Press, Chestnut Hill MA, 1999 </li></ul><ul><li>G. Strang, T. Nguyen, «Wavelets and filter banks», Wellesley-Cambridge Press, Wellesley MA, 1996. </li></ul><ul><li>F. Truchetet, “Ondelettes pour le signal numérique”, Hermès, Paris, 1998. </li></ul><ul><li>F. Truchetet, O. Laligant, “ Industrial applications of the wavelet and multiresolution based signal-image processing, a review”, proc. of QCAV 07, SPIE, vol. 6356, may 2007 </li></ul><ul><li>M. Vetterli, J. Kovacevic, « Wavelets and Subband Coding », Prentice Hall, Englewood Cliffs, NJ, 1995. </li></ul>
    46. 46. Which wavelet? <ul><li>Freedom to choose a wavelet </li></ul><ul><ul><li>Blessing or Curse? </li></ul></ul><ul><li>How much efforts need to be made for finding a good wavelet? </li></ul><ul><ul><li>Any wavelet will do? </li></ul></ul><ul><li>What properties of wavelets need to be considered? </li></ul><ul><li>Symmetry, regularity, vanishing moments, compacity </li></ul>
    47. 47. Symmetry <ul><li>In some applications the analyzing function needs to be symmetric or antisymmetric: </li></ul><ul><li>Real world images </li></ul><ul><li>This is related to phase linearity </li></ul><ul><li>Symmetric : Haar, Mexican hat, Morlet </li></ul><ul><li>Non symmetric : Daubechies, 1D compact support orthogonal wavelets </li></ul>
    48. 48. Regularity <ul><li>The order of regularity of a wavelet is the number of its continuous derivatives. </li></ul><ul><li>Regularity can be expanded into real numbers. (through Fourier Transform equivalent of derivative) </li></ul><ul><li>Regularity indicates how smooth a wavelet is </li></ul>
    49. 49. Vanishing Moment <ul><li>Moment: j’s moment of the function </li></ul><ul><li>When the wavelet’s k+1 moments are zero </li></ul><ul><li>i.e. </li></ul><ul><li>the number of Vanishing Moments of the wavelet is k. </li></ul><ul><li>Weakly linked to the number of oscillations. </li></ul>
    50. 50. Vanishing moments <ul><li>When a wavelet has k vanishing moments, WT leads to suppression of signals that are polynomial of degree lower or equal to k…. (whatever the scale) </li></ul><ul><li>… or detection of higher degree components: singularities </li></ul><ul><li>If a wavelet is k times differentiable, it has at least k vanishing moments </li></ul>Show that from
    51. 51. Compacity (size of the support) <ul><li>The number of FIR filter coefficients. </li></ul><ul><li>The number of vanishing moments is proportional to the size of support. </li></ul><ul><li>Trade-off between computational power required and analysis accuracy </li></ul><ul><li>Trade-off between time resolution and frequency resolution </li></ul><ul><li>A compact orthogonal wavelet cannot be symmetric in 1D </li></ul>
    52. 52. Which wavelet: examples for DWT Db1 (Haar) Db2 (D4) Db5 (D10) Db10 (D20) R=NA R=0.5 R=1.59 R=2.90 VM=1 VM=2 VM=5 VM=10 SS=2 SS=4 SS=10 SS=20
    53. 53. Discrete wavelet transform Multiresolution Analysis: orthogonal basis
    54. 54. Multi Resolution Analysis of L 2 (R) <ul><li>Approximation spaces </li></ul><ul><ul><li>Working space: L 2 (R), for continuous functions, f(x), on R with finite norm (finite energy) </li></ul></ul><ul><ul><li>An analysis at resolution j of f is obtained by a linear operator : </li></ul></ul><ul><ul><li>V j is a subspace of L 2 (R), A j is a projection operator (idempotent) </li></ul></ul><ul><ul><li>A multiresolution analysis (MRA) is obtained with a set of embedded subspaces V j , such that going from one to the next one is performed by dilation: </li></ul></ul><ul><ul><li>In the dyadic case for instance, the dilation factor is 2. </li></ul></ul><ul><ul><li>The functions in subspace V j+1 are coarser than in subspace V j and </li></ul></ul><ul><ul><li>If j goes to - infinity, the subspace must tend toward L 2 (R). </li></ul></ul>
    55. 55. Multi Resolution Analysis of L 2 (R) <ul><li>Set of axioms for dyadic MRA (S. Mallat, Y. Meyer): </li></ul>The last property allows the invariance for translation by integer steps
    56. 56. Multi Resolution Analysis of L 2 (R) <ul><li>In these conditions there exists a function  (x) called scaling function from which, by integer translation, a basis of V 0 can be built. </li></ul><ul><li>Then a basis can be obtained for each subspace by dilating  (x) </li></ul><ul><li>The basis is orthogonal if </li></ul>
    57. 57. Multi Resolution Analysis of L 2 (R) The approximation at scale j of the function f is given by: The approximation coefficients constitutes a discrete signal. If the basis is orthogonal, then
    58. 58. Multi Resolution Analysis of L 2 (R) For each subspace V j its orthogonal complement W j in V j-1 can be defined. It is called the detail subspace at scale j As W j is orthogonal to V j , it is also orthogonal to W j+1 which is in Vj. Therefore, all the W j are orthogonal
    59. 59. Multi Resolution Analysis of L 2 (R) In these conditions there exists a function  (x) called wavelet function from which, by integer translation, a basis of W 0 can be built. Then a basis can be obtained for each subspace by dilating  (x) The basis is orthogonal if And the complement of the approximation at scale j can be computed by
    60. 60. Multi Resolution Analysis of L 2 (R) The details of f at scale j are obtained by a projection on W j as These coefficients are the wavelet coefficients or the coefficients of the discrete wavelet transform DWT associated to this MRA. They constitute a discrete signal.
    61. 61. Multi Resolution Analysis of L 2 (R) <ul><li>Set of axioms for dyadic MRA (S. Mallat, Y. Meyer): </li></ul>
    62. 62. MRA and orthogonal wavelet basis with n integer, constitutes an orthogonal basis of V i , the scaling functions are not admissible wavelets! with n integer, constitutes an orthogonal basis of W i All W i are orthogonal and the direct sum of all these subspaces is equal to L 2 ( R ): for i and n integers constitutes an orthogonal basis of L 2 ( R ) Scaling function family : Wavelet family:
    63. 63. Multiresolution analysis Detail signal and approximation signal are characterized by the discrete sequences of wavelet and scale coefficients: Sampling is a consequence of MRA
    64. 64. Discrete Wavelet Transform: Mallat’s algorithm <ul><li>Recursive algorithm: MRA </li></ul><ul><li> A pproximation + D etail </li></ul><ul><li>(wavelet coefficients) </li></ul>Question: initialization? What are the first approximation coefficients?
    65. 65. Wavelet Transform
    66. 66. Example of MRA: Haar basis The scale function The wavelet function Verify invariance, normality and describe the functions of V j and W j and give the Haar analysis of f(x)=x.
    67. 67. MRA: example of Haar analysis x A 0 x A 1 x D 1 x A 2 x D 2 x  
    68. 68. MRA: general case  2  2  1  1 Scale function wavelet
    69. 69. MRA: general case <ul><li>Example of approximations and details of f </li></ul>f P V 0 f P V 1 f P W 1 f P V 2 f P W 2 f P V 3 f P W 3 f
    70. 70. Mallat’s algorithm: analysis By definition,  (x) is a function of V 0 and as ,  (x) can be decomposed on the basis of V -1 and a discrete sequence with can be found such that With and or Show that
    71. 71. Mallat’s algorithm: analysis The approximation coefficients a j : can be computed following a recursive algorithm: then If h is considered as the impulse response of a discrete filter, we have a convolution followed by a downsampling by two: 2
    72. 72. Mallat’s algorithm: analysis In the same way, W 0 is in V -1 and a discrete sequence g[n] can be found by projecting the wavelet function on the basis of V -1 : or Show that If g is considered as the impulse response of a discrete filter, we have a convolution followed by a down sampling by two: 2
    73. 73. Mallat’s algorithm <ul><li>Analysis: recursive algorithm </li></ul><ul><li>Linear and invariant digital filtering. </li></ul><ul><li>Two filters h[n] (low pass) and g[n] (high pass)  </li></ul>
    74. 74. Mallat’s algorithm: synthesis The analysis at scale j-1 gives two components, one in V j and the other in W j with As A j is a projection operator (idempotent): then and therefore
    75. 75. Mallat’s algorithm: synthesis We have seen that As the basis of V j-1 is orthogonal then and Therefore from a synthesis equation can be written:
    76. 76. Mallat’s algorithm: synthesis This equation can be seen as the sum of two convolution products (digital linear filtering) if two up sampled versions of a j and d j are introduced:
    77. 77. Dyadic Discrete Wavelet Transform <ul><li>Fast Transform: Mallat’s algorithm </li></ul><ul><li>Recursive algorithm driving through scales; from scale j to scale j-1 </li></ul>
    78. 78. Example of DWT: Haar basis Find the filters h and g for the Haar analysis Verify the algorithm of Mallat for f(x)=x and one scale
    79. 79. Mallat’s algorithm: building recursively the basis functions The mother scale function belongs to V 0 and the basis is orthogonal: and Then for the mother scale function  : Then an approximation at scale j of  can be obtained by cranking the machine up to scale j with a Dirac as approximation coefficient at scale 0 as only input
    80. 80. Mallat’s algorithm: building recursively the basis functions: the cascade algorithm Verify this result for the Haar basis A similar result can be obtained for the wavelets: therefore The only detail coefficient sequence is a Dirac at scale 0
    81. 81. Synthesis of a projection on V j or W j More generally, an approximation or a detail function at scale j can be obtained by following the synthesis algorithm
    82. 82. Projection on V 0 a 0 a 1 d 1 a 2 d 2 a 3 d 3 Coefficients of the analysis Example of coefficients projections on V 0 for some approximations and details A 0 a 2 . 0 a 2 0 . 0 A 0 d 3 . 0 0 d 3 A 0 d 2 . 0 0 d 2
    83. 83. Example of synthesis of a detail signal Analysis Synthesis Approximation Détail
    84. 84. Projected transform: example d 1 d 2 d 3 Wavelet coefficients Detail approximation A 0 d 1 A 0 d 2 A 0 d 3
    85. 85. Example of approximations of the scale function for the basis Daubechies with N=2
    86. 86. Orthogonal MRA Properties and building
    87. 87. DWT : Properties of the basis functions and of the associated filters <ul><li>Orthogonality of the scale function and of the associated filter </li></ul><ul><li>Orthogonality of the wavelet function and of the associated filter </li></ul><ul><li>Scale functions  and filters associated h in the Fourier domain </li></ul><ul><li>Wavelet functions  and filters g associated in the Fourier domain </li></ul>
    88. 88. Orthogonality of the functions and of the associated filters For the scale function: Therefore ? For n=0
    89. 89. For the wavelets Between W j and V j Between wavelets within the same scale Generally as Therefore and
    90. 90. Scale functions  and associated filters h in the Fourier domain h[n] is considered as the impulse response of a discrete linear filter: Transfer function: Frequency response: and therefore or (2  -periodic)
    91. 91. Scale functions  and associated filters h in the Fourier domain Orthogonality in the Fourier domain Show that using autocorrelation in the Fourier domain and the Poisson formula Analyzing a function with a non zero mean value shows that we must have:
    92. 92. Poisson equation autocorrelation In Fourier sampling In Fourier or As Fourier transform of Dirac is 1 and
    93. 93. Scale functions  and associated filters h in Fourier domain as is 2  -periodic Separating odd and even terms: or
    94. 94. Scale functions  and associated filters h in the Fourier domain as For  =0 in this equation and in the previous one, it comes Therefore, h is a low pass filter giving a low resolution version of the signal and
    95. 95. Wavelet functions  and associated filters g in the Fourier domain g[n] is considered as the impulse response of a discrete linear filter: Transfer function: Frequency response: and therefore Intra scale wavelet orthogonality
    96. 96. Wavelet functions  and associated filters g in the Fourier domain Wavelet-scaling function orthogonality For  =0 and Therefore
    97. 97. Wavelet functions  and associated filters g in the Fourier domain From Show that or Therefore
    98. 98. Wavelet functions  and associated filters g in the Fourier domain  is an admissible wavelet function g is a high pass filter keeping the high frequency components, i.e. the details How to deduce g from h?
    99. 99. Relationship between h and g in orthogonal bases From  with The simplest solution with linear phase : For example
    100. 100. Relationship between h and g in orthogonal bases From Show that Or more generally Such a pair of filters is called QMF : Quadrature Mirror Filters
    101. 101. Building an MRA <ul><li>Begin with the scaling function or the approximation subspaces </li></ul><ul><li>Determine h filters </li></ul><ul><li>Deduce g filters </li></ul><ul><li>Finally deduce the wavelet functions </li></ul>1 and 2 can be switched round
    102. 103. Low frequencies High frequencies 0 % Mallat’s algorithm x(n) 25% 50% 12.5% g ( n ) 2 a h ( n ) 2 h ( n ) 2 g ( n ) 2 b h ( n ) 2 d g ( n ) 2 c
    103. 104. Examples of wavelets for orthogonal MRA Haar, Littlewood-Paley, Spline, Daubechies
    104. 105. Examples of orthogonal MRA: Haar Mother scaling function: Approximation subspaces: Projection on a finer subspace From It comes or and with the QMF property
    105. 106. Examples of orthogonal MRA: Haar From We have Therefore Scaling and wavelet functions in the Fourier domain:
    106. 107. Examples of orthogonal MRA: Haar Very compact in space, very bad localized in frequency Symmetric, no regularity, 1 vanishing moment
    107. 108. Examples of orthogonal MRA: Littlewood-Paley It comes from the same idea: the approximation subspaces in Fourier domain are piecewise constant. Kind of dual basis to the Haar’s the orthogonality property in Fourier is clearly verified: If To have symmetry: a zero-phase condition is set, show that :
    108. 109. Examples of orthogonal MRA: Littlewood-Paley
    109. 110. Examples of orthogonal MRA: Littlewood-Paley The associated filters from It comes Therefore and with the QMF relationship These filters are IIR
    110. 111. Examples of orthogonal MRA: Littlewood-Paley The wavelet From
    111. 112. Example of MRA: Spline bases (Battle-Lemarié) <ul><li>Improve the Haar basis for a better piecewise approximation using polynomial functions </li></ul><ul><li>Keep the symmetry (linear phase) </li></ul><ul><li>Use the B-spline basis properties in connection with </li></ul><ul><li>The B-spline functions are a basis for piecewise polynomial functions but not an orthogonal basis in </li></ul><ul><li>An orthogonalization process is required </li></ul>
    112. 113. Example of MRA: Spline bases (Battle-Lemarié) The approximation subspace V j is defined as the set of piecewise polynomial functions on 2 j width segments. The B-spline basis of order n is built by autoconvolution of a box function: Therefore
    113. 114. Example of MRA: Spline bases (Battle-Lemarié) Examples of B-spline with n=1 and n=2 Compact support but not orthogonal
    114. 115. Example of MRA: Spline bases (Battle-Lemarié) Therefore The orthogonalization process is based on the following property of orthogonal bases: It can be shown that if f(t) is a basis, an orthogonal basis is obtained by:
    115. 116. Example of MRA: Spline bases (Battle-Lemarié) The orthogonal scaling function basis is given by It can be shown that the normalization factor can be computed with discrete B-splines: with Therefore finally
    116. 117. Example of MRA: Spline bases (Battle-Lemarié) Infinitely supported but orthogonal n=1 n=2 In Fourier
    117. 118. Example of MRA: Spline bases (Battle-Lemarié) Filters Cubic spline basis: Battle-Lemarié
    118. 119. Example of MRA: Spline bases (Battle-Lemarié) Wavelets n=1 n=2 In Fourier Compute an approximation of the Battle-Lemarié wavelet with the matlab wavelet toolbox