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- 1. Wavelet and multiresolution based signal-image processing By Fred Truchetet Le2i, UMR 5158 CNRS-Université de Bourgogne, France [email_address]
- 2. Overview
- 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. 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. A musical sound: a function of time, a signal Separate notes and chord
- 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. Distinguish the frequencies In a chord
- 8. Distinguish the times and the frequencies for series of notes
- 9. A sound: a wave
- 10. A sound: a function of time and frequency
- 11. Wave and impulse
- 12. A wavelet ? <ul><li>Oscillating mother function, well localized both in time and frequency : </li></ul><ul><li> (t) </li></ul>
- 13. Wavelet ? <ul><li>A family built by dilation </li></ul><ul><li> (t) (t/2) (t/4) </li></ul>
- 14. Wavelets ? <ul><li>and translation : </li></ul><ul><li> (t) (t-20) (t-40) </li></ul>
- 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. Why the wavelets?
- 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. 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. 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. 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. The wavelets, why? Example of “gaboret” for two frequencies (real part) The window size does not depend on the frequency
- 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. 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. 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. A wavelet, what is it? <ul><li>A mother function oscillating, localized: </li></ul><ul><li> (t) </li></ul>
- 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. A wavelet, what is it? <ul><li>and translation: </li></ul><ul><li> (t) (t-20) (t-40) </li></ul>
- 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. 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. 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. Continuous wavelet transform
- 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. 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. 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. Wavelet Transform <ul><li>Synthesis </li></ul><ul><li>Add the wavelets weighted by their respective weights </li></ul>
- 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. 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. Wavelet Transform as time-frequency analysis
- 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. 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. 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. 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. 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. <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. 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. 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. 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. 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. 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. 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. 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. 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. Discrete wavelet transform Multiresolution Analysis: orthogonal basis
- 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. 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. 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. 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. 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. 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. 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. Multi Resolution Analysis of L 2 (R) <ul><li>Set of axioms for dyadic MRA (S. Mallat, Y. Meyer): </li></ul>
- 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. 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. 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. Wavelet Transform
- 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. MRA: example of Haar analysis x A 0 x A 1 x D 1 x A 2 x D 2 x
- 68. MRA: general case 2 2 1 1 Scale function wavelet
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Example of synthesis of a detail signal Analysis Synthesis Approximation Détail
- 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. Example of approximations of the scale function for the basis Daubechies with N=2
- 86. Orthogonal MRA Properties and building
- 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. Orthogonality of the functions and of the associated filters For the scale function: Therefore ? For n=0
- 89. For the wavelets Between W j and V j Between wavelets within the same scale Generally as Therefore and
- 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. 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. Poisson equation autocorrelation In Fourier sampling In Fourier or As Fourier transform of Dirac is 1 and
- 93. Scale functions and associated filters h in Fourier domain as is 2 -periodic Separating odd and even terms: or
- 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. 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. Wavelet functions and associated filters g in the Fourier domain Wavelet-scaling function orthogonality For =0 and Therefore
- 97. Wavelet functions and associated filters g in the Fourier domain From Show that or Therefore
- 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. Relationship between h and g in orthogonal bases From with The simplest solution with linear phase : For example
- 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. 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
- 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
- 104. Examples of wavelets for orthogonal MRA Haar, Littlewood-Paley, Spline, Daubechies
- 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
- 106. Examples of orthogonal MRA: Haar From We have Therefore Scaling and wavelet functions in the Fourier domain:
- 107. Examples of orthogonal MRA: Haar Very compact in space, very bad localized in frequency Symmetric, no regularity, 1 vanishing moment
- 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 :
- 109. Examples of orthogonal MRA: Littlewood-Paley
- 110. Examples of orthogonal MRA: Littlewood-Paley The associated filters from It comes Therefore and with the QMF relationship These filters are IIR
- 111. Examples of orthogonal MRA: Littlewood-Paley The wavelet From
- 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>
- 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
- 114. Example of MRA: Spline bases (Battle-Lemarié) Examples of B-spline with n=1 and n=2 Compact support but not orthogonal
- 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:
- 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
- 117. Example of MRA: Spline bases (Battle-Lemarié) Infinitely supported but orthogonal n=1 n=2 In Fourier
- 118. Example of MRA: Spline bases (Battle-Lemarié) Filters Cubic spline basis: Battle-Lemarié
- 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

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