WAVELET ANALYSIS by wijethungashdp

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            WAVELET ANALYSIS AND SOME OF ITS APPLICATIONS
                             IN PHYSICS



                                                J.-P. ANTOINE
                                             e                 e
                      Institut de Physique Th´orique, Universit´ Catholique de Louvain
                                     B-1348 Louvain-la-Neuve, Belgium
                                       E-mail: Antoine@fyma.ucl.ac.be


                 We review the general properties of the wavelet transform, both in its continuous
                 and its discrete versions, in one or two dimensions, and we describe some of its
                 applications in signal and image processing. We also consider its extension to
                 higher dimensions.



           1. What is wavelet analysis?
           Wavelet analysis is a particular time- or space-scale representation of signals
           which has found a wide range of applications in physics, signal processing
           and applied mathematics in the last few years. In order to get a feeling
           for it and to understand its success, let us consider first the case of one-
           dimensional signals.
               It is a fact that most real life signals are nonstationary. They often
           contain transient components, sometimes very significant physically, and
           mostly cover a wide range of frequencies. In addition, there is frequently a
           direct correlation between the characteristic frequency of a given segment
           of the signal and the time duration of that segment. Low frequency pieces
           tend to last a long interval, whereas high frequencies occur in general for
           a short moment only. Human speech signals are typical in this respect.
           Vowels have a relatively low mean frequency and last quite long, whereas
           consonants contain a wide spectrum, up to very high frequencies, especially
           in the attack, but they are very short.
               Clearly standard Fourier analysis is inadequate for treating such signals,
           since it loses all information about the time localization of a given frequency
           component. In addition, it is very uneconomical. If a segment of the
           signal is almost flat, thus carries little information, one still has to sum an
           (alternating) infinite series for reproducing it. Worse yet, Fourier analysis is

                                                        1
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           2




                 2
                 4


           Figure 1. A traditional time-frequency representation of a signal (from Mozart’s Don
           Giovanni, Act 1).


           highly unstable with respect to perturbation, because of its global character.
           For instance, if one adds an extra term, with a very small amplitude, to a
           linear superposition of sine waves, the signal will barely be modified, but
           the Fourier spectrum will be completely perturbed. Such pathologies do
           not occur if the signal is represented in terms of localized components.
               Therefore, signal analysts turn to time-frequency (TF) representations.
           The idea is that one needs two parameters. One, called a, characterizes
           the frequency, the other one, b, indicates the position in the signal. This
           concept of a TF representation is in fact quite old and familiar. The most
           obvious example is simply a musical score (Figure 1)!
               If one requires, in addition, the transform to be linear, a general TF
           transform will take the form:
                                                       ∞
                                 s(x) → S(b, a) =            ψb,a (x) s(x) dx,                (1.1)
                                                      −∞

           where s is the signal and ψb,a the analyzing function (we denote the time
           variable by x, in view of the extension to higher dimensions). The as-
           sumption of linearity is actually nontrivial, for there exists a whole class
           of quadratic or, more properly, sesquilinear time-frequency representations.
           The prototype is the so-called Wigner-Ville transform, introduced origi-
           nally by E.P. Wigner1 in quantum mechanics (in 1932!) and extended by
           J. Ville2 to signal analysis:
                                       +∞
                                                             x        x
                        Ws (b, ξ) =         dx e−iξx s(b −     ) s(b + ),        ξ = 1/a.     (1.2)
                                      −∞                     2        2
           Notice that, contrary to the linear version (1.1), this transform is intrinsic,
           it contains no external probe ψ, which inavoidably influences the result.
               Within the class of linear TF transforms, two stand out as particularly
           simple and efficient, the windowed or short time Fourier transform (WFT)
           and the wavelet transform (WT). For both of them, the analyzing function
           ψb,a is obtained by a group action on a basic (or mother) function ψ, only
           the group differs. The essential difference between the two is in the way
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                                                                                                     3

                 1/a ≈ frequency


                        ψb,a (x)
                                                    a<1
            high




                                                    a=1
          medium




                                                    a>1
            low




                                                                                            x


           Figure 2. The function ψb,a (x) for different values of the scale parameter a : in the case
           of the Windowed Fourier Transform (left); in the case of the wavelet transform (right).


           the frequency parameter a is introduced. For the WT, one takes:
                                                    1         x−b
                                        ψb,a (x) = √ ψ                 .                        (1.3)
                                                     a         a
           The action of a on the function ψ is a dilation (a > 1) or a contraction
           (a < 1): The shape of the function is unchanged, it is simply spread out
           or squeezed. As for b, it is simply a translation. By contrast, the WFT
           takes for ψb,a the function ψb,a (x) = eix/a ψ(x − b). This means that the
           a-dependence is a modulation (1/a ∼ frequency); the window has constant
           width, but the lower a, the larger the number of oscillations in the window
           ψ. All this is illustrated on Figure 2.
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           4


               Actually one should distinguish between two radically different versions
           of the wavelet transform, the continuous WT (CWT) and the discrete WT
           (DWT), although they are based on the same basic formula, namely:
                                        ∞
                                                x−b
                    S(b, a) = a−1/2         ψ           s(x) dx,    a > 0, b ∈ R.       (1.4)
                                       −∞        a
           In this relation, s is a finite energy signal and S(b, a) is its WT with respect
           to the analyzing wavelet ψ. Now, in the CWT, all values of a and b are
                                                                 o
           considered. The CWT therefore plays the same rˆle as the Fourier trans-
           form and is mostly used for analysis and feature detection in signals. Of
           course, the CWT must be discretized for its numerical implementation, but
           the user may choose the sampling grid according to his needs. The price to
           pay, however, is that no orthonormal bases are obtained in this way, only
           frames: the CWT is a redundant representation. This may be seen as a
           defect, but redundancy actually has the advantage of ensuring the stability
           of the representation.
               The DWT, on the other hand, is based on a preselected grid (often
           dyadic) and is explicitly designed for generating (bi)orthonormal bases,
           starting from multiresolution analysis. It may be viewed as the analogue
           of the Discrete Fourier Transform (see, for instance, Ref. 3) and is more
           appropriate for data compression and signal reconstruction. As a matter of
           fact, the (discretized) CWT is incompatible with the DWT, they are based
           on totally different philosophies and also have different purposes.
               In these lectures, we will review the CWT, both from the theoretical
           side and in its practical applications, in one and two dimensions, plus some
           extensions. We will also give some indications on the Discrete WT.
               Finally a word about references. As a first contact, introductory articles
           such as Refs. 3 or 4 may be a good suggestion, followed by the popular
           book of B. Burke Hubbard5 or the elementary book of Y. Meyer.6 As for
           textbooks, we note in particular the celebrated volume of I. Daubechies7
           and the treatise of S. Mallat.8 Our main source of information, however,
           especially concerning the applications of the CWT in physics, will be van
           den Berg’s survey9 and our own upcoming monograph.10
               In addition, we ought to mention the popular newsletter The Wavelet
           Digest, originally founded 12 years ago by Wim Sweldens, and nowadays
           edited by Michael Unser at the Swiss Federal Institute of Technology in
           Lausanne, Switzerland (EPFL). The circulation has by now passed the
           20,000 mark and is still growing! For further information, we refer to the
           website <http://www.wavelet.org/>.
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                                                                                               5


           2. The one-dimensional CWT
           2.1. Basic formulas
           As announced above, the basic formulas read, in time domain and in fre-
           quency domain, respectively,
                                                  ∞
                              S(b, a) = a−1/2         dx ψ (a−1 (x − b)) s(x)           (2.1)
                                                 −∞
                                                 ∞
                                      = a1/2          dω ψ(aω) s(ω) eiωb ,              (2.2)
                                                −∞

           where a > 0 is a scale parameter, b ∈ R is a translation parameter and
           the hat denotes a Fourier transform. Thus the pair (b, a) runs over the
           time-scale half-plane R2 . Sometimes one uses also the full range a = 0, but
                                    +
           this is physically less natural.
               In these relations, s is a square integrable function and the analyzing
           wavelet ψ assumed to be well localized both in the space (or time) domain
           and in the frequency domain. Here the minimal requirement is that ψ ∈
           L1 ∩L2 , but in practice stronger conditions are usually imposed (like ψ ∈ S,
           Schwartz’s space of fast decreasing functions). In consequence, the CWT
           provides good bandpass filtering both in x and in ω.
               Moreover, ψ must satisfy the following admissibility condition, which
           guarantees the invertibility of the WT (see (2.13) below):
                                                 ∞
                                                          |ψ(ω)|2
                                     cψ ≡ 2π         dω           < ∞.                  (2.3)
                                                −∞          |ω|
           In most cases, this condition may be reduced to the requirement that ψ has
           zero mean (hence it must be oscillating):
                                                          ∞
                                   ψ(0) = 0 ⇐⇒                dx ψ(x) = 0.              (2.4)
                                                       −∞

           In addition, ψ is often required to have a certain number of vanishing
           moments:
                                    ∞
                                        dx xn ψ(x) = 0, n = 0, 1, . . . N.              (2.5)
                                   −∞

           This property improves the efficiency of ψ at detecting singularities in the
           signal, since it is blind to polynomials up to order N .
               Finally, it is often useful, but not essential, that ψ be progressive, which
           means that ψ is real and ψ(ω) = 0 for ω < 0 (in the language of signal
           processing, ψ is an analytic signal).
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           6




                              −10          0       10      −10                    0         10


           Figure 3. (left) The Mexican hat or Marr wavelet; (right) The real part of the Morlet
           wavelet, for ωo = 5.6.


              In order to fix ideas, we exhibit here two commonly used wavelets (see
           Figure 3).
                 (1) The Mexican hat or Marr wavelet
                     This wavelet is simply the second derivative of a Gaussian:
                                                             2                                   2
                                    ψH (x) = (1 − x2 ) e−x       /2
                                                                      ,     ψH (ω) = ω 2 e−ω         /2
                                                                                                          .   (2.6)
                     It is real and admissible, it has 2 vanishing moments n = 0, 1, but
                     it is not progressive.
                 (2) The Morlet wavelet
                     This wavelet is essentially a plane wave within a Gaussian window:
                                                                      2      2
                                          ψM (x) = eiko x e−x             /2σo
                                                                                 + c(x)                       (2.7)
                                                          −[(ω−ωo )σo ]2 /2
                                          ψM (ω) = σo e                               + c(ω).                 (2.8)
                        It is complex, but not progressive, strictly speaking (numerically it
                        is). Here the correction term c must be added in order to satisfy
                        the admissibility condition (2.4), but in practice one will arrange
                        that this term be numerically negligible ( 10−4 ) and thus can be
                        omitted (it suffices to choose the basic frequency |ωo | large enough,
                        namely, one has to take |σo ωo | > 5.5).
           These two wavelets have very different properties and, naturally, they will
           be used in quite different situations. Typically, the Mexican hat is sensitive
           to singularities in the signal, and it yields a genuine time-scale analysis.
           On the other hand, since the Morlet wavelet is complex, it will catch the
           phase of the signal, hence will be sensitive to frequencies, and will lead to
           a time-frequency analysis, somewhat closer to a Gabor analysis. In both
           cases, additional flexibility is obtained by adding a width parameter to the
           Gaussian.
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                                                                                                   7

                                       ω ∼ 1/a
                                           6


                        a<1:      ωo /a –                   Ω/a




                                                       aL


                        a=1:         ωo –                     Ω

                                                        L

                        a>1:      ωo /a –                         Ω/a
                                                       aL
                                                                         -
                                                        b                 x
                               Figure 4.                                   d
                                            Support properties of ψb,a and ψb,a .



           2.2. Localization properties and interpretation

           We must now make more precise the localization conditions on the wavelet
           ψ. Assume that the numerical support of ψ(x) is an interval of length L
           around 0, and that of ψ(ω) is an interval of length Ω, centered around
           the mean frequency ωo (by numerical support, we mean the smallest set
           outside of which the function is numerically negligible). Then the numerical
           support of ψb,a (x) is an interval of length aL around b, while that of ψ(ω)
           is an interval of length Ω/a, centered around ωo /a.
               Therefore (see Figure 4):

                  • if a    1, ψb,a is a wide window (long duration) and ψb,a is peaked
                    around the small frequency ωo /a: for large scales, the CWT is
                    sensitive to low frequencies, and thus yields a rough analysis.
                  • if a     1, ψb,a is a narrow window (short duration), and ψb,a is
                    wide and centered around the high frequency ωo /a: for very small
                    scales, the CWT is sensitive to high frequencies (small details).

               These support properties are illustrated in Figure 5 for the case of the
           Morlet wavelet ψM .
               Combining now these localization properties with the fact that the CWT
           is a convolution with a zero mean filter, we conclude that:
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           8




                 −10         0        10     −10         0       10   −10          0        10




                 0         10        20       0         10      20      0        10         20


           Figure 5. Support properties of the Morlet wavelet ψM : for a = 0.5, 1, 2 (left to right),
                                                              d
           ψb,a has width 3, 6, 12, respectively (top), while ψb,a has width 3, 1.5, 0.75, and peaks
           at 12, 6, 3 (bottom).


                     • the CWT provides a local filtering in time (b) and scale (a):
                                          S(b, a) ≈ 0    ⇐⇒     ψb,a (x) ≈ s(x).
                     • the CWT may be interpreted as a mathematical microscope, with
                       optics ψ, position b, and magnification 1/a.
                     • the CWT works at constant relative bandwidth : ∆ω/ω = const.
           As a consequence, one may consider the CWT as a singularity detector and
           analyzer.

           2.3. Mathematical properties
           Given an admissible wavelet ψ, the CWT Wψ : s(x) → S(b, a) is a linear
           map, with the following properties:
              (1) Covariance under translation and dilation:
                                     Wψ : s(x − xo ) → S(b − xo , a)                             (2.9)

                                           1  x      b a
                                     Wψ : √ s    →S   ,   .                                   (2.10)
                                           ao ao    ao ao
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                                                                                                       9


                 (2) Energy conservation :
                                 ∞
                                                                   da db
                                     dx |s(x)|2 = c−1
                                                   ψ                     |S(b, a)|2 .         (2.11)
                               −∞                             R2
                                                               +
                                                                    a2

           Thus, |S(b, a)|2 may be interpreted as the energy density in the time-scale
           half-plane.
               The relation (2.11) means that Wψ is an isometry from the space of
           signals L2 (R) onto a closed subspace Hψ of L2 (R2 , da db/a2 ), namely, the
                                                             +
           space of wavelet transforms. An equivalent statement is that the wavelet
           ψ generates a resolution of the identity:
                                                    da db
                                     c−1
                                      ψ                   |ψba ψba | = I.                     (2.12)
                                               R2
                                                +
                                                     a2

               (3) Reconstruction formula : As a consequence, Wψ is invertible on
           its range Hψ by the adjoint map, i.e., we obtain an exact reconstruction
           formula:
                                                          da db
                                 s(x) = c−1
                                         ψ                      ψb,a (x) S(b, a).             (2.13)
                                                    R2
                                                     +
                                                           a2

           In other words, we have a representation of the signal s(x) as a linear
           superposition of wavelets ψb,a with coefficients S(b, a).

              (4) The projection Pψ : L2 (R2 , da db/a2 ) → Hψ is an integral operator,
                                           +
           with kernel

                                      K(b , a ; b, a) = c−1 ψb a |ψba .
                                                         ψ                                    (2.14)

           The kernel K is the autocorrelation function of ψ and it is a reproducing
           kernel. Indeed, the function f ∈ L2 (R2 , da db/a2 ) is the WT of a certain
                                                    +
           signal iff it satisfies the reproduction property
                                                         da db
                             f (b , a) = c−1
                                          ψ                    ψb ,a |ψb,a f (b, a).          (2.15)
                                                    R2
                                                     +
                                                          a2

           This means that the CWT is a highly redundant representation!
               The last statement implies that the full information must be contained
           is a small subset of half-plane. This property may be exploited in two ways:
           either one takes a discrete subset, which leads to the theory of frames, or
           one considers only the lines of local maxima, called ridges. We will explore
           these two approaches in succession.
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           10


           2.4. Discretization of the CWT : Frames
           We will say that a discrete lattice Γ = {aj , bk , j, k ∈ Z} yields a good
           discretization if
                                  s(x) =             ψjk |s ψjk (x), ∀ s ∈ L2 (R),            (2.16)
                                           j,k ∈ Z

           where ψjk ≡ ψbk ,aj and ψjk is explicitly constructible from ψjk (the relation
           is still exact).
               The relevant concept here is that of a frame, that is, a family {ψjk } for
           which there exists two constants A > 0, B < ∞ such that
                              2
                        A s                   | ψjk |s |2      B s 2 , ∀ s ∈ L2 (R).          (2.17)
                                    j,k ∈ Z

           The upper bound means that the map s → { ψjk |s } is continuous from
           L2 to l2 , whereas the lower bound guarantees the numerical stability of the
           inverse map.7 The constants A, B are called frame bounds. If A = B = 1,
           the frame is said to be tight. Of course, if A = B = 1 and ψjk = 1, ∀ j, k,
           we simply get an orthonormal basis. A detailed analysis then shows that
           the expansion (2.16) converges essentially as a power series in the quantity
           B/A − 1.
               Now the question is, given the wavelet ψ, can one find a lattice Γ such
           that {ψjk } is a good frame ? By good frame, we mean a frame such that
           (2.16) converges fast, and thus can be truncated after very few terms. This
           requires B/A − 1        1. In order to achieve this, one usually chooses a
           lattice adapted to the geometry of the time-scale half-plane, for instance,
           the dyadic lattice
                                   ψjk (x) = 2j/2 ψ(2j x − k),         j, k ∈ Z.
           A direct estimation shows that both the Mexican hat and the Morlet
           wavelets give good, but nontight frames.7

           2.5. Reducing the computational cost : Ridges
           Real life signals are often entangled and noisy, which makes the WT difficult
           to interpret. But the energy density |S(b, a)|2 is usually well concentrated,
           around lines of local maxima, called ridges. A fundamental result (based on
           stationary phase arguments, the integral in (1.4) being rapidly oscillating)
           is that the restriction of the WT S(b, a) to the set of its ridges contains
           essentially the whole information. Thus it sufficient to consider this re-
           striction, called the skeleton of the WT, which reduces substantially the
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                                                                                                                                                                               11


                         1                                                                                    1



                        1.5                                                                                  1.5
            Log Scale




                                                                                                 Log Scale
                         2                                                                                    2



                        2.5                                                                                  2.5



                         3                                                                                    3




                               100   200   300   400    500       600   700   800   900   1000                     100    200   300   400     500      600   700   800   900   1000
                                                       Position                                                                             Position




           Figure 6. CWT of an academic discontinous signal (left) and the corresponding skele-
           ton, showing the vertical ridges pointing towards each singularity (right); the signal is
           shown at the bottom of each panel.


           computational cost, and also simplifies considerably the interpretation of
           the WT.
               Actually one may distinguish several types of ridges. Two particular
           cases are the vertical ridges, characteristic of singularities in the signal, and
           horizontal ridges, that signal the appearance of given prominent frequencies
           (see Refs. 11 and 12 for a full discussion). Let us discuss the first type.
               Assume s(x) has a singularity of order α          −1 at xo (here we take
           α ∈ Z, but the analysis can be extended to any α ∈ R):

                                                       s(x) ∼ θ(x − xo ) (x − xo )α for x ∼ xo ,

           so that dα+1 s(x) ∼ δ(x − xo ) (most singularities may be modeled in this
                    x
           way). Let the wavelet be the nth derivative of a smooth function (e.g. a
           Gaussian), ψ = dxn φ, with n α + 1. Then the CWT of s(x) reads
                                                                                                                         xo − b
                                                              S(b, a) ∼ aα dxn−α−1 φ                                            .
                                                                                                                           a
           Assume now that the modulus of dxn−α−1 φ has N maxima {φm , m =
           1, . . . , N } at positions {xm , m = 1, . . . , N }. Then, for each a, the mod-
           ulus |S(b, a)| has N maxima localized at positions {bm = axm + xo , m =
           1, . . . , N }, which converge toward xo as a → 0. Furthermore, the maxima of
           |S(b, a)| lie on N lines, called vertical ridges {b = axm + xo , m = 1, . . . , N },
           which converge toward the singularity xo of the signal, and the modulus of
           |S(b, a)| along the mth ridge behaves as aα :

                                                       |S(b = axm + xo , a)| = Γ(α + 1) aα φm .                                                                          (2.18)
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           12
                                                                        250
                                                                                                                                      (a)

                                                                        200


                                                        Force [Volts]
                                                                        150



                                                                        100



                                                                        50




                                                                         500   600   700   800    900 1000 1100 1200 1300 1400 1500
                                                                                                   Time [10−4 s]



                                                                                            (b)                                                                           (c)
                    1.86                                                                                         1.86



                     3.59                                                                                         3.59



                     6.90                                                                                         6.90
                                                                                                         Scale
            Scale




                    13.29                                                                                        13.29



                    25.58                                                                                        25.58



                    49.25                                                                                        49.25


                              600    700   800   900   1000 1100 1200 1300 1400 1500                                     600   700   800    900 1000 1100 1200 1300 1400 1500
                                                                    −4                                                                                 −4
                                                 Time [10                s]                                                                 Time [10        s]


           Figure 7. Analysis of the rebound signal, with a Mexican hat wavelet: (a) The signal
           and the points detected by the respective ridges; (b) The modulus of the CWT; and (c)
           The corresponding skeleton (from Ref. 12).



           Hence the strength α of the singularity may be read off a log-log plot:
                                                       ln |S(axm + xo , a)| ∼ α ln a + ln φm .                                                                          (2.19)
                                                          o
           This is, of course, the analysis of the local H¨lder regularity of the signal,
           and it is based on the covariance property (2.10) of the CWT under dilation.
               We illustrate this situation with two examples. The first one (Figure
           6) is an academic signal, with singularities of various strengths, analyzed
           with a Mexican hat wavelet. The skeleton of the CWT clearly shows the
           vertical ridges pointing towards each singularity.
               The second example comes from the analysis of the behavior of a ma-
           terial under impact, as made in Ref.12. The physical context is that of a
           so-called ‘instrumented falling weight impact’ testing. During such a test,
           a striker falls from a certain height on a clamped disk, so that either the
           striker rebounds or the disk breaks. In both cases, one records the time
           and the force on the striker. This type of event occurs on a very short time
           scale and is thus essentially transient, so that a time-frequency method
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                                                                                          13


           is required for the analysis. We focus here on the case of a rebound and
           a wavelet analysis of the force signal. The Mexican hat detects precisely
           three discontinuity points, namely, first contact, maximal penetration and
           last contact. The results are shown in Figure 7. The signal is given in panel
           (a), whereas the CWT and its skeleton are presented in (b) and (c). The
           latter, in particular, shows the three vertical ridges that point towards the
           three instants mentioned. A further analysis exploits the behavior of the
           modulus of the CWT along each ridge. This yields precious insight into
           the physics of the phenomenon, particularly in the case of the rupture of
           the sample, not shown here.


           2.6. The discrete WT (DWT)

           One of the successes of the WT was the discovery that it is possible to
           construct functions ψ for which {ψjk , j, k ∈ Z} is indeed an orthonormal
           basis of L2 (R), while keeping the good properties of wavelets, including
           space and frequency localization. In addition, this approach yields fast
           algorithms, and this is the key to the usefulness of wavelets in many appli-
           cations.
                The construction is based on two facts. First, almost all examples of
           orthonormal bases of wavelets can be derived from a multiresolution analy-
           sis, and then the whole construction may be transcripted into the language
           of Quadrature Mirror Filters (QMF), familiar in the signal processing lit-
           erature.
                A multiresolution analysis of L2 (R) is an increasing sequence of closed
           subspaces

                                  . . . ⊂ V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ . . .

           with     j ∈ Z Vj   = {0} and   j∈Z   Vj dense in L2 (R), and such that

                 (1) f (x) ∈ Vj ⇔ f (2x) ∈ Vj+1 ;
                 (2) There exists a function φ ∈ V0 , called a scaling function, such that
                     the family {φ(x − k), k ∈ Z} is an orthonormal basis of V0 .

           Combining conditions (1) and (2), one sees that {φjk (x) ≡ 2j/2 φ(2j x − k),
           k ∈ Z} is an orthonormal basis of Vj . The space Vj can be interpreted as
           an approximation space at resolution 2j . Defining now

                                             Vj ⊕ Wj = Vj+1 ,                        (2.20)
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           14


                                                           Signal
                           2
                           1
                           0
                           1
                           0
                          −1
                           1
                           0
                          −1
                           1
                           0
                          −1
                           2
                           0
                          −2
                           2
                           0
                          −2
                           2
                           0
                          −2
                          10
                           0
                         −10
                                100   200    300     400   500      600     700    800   900   1000




           Figure 8. Five level decomposition of the academic signal on an orthonormal basis
           of Daubechies d6 wavelets. The low resolution approximation (∈ V−5 ) is shown on
           the bottom panel and the five levels of details with increasing resolution (∈ Wj , j =
           −5, . . . , −1) in the next five panels.


           we see that Wj contains the additional details needed to improve the reso-
           lution from 2j to 2j+1 . Thus one gets the decomposition
                                                   L2 (R) =          Wj .                                (2.21)
                                                              j∈ Z

           Equivalently, one may choose a lowest resolution level jo and obtain
                                                         
                                                                     ∞
                                        L2 (R) = Vjo ⊕                    Wj  ,                        (2.22)
                                                                    j=jo

               The crucial theorem then asserts the existence of a function ψ, some-
           times called the mother wavelet, explicitly computable from φ, such that
           {ψjk (x) ≡ 2j/2 ψ(2j x − k), j, k ∈ Z} constitutes an orthonormal basis of
           L2 (R): these are the orthonormal wavelets. Well-known examples include
           the Haar wavelets, the B-splines, and the various Daubechies wavelets.
               In practice, one starts from a sampled signal, taken in some VJ , and
           then the decomposition (2.22) is replaced by the finite representation
                                                         
                                                                 J−1
                                            VJ = Vjo ⊕                   Wj  .                         (2.23)
                                                                 j=jo
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                                                                                       15


              Figure 8 shows an example of a decomposition of order 5, namely
           the academic signal of Figure 6 decomposed over an orthonormal basis
           of Daubechies d6 wavelets.7 Thus we take J = 0 and jo = −5 in (2.23):
                            V0 = V−5 ⊕ W−5 ⊕ W−4 ⊕ W−3 ⊕ W−2 ⊕ W−1 .              (2.24)
               The construction of orthogonal wavelets then proceeds with (fast) al-
           gorithms for generating in a hierarchical way the various coefficients in a
           decomposition like (2.24), exploiting standard procedures from signal pro-
           cessing. The resulting tool is quite efficient, but in fact too rigid. Indeed,
           once the scaling function is given, everything is fixed, in particular, one has
           no freedom left in the design of the wavelet ψ. In addition, the DWT is no
           longer covariant under (discrete) translations, the so-called shift invariance
           is lost, which is a seroius flaw for pattern recognition. Therefore, vari-
           ous generalizations have been proposed in the literature. To name a few:
           biorthogonal wavelet bases, wavelet packets and the Best Basis Algorithm,
           the lifting scheme and second generation wavelets, the redundant WT (here
           one uses a rectangular lattice, instead of the dyadic one, so as to restore a
           (discrete) translation covariance). We shall refrain from describing these,
           for lack of space. Further information may be found in Ref.6.

           2.7. Applications of the 1-D CWT
           The CWT has found a wide variety of applications in various branches of
           physics and/or signal processing. We will list here a representative selection
           of one-dimensional applications, in order to convey to the reader a feeling
           about the scope and richness of the field. Most of the early applications,
           and the original references, may be found in the proceedings volumes Refs.
           13–15. In all cases, the CWT is primarily used for analyzing transient
           phenomena, detecting abrupt changes in a signal or comparing it with a
           given pattern. Here is the list:

                  • Sound and acoustics: musical synthesis, speech analysis (formant
                    detection), disentangling of an underwater acoustic wavetrain.
                  • Geophysics: analysis of microseisms in oil prospection, gravime-
                    try (fluctuations of the local gravitational field), seismology, geo-
                    magnetism (fluctuations of the Earth magnetic field), astronomy
                    (fluctuations of the length of the day, variations of solar activity,
                    measured by the sunspots, etc).
                  • Fractals, quasicrystals, turbulence (1-D and 2-D): diffusion limited
                    aggregates, arborescent growth phenomena, structure of quasiperi-
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           16


                        odic patterns, identification of coherent structures in developed
                        turbulence.
                   •    Atomic physics: analysis of harmonic generation in laser-atom in-
                        teraction.
                   •    Spectroscopy: in NMR spectroscopy, subtraction of spectral lines,
                        noise filtering.
                   •    Medical and biological applications: analyzing or monitoring of
                        EEG, VEP, ECG; long-range correlations in DNA sequences.
                   •                                                             o
                        Analysis of local singularities: determination of local H¨lder expo-
                        nents of functions.
                   •    Shape characterization: in robotic vision, contour of an object
                        treated as a complex curve in the plane.
                   •    Industrial applications: monitoring of nuclear, electrical or me-
                        chanical installations; quality assessment of telephone lines; analy-
                        sis of behavior of materials under impact.

           3. The two-dimensional CWT
           3.1. Basic formulas
           Now we turn to the two-dimensional CWT, which has become a major tool
           in image processing. In this context, an image is a two-dimensional signal
           of finite energy, represented by a complex-valued function s ∈ L2 (R2 , d2 x):
                                             2
                                         s       =        d2 x |s(x)|2 < ∞.             (3.1)
                                                     R2

           (sometimes it is useful to take s integrable as well). In practice, a black
           and white image will be represented by a bounded non-negative function:
                                     0   s(x)        M, ∀x ∈ R2 (M > 0),                (3.2)
           the discrete values of s(x) corresponding to the level of gray of each pixel.
               Given an image s, all the geometric operations we want to apply to it
           are obtained by combining three elementary transformations of the plane,
           namely, rigid translations in the plane of the image, dilations or scaling
           (global zooming in and out) and rotations. Explicitly, the transformations
           act on x ∈ R2 in the familiar way:
                        (i) translation by b ∈ R2 : x → x = x + b,
                        (ii) dilation by a factor a > 0 : x → x = ax,
                        (iii) rotation by an angle θ : x → x = rθ (x),
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                                                                                                  17


           where
                                            cos θ − sin θ
                                   rθ ≡                             ,0       θ < 2π,
                                            sin θ cos θ
           is the familiar 2 × 2 rotation matrix.
               The combined action of these three types of transformations is real-
           ized by the following unitary map in the space L2 (R2 , d2 x) of finite energy
           signals:

                           U (b, a, θ)s (x) ≡ sb,a,θ (x) = a−1 s(a−1 r−θ (x − b)).             (3.3)

           In terms of this action, the basic formulas for the 2-D CWT read

                            S(b, a, θ) = a−1           d2 x ψ(a−1 r−θ (x − b)) s(x)            (3.4)
                                                  R2

                                       =a         d2 k eib·k ψ(ar−θ (k)) s(k)                  (3.5)
                                             R2

           As in 1-D, we have to impose an admissibility condition on the wavelet ψ,
           namely,
                                                                   |ψ(k)|2
                                   cψ ≡ (2π)2               d2 k              < ∞,             (3.6)
                                                       R2           |k|2
           which again may be replaced in practice by the following necessary condi-
           tion:

                                   ψ(0) = 0 ⇐⇒                     d2 x ψ(x) = 0.              (3.7)
                                                              R2

           The important observation to make here is that all the formulas are almost
           identical in 1-D and in 2-D ! As a consequence, the interpretation of the
           CWT as a singularity analyzer still holds, and the mathematical properties
           of the 2-D transform strictly parallel those of its 1-D counterpart.

           3.2. Group-theoretical justification
           However, before proceeding, we should pause and ask the reason of this
           situation. As can be expected, the answer lies in group theory.
           (1) In one dimension
               Observe that dilations and translations are affine transformations of the
           line:
                                  x = (b, a)y ≡ ay + b,              a > 0, b ∈ R,
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           18


           and they obey the following composition rule: (b, a)(b , a ) = (b + ab , aa ).
           Thus, the set of all these affine transformations constitutes a group, the
           (connected) affine group of the line, {(b, a)} ≡ G+ .
                                                           aff
                 The natural action of (b, a) on a signal, ψ → U (b, a)ψ, reads
                                                                     x−b
                                    (U (b, a)ψ)(x) = a−1/2 ψ                                    (3.8)
                                                                      a
           and U is a unitary irreducible representation of G+ in L2 (R). In addition,
                                                             aff
           U is square integrable:
                                                           db da
                         ψ admissible ⇐⇒                         | U (b, a)ψ|ψ |2 < ∞           (3.9)
                                                  G   +     a2
                                                      aff
           This explains the mathematical meaning of the admissibility condition of
           wavelets.
           (2) In two dimensions
               In two dimensions, dilations, translations, and rotations together con-
           stitute the similitude group of the plane: SIM(2) = R2 (R+ × SO(2)),
                                                                          ∗
           with action
                                        x = (b, a, θ)y ≡ arθ y + b.
           This transformation is realized by the following unitary map on finite energy
           signals:

                                 U (b, a, θ)s (x) = a−1 s(a−1 r−θ (x − b))                     (3.10)

           and U is a unitary irreducible representation of SIM(2) in L2 (R2 ). In
           addition, U is square integrable:
                                                           da                       2
                 ψ admissible ⇐⇒                  d2 b        dθ   U (b, a, θ)ψ|ψ       <∞     (3.11)
                                         SIM(2)            a3

           3.3. Interpretation of the 2-D CWT
           The 2-D formulas being the exact analogues of the ones in 1-D, in particular
           the admissibility condition (3.6) or (3.7), clearly the interpretation of the
           CWT will be exactly the same as in 1-D. Thus, combining the localization
           properties of ψ with the fact that the CWT is a convolution with a zero
           mean function, we conclude again that the 2-D CWT
                   • yields a local filtering, this time in all three variables b, a, θ, in
                     particular, a directional filtering;
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                                                                                                      19


                   • may be seen as a mathematical directional microscope with optics
                     ψ, global magnification 1/a, and orientation tuning parameter θ;
                   • works with constant relative bandwidth: ∆k/k = const, k = |k|,
                     thus it is particularly efficient for large spatial frequencies.
           Therefore we may say that the 2-D CWT is a detector and analyzer of
           singularities (edges, contours, corners, . . . ).

           3.4. Main properties of the 2-D CWT
           For the same reason, the mathematical properties of the 2-D CWT are
           essentially the same as in the 1-D case, so we list them without further
           comment:
                 (1) Energy conservation
                                               da
                            c−1
                             ψ          d2 b      dθ|S(b, a, θ)|2 =       d2 x |s(x)|2 .         (3.12)
                                               a3

                 (2) Reconstruction formula
                                                da
                             s(x) = c−1
                                     ψ             d2 b
                                                    dθ ψb,a,θ (x) S(b, a, θ),      (3.13)
                                                 a3
           i.e., we have a decomposition of the signal in terms of the analyzing wavelets
           ψb,a,θ , with coefficients S(b, a, θ).
                 (3) Reproduction property (reproducing kernel)
                                                       da
                  S(b , a , θ ) = c−1
                                   ψ            d2 b      dθ ψb ,a ,θ |ψb,a,θ S(b, a, θ).        (3.14)
                                                       a3

              (4) the CWT is covariant under translations, dilations and rotations,
           that is, under SIM(2).

           3.5. Choice of the analyzing wavelet
           Even more than in 1-D, it is important to choose a wavelet that is well
           adapted to the problem at hand. One can distinguish two classes, isotropic
           wavelets and direction sensitive wavelets.
           (i) Isotropic wavelets
               If one decides to perform a pointwise analysis, or if directions are ir-
           relevant, it is sufficient to use a rotation invariant wavelet. Two standard
           examples are:
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           20



                                                                                       20



                       2
                                                                                       40

                  1.5


                       1                                                               60


                  0.5

                                                                                       80
                       0


                 −0.5                                                            5
                   −5                                                                 100


                                                               0
                                 0
                                                                                      120

                                           5    −5                                               20    40   60   80        100   120




                                                                                       20



                 0.8
                                                                                       40


                 0.6

                                                                                       60
                 0.4



                 0.2                                                                   80



                  0                                                              5
                 −5                                                                   100


                                                           0
                                0
                                                                                      120

                                           5    −5                                              20     40   60   80        100   120




           Figure 9. An isotropic wavelet: The 2-D Mexican hat: (top) In position space; (bottom)
           In spatial frequency space.


                  (1) The 2-D Mexican hat wavelet
                           This wavelet (originally introduced by Marr in his pioneering work
                           on vision16 ) is simply the Laplacian of a Gaussian (Figure 9):
                                                                    2                                             2
                               ψH (x) = (2 − |x|2 ) e−|x|               /2
                                                                             ,       ψH (k) = |k|2 e−|k|              /2
                                                                                                                                 (3.15)
                  (2) The Difference-of-Gaussians or DOG wavelet
                                                               2
                                                                   /2α2                     2
                               ψD (x) =    1
                                          2α2        exp−|x|                 − exp−|x|          /2
                                                                                                      (0 < α < 1)                (3.16)
                           This is a very good substitute to the Mexican hat; for α−1 = 1.6,
                           there are almost undistinguishable.
           (ii) Directional wavelets
               On the other hand, if the goal is to detect directional features in an im-
           age, or to perform directional filtering, clearly one should resort to a direc-
           tion sensitive wavelet. The most efficient solution is a directional wavelet,
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                                                                                                                                               21

                                                                                             20


              0.08
                                                                                             15

              0.06

                                                                                             10
              0.04


              0.02
                                                                                              5
                 0


             −0.02                                                                            0

             −0.04
                                                                                            −5
             −0.06


             −0.08
                                                                                            −10
                 0

                     10
                                                                                            −15
                          20

                               30
                                                                         30   35   40
                                    40                  15    20    25                      −20
                                         0    5   10                                         −20        −15   −10    −5   0   5     10   15   20




           Figure 10. The 2-D Morlet wavelet, for                              = 2, ko = (0, 6): (left) In position space (real
           part); (right) In spatial frequency space.


           that is, a wavelet ψ(x) such that the numerical support of its Fourier trans-
           form ψ(k) is contained in a convex cone with apex at the origin. Two useful
           examples are as follows:

                     (1) The 2-D Morlet wavelet
                                                                                              2
                                                                   ψM (x) = eiko ·x e|x|           /2
                                                                                                          + corr.                        (3.17)
                                                                                        2
                                                                   ψM (k) = e|k−ko |        /2
                                                                                                   + corr.                               (3.18)

                                     As in 1-D, the correction term is required in order to satisfy the
                                     admissibility condition, and here too, it is negligible for |ko | 5.6.
                                     The 2-D Morlet wavelet is shown in Figure 10 for = 2, ko = (0, 6).


                     (2) Conical wavelets, with support in the convex cone

                                                  C(−α, α) ≡ {k ∈ R2 | − α                        arg k             α, α < π/2}


                                                                                                  1   2
                                                             (k · e−α )m (k · eα )m e− 2 kx , k ∈ C(−α, α)
                                                                    ˜          ˜
                                             ψC (k) =                                                                                    (3.19)
                                                             0, otherwise

                                                               ˜
           where eφ is the unit vector in the direction φ and α = −α + π/2, so that
           e−α · eα = eα · e−α = 0. The frequency space version of this so-called
                  ˜           ˜
           Gaussian conical wavelet is shown in Figure 11, in the case m = 4, α = 10◦ .
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           22




                 Figure 11.   The Gaussian conical wavelet, in frequency space, for m = 4, α = 10◦ .



           3.6. Ridges in the 2-D CWT

           As in 1-D, the 2-D CWT is highly redundant, so that the full information
           can be retrieved from small subsets of the parameter space. Here too, one
           can either choose a discrete subset and get a frame, or focus on the regions
           where most of the energy is concentrated. Thus one considers the energy
           density of the CWT (written here for the isotropic case):


                                              E[s](b, a) ≡ |S(b, a)|2 ,                        (3.20)


           and restricts the attention to the lines of local maxima of E[s](b, a), that
           is, the ridges of the CWT. It should be emphasized, however, that several
           definitions have been proposed in the literature. The important point is
           that, as in 1-D, the restriction of the CWT to the set of its ridges (the
           skeleton of the CWT) characterizes the signal completely.
                Here we will define a (vertical) ridge R as a 3-D curve (r(a), a) such
           that, for each scale a ∈ R+ , E[s](r(a), a) is locally maximum in space and r
           is a continuous function of scale. Figure 12 gives a concrete example. The
           signal (left panel) represents a set of singularities on a smooth background,
           simulating a set of bright points on the surface of the Sun and modelled
           by a random distribution of Gaussians of small (but random) width. The
           corresponding vertical ridges of the CWT of that signal are shown on the
           right panel, clearly each ridge points towards a singularity.
                Given such a vertical ridge, one may distinguish three characteristic
           features:
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                                                                                                   23




           Figure 12. An example of a 2-D ridge: (left) The signal: a field of singularities, simu-
           lating a set of bright points on the surface of the Sun; (right) The corresponding vertical
           ridges of the CWT of that signal.


                   . The amplitude of the ridge
                                              AR = lim E[s](r(a), a).
                                                          a→0

                   . The slope of E[s] on the ridge
                                                            d ln E[s](r(a), a)
                                            SR = lim                           .
                                                     a→0          d ln a
                   . The energy of the ridge
                                                         amax
                                                                da
                                          ER =                     E[s](r(a), a),
                                                     0          a3
           where [0, amax ] is the interval of definition of r(a). All three features may
           be used for extracting information on the underlying physical process. The
           first two, in particular, have been exploited for discriminating between sev-
           eral classes of small features (bright points, cosmic impacts) in images of
           the Sun.17

           3.7. Applications of the 2-D CWT
           As in 1-D, even more so, the 2-D CWT has been applied to a considerable
           number of problems. It is useful to distinguish between two types of appli-
           cations, namely those pertaining to image processing and those belonging
           to genuine physical problems. Detailed information and references may be
           found in Ref. 10. In addition, two novel applications, one for each type,
           are presented in separate paper in this volume.18
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           24




                              1                                                             1



                             0.5                                                           0.5



                              0                                                             0
                        Z




                                                                                      Z




                            −0.5                                                          −0.5



                             −1                                                            −1
                             −1                                                            −1
                                                                                 −1                                                             −1
                                   −0.5                                                          −0.5
                                                                          −0.5                                                           −0.5
                                          0                                                             0
                                                                      0                                                              0
                                                  0.5           0.5                                             0.5           0.5
                                                        1   1                                                         1   1
                                                                      X                                                              X
                                              Y                                                             Y




           Figure 13. Two spherical wavelets, in different positions : (top) The DOG wavelet;
           (bottom) The real part of the spherical Morlet wavelet at scale a = 0.03.


           (a) Applications in image processing

                   • Contour detection, character recognition: detection of edges, con-
                     tours, corners . . .
                   • Object detection and recognition in noisy images: automatic target
                     recognition (ATR), application to infrared radar imagery, using
                     both position and scale-angle features.
                   • Image retrieval: recognition of a particular image in a large data
                     bank, characterization of images by particular features.
                   • Medical imaging: Magnetic resonance imaging (MRI), contrast en-
                     hancement, segmentation.
                   • Detection of symmetries in 2-D patterns: detection of discrete infla-
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                                                                                      25


                    tion (rotation + dilation) symmetries, quasicrystals (mathematical
                    and genuine), quasiperiodic point sets.
                  • Image denoising: removal of noise in images using directional
                    wavelets.
                  • Watermarking of images: adding a robust, but invisible, signature
                    in images (e.g. with directional wavelets).

           (b) Physical applications

                  • Astronomy and astrophysics: : structure of the Universe, cosmic
                    microwave background (CMB) radiation, feature detection in im-
                    ages of the Sun, detection of gamma-ray sources in the Universe.
                  • Geophysics: fault detection in geology, detection of arrival time
                    of individual seismic waves in seismology, hurricane detection in
                    climatology.
                  • Fluid dynamics: detection of coherent structures in turbulent flu-
                    ids, measurement of a velocity field, disentangling of an underwater
                    acoustic wave train.
                  • Fractals and the thermodynamical formalism: analysis of 2-D frac-
                    tals by the WTMM method, determination of fractal dimension,
                    unraveling of universal laws, shape recognition and classification of
                    patterns.
                  • Texture analysis: classification of textures, “Shape from texture”
                    problem.

           3.8. Generalizations
           The transition from one to two dimensions gives us a hint as how to gener-
           alize further the CWT: given a class of (finite energy) signals, one specifies
           the type of transformations one wants to apply to these signals. Then, if the
           latter form a group and if the action of that group on the signals is a uni-
           tary, square integrable representation, the formalism applies exactly in the
           same way (square integrability may be relaxed somewhat) and an adapted
           wavelet transform may be constructed (the formalism in question stems
           from the general construction of coherent states in quantum physics19 ). In
           this section, we will list very briefly a number of such extensions.
           (1) Three-dimensional wavelets
              This is the most immediate extension, since all formulas of the CWT
           extend with obvious adaptations to three or more dimensions. The new
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           26


           aspect here is that axisymmetric wavelets play a prominent role, in which
           the parameter space is now X = R3 (R+ × S 2 ) ∼ R3 × R3 , thus we have
                                                   ∗                 ∗
           again a doubling of dimensions (and X may be interpreted as a phase space
           in the sense of Hamiltonian mechanics). The same situation holds true in
           any dimension d 3.
           (2) Wavelets on the 2-sphere
           The CWT on the 2-sphere S 2 may be obtained, using group-theoretical
           techniques, from the usual 2-D CWT on the tangent plane at the north
           Pole. The precise link is the inverse stereographic projection
                                                           θ
                               S2 (θ, ϕ) ⇐⇒ (r, ϕ) ≡ (2 tan , ϕ) ∈ R2 .
                                                           2
           In particular, all the 2-D wavelets may be lifted to the 2-sphere. We
           present in Figure 13 two examples of spherical wavelets obtained in this way,
           the spherical DOG wavelet and the spherical Morlet wavelet, respectively.
           Clearly the spherical CWT will find applications in geography (maps) and
           in astrophysics, whenever the of the Earth, resp. the universe, has to be
           taken into account. An example of the latter type is the analysis of the
           Cosmic Microwave Background (CMB) radiation (see Refs. 10, 17 and 20
           for further details and references).
           (3) Spatio-temporal wavelets and motion estimation
               Wavelets that depend on space and time can be designed starting from
           separate dilations and translations in both variables. Then, by a simple
           change of variables, the two dilations are replaced with two new opera-
           tions: global dilation (space-time zooming) and speed tuning; the former
           will catch the global size of objects, whereas the latter will detect their
           velocity. In more than one dimension, one simply adds the usual rotation
           parameter. In this way one obtains a tool for analyzing moving objects
           (video sequences). A systematic use of the partial energy densities then
           yields a tracking algorithm that outperforms many standard ones; in par-
           ticular, it does not lose track of the target when the latter changes its shape,
           like a maneuvering airplane.10

           4. The 2-D discrete wavelet transform
           The basic recipe of the 2-D discrete wavelet transform is simply to take
           the tensor product of two 1-D transforms: 2-D = 1-D ⊗ 1-D. Thus, by
           definition, one works in Cartesian coordinates.
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                                                                                            27




                                c3     v
                                      d3
                                                v
                                               d2
                                 h  d
                                d3 d3
                                                             v
                                                            d1
                                      h         d
                                     d2        d2




                                           h                 d
                                          d1                d1




           Figure 14. Schematic three level decomposition of an image into a low resolution ap-
           proximation, plus increasingly finer details, of the 3 types (h, v, d).


               The first step is to design a multiresolution analysis in 2-D along these
           lines, that is, from a 1-D multiresolution analysis {Vj , j ∈ Z} of L2 (R), one
                                                    (2)
           builds a 2-D multiresolution analysis { Vj = Vj ⊗ Vj , j ∈ Z} of L2 (R2 ).
           The net result is that one needs now
                   . one scaling function Φ(x, y) = φ(x) φ(y);
                   . three wavelets

                              Ψh (x, y) = φ(x) ψ(y)    : detects horizontal features
                                v
                              Ψ (x, y) = ψ(x) φ(y)     : detects vertical features
                                d
                              Ψ (x, y) = ψ(x) ψ(y)     : detects oblique features.

               Using this approach, one is led to the standard presentation of the
           decomposition of images schematized in Figure 14. A concrete example,
           using the familiar image lena, is shown in Figure 15.
               Obviously, despite its widespread use, this technique has a number of
           drawbacks. On one hand, it offers a poor sensitivity to directions, since
           one is bound to the Cartesian coordinates (x, y). This is natural in certain
           cases, like TV, but often it is a nuisance: even displaying isotropic features
           is nontrivial! Worse, the discrete scheme lacks “shift invariance” (properly,
           translation covariance), even a discrete one, when one uses the dyadic grid.
           This is often disastrous, in particular in the context of form recognition.
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           28




           Figure 15. Typical three level decomposition of an image into a low resolution approx-
           imation, plus increasingly finer details, of the 3 types (h, v, d).


           Thus, quite naturally, a number of generalizations have been proposed in
           the literature, exactly as in 1-D:
               . Redundant transforms, using a Cartesian lattice
               . Biorthogonal wavelet bases
               . Wavelet packets
               . The lifting scheme : second generation wavelets.
           Actually there is a worse aspect. Indeed some features in an image are not
           really two-dimensional. a curve is in fact a one-dimensional structure, albeit
           not straight. The standard 2-D DWT described above completely misses
           that aspect, thus it is a terrible waste of computing time. This is simply
           visible by estimating the number of squares it takes to cover a portion of
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                                                                                           29


           a curve at resolution 2−j , for increasing j. This phenomenon, dubbed the
           “curse of dimension”, may be avoided by defining various new transforms,
           better adapted to geometry : ridgelets, curvelets, complex wavelets, non-
           linear approximations. There has been a rich supply in proposals in the
           recent years. A look at the most recent proceedings volume is eloquent.21



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                      o                                                             u
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