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`IEEE TRANSACTIONS ON IMAGE PROCESSING. VOL.
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`I. NO. 2. APRIL 1992
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`Image Coding Using Wavelet Transform
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`Marc Antonini, Michel Barlaud, Member, IEEE, Pierre Mathieu, and Ingrid Daubechies, Member, IEEE
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`V t
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`Abstract—Image compression is now essential for applica-
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`tions such as transmission and storage in data bases. This paper
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`proposes a new scheme for image compression taking into ac-
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`count psychovisual features both in the space and frequency
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`domains; this new method involves two steps. First, we use a
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`wavelet transform in order to obtain a set of biorthogonal sub-
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`classes of images; the original image is decomposed at different
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`scales using a pyramidal algorithm architecture. The decom-
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`position is along the vertical and horizontal directions and
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`maintains constant the number of pixels required to describe
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`the image. Second, according to Shannon’s rate distortion the-
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`ory, the wavelet coefficients are vector quantized using a multi—
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`resolution codebook. Furthermore, to encode the wavelet coef-
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`ficients, we propose a noise shaping bit allocation procedure
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`which assumes that details at high resolution are less visible to
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`the human eye. Finally, in order to allow the receiver to rec-
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`ognize a picture as quickly as possible at minimum cost, we
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`present a progressive transmission scheme. It is shown that the
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`wavelet transform is particularly well adapted to progressive
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`transmission.
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`Keywords—Wavelet, biorthogonal wavelet, multiscale py-
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`ramidal algorithm, vector quantization, noise shaping, pro-
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`gressive transmission.
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`I.
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`INTRODUCTION
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`N many different fields, digitized images are replacing
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`conventional analog images as photograph or xrrays.
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`The volume of data required to describe such images
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`greatly slow transmission and makes storage prohibitively
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`costly. The information contained in the images must,
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`therefore, be compressed by extracting only the visible
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`elements, which are then encoded. The quantity of data
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`involved is thus reduced substantially.
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`A fundamental goal of data compression is to reduce
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`the bit rate for transmission or storage while maintaining
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`an acceptable fidelity or image quality. Compression can
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`be achieved by transforming the data. projecting it on a
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`basis of functions, and then encoding this transform. Be-
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`cause of the nature of the image signal and the mecha—
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`nisms of human vision,
`the transform used must accept
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`nonstationarity and be well localized in both the space and
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`frequency domains. To avoid redundancy, which hinders
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`compression, the transform must be at least biorthogonal
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`and lastly, in order to save CPU time, the corresponding
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`algorithm must be fast. The two—dimensional wavelet
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`transform defined by Meyer and Lemarie;[31], [24], [25],
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`Manuscript received February 7, 1990; revised March 26. 1991.
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`M. Antonini, M. Barlaud. and P. Mathieu are with LASSY 138 CNRS.
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`Universite de Nicc»Sophia Antipolis. 06560 Valbonne. France.
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`I. Daubeehies is with AT&T Bell Laboratories, Murray Hill. N107974.
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`IEEE Log Number 9106073.
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`together with its implementation as described by Mallat
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`[27], satisfies each of these conditions.
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`The compression method we have developed associates
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`a wavelet
`transform and a vector quantization coding
`scheme. The wavelet coefficients are coded considering a
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`noise shaping bit allocation procedure. This technique ex—
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`ploits the psychovisual as well as statistical redundancies
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`in the image data, enabling bit rate reduction.
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`Section II describes the wavelet transforms used in this
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`paper. After a quick review of wavelets in general, we
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`explain in more detail the properties and construction of
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`regular biorthogonal wavelet bases. We then extend this
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`construction
`two-dimensional
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`scheme with separable filters. The new coding scheme is
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`next presented in Section III. We focus particularly in this
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`section on the statistical properties of wavelet coefii-
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`cients, on the asymptotic coding gain that can be achieved
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`using vector quantization in the subimages, and on the
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`optimal allocation across the subimages. Experimental re—
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`sults are given in Section IV for images taken within and
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`outside of the training set.
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`II. WAVELETS
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`A. A Short Review of Wavelet Analysis
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`Wavelets are functions generated from one single func—
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`tion 11/ by dilations and translations
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`¢a.h(t)=|a|71/k¢<l
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`is a one—dimenr
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`11/ has to satisfy
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`5 dx tux) : O, which implies at least some oscillations.
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`(Technically speaking.
`the condition on 11/ should be
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`S dw l‘l’(cu)lZ |cu|7l < 00, where \I/ is the Fourier trans4
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`form of 11/; if Mt) decays faster than |tl" fort a 00, then
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`this condition is equivalent to the one above). The defi—
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`high frequency wavelets correspond to a < l or narrow
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`width, while low frequency wavelets have a > 1 or wider
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`The basic idea of the wavelet transform is to represent
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`any arbitrary function f as a superposition of wavelets.
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`Any such superposition decomposesfinto different scale
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`levels, where each level is then further decomposed with
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`a resolution adapted to the level. One way to achieve such
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`a decomposition writes fas an integral over a and b of
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`”'b with appropriate weighting coefficients [22]. In prac-
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`ice. one prefers to writefas a discrete superposition (sum
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`rather than integral). Therefore, one introduces a discrer
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`[05771499239300 «C 19921EEE
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`IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 1. NO. 2, APRIL 1992
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`for decomposition. Such filters are well known since the
`work of Smith and Bamwell [35l and of Vetterli [37]. The
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`extra ingredient in the orthonormal wavelet decomposi—
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`tion is that it writes the signal to be decomposed as a su—
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`perposition of reasonably smooth elementary building
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`blocks. The filters must satisfy the additional condition:
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`th’ko
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`TI
`k;1
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`(1)
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`tization, a = a5", b = nboaf)", with m, n e Z, and a0 > 1,
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`120 > 0 fixed. The wavelet decomposition is then
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`f : Z ammo...
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`with two) = Mamie) = arm/Wag": — "b0)» De-
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`compositions of this type were studied in [l4], [15]. For
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`do = 2, [)0 = 1 there exist very special choices of 11/ such
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`that the 11/,” constitute an orthonormal basis, so that
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`Cm.n(f) = (ll/mmsf) = de ll’m,n(x) f(x)
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`in this case. Different bases of this nature were con—
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`structed by Stromberg [36], Meyer [31], Lemarié [24],
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`Battle [7], and Daubechies [16]. All these examples cor—
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`respond to a multiresolution analysis, a mathematical tool
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`invented by Mallat [27], which is particularly well adapted
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`to the use of wavelet bases in image analysis, and which
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`gives rise to a fast computation algorithm.
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`In a multiresolution analysis, one really has two func-
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`tions: the mother wavelet 11/ and a scaling function (1). One
`also introduces dilated and translated versions of the seal—
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`ing function, ¢m~n(x) : 27m/2¢(27mx — n). For fixed m,
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`the 45".," are orthonormal. We denote by V,,,
`the space
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`spanned by the ¢,,,,,,; these spaces Vm describe successive
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`approximation spaces,
`- V2 C V, C V0 C V_1 C V_2
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`, each with resolution 2’". For each m, the it,“ span
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`a space Wm which is exactly the orthogonal complement
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`in V,,,~1 of V,,,; the coefficients (114“, f >, therefore, de-
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`scribe the information lost when going from an ap—
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`proximation off with resolution 2"“1 to the coarser ap-
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`proximation with resolution 2’". All
`this is translated
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`into the following algorithm for the computation of the
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`cm_,,(f) = <¢m_,,,f) (for more details, see [27]):
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`Cm,n(f) = g: g2n—kam—l,k(f)
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`(2)
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`am,n(f)
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`; h2n k0,", l,k(f)
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`where g, = (—1)'h_,+1and h" = 21/2 I dx ¢(x — n) ¢(2x).
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`In fact the am‘"( f ) are coefficients characterizing the pro-
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`jection off onto Vm. If the functionfis given in sampled
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`form, then one can take these samples for the highest or-
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`der resolution approximation coefficients am, and (2) de-
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`scribes a subband coding algorithm on these sampled val-
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`ues, with low—pass filter h and high—pass filter g. Because
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`filters give exact reconstruction, i.e.:
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`(3)
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`amil,l(f) = § [hZIz-Iam,rr(f) + an—lcm,n(f)]‘
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`Most of the orthonormal wavelet bases'have infinitely
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`supported 30, corresponding to filters h and g with infi-
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`nitely many taps. The construction in [16] gives tl with
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`finite support, and therefore, corresponds to FIR filters.
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`It follows that the orthonormal bases in [16] correspond
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`to a subband coding scheme with exact reconstruction
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`property, using the same FIR filters for reconstruction as
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`decay faster than C(l + I E [)7“()'5 as | E I —> 00, for some
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`HQ) = 24/2 Z} knew.
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`This extra regularity requirement is usually not satisfied
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`by the exact reconstruction filters in the ASSP literature.
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`B. Applications of Wavelet Bases to Image Analysis
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`I) Biorthogonal Wavelet Bases: Since images are
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`mostly smooth (except for occasional edges) it seems ap-
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`propriate that an exact reconstruction subband coding
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`scheme for image analysis should correspond to an or—
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`thonormal basis with a reasonably smooth mother wave-
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`let. In order to have fast computation, the filters should
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`so they cannot be too short). On the other hand it is de-
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`sirable that the FIR filters used be linear phase, since such
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`filters can be easily cascaded in pyramidal filter structures
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`without the need for phase compensation. Unfortunately,
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`there are no nontrivial orthonormal linear phase FIR fil-
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`ters with the exact reconstruction property [35], regard~
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`less of any regularity considerations. The only symmetric
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`exact reconstruction filters are those corresponding to the
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`Haar basis, i.e., ho : h1 = 2”2 and g0 = —g, = 21/2,
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`with all other h”, g" = 0.
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`One can preserve linear phase (corresponding to sym—
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`metry for the wavelet) by relaxing the orthonormality re-
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`quirement, and using biorthogonal bases. It is then still
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`possible to construct examples where the mother wavelets
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`In such a scheme, we still decompose as in (2). but
`reconstruction becomes
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`amm = § [fizniiamm + swam (4)
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`where the filters H, g may be different from h, g. In order
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`to have exact reconstruction, we impose:
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`gr: : (_1)nh— 4-1
`. - "
`gn = (*1) h—n+l
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`..
`Em...“ = 5m.
`n
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`(5)
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`So far, we have not performed anything differently from
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`the usual exact reconstruction subband coding schemes
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`with synthesis filters different from the decomposition fil-
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`ters. If the filters satisfy the additional condition that:
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`k1}
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`"(2’53 and
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`I},
`lie—ks)
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`(6a)
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`Page 356 0f448
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`Unified Patents Exhibit 1005 App'x A-N
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`Page 356 of 448
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`Unified Patents Exhibit 1005 App'x A-N
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`

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`207
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`Fig. 1. Filter batik structure and the associating wavelets.
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`Feauveau explores the construction from the point of view
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`of multiresolution spaces rather than from the filters. Bas—
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`ically one has two hierarchies of spaces in the bior-
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`thogonal case, each corresponding to one pair of filters.
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`It is shown in [12] that arbitrarily high regularity can
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`be achieved by both if and (I, provided one chooses suf—
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`ficiently long filters. In particular, if the functions (0 and
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`CL are, respectively, (k — 1) and (k — 1) times continu~
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`ously differentiable, then the trigonometric polynomials
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`H(£) and [3(5) have to be divisible by (1 + (”)1‘ and
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`(1 + e'15)1‘, respectively, so that the length of the corre
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`sponding filters h, 5 has to exceed k, k.
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`By (5), divisibility of 9(5) by (1 + [(5)1 means that (L
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`will have kconsecutive moments zero:
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`dex’w) :0,
`for z=0,1,--- ,k— 1.
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`For more details concerning this discussion. see [12].
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`It is well known (and it can easily be checked by using
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`Taylor expansions) that if 39 has k moments zero, then the
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`coefficients (([xmm f) will represent functions f, which
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`are ktimes differentiable, with a high compression poten-
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`tial (many coefficients will be negligibly small).
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`Many examples of biorthogonal wavelet bases with rea—
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`sonably regular 11/ and 17/ can be constructed; for our ap-
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`plications, the regularity of the elementary building blocks
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`171,”, which is linked to the number of zero moments of
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`ill, is more important than the regularity of the 11",.” or the
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`number of zero moments of 17/. Within the limits imposed
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`by the support widths, we will, therefore, try to choose
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`k as large as possible.
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`In terms of trigonometric polynomials H(E) and 1:1(5),
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`the exact reconstruction requirement condition on It and
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`Ii given in (5) reduces to (for symmetric filters)
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`Htémtt) + HQ + «NM + 7r) : 1.
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`Together with divisibility of H and [-7, respectively, by
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`(1 + e'jE)k and (1 + e_/£)k, this leads to (see [12])
`"" I— 1 +
`H(5)H(g) = cos (g/zf'l Z <
`p)
`(9)
`~ sin (5/2)” + sin (s/zf’mol
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`where R(£) is an odd polynomial in cos (5), and where
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`21 = k + k(symmetry ofh and k7 forces k + k to be even).
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`P:0
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`p
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`(8)
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`ANTONINI ('1 al.‘
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`IMAGE CODING USING WAVELET TRANSFORM
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`decay faster than C(l + 151175 7 0'5 as |£1 —> Go, for some
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`6 > 0, where
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`mg) = 2’”2 2 line W HQ) 2 2’”2 21 knew
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`(6b)
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`then we can give the following interpretation to (2) and
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`(4). Define functions o and <7) by
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`(bot) = Z hnq>(2x e n)
`and (Mr) I Z Enéax — n).
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`Their Fourier transforms are exactly the infinite products
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`(6a), and they are,
`therefore, well-defined square inte—
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