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`IEEE TRANSACTIONS ON IMAGE PROCESSING. VOL.
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`|. NO. 2. APRIL 1992
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`205
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`Image Coding Using Wavelet Transform
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`Marc Antonini, Michel Barlaud, Member, JEEE, Pierre Mathieu, and Ingrid Daubechies, Member, IEEE
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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
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`scheme. The wavelet coefficients are coded considering a
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`noise shapingbit 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 wavelcts 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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`one-dimensional
`construction
`two-dimensional
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`schemewith separable filters. The new coding schemeis
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`next presented in Section II]. We focus particularly in this
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`section on the statistical properties of wavelet coeffi-
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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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`optimalallocation 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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`Il. 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 y by dilations and translations
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`/2
`~b
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`Yra = ja} 0(!
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`}a
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`is a one-dimen-
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`sional variable). The mother wavelet w has to satisfy
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`| dx ¥(x) = 0, which implies at least some oscillations.
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`(Technically speaking,
`the condition on y should be
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`J da | ¥(w)|? |w|~' < o, where W is the Fourier trans-
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`form of y; if ¥(4) decays faster than |r|~! fort > oo, then
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`this condition is equivalent to the one above). The defi-
`nition of wavelets as dilates of one function means that
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`high frequency wavelets correspond to a < | 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 decomposes f into 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 wayto achieve such
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`a decomposition writes f as an integral over a and b of
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`)*-? with appropriate weighting coefficients [22]. In prac-
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`ice, one prefers to write fas a discrete superposition (sum
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`rather than integral). Therefore, one introduces a discre-
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`¥ t
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`Abstract—Image compression is now essential for applica-
`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 decomposedat 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 numberof pixels required to describe
`the image. Second, according to Shannon’s rate distortion the-
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`ory, the wavelet coefficients are vector quantized using a multi-
`resolution codebook. Furthermore, to encode the wavelet coef-
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`ficients, we propose a noise shaping bit allocation procedure
`which assumesthat 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
`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-
`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 manydifferent fields, digitized images are replacing
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`conventional analog images as photograph or x-rays.
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`The volume of data required to describe such images
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`greatly slow transmission and makesstorage 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 goul 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 mustbe 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 Lemarié[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 Nice-Sophia Antipolis. 06560 Valbonne. France.
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`I. Daubechies is with AT&T Bell Laboratories, Murray Hill, NJ 07974.
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`IEEE Log Number 9106073.
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`1057-7149/92$3.00 © 1992 IREE
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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
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`work of Smith and Barnwell [35| 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. Thefilters must satisfy the additional condition:
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`decay faster than C(1 + |£1)~¢~ °° as |E| > &, for some
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`Tl Wo“)
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`Meg) = 2°'? Dh.
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`This extra regularity requirementis 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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`1) 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 handit 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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`less of any regularity considerations. The only symmetric
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`exact reconstruction filters are those corresponding to the
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`Haarbasis, i.e., Ay = hy = 2'/? and gy = —g, = 2'”,
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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, westill decompose as in (2), but
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`Om — CF) = 2a an 1aLY + Bon 16m FI A)
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`wherethe filters #, & may be different from h, g. In order
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`tization, ad = aj, b = nbgag, with m, ne Z, and a) > 1,
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`bo > O fixed. The wavelet decomposition is then
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`f= Dena fein
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`With Yor nl) = YrB(r) = ag”’?Y(ag"t — nby). De-
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`compositions of this type were studied in [14], [15]. For
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`do = 2, bo = 1 there exist very special choices of y such
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`Cnnf) = {Vinens f> = ( dx Wmnl) FQ)
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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], whichis 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 y and a scaling function ¢. One
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`also introduces dilated and translated versions of the scal-
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`ing function, },, ,(x) = 27/7 6(2-"x — n). Forfixed m,
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`the ¢,, ate orthonormal. We denote by V,,
`the space
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`spannedby the @,, ,; these spaces V,, describe successive
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`- ++ V¥; CV; C Vp C V_;} C V_»
`approximation spaces,
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`> +, each with resolution 2”. For each m, the y,, , span
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`a space W,, which is exactly the orthogonal complement
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`in V,,, of V,,; the coefficients <¥,,,, f >, therefore, de-
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`scribe the information lost when going from an ap-
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`proximation of f with resolution 2”~' 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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`Conf) = bm.ny f> (for more details, see [27]):
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`malt) = 2 Bon — am —14f)
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`(2)
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`annf) = 2 hoy kn — wef)
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`where g, = (—1)'A_; 4, andh, = 2'/? [dx d(x — n) ¢(2x).
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`In fact the a,,,( f) are coefficients characterizing the pro-
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`jection of f onto V,,. If the function fis 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 ag_,, 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 A and high-pass filter g. Because
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`of their association with orthonormal wavelet bases, these
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`filters give exact reconstruction, i.e.:
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`(3)
`Qm—1i(f) = a [Aon — 14m.nf) + Bon -1€m nF)I-
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`Most of the orthonormal wavelet bases have infinitely
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`supported ¥, corresponding to filters hk and g with infi-
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`nitely many taps. The construction in [16] gives ¥ with
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`finite support, and therefore, corresponds to FIRfilters.
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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 FIRfilters for reconstruction as
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`~
`En = (-1)"h_ +]
`a htntin coe = Oa)
`ce
`Bn = (-1) has a
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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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`TT AQ"&)
`and TT HQ~'8)
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`(6a)
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`Page 356 of 448
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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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`207
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`Feauveau explores the construction from the point of view
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`of multiresolution spaces rather than from thefilters. Bas-
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`ically one has two hierarchies of spaces in the bior-
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`thogonal case, each corresponding to onepair offilters.
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`It is shown in [12] that arbitrarily high regularity can
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`be achieved by both y and yy, provided one choosessuf-
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`ficiently long filters. In particular, if the functions y and
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`y are, respectively, (k — 1) and (& — 1) times continu-
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`ously differentiable, then the trigonometric polynomials
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`H(&) and A(é) have to be divisible by (1 + e /*)* and
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`(1 + e~®)‘, respectively, so that the length of the corre-
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`spondingfilters h, A has to exceed k, ko
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`By (5), divisibility of H(£) by (1 + e~*)' meansthat
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`will have & consecutive moments zero:
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`| axx'yon =0,
`for 7/=0,1,°-:,k-1.
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`For more details concerning this discussion, see [12].
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`It is well known (andit can easily be checked by using
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`Taylor expansions) that if y has k moments zero, then the
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`coefficients (¥,,,, /> will represent functions f, which
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`are k times differentiable, with a high compression poten-
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`tial (many coefficients will be negligibly small).
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`Many examplesof biorthogonal wavelet bases with rea-
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`sonably regular y and y¥ can be constructed; for our ap-
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`plications, the regularity of the elementary building blocks
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`Vinns which is linked to the number of zero moments of
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`y, is more important than the regularity of the y,,.,, or the
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`numberof zero moments of y. 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 A(é) and Al).
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`the exact reconstruction requirement condition on A and
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`fh given in (5) reduces to (for symmetricfilters)
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`(8)
`A(E)A(E) + WE + wm) ACE + 3) = 1.
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`Together with divisibility of H and H, respectively, by
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`d+ e*)* and (1 + e%*)*, this leads to (see [12])
`“t-1
`H(E)A(E) = cos (€/2y" = (
`woe)
`/2)°? + sin /2y"me|
`- sin (€
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`where R(£) is an odd polynomial in cos (&), and where
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`21 =k + k (symmetry of A and A forces k + Kk to be even).
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`(9)
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`Pp
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`Page 357 of 448
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`Unified Patents Exhibit 1005 App'x A-N
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` y
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`Y
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`Filter bank structure and the associating wavelets.
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`Fig. 1.
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`ANTONINI et ai.:
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`IMAGE CODING USING WAVELET TRANSFORM
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`decayfaster than C(1 + |&|) ~*~ °° as || > ©, for some
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`€ > 0, where
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`Mg) = 2°? he ™ He) = 2717 Dae in
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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 ¢ and ¢ by
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`o(x) = VA, e2x —n)
`and $0) = Dh, d2x — ni.
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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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`grable functions, compactly supported if the filters A and
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