674 IEEE TRANSACTIONSON PATTERNANALYSISAND MACHINE INTELLIGENCE.VOL. II, NO. 7. JULY 1%') A Theory for Multiresolution Signal Decomposition: The Wavelet Representation STEPHANE Abstract-Multiresolution representations are very effective for analyzing the information content of images. We study the properties of the operator which approximates a signal at a given resolution. We show that the difference of information between the approximation of a signal at the resolutions 2’ + ’ and 2jcan be extracted by decomposing this signal on a wavelet orthonormal basis of L*(R”). In LL(R ), a wavelet orthonormal basis is a family of functions ( @ w (2’~ n)) ,,,“jEZt, which is built by dilating and translating a unique function t+r(xl. This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror lilters. For images, the wavelet representation differentiates several spatial orientations. We study the application of this representation to data compression in image coding, texture discrimination and fractal analysis. Index Terms-Coding, fractals, multiresolution pyramids, ture mirror filters, texture discrimination, wavelet transform. quadra- I. INTRODUCTION N computer vision, it is difficult to analyze the information content of an image directly from the gray-level intensity of the image pixels. Indeed, this value depends upon the lighting conditions. More important are the local variations of the image intensity. The size of the neighborhood where the contrast is computed must be adapted to the size of the objects that we want to analyze [41]. This size defines a resolution of reference for measuring the local variations of the image. Generally, the structures we want to recognize have very different sizes. Hence, it is not possible to define a priori an optimal resolution for analyzing images. Several researchers [ 181, [3 11, [42] have developed pattern matching algorithms which process the image at different resolutions. For this purpose, one can reorganize the image information into a set of details appearing at different resolutions. Given a sequence of increasing resolutions (rj )jEz, the details of an image at the resolution rj are defined as the difference of information between its approximation at the resolution rj and its approximation at the lower resolution rj _ , I Manuscript received July 30. 1987: revised December 23. 1988. This work was supported under the following Contracts and Grants: NSF grant IODCR-84 1077 1. Air Force Grant AFOSR F49620-85-K-0018. Army DAAG-29-84-K-0061. NSF-CERiDC82-19196 Ao2. and DARPAiONR ARPA N0014-85-K-0807. The author is with the Department of Computer Science Courant Institute of Mathematical Sciences. New York University. New York, NY 10012. IEEE Log Number 8928052. G. MALLAT A multiresolution decomposition enables us to have a scale-invariant interpretation of the image. The scale of an image varies with the distance between the scene and the optical center of the camera. When the image scale is modified, our interpretation of the scene should not change. A multiresolution representation can be partially scale-invariant if the sequence of resolution parameters ( ‘; ),E~ varies exponentially. Let us suppose that there exists a resolution step cy E R such that for all integers j, rj = oJ. If the camera gets 01times closer to the scene, each object of the scene is projected on an area cy2times bigger in the focal plane of the camera. That is, each object is measured at a resolution cx times bigger. Hence, the details of this new image at the resolution olj correspond to the details of the previous image at the resolution cy/ +I. Resealing the image by cx translates the image details along the resolution axis. If the image details are processed identically at all resolutions, our interpretation of the image information is not modified. A multiresolution representation provides a simple hierarchical framework for interpretating the image information [22]. At different resolutions, the details of an image generally characterize different physical structures of the scene. At a coarse resolution, these details correspond to the larger structures which provide the image “context”. It is therefore natural to analyze first the image details at a coarse resolution and then gradually increase the resolution. Such a coarse-to-fine strategy is useful for pattern recognition algorithms. It has already been widely studied for low-level image processing such as stereo matching and template matching [ 161, [ 181. Burt [5] and Crowley [8] have each introduced pyramidal implementation for computing the signal details at different resolutions. In order to simplify the computations, Burt has chosen a resolution step Q! equal to 2. The details at each resolution 2’ are calculated by filtering the original image with the difference of two low-pass filters and by subsampling the resulting image by a factor 2 ‘. This operation is performed over a finite range of resolutions. In this implementation, the difference of low-pass filters gives an approximation of the Laplacian of the Gaussian. The details at different resolutions are regrouped into a pyramid structure called the Laplacian pyramid. The Laplacian pyramid data structures, as studied by Burt and Crowley, suffer from the difficulty that data at separate levels are correlated. There is no clear model 0162-8828/89/0700-0674$01 .OO 0 1989 IEEE 675 MALLAT. THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSlTlON which handles this correlation. It is thus difficult to know whether a similarity between the image details at different resolutions is due to a property of the image itself or to the intrinsic redundancy of the representation. Furthermore, the Laplacian multiresolution representation does not introduce any spatial orientation selectivity into the decomposition process. This spatial homogeneity can be inconvenient for pattern recognition problems such as texture discrimination [2 11. In this article, we first study the mathematical properties of the operator which transforms a function into an approximation at a resolution 2 J. We then show that the difference of information between two approximations at the resolutions 2 J + ’ and 2 J is extracted by decomposing the function in a wavelet orthonormal basis. This decomposition defines a complete and orthogonal multiresolution representation called the wavelet representation. Wavelets have been introduced by Grossmann and Morlet [ 171 as functions $(x) whose translations and dilations (A rl/(.= - t)) ,u__,)ER+xR can be used for expansions of L’(R) functions. Meyer [35] showed that there exists wavelets +(x) such that ( J2/ 1c/(2 ‘x k))i,,I;)Ez? is an orthonormal basis of L’( R ). These bases generalize the Haar basis. The wavelet orthonormal bases provide an important new tool in functional analysis. Indeed, before then, it had been believed that no construction could yield simple orthonormal bases of L’ (R ) whose elements had good localization properties in both the spatial and Fourier domains. The multiresolution approach to wavelets enables us to characterize the class of functions II/(x) E L*(R) that generate an orthonormal basis. The model is first described for one-dimensional signals and then extended to two dimensions for image processing. The wavelet representation of images discriminates several spatial orientations. We show that the computation of the wavelet representation may be accomplished with a pyramidal algorithm based on convolutions with quadrature mirror filters. The signal can also be reconstructed from a wavelet representation with a similar pyramidal algorithm. We discuss the application of this representation to compact image coding, texture discrimination and fractal analysis. In this article, we omit the proofs of the theorems and avoid mathematical technical detals. Rather, we try to illustrate the practical implications of the model. The mathematical foundations are more thoroughly described in w31. A. Notation Z and R denote the set of integers and real numbers respectively. L’( R ) denotes the vector space of measurable, square-integrable one-dimensional functions f (x ). Forf(x) E L*(R) and g(x) E L’(R), the inner product off(x) with g(x) is written The norm of f(x) in L’ (R ) is given by I/ flj* = 11, 1.f’(41* du. We denote the convolution of two functionsf(x) and g(x) E L*(R) by f* g(x) = (f(u) * g(u)) (x> +CC J-W g(x - u) du. =I’ The Fourier transform off(x) and is defined by f(w) E L* (R ) = II,f(x) E L* (R ) is written p( w ) Cwr dx. Z* (Z ) is the vector space of square-summable sequences L* (R*) is the vector space of measurable, square-integrable two dimensional functionsf(x, y). Forf(x, y) E L*(R*) and g(x, y) E L’(R’), the inner product off(x, y) with g(x, y) is written ( f(X, y), g(x, ?1,> = 11,” sI_ m. The Fourier transform of f(x, P(% w,.) and is defined by &,, w,.) = i+m s+mf(x, -cc -m II. MULTIRESOLUTION Y> g(x, Y) dx dJ. y) E L*( R2) is written y) ,-‘(wl’+uixl dx dy. TRANSFORM In this section, we study the concept of multiresolution decomposition for one-dimensional signals. The model is extended to two dimensions in Section IV. A. Multiresolution Approximation of L’(R ) Let A?, be the operator which approximates a signal at a resolution 2 j. We suppose that our original signal f(x) is measurable and has a finite energy: f(x) E L’(R). Here, we characterize AZ, from the intuitive properties that one would expect from such an approximation operator. We state each property in words, and then give the equivalent mathematical formulation. 1) A*, is a linear operator. If A?,f(x) is the approximation of some function f(x) at the resolution 2 ‘, then A2,f(x) is not modified if we approximate it again at the resolution 2 ‘. This principle shows that A*,0 Azi = A2 ,. The operator A,, is thus a projection operator on a particular vector space V,, C L’ (R ). The vector space V2, can be interpreted as the set of all possible approximations at the resolution 2 J of functions in L’ (R) . 2) Among all the approximated functions at the resolution 2”, A?,f(x) is the function which is the most similiar tof(x). 676 IkEE TRANSACTIONS ON PATTkRN ANALYSIS AND MACHINE INTELLIGENCE. VOL. Il. NO. 7. JULY I%39 vg(x> E v21, jjdx) -f(x)/1 2 II&Sb) -S(x)/I. (1) Hence, the operator A*, is an orthogonal projection on the vector space VI,. 3) The approximation of a signal at a resolution 2”’ contains all the necessary information to compute the same signal at a smaller resolution 2 J. This is a causality property. Since Al, is a projection operator on VI, this principle is equivalent to VjEZ, v*, c v*,i1. (2) 4) An approximation operation is similar at all resolutions. The spaces of approximated functions should thus be derived from one another by scaling each approximated function by the ratio of their resolution values Vj E Z, f(x) E v,, * f(2x) E V2,’ 1. (3) 5) The approximation AZ/f(x) of a signal f(x) can be characterized by 2.’ samples per length unit. Whenf(x) is translated by a length proportional to 2 I, A?, f (x ) is translated by the same amount and is characterized by the same samples which have been translated. As a consequence of (3), it is sufficient to express the principle 5) for the resolution j = 0. The mathematical translations consist of the following. l Discrete characterization: There exists an isomorphism Z from V, onto Z* (Z ) . (4) l Translation of the approximation: vk E Z, A,fx(x) = A,f(x l - k), whereh(x) =f(x - k). (5) Translation of the samples: Z(A,f(-~)) = (my,)i~z * Z(Mi(x)) = (vn>,,,. (6) 6) When computing an approximation of f(x) at resolution 2 ‘, some information aboutf (x) is lost. However, as the resolution increases to + 00 the approximated signal should converge to the original signal. Conversely as the resolution decreases to zero, the approximated signal contains less and less information and converges to zero. Since the approximated signal at a resolution 2 ’ is equal to the orthogonal projection on a space V2,, this principle can be written lim V2, = ;” V?, is dense in L2 (R ) ,++CC ,=-'72 (7) lim J--m (8) and V,, = 77 V,, = (0). J= -cc We call any set of vector spaces ( V21),Ez which satisfies the properties (2)-(8) a multiresolution approximation of L’ (R ). The associated set of operators A*, satisfying l)6) give the approximation of any L’ ( R ) function at a res- olution 2 I. We now give a simple example of a multiresolution approximation of L’ ( R ). Example: Let V, be the vector space of all functions of L’( R ) which are constant on each interval ] k, k + 1 [, for any k E Z. Equation (3) implies that V2, is the vector space of all the functions of L’ (R ) which are constant on each interval ] k2 -j, (k + 1)2-’ [, for any k E Z. The condition (2) is easily verified. We can define an isomorphism Z which satisfies properties (4), (5), and (6) by associating with any function f(x) E V, the sequence ( CY~)~~~such that (Ye equals the value of f(x) on the interval ] k, k + 1 [. We know that the vector space of piecewise constant functions is dense in L? (R ). Hence, +Oa we can derive that U I’,, is dense in L’ (R >. It is clear +CO J’-m n VI,= (0) , so the sequence of vector spaces ,=-cc ( Vr,),EZ is a multiresolution approximation of L’(R). Unfortunately, the functions of these vector spaces are neither smooth nor continuous, making this multiresolution approximation rather inconvenient. For many applications we want to compute a smooth approximation. In Appendix A, we describe a class of multiresolution approximations where the functions of each space Vz , are n times continuously differentiable. We saw that the approximation operator A?, is an orthogonal projection on the vector space Vz,. In order to numerically characterize this operator, we must find an orthonormal basis of Vz,. The following theorem shows that such an orthonormal basis can be defined by dilating and translating a unique function 4 (x). Theorem 1: Let ( V2,),Ez be a multiresolution approximation of L’ (R ). There exists a unique function 4 (x) E L’ (R ), called a scaling function, such that if we set &!(x) = 2’$(2’x) forj E Z. (the dilation of 4(x) by 2’), then that ( &? &,(x - 2m’n)),rGz is an orthonormal basis of I’2 1. n (9) Indications for the proof of this theorem can be found in Appendix B. The theorem shows that we can build an orthonormal basis of any V,, by dilating a function $ (x) with a coefficient 2 ’ and translating the resulting function on a grid whose interval is proportional to 2-‘. The functions &,(x) are normalized with respect to the L’ (R ) norm. The coefhcient J2-’ appears in the basis set in order to normalize the functions in the L’( R ) norm. For a given multiresolution approximation ( V21)JEz, there exists a unique scaling function 4(x) which satisfies (9). However, for different multiresolution approximations, the scaling functions are different. One can easily show that the scaling function corresponding to the multiresolution described in the previous example is the indicator function of the interval [0, I]. In general, we want to have a smoother scaling function. Fig. 1 shows an example of a continuously differentiable and exponentially decreasing scaling function. Its Fourier transform has the shape of a 677 MALLAT: THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSITION ok) ; Hence, we can rewrite Af,f: A&f = ((f(u) -5 (a) ao, i 1 t i B. Implementation \ 0 i __iJ I -70 -Jc 0 7T w 10 (b) Fig. 1. (a) Example of scaling function 4(x). This function ia computed in Appendix A. (b) Fourier transform i(w). A scaling function ia a lowpass filter. low-pass filter. The corresponding multiresolution approximation is built from cubic splines. This scaling function is described further in Appendix A. The orthogonal projection on V2, can now be computed by decomposing the signal f(x) on the orthonormal basis given by Theorem 1. Specifically, vf(x) E L’(R), = 2-” AZ/~(X) !iY (f(u), ,1=-C= - 2-‘n)) &,(x - 2-‘n). ( 10) The approximation of the signalf(x) at the resolution 2 ‘, A,,f(x), is thus characterized by the set of inner products which we denote by A&f = (( f(u), hi(u - 2y’n))),lFZ. (11) A!, f is called a discrete approximation off(x) at the resolution 2 .‘. Since computers can only process discrete signals, we must work with discrete approximations. Each inner product can also be interpreted as a convolution product evaluated at a point 2:‘n (f(u), &,(u = (f(u) * bi( -u)) of a Multiresolution Transform qb2,(x - 2-‘n) 2-‘-l ,z m = (&,(u - 2:/n), &,+I(u - 2-‘-‘k)) - 2-‘+k). (13) By changing variables in the inner product integral, one can show that 2-‘-I( &,(u - 2-‘n), &,+~(u - 2-‘-‘k)) $(u - (k - 2n))). (14) When computing the inner products off(x) sides of (13) we obtain with both = (42-1(u), ( f (UL +2,(u = ,=g, - 2+n) > (5% I(U), +(u - (k - 2n))) . (f(u), 42,+,(~ - 2-‘-‘k)). Let H be the discrete filter whose impulse response is given by - 2-44) *cc em f(u) bi(u =s (12) (2-/n))ntz. In practice, a physical measuring device can only measure a signal at a finite resolution. For normalization purposes, we suppose that this resolution is equal to 1. Let Aff be the discrete approximation at the resolution 1 that is measured. The causality principle says that from Af’f we can compute all the discrete approximations A:, f for j < 0. In this section. we describe a simple iterative algorithm for calculating these discrete approximations. Let (V2,),d be a multiresolution approximation and 4 (x) be the corresponding scaling function. The family is an orof functions (m +*,+1(x - 2P”P’k)),,, thonormal basis of V2,,+1. We know that for any n E Z, the function $*,(x - 2-‘n) is a member of V?, which is included in V2,+ I. It can thus be expanded in this orthonormal basis of V,,, , : * &t+,(x &,(u -u)> Since 4(x) is a low-pass filter, this discrete signal can be interpreted as a low-pass filtering off(x) followed by a uniform sampling at the rate 2 ‘. In an approximation operation, when removing the details off(x) smaller than 2-‘, we suppress the highest frequencies of this function. The scaling function 4 (x) forms a very particular lowpass filter since the family of functions (fl &, (x - 2Pin))REz is an orthonormal family. In the next section we show that the discrete approximation off(x) at the resolution 2 ’ can be computed with a pyramidal algorithm. X 5 0 * h( - 2-/n) du (2-/n). Vn E Z, h(n) = (4+(u), 6(u - n)). (15) Let fi be the mirror filter with impulse response i; (n ) = h( -n). By inserting (15) in the previous equation, we IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE. VOL. II, NO. 7. JULY IYXY 678 obtain (f(u), Al-:fd - 2-/n) > h(u = kjYm h(2n - k) (f(U), (f12,Tl(u - 2-‘-h)). (16) ,..‘. A2mzfd ;""'~'~~~"~~,.....,,,~, Equation (16) shows that Ai,f can be computed by convolving A:,+ of with fi and keeping every other sample of the output. All the discrete approximations A,d,f, forj < 0, can thus be computed from Affby repeating this process. This operation is called a pyramid transform. The algorithm is illustrated by a block diagram in Fig. 5. In practice, the measuring device gives only a finite number of samples: A:f = (a,,), i ,, I N. Each discrete signal Af,,f(j < 0) has 2 ‘N samples. In order to avoid border problems when computing the discrete approximations Af,f, we suppose that the original signal Af’f is symmetric with respect to n = 0 and n = N (a-,, if-N<n<O i %.Y-,I if0 (4, = and 14’(x)/ = 0(x-“). = .".."' _,,_ ___. ----. ,. ?-;f d _.’-...x..-_.___,___ ;. ,____.. ‘.-.,.... .----.I,_...._ *,/*. Y-----1...- A Lfd 3- 5: .__. 13: .-.,: 152 '.,,---zoo ', 25, (a) E h(n)ep”‘“. ,1=--m Fig. 2. (a) Discrete approximations A: ,fat the resolutions I. l/2. I /4. I /8. I / 16. and l/32. Each dot gives the amplitude of the inner product ( ,f( u), & (u - 2 -‘?I ) ) depending upon 2-‘,1. (b) Continuous approximations A,,f(.r) at the resolutions 1. l/2. I /4. I /8. I /l6. and I /32. These approximations are computed by interpolating the discrete approximations with (IO). H(w) satisfies the following iH( IH(,)/’ = 1 and + (H(w h(n) (17) two properties: = O(n-‘) at infinity. + a)/’ = 1. (17a) (17b) Conversely let H( w ) be a Fourier series satisfying (17a) and (17b) and such that IH( # 0 for w E [0, 7r/2]. (17c) The function defined by J(w) The following theorem gives a practical characterization of the Fourier transform of a scaling function. Theorem 2: Let 4(x) be a scaling function, and let H be a discrete filter with impulse response h(n) = ($2 I(U), 4(u - n)>. Let H(w) be the Fourier series defined by H(o) '...,,.. < II < 0. If the impulse response of the filter fi is even ( fi = H ), each discrete approximation At,fwill also be symmetric with respect to n = 0 and n = 2-/N. Fig. 2(a) shows the discrete approximated signal Af,f of a continuous signal f(x) at the resolutions 1, l/2, l/4, l/8, l/16, and l/32. These discrete approximated signals have been computed with the algorithm previously described. Appendix A gives the coefficients of the filter H that we used. The continuous approximated signals AIlf(x) shown in Fig. 2(b) have been calculated by interpolating the discrete approximations with (10). As the resolution decreases, the smaller details off(x) gradually disappear. Theorem 1 shows that a multiresolution approximation ; b),ez is completely characterized by the scaling function 4 (x). A scaling function can be defined as a function 4(x) E L’(R) such that, for all j E Z, (J2-’ &, (x - 2-in)),,,, is an orthonormal family, and if VI, is the vector space generated by this family of functions, then is a multiresolution approximation of L’( R ). We (V&z also impose a regularity condition on scaling functions. A scaling function 4 (x) must be continuously differentiable and the asymptotic decay of 4(x) and 4’ (x) at infinity must satisfy (45(x)1 = O(F) A .'. = ,g, H(2-“w) (18) is the Fourier transform of a scaling function. n Indication for the proof of this theorem are given in Appendix C. The filters that satisfy property (17b) are called conjugare filters. One can find extensive descriptions of such filters and numerical methods to synthesize them in the signal processing literature [lo], [36], [40]. Given a conjugate filter H which satisfies (17a)-( 17c), we can then compute the Fourier transform of the corre- 679 MALLAT: THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSITION h(n) sponding scaling function with (16). It is possible to choose H(o) in order to obtain a scaling function 4 (x) which has good localization properties in both the frequency and spatial domains. The smoothness class of 4 (x) and its asymtotic decay at infinity can be estimated from the properties of H( w ) [9]. In the multiresolution approximation given in the example of Section II-A, we saw that the scaling function is the indicator function of the interval [0, 11. One can easily show that the corresponding function H ( w ) satisfies H(w) = e-‘” cos 0 0.6 t 4 III. THE WAVELET REPRESENTATION As explained in the introduction, we wish to build a multiresolution representation based on the differences of information available at two successive resolutions 2 J and 2JC’. This section shows that such a representation can be computed by decomposing the signal using a wavelet orthonormal basis. A. The Detail Signal Here, we explain how to extract the difference of information between the approximation of a function f(x) at the resolutions 2 J + ’ and 2 ‘. This difference of information is called the detail signal at the resolution 2,‘. The approximation at the resolution 2j’ ’ and 2’ of a signal are respectively equal to its orthogonal projection on Vz,+ I and Vz,. By applying the projection theorem, we can easily show that the detail signal at the resolution 2’ is given by the orthogonal projection of the original signal on the orthogonal complement of V,, in Vz,+ 1. Let 02, be this orthogonal complement, i.e., V2,, 02, 0 v2, = V2,+, To compute the orthogonal projection of a functionf(x) we need to find an orthonormal basis of O2 ,. Much like Theorem 1, Theorem 3 shows that such a basis can be built by scaling and translating a function $ (x). Theorem 3: Let (V?,)jEZ be a multiresolution vector space sequence, 4 (x) the scaling function, and H the corresponding conjugate filter. Let +(x) be a function whose Fourier transform is given by on 02,, $(w)=G; 4 i) ; 0 with I I -20 I -10 ,. 0 I.... 10 20 n (a) Appendix A describes a class of symmetric scaling functions which decay exponentially and whose Fourier transforms decrease as 1 /w”, for some n E N. Fig. 3 shows the filter H associated with the scaling function given in Fig. 1. This filter is further described in Appendix A. O?, is orthogonal to r G(w) = e-la H(w + T) . (19) :g ;'---I 0 -J -n LI , 1 -2 0 2 I.. 7Tm (b) Fig. 3. (a) Impulse response of the filter H associated to the scaling func;ion shown in Fig. I .-The coefficients of this filter are given in Appendix A. (b) Transfer function H(w) of the filter H. Let &,(x) 2,‘. Then (@ = 2’$(2’x) $2/(X denote the dilation of +5(x) by - 2-Jn)),*pzis an orthonormal basis of 02, and is an orthonormal basis of L* (R ). H $(x) is called an orthogonal wavelet. Indications for the proof of this theorem can be found in Appendix D. An orthonormal basis of 02, can thus be computed by scaling the wavelet $(x) with a coefficient 2 J and translating it on a grid whose interval is proportional to 2-j. The wavelet function corresponding to the example of multiresolution given in Section II-A is the Haar wavelet 1 rc/(x) = i -1 L 0 ifOlx<i if$lx< 1 . otherwise This wavelet is not even continuous. In many applications, we want to use a smooth wavelet. For computing a wavelet, we can define a function H(w) which satisfies the conditions (17a)-(17c) of Theorem 2, compute the corresponding scaling function 4 (x) with equation (18) 680 IEEE ‘TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE. VOL. II. NO 7. JULY 1989 and the wavelet $(x) with (19). Depending upon choice of H( w), the scaling function 4(x) and the wavelet $(x) can have good localization both in the spatial and Fourier domains. Daubechies [9] studied the properties of 6 (x) and $(x) depending upon H( w ). The first wavelets found by Meyer [35] are both C” and have an asymptotic decay which falls faster than the multiplicative inverse of any polynomial. Daubechies shows that for any II > 0, we can find a function H(w) such that the corresponding wavelet $(x) has a compact support and is n times continuously differentiable [9]. The wavelets described in Appendix A are exponentially decreasing and are in C” for different values of n. These particular wavelets have been studied by Lemarie [24] and Battle 131. The decomposition of a signal in an orthonormal wavelet basis gives an intermediate representation between Fourier and spatial representations. The properties of the wavelet orthonormal bases are discussed by Meyer in an advanced functional analysis book [34]. Due to this double localization in the Fourier and the spatial domains, it is possible to characterize the local regularity of a function f(-r) based on the coefficients in a wavelet orthonormal basis expansion [25]. For example, from the asymptotic rate of decrease of the wavelet coefficients, we can determine whether a function f(x) is n times differentiable at a point x0. Fig. 4 shows the wavelet associated with the scaling function of Fig. 1. This wavelet is symmetric with respect to the point x = l/2. The energy of a wavelet in the Fourier domain is essentially concentrated in the intervals [ -2~, -ir] U [T, 27r]. Let PO:, be the orthogonal projection on the vector space O?,. As a consequence of Theorem 3, this operator can now be written cm PO,f(x) = 2-’ ,,=F, (f(u), IcZl(U- 2-‘n) > . l&,(.x - 2-/n). (20) PO2f(.~) yields to the detail signal off(x) at the resolution 2 ‘. It is characterized by the set of inner products Dzif = (( f(u) 1 iczi(u - 2-‘4 ,),l,z. (21) Dz,f is called the discrete detail signal at the resolution 2 J. It contains the difference of information between A:,- 1f and A$ f. As we did in (12), we can prove that each of these inner products is equal to the convolution off(x) with &,( -x) evaluated at 2-/n (f(u), $?,(U - 2-52)) = (f(u) * ic?,(-u))(2-'n). (22) Equations (21) and (22) show that the discrete detail signal at the resolution 2’ is equal to a uniform sampling of (f(u) * rl/?,( -u)) (x) at the rate 2’ &if = ((f(u) * h-4) (2-/")),,Ez. The wavelet $(x) can be viewed as a bandpass filter whose frequency bands are approximatively equal to vJv(x 1 1 0.5 0 -0.5 :+ 0.4 1 0.2 0 1 t Fig. 4. (a) Wavelet $(I) associated to the scaling function of Fig. 1. (b) Modulus of the Fourier transform of I/J(X). A wavelet is a band-pass filter. [ -277, -x] U [K, 2~1. Hence, the detail signal, D?,f describes f(x) in the frequency bands [ -2 -‘+ ’x, -2-‘7r] u [2-JT, 2-‘+‘7r]. We can prove by induction that for any J > 0, the original discrete signal A:f measured at the resolution 1 is represented by (AP Jf, (Dz~fIp~~j<-l). (23) This set of discrete signals is called an orthogonal wavelet representation, and consists of the reference signal at a coarse resolution A: if and the detail signals at the resolutions 2’ for -J I j I - 1. It can be interpreted as a decomposition of the original signal in an orthonormal wavelet basis or as a decomposition of the signal in a set of independent frequency channels as in Marr’s human vision model [30]. The independence is due to the orthogonality of the wavelet functions. It is difficult to give a precise interpretation of the model in terms of a frequency decomposition because the overlap of the frequency channels. However, we can control this overlap thanks to the orthogonality of our decomposition functions. That is why the tools of functional analysis give a better understanding of this decomposition. If we ignore the overlapping spectral supports, the interpretation in the frequency domain provides an intuitive approach to the model. In analogy with the Laplacian pyramid data structure, Ai rfprovides the top-level Gaussian pyramid data, and the D?,,f data provide the successive 681 MALLAT: THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSITION Laplacian pyramid levels. Unlike the Laplacian pyramid, however, there is no oversampling, and the individual coefficients in the set of data are independent. B. Implementation of an Orthogonal Wavelet Representation In this section, we describe a pyramidal algorithm to compute the wavelet representation. With the same derivation steps as in Section II-B, we show that D2, f can be calculated by convolving A;‘,+, f with a discrete filter G whose form we will characterize. For any n E Z, the function &, (X - 2-’ n ) is a member of 02, c Vz,- I. In the same manner as (13)) this function can be expanded in an orthonormal basis of V, I + I I&,(X - 2-/n) = keep one sample out of Iwo l-7-j * &,*l(X ,t”.. ($,,(u - 2-/n), qlh,.I(U - 2p’-‘k)) - 2p’eq. (24) As we did in (14), by changing variables in the inner product integral we can prove that 2-‘-I( &!(u - 2-jn), &,+I(u - 2-‘-‘k)) (25) Hence, by computing the inner product off(x) functions of both sides of (24), we obtain = ,jfm (bl(u>, . (f(u), uses a symmetry with respect to the first and the last sample as in Section II-B. Equation (19) of Theorem 3 implies that the impulse response of the filter G is related to the impulse response of the filter H by +(u - (k - 2n))) &,+,(u - 2-‘-Q)). g(n) = (4&l(u), 4(u - n)), (26) (27) and G be the symmetric filter with impulse response g (n ) = g ( -n ). We show in Appendix D that the transfer function of this filter is the function G(w) defined in Theorem 3, equation (19). Inserting (27) to (26) yields bh,(u - 2:'n)) = jY,"(Zn (-l)‘-” = h(1 - n). (29) This equation is provided in Appendix D. G is the mirror filter of H, and is a high-pass filter. In signal processing, G and H are called quudrature mirror filters [lo]. Equation (28) can be interpreted as a high-pass filtering of the discrete signal A;‘, +of. If the original signal has N samples, then the discrete signals D2, f and A:, f have 2’ N samples each. Thus, the wavelet representation with the Let G be the discrete filter with impulse response (f(u), X Fig. 5. Decomposition of a discrete approximation A$, , ,f into an approximation at a coarser resolution A$ j’and the signal detail LLf. By repeating in cascade this algorithm for -1 2 j 2 -J, we compute the wavelet representation of a signal A‘:fon J resolution levels. g(n) 2-‘-l convolve with IMer - k) (f(u),c#Q,+,(u - 2-‘-‘/c)). (28) Equation (28) shows that we can compute the detail signal D,, f by convolving At,- ,f with the filter G and retaining every other sample of the output. The orthogonal wavelet representation of a discrete signal Aff can therefore be computed by successively decomposing At,, ,f into At, f and Dz,f for -J 5 j I - 1. This algorithm is illustrated by the block diagram shown in Fig. 5. In practice, the signal A;‘f has only a finite number of samples. One method of handling the border problems has the same total number of samples as the original approximated signal A: f. This occurs because the representation is orthogonal. Fig. 6(b) gives the wavelet representation of the signal Ayfdecomposed in Fig. 2. The energy of the samples of D,, f gives a measure of the irregularity of the signal at the resolution 2’+‘. Whenever AI, f (x) and A2,+ If(x) are significantly different, the signal detail has a high amplitude. In Fig. 6, this behavior is observed in the textured area between the abscissa coordinates of 60 and 80. C. Signal Reconstruction from an Orthogonal Wavelet Representation We have seen that the wavelet representation is complete. We now show that the original discrete signal can also be reconstructed with a pyramid transform. Since Oz, is the orthogonal complement of V, I in VI,+ I, ( fi &)(x - 2-jn), J2-l Ic/?,(x - 2Pin)),,EZ is an orthonorma1 basis of VI ,+ I. For any n > 0, the function &,+ I (X - 2-j- ’n ) can thus be decomposed in this basis &,+I(x - 2-‘-In) = 2-‘,:<- (&,(u - 2-/k), &,-I(U - 2:‘-‘n)) . $2,(x - 2-‘/k) + 2-j j!Zy, ($?,(u - 2-/k), * y&,(x - 2-/i?). &,+I(u - 2:‘-‘n)) (30) 682 IEEE TRANSACTIONS O N PATTERN ANALYSIS AND MACHINE INTELLIGENCE. VOL. Il. NO 7. JULY -+Ajl;fs$+,f IT?I + put one zero between each sample convolve wlh tiller a: : multiplication by 1. 19X9 X 2 Fig. 7. Reconstruction of a discrete approximation A!, I ffrom an approximation at a coarser resolution &,f and the signal detail D,, f. By repeating in cascade this algorithm for -.I I j 5 - 1, we reconstruct A:f from its wavelet representation. yields 4*,+1(U - 2-‘-w (f(u), (a) = 2 ,g Az-sfd m + 2 &+ h(n - 2k) (f(u),+& - 2k)(f(u), - $*,(u 2:+)) - 2-JJk)). ' Dpf_ (32) D2-6.’ ..‘:_,, D2-:fe Dpf ..,; 7’ .:, . : ,‘. ,: .::: ‘.. ; Y..,.. .._. Fig. 6. (a) Multiresolution continuous approximationsAz,f(X). (b) Wavelet representation of the signal A,f(x). The dots give the amplitude of the inner products ( f(u), $?, (u - 2 -‘n) ) of each detail signal Dz,f depending upon 2-‘n. The detail signals samples have a high amplitude when the approximationsAzrf(X) andA,,+,f(*) shown in (a) are locally different. The top graph gives the inner products ( f( u ). 4 J (u - 2 ‘II) ) of the coarse discrete approximation A$,f. By computing the inner product of each side of equation (30) with the functionf(x), we have This equation shows that A$, + I f can be reconstructed by putting zeros between each sample of A&f and &, f and convolving the resulting signals with the filters H and G, respectively. A quite similar process can be found in the reconstruction algorithm of Burt and Adelson from their Laplacian pyramid [5]. The block diagram shown in Fig. 7 illustrates this algorithm. The original discrete signal A;‘fat the resolution 1 is reconstructed by repeating this procedure for -1 5 j < 0. From the discrete approximation Aff, we can recover the continuous approximation A, f(x) with equation (10). Fig. 8(a) is a reconstruction of the signal A, f(x) from the wavelet representation given in Fig. 6(b). By comparing this reconstruction with the original signal shown in Fig. 8(b), we can appreciate the quality of the reconstruction. The low and high frequencies of the signal are reconstructed well, illustrating the numerical stability of the decomposition and reconstruction processes. IV. EXTENSION OF THE ORTHOGONAL REPRESENTATION (f(u), q!J*,+,(u = 2-j&y (qb*,(u * (f(u), + 2-j j<, 2-'-51)) ($,,(u * (f(u), h(u - 2%c), - - - 2-'%I)) 2-w - 2-'k), $r,(u 4*,+1(u c#Q,+,(u - 2-'-'4) 2-q). (31) Inserting (14) and (25) in this expression and using the filters H and G, respectively, defined by (15) and (27) WAVELET TO IMAGES The wavelet model can be easily generalized to any dimension y1 > 0 [33]. In this section, we study the twodimensional case for image processing applications. The signal is now a finite energy functionf(x, y) E L*(R’). A multiresolution approximation of L2 (R*) is a sequence of subspaces of L’ (R’) which satisfies a straightforward two-dimensional extension of the properties (2) to (8). Let (Vz, )jez be such a multiresolution approximation of L* ( R2). The approximation of a signal f(x, y ) at a resolution 2 J is equal to its orthogonal projection on the vector space V,,. Theorem 1 is still valid in two dimensions, and one can show that there exists a unique scaling function @(x, y) whose dilation and translation given an or- MALLAT. THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSITION Fig. 9. Approximations l/8 of an image at the resolutions 1, l/2, (j = 0, -I, -2, -3). l/4, and The approximation of a signal f(x, y) at a resolution 2 ’ is therefore characterized by the set of inner products A&f = (( f(x, (b) Fig. 8. (a) Original sIgnal A,,~(.I ) approximated at the resolution I. (b) Reconstruction of A, f(.r) from the wavelet representation shown in Fig. 6(b). Bj comparing both figures. we can appreciate the quality of the reconstruction. thonormal basis of each space I’?,, . Let @ ‘2,(x, y) = 2” @ (2”x, 2’~). The family of functions pb(x - 2-,/n, \’ - 2-‘m))(n,,,,)E72 forms an orthonormal basis of V,,. The factor 2-’ normalizes each function in the L2(R2) norm. The function + (x, .v) is unique with respect to a particular multiresolution approximation of L’ (R 2 ). W e next describe the particular case of separable multiresolution approximations of L’( R2) studied by Meyer [35]. For such multiresolution approximations, each vector space If,, can be decomposed as a tensor product of two identical subspaces of L2 (R ) v,, = v:, 0 v:,. The sequence of vector spaces ( V2,),Ez forms a multiresolution approximation of L*( R2) if and only if (VL),,EZ is a multiresolution approximation of L* (R ). One can then easily show that the scaling function + (x, y ) can be written as Y>? 42,(x - 2w (2p%‘b - 2-h = phh(~ ))i,z,,,,irZl @(x, y) = 4,(x> G(y), *‘(x, **(x3 .v> = $(x) 4,(y), vv) = $(x> $(Y) are such that - 2-/n, y - 2:/m). 2:‘*:,(x - 2:ln, y - 2:jrn), 2-9:,(x - 2-/n, ?’ - 2-‘rn))(,~,,,~)Fz’ (34) is an orthonormal basis of 02, and (2-‘9j,(x - 2-‘n, y - 2-/m), 2-Q!;, (x - 2-/n, y - 2-/m), ?’ - 2-‘mj)(,,,,,,,Ez2 - 2-/n) &I(? - 2-w Let us suppose that the camera measures an approximation of the irradiance of a scene at the resolution 1. Let Affbe the resulting image and N be the number of pixels. One can easily show that, for j < 0, a discrete image approximation At,f has 2 JN pixels. Border problems are handled by supposing that the original image is symmetric with respect to the horizontal and vertical borders. Fig. 9 gives the discrete approximations of an image at the resolutions 1, l/2, l/4, and l/8. As in the one-dimensional case, the detail signal at the resolution 2’ is equal to the orthogonal projection of the signal on the orthogonal complement of V2, in V2, + I. Let 01, be this orthogonal complement. The following theorem gives a simple extension of Theorem 3, and states that we can build an orthonormal basis of 02, by scaling and translating three wavelets functions, * ’(x, y), q2 (x, y), and *j(x, y). Theorem 4: Let ( V2,)jGz be a separable multiresolution approximation of L2(R2). Let Q(.x, y) = C#I(X) $(y) be the associated two-dimensional scaling function. Let \k (x) be the one-dimensional wavelet associated with the scaling function C#I(x). Then, the three “wavelets” (2-‘Ql,(x where C#I (x) is the one-dimensional scaling function of the multiresolution approximation ( V:,)jEz. W ith a separable multiresolution approximation, extra importance is given to the horizontal and vertical directions in the image. For many types of images, such as those from man-made environments, this emphasis is appropriate. The orthogonal basis of V2, is then given by h(Y - 2-‘m))(,l,,,l,tz1. (33) 2-)*G,(x - 2-h, Y - 2e’m))(,,,,n,Ez3 is an orthonormal basis of L’( R’). (35) 684 IEEE TRANSACTIONS O N PATTERN ANALYSIS AND MACHINE INTELLIGENCE. VOL Il. NO. 7. JULY 19X’) Appendix E gives a proof of this theorem. The difference of information between A$ + I f and A$ f is equal to the orthonormal projection of f(x) on 02,, and is characterized by the inner products of f(x) with each vector of an orthonormal basis of O,, . Theorem 4 says that this difference of information is given by the three detail images D;,f = (( j-(x, Y), *:,cx - 2-k Y - 2-“m) ))ln,m)FZ?: (36) D&f = ((j-(x, y), *:,(X - 2-k Y - 2-Jm)))i,,,,n)FZ? D;,f = (( f(x, Y), ‘P;,(x - 2-k Y - 2-44 (37) ))ir~,,,I~EZ~. (38) Just as for one-dimensional signals, one can show that in two dimensions the inner products which define A’:,f, 0: ,f, 0; ,f, and D:, f are equal to a uniform sampling of two-dimensional convolution products. Since the three wavelets \k,(x, y), \k>(x, y), and \k3(x, y) are given by separable products of the functions 4 and $, these convolutions can be written A$,f = ((fb Y> * +21(-x) d)*d-Y)) (b) Fig. 10. (a) Decomposition of the frequency support of the image A!, 1f into A’,‘,fand the detail images LIl f. The image A’,‘,fcorresponds to the lower horizontal and vertical frequencies of A,, ,_ Di,fgives the vertical high frequencies and horizontal low frequencies. L2-, J‘the horizontal high frequencies and vertical low frequencies and Dl,fthe high frequencies in both horizontal and vertical directions. (b) Disposition of the Di,f and A$ , f’ images of the image wavelet representations shown in this article. F’n, 2-‘m))(,l.,n)Ez1 (39) D;,f = 2-‘n, 2-Jm))(,1.,r,,Ez1 (40) Dfif = ((fb (a) ~9 * Ir/2,( -4 h( -Y)> (2-k 2pJm)),,l,,n)Ez2 (41) D;,f = The expressions (39) through (42) show that in two dimensions, A$, f and the Di, f are computed with separable filtering of the signal along the abscissa and ordinate. The wavelet decomposition can thus be interpreted as a signal decomposition in a set of independent, spatially oriented frequency channels. Let us suppose that C#I (x) and $(x) are, respectively, a perfect low-pass and a perfect bandpass filter. Fig. 10(a) shows in the frequency domain how the image A$+ I f is decomposed into A,“, f, 04, f, D:, f, and D:, f. The image A:, f corresponds to the lowest frequencies, D:, f gives the vertical high frequencies (horizontal edges), Ds, f the horizontal high frequencies (vertical edges) and D:, f the high frequencies in both directions (the comers). This is illustrated by the decomposition of a white square on a black background explained in Fig. 11(b). The arrangement of the Di,f im- Cc) Fig. I I. (a) Original image. (b) Wavelet representation on three resolution levels. The black, grey. and white pixels correspond respectively to negative, zero. and positive wavelet coefficients. The disposition of the detatl images is explained in Fig. IO(b). (c) These images show the absolute value of the wavelet coefficients for each detail images Di, shown in (b). Black and white pixels correspond respectively to zero and high amplitude coefficients. The amplitude is high along the edges of the square for each orientation. 685 MALLAT: THEORY FOR MULTIRESOLUTIONSIGNAL DECOMPOSITION ages is shown in Fig. 10(b). The black, grey, and white pixels respectively correspond to negative, zero, and positive coefficients. Fig. 1 l(c) shows the absolute value of the detail signal samples. The black pixels correspond to zero whereas the white ones have a high positive value. As expected, the detail signal samples have a high amplitude on the horizontal edges, the vertical edges and the comers of the square. For any J > 0, an image A;(fis completely represented by the 3J + 1 discrete images m : keeo one row 11121: This set of images is called an orthogonal wavelet representation in two dimensions. The image Af-,f is the coarse approximation at the resolution 2PJ and the D:,f images give the detail signals for different orientations and resolutions. If the original image has N pixels. each image Af?‘,f, DJ,f, D<,f, Dz,fhas 2/N pixels ( j < 0). The total number of pixels in this new representation is equal to the number of pixels of the original image, so we do not increase the volume of data. Once again, this occurs due to the orthogonality of the representation. In a correlated multiresolution representation such as the Laplacian pyramid. the total number of pixels representing the signal is increased by a factor of 2 in one dimension and of 4/3 in two dimensions. A. Decomposition and Reconstruction Algorithms in TN)ODimerzsiorzs In two dimensions, the wavelet representation can be computed with a pyramidal algorithm similar to the onedimensional algorithm described in Section III-B. The two-dimensional wavelet transform that we describe can be seen as a one-dimensional wavelet transform along the s and y axes. By repeating the analysis described in Section III-B, we can show that a two-dimensional wavelet transform can be computed with a separable extension of the one-dimensional decomposition algorithm. At each step we decompose A&+ of into A;,f, D$, f, D$, f, and 02) f. This algorithm is illustrated by a block diagram in Fig. 12. We first convolve the rows of A;, , , f with a onedimensional filter, retain every other row, convolve the columns of the resulting signals with another one-dimensional filter and retain every other column. The filters used in this decomposition are the quadrature mirror filters fi and e described in Sections II-B and III-B. The structure of application of the filters for computing A:, . 04, , Di, , and D$, is given in Fig. 12. We compute the wavelet transform of an image A: f by repeating this process for - 1 2 j 2 -J. This corresponds to a separable conjugate mirror filter decomposition [44]. Fig. 14(b) shows the wavelet representation of a natural scene image decomposed on 3 resolution levels. The pattern of arrangement of the detail images is as explained in Fig. 10(b). Fig. 14(c) gives the absolute value of the wavelet coefficients of each detail image. The wavelet convolve (rows or columns) wilh the fuller X keep one column ouL of two I?-I’ out of two Fig. 12. Decomposition of an image A:,. f into A$,f. Dlflf. Di,f. and D:,f: This algorithm is based on one-dimensional convolutions of the rows and columns of A’,‘.. fwith the one dimensional quadrature mirror filters fi and G. coefficients have a high amplitude around the images edges and in the textured areas within a given spatial orientation. The one-dimensional reconstruction algorithm described in Section III-C can also be extended to two dimensions. At each step, the image A$+ of is reconstructed from A;‘, f, Di,,f, Dz, f, and 02, f. This algorithm is illustrated by a block diagram in Fig. 13. Between each column of the images A$ f, D:, f, Ds, f, and Dz, f, we add a column of zeros, convolve the rows with a one dimensional filter, add a row of zeros between each row of the resulting image, and convolve the columns with another one-dimensional filter. The filters used in the reconstruction are the quadrature mirror filters Hand G described in Sections II-B and III-B. The image Ayf is reconstructed from its wavelet transform by repeating this process for -J i ,j i - 1. Fig. 14(d) shows the reconstruction of the original image from its wavelet representation. If we use floating-point precision for the discrete signals in the wavelet representation, the reconstruction is of excellent quality. Reconstruction errors are more thoroughly discussed in the next section. V. APPLICATIONS OF THE ORTHOGONAL WAVELET REPRESENTATION A. Compact Coding of Wavelet Image Representations To compute an exact reconstruction of the original image, we must store the pixel values of each detail image with infinite precision. However, for practical applications, we can allow errors to occur as long as the relevant information is not destroyed for a human observer. In this section, we show how to use the sensitivity of the human visual system as well as the statistical properties of the image to optimize the coding by the wavelet representation. The conjugate mirror filters which implement the wavelet decomposition have also been studied by Woods [44] and Adelson et al. [l] for image coding. Let (A$,f, (D;if LJs,,s -17 (D:,f >pJs,s -13 (D:, f ) -.,<, 5 ~I ) be the wavelet representation of an image A;‘f. Let (E-J, (+Js,+l~ ($-Js,+l? ( E: ) -J <, 5 ~, ) be the mean square errors introduced when coding each image component of the wavelet representa- 6X6 IEEE TRANSACTIONS O N PATTERN ANALYSIS prove that AND MACHINE INTELLIGENCE. 3 Eo = 22Jt-J + c k=I A$+I f convoive (row or column) with filter ;, gj X put one row of zeros between each row put one column of zeros between each column multiply by 4 Fim C’ 13. Reconstruction of an image A$ f from A$f, Di,f; Df,f. and 0: jY The row and columns of these images are convolved with the one dimensional quadrature mirror filtera H and C. (a) (b) (4 Fig. 14. (a) Original image. (b) Wavelet representation on three resolution levels. The arrangement of the detail images ia explained in Fig. IO(b). Cc) These images show the absolute value of the wavelet coefficients for each detail images Di, shown in (b). The amplitude is high along the edges and the textured area for each orientation. (d) Reconstruction of the original image from the wavelet representation given in (b). tion. Let cObe the mean square error of the image reconstructed from the coded wavelet representation. Since the wavelet representation is orthogonal in L’(R’), one can VOL. II. NO 7. JULY lY8Y -1 c 2-y. ,,=J The factors 2-2’ are due to the normalization factor 2-j which appears in (33) and (34). Psychological experiments on human visual sensitivity show that the visible distortion on the reconstructed image will not only depend on the total mean square error co, but also on the distribution of this error between the different detail images Di, f. The contrast sensitivity function of the visual SYStern [6] shows that the perception of a contrast distortion in the image depends upon the frequency components of the modified contrast. Visual sensitivity also depends upon the orientation of the stimulus. The results of Campbell and Kulikowski [7] show that the human visual system has a maximum sensitivity when the contrast is horizontal or vertical. When the contrast is tilted at 45’) the sensitivity is minimum. A two-dimensional wavelet representation corresponds to a decomposition of the image into independent frequency bands and three spatial orientations. Each detail image Di, gives the image contrast in a given frequency range and along a particular orientation. It is therefore possible to adapt the coding error of each detail image to the sensitivity of human perception for the corresponding frequency band and spatial orientation selectivity. The more sensitive the human visual system, the less coding error we want to introduce in the detail image 05, f. Watson [43] has made a particularly detailed study of subband image coding adapted to human visual perception. Given an allocation of the coding error between the different resolutions and orientations of the wavelet representation, we must then code each detail image with a minimum number of bytes. In order to optimize the coding, one can use the statistical properties of the wavelet coefficients for each resolution and orientation. Natural images are special kinds of two-dimensional signals. This shows up clearly when one looks at the histogram of the detail images D$ If. Since the pixels of these detail images are the decomposition coefficients of the original image in an orthonormal family, they are not correlated. The histogram of the detail images could therefore have any distribution. Yet in practice, for all resolutions and orientations, these histograms are symmetrical peaks centered in zero. Natural images must therefore belong to a particular subset of L’( R’). The modeling of this subset is a well known problem in image processing [26]. The wavelet orthonormal bases are potentially helpful for this purpose. Indeed, the statistical properties of an image decomposition in a wavelet orthonormal basis look simpler than the statistical properties of the original image. Moreover, the orthogonality of the wavelet functions can simplify the mathematical analysis of the problem. We have found experimentally that the detail image histograms can be modeled with the following family of histograms: /l(u) = Ke-‘l”l/,)J. (44) MALLAT: THEORY FOR MULTIRESOLUTION SIGNAL The parameter 0 modifies the decreasing rate of the peak and CYmodels the variance. This model was built by studing the histograms of seven different images decomposed on four resolution levels each. Our goal was only to define a qualitative histogram model. The constant K is adjusted in order to have ST, h(u) du = N where N is the total number of pixels of the given detail image. By changing variables in the integral, one can derive that K=- NP, where I’(r) = m s e-Q-’ 0 687 DECOMPOSITION du. F-‘(x) t 1.5 1 0.5 :_; (45) OA 0.2 0.4 0 Fig. 15. Graph of the function F-‘(x) The coefficients CYand @ of the histogram model can be computed by measuring the first and second moment of the detail image histogram: X characterized by (49) h(u) r 03 ml = -m lulh(u)du and m2 = s m u2h(u) du. c --co 15000 (46) By inserting (44) of the histogram model and changing variables in these two integrals, we obtain ml = and m 2- -2Koi71’ (47) 10000 5000 0 P Thus. (a) where (YIP (48) 0 1 m217 P \ 0 , (49) -10 -5 0 ,,,,,,,,j 5 10 u (b) Nr ; 0 The function F-‘(x) is shown in Fig. 15. Fig. 16(a) gives a typical example of a detail image histogram obtained from the wavelet representation of a real image. Fig. 16(b) is the graph of the model derived from (44). The a priori knowledge of the detail signal’s statistical distribution given by the histogram model (44) can be used to optimize the coding of these signals. W e have developed such a procedure [27] using Max’s algorithm [32] to minimize the quantization noise on the wavelet representation. Predictive coding procedures are also effective for this purpose. Results show that one can code an image using such a representation with less than 1.5 bits per pixel with few visible distortions [ 11, [27], [44]. B. Texture Discrimination and Fractal Analysis W e now describe the application of the wavelet orthogonal representation to texture discrimination and fractal Fig. 16. (a) Typical example of a detail image histogram h(u). (b) Modeling of h ( U) obtained from equation (44). The parameters oi and 0 have been computed from the first two moments of the original histogram (01 = 1.39andP = 1.14). analysis. Using psychophysics, Julesz [21] has developed a texture discrimination theory based on the decomposition of textures into basic primitives called textons. These textons are spatially local; they have a particular spatial orientation and narrow frequency tuning. The wavelet representation can also be interpreted as a texton decomposition where each texton is equal to a particular function of the wavelet orthonormal basis. Indeed, these functions have all the discriminative abilities required by the Julesz theory. In the decomposition studied in this article, we have used only three orientation tunings. However, one can build a wavelet representation having as many orientation tunings as desired by using non-separable wavelet orthonormal bases [33]. 688 IEEE TRANSACTIONS O N PATTERN Fig. 17(a) shows three textures synthesized by Beck. Humans cannot preattentively discriminate the middle from the right texture but can separate the left texture from the others. In this example, human discrimination is based mainly on the orientation of these textures as their frequency content is very similar. With a first-order statistical analysis of the wavelet representation shown in Fig. 17(b), we can also discriminate the left texture but not the two others. This example illustrates the ability of our representation to differentiate textures on orientation criteria. This is of course only one aspect of the problem, and a more sophisticated statistical analysis is needed for modeling textures [ 131. Although several psychophysical studies have shown the importance of a signal decomposition in several frequency channels [4], [ 151, there still is no statistical model to combine the information provided by the different channels. From this point of view, the wavelet mathematical model might be helpful to transpose some of the tools currently used in functional analysis to characterize the local regularity of functions [25]. Mandelbrot [29] has shown that certain natural textures can be modeled with Brownian fractal noise. Brownian fractal noise F(x) is a random process whose local differences (F(x) ANALYSIS AND MACHINE INTELLIGENCE. VOL. II. NO. 7. JULY 1%‘) (a) (b) Fig. 17. (a) .I. Beck textures: only the left texture 15 preattcntively discrlminable by a human observer. (h) These Images show the alxolutc value of the wavelet coefficients of image (a). computed on three resolution levels. The left texture can be discrimmated with a first-order statistical analysis of the detail signals amplitude. The two other textures can not be dlscriminatcd with such a technic. - F(x + Ax) / I/Ax/IH has a probability distribution function g(x) which Gaussian. Such a random process is self-similar, i.e.. tlr > 0, F(x) and rHF(rx) is are statistically identical. (a) Hence, a realization of F(x) looks similar at any scale and for any resolution. Fractals do not provide a general model which can be used for the analysis of any kind of texture, but Pentland [39] has shown that for a fractal texture, the psychophysical perception of roughness can be quantified with the fractal dimension. Fig. 18(a) shows a realization of a fractal noise which looks like a cloud. Its fractal dimension is 2.5. Fig. 18(b) gives the wavelet representation of this fractal. As expected, the detail signals are similar at all resolutions. The image Ai-if gives the local dc component of the original fractal image. For a cloud, this would correspond to the local differences of illuminations. Let US show that the fractal dimension can be computed from the wavelet representation. We give the proof for one-dimensional fractal noise, but the result can be easily extended to two dimensions. The power spectrum of fractal noise is given by [29] P(w) (b) Fig. 18. (a) Brownian fractal image. (b) Wavelet representation on three resolution levels of image (a). A\ expected. the detail signals are similar at all resolutions. interpreted in the classical sense. Flandrin [ 121 has shown how to define precisely this power spectrum formula with a time-frequency analysis. We saw in equation (22) that the detail signals Dz,fare obtained by filtering the signal with &, ( -x) and sampling the output. The power spectrum of the fractal filtered by &, ( -x) is given by = kcC2Hp’. (50) The fractal dimension is related to the exponent H by D=T+l-H (51) where T is the topological dimension of the space in which x varies ( for images T = 2 ). Since Brownian fractal noise is not a stationary process, this power spectrum cannot be P2#(W) = P(w) I5(2-‘w)~?. (52) After sampling at a rate 2’, the power spectrum of the discrete detail signal becomes [37] P;,(w) = 2J ,;<m P,,(w + 2/2kr). (53) VALI.&T THEORY FOR MULTIRESOLUTION SIGNAL DECOMPOSITION Let ai, be the energy of the detail signal Dz, f *217r 2-J o;, = -2T -2,n mJJ) h. \’ (54) By inserting (52) and (53) into (54) and changing variables in this integral, we obtain that 2 = 2’Ho5 / 1 1. 02, (55) For a fractal, the ratio ai,/ai,&r is therefore constant. From the wavelet representation of a Brownian fractal, we compute H from equation (55) and derive the fractal dimension D with (51). This result can easily be extended to two dimensions in order to compute the fractal dimension of fractal images. Analogously, we compute the ratios of the energy of the detail images within each direction, and derive the value of the coefficient H. A similar algorithm has been proposed by Heeger and Pentland for analyzing fractals with Gabor functions [ 191. For the fractal shown in Fig. 18(a), we calculated the ratios of the energy of the detail images within each orientation for the resolutions l/2. l/4, and l/8. We recovered the fractal dimension of this image from each of these ratios with a 3 percent maximum error. Much research work has recently concentrated on the analysis of fractals with the wavelet transform [2]. This topic is promising because multiscale decompositions, such as the wavelet transform, are well adapted to evaluate the self-similarity of a signal and its fractal properties. VI. CONCLUSION This article has described a mathematical model for the computation and interpretation of the concept of a multiresolution representation. We explained how to extract the difference of information between successive resolutions and thus define a new (complete) representation called the wavelet representation. This representation is computed by decomposing the original signal using a wavelet orthonormal basis, and can be interpreted as a decomposition using a set of independent frequency channels having a spatial orientation tuning. A wavelet representation lies between the spatial and Fourier domains. There is no redundant information because the wavelet functions are orthogonal. The computation is efficient due to the existence of a pyramidal algorithm based on convolutions with quadrature mirror filters. The original signal can be reconstructed from the wavelet decomposition with a similar algorithm. We discussed the application of the wavelet representation to data compression in image coding. We showed that an orthogonal wavelet transform provides interesting insight on the statistical properties of images. The orientation selectivity of this representation is useful for many applications. We reviewed in particular the texture discrimination problem. A wavelet transform is particularly well-suited to analyze the fractal properties of images. Specifically, we showed how to compute the fractal dimension of a Brownian fractal from its wavelet represen- 689 tation. In this article, we emphasized the computer vision applications, but this representation can also be helpful for pattern recognition in other domains. Grossmann and Kronland-Martinet [23] are currently working on speech recognition applications, and Morlet [ 141 studies seismic signal analysis. The wavelet orthonormal bases are also studied in both pure and applied mathematics [20], [25], and have found applications in Quantum Mechanics with the work of Paul [38] and Federbush [ 111. APPENDIX A AN EXAMPLE MULTIRESOLUTION APPROXIMATION In this appendix, we describe a class of multiresolution approximations of L’(R) studied by Lemarie [24] and Battle [3]. We explain how to compute the corresponding scaling functions 4 (x), wavelets G(x), and quadrature filters H. These multiresolution approximations are built from polynomial splines of order 2p + 1. The vector space V, is the vector space of all functions of L’( R) which are p times continuously differentiable and equal to a polynomial of order 2p -t 1 on each interval [k, k + 11, for any k E Z. The other vector spaces V,, are derived from V, with property (3). Lemarie has shown that the scaling function associated with such a multiresolution approximation can be written . 4(w) where = d&j n = 2 + 2p, (56) and where the function C,,(w) is given by C,,(w) = 1 E h=-w (w + 2kr)“’ (57) We can compute a closed form of C,, (0) by calculating the derivative of order y1 - 2 of the equation C*(w) = l 4 sin* (w/2)’ Theorem 2 says that a(w) is related to the transfer function H( w ) of a quadrature mirror filter by $(2w) = H(w) $(w). From (56) we obtain H(w) = C*,,(w) \i 2?17c2,,(2w)’ (58) The Fourier transform of the corresponding orthonormal wavelet can be derived from the property (19) of Theorem 3 690 IEEE TRANSACTIONS O N PATTERN ANALYSIS AND MACHINE INTELLIGENCE. = 5 + 30 ( cos 2-)‘+ II. NO 7. JULY 1989 TABLE1 The wavelet $(x) defined by (59) decreases exponentially. The scaling function shown in Fig. 1 was obtained with p = 1, and thus n = 4. It corresponds to a multiresolution approximation built from cubic splines. Let N,(o) VOL. h(n) 3O(sin~)?(cos~j 0.006 0.006 and -0.003 N*(a) = 2(sini)i(iorq)? + 7O(cOs~) -0.002 By inserting (62) in (60), we obtain / +CX? M(w) = The function Cs( w) is given by c,(w) = N,(w) + h(w) 105 sin i c Y. One can show that the continuity of the isomorphism Z implies that there exists two constants C, and C2 such that For this multiresolution approximation based on cubic splines, the functions 6 (w ) and 5 (w) are computed from (56) and (59) with n = 4. The transfer function H(w) of the quadrature mirror filter is given by equation (58). Table I gives the first 12 coefficients of the impulse response This filter is symmetrical. The impulse re(h(n))n,z. sponse of the mirror filter G is obtained with (29). APPENDIX B 1 This appendix gives the main steps of the proof of Theorem 1. More details can be found in [28]. We prove Theorem 1 forj = 0. The result can be extended for any j E Z using the property (3). From the properties (5) and (6) of the isomorphism Z from VI onto Z*(Z ), one can prove that there exists a function g (x) such that ( g (X k))kEz is a basis of VI. We are looking for a function 4(x) E V, suchthat (4(x - k))kEZ is an orthonormal basis of V, Let 4 (w ) be the Fourier transform of $ (x). With the Poisson formula, we can show that the family of functions ($(x - k))kEZ is orthonormal if and only if I/* C, I x 2ka)(* = 1. (60) Since 4 (x) E V,, it can be decomposed in the basis ( g (X - k)),,,: +-x)kEZ E z2w such that $(x) = k: akg(x I c*. ! Hence, (63) is defined for any simple to prove that equations Fourier transform of a function k))kcz is an orthonormal family PROOF OF THEOREM ,g, I$(w + Ig(w + 2ka)l* w E R. Conversely, it is (62) and (63) define the 4 (x) such that (4(x that generates VI. C APPENDIX PROOF OF THEOREM 2 This appendix gives the main steps of the proof of Theorem 2. More details can be found in [28]. Let us first prove property (17a). Since $2m1(~) E V-, C V,, it can be decomposed in the orthogonal basis (4 (x - k) ) kc~ 42-'(x) = ,g, (+2-W 4(u - 4) +(u The Fourier transform of this equation yields 6P-4 = H(w) 6(w) From (64) we obtain 1H(0) 1 = 1. Since the asymptotic decay of 4 (x) at infinity satisfies )4(x)) = w-x-2) we can also derive that with M(w) = ,Iz’, (64) where H( w ) is the Fourier series defined by (17). One can show that the property (7) of a multiresolution approximation implies that any scaling function satisfies The Fourier transform of (61) can be written = M(w) g(w) k). - k). VW $(a) - Coke’““. (62) h(n) = (q!-‘(u), 4(u - rl)) = 0(n-‘) at infinity. 691 MALLAT. THEORY FOR MULTIRESOLUTlONSIGNAL DECOMPOSITION APPENDIX D PROOF OF THEOREM 3 Let us now prove property ( 17b). We saw in Appendix A that a scaling function must satisfy ,g., I$( w + 2k7r)12 = 1. Since H(w) is 2~ periodic, (H(w)(2 (66) (64) and (66) yield + /H((w + 7r)f = 1. This appendix gives the main steps of the proof of Theorem 3. More details can be found in [28]. This theorem is proved for j = - 1. We are looking for a function $(x) E L’(R) such that (&?lr/> ,(x - 2-‘n)),,Z is an orthonormal basis of O,-, The orthogonality of this family can be expressed with the Poisson formula Let us write (71) (67) We will show that this equation defines the Fourier transform of a scaling function. We need to prove that (a) 4(w) E L’(R) and (4(x - n)),,,Z is an orthonormal family. (p) If V,, is the vector space generated by the family of functions ( &, (X - 2:‘~)) nEZ, then the sequence of vector spaces ( Vz I),i,z is a multiresolution approximation ofL’(R). Let us first prove property ((Y). With the Parseval theorem, we can show that this statement is equivalent to Since &,(x) orthonormal E O2 I C V,, we can decompose it on the basis ( 4 (x - n ) ) ,,Ez: li/2~l(x) = ,,g, (&I(U), 4(u - n)) 4(x - n). (72) Let us define G(w) = E ( G2 i(u), $(u ,,= -cc - n)) emino. (73) The Fourier transform of (73) yields $(2w) = G(w) J(w). (74) As in Appendix C, (74) and (71) give Let us define the sequence of functions ( g,(w)), that g,(o) = for (w ( < 2qr fi H(2-“w) P=l for Iw[ 1 2qn. 0 , , such Vn E Z, 5 erkwdw = ar i ifk ifk (69) Since H(w) = H(w) f a)(’ = 1. (75) each function of the family ( &?+2m1(x - 2-‘n)),Ez. The Poisson formula shows that this property is equivalent to Vn E Z, 5 q+(2w + 2n7r) $(2w ,1=--03 + 2n7r) = 0. (76) = 0 # 0. (70) With hypothesis (17~) of Theorem 2, it is then possible to apply the dominated convergence theorem to the seto derive from (70) that 4 (o) satisquence ( gq(w))y,l fies (68). Let us now prove property ( 0). In order to prove that ( W,,EZ is a multiresolution approximation of L’(R), we must show that assertions (2)-(8) apply. The properties (2)-(6) can be derived from the equation $(2w) + (G(w Since 02- 1 is orthogonal to VI I, each function of the family ( @$2m1(x - 2-l n) ) ,rcZ should be orthogonal to As q tends to + 00, the sequence (g,(u)),, I converges towards (4(w) (’ almost everywhere. We can also prove +CZ ~Q1g,(w) (G(wf J(w). satisfies (17a), we can show that By inserting (64), (66), and (74) in (77), we obtain H(w) G(o) + r) G(w + T) = 0. (77) We can prove that the necessary conditions (75) and (76) on G(w) are sufficient to ensure that ( fl&-~(x 2p’n)),,Ez is an orthogonal basis of 02-1. An example of such function G(w) is given by G(w) = e -‘“H(w + 7r). (78) The functions G(w) and H(w) can be viewed as the transfer functions of a pair of quadrature mirror filters. By taking the inverse Fourier transform of (79), we prove that the impulse responses ( g (n ) ) nEZ and (h (n ) ) ,lEZ of these filters are related by g(n) From this equation, one can prove that the sequence of vector spaces ( Vli)jGz defined in (0) do satisfy the last two properties (7) and (8) of a multiresolution representation. + H(o By definition, satisfies = (-I)‘-“h( a multiresolution lim Vz, = L’(R) ./++a and 1 - n). approximation lim p--m (79) of L’( R) V,, = (0). 692 lEtE TRAKSACTIONS O N PATTERN Since O? , is the orthogonal complement of V2, in V?, + I, we can derive that that for any j # k, 02, is orthogonal to O?h and L’(R) = it Oz,. ,,= -cc (80) We proved that for anyj E Z, (G&,(x - 2-Jn)),,.z is an orthonormal basis of 02, . The family of functions ( m~3r(X - 2-/n)) C,I, IEZ: is therefore an orthonormal basis of L’( R ). E APPENDIX PROOF OF THEOREM This appendix gives the main steps of Theorem 4 proof. More details can be found in [35]. Let ( V2,)iGz be a multiresolution approximation of L’( R ) such that for any j E Z, (81) approximation where ( Vi, licz is a multiresolution L’(R). We want to prove that the family of functions (2-‘$&(x 2-‘&,(x - 2p’ll) &,(y - 2-h), - 2-/n) &,(.v - 2+n), 2-‘jc7iCer- 231) bi(Y - 2-‘m))(,i,,,r)Ez2 8 vi,*, This can be rewritten v,,-1 = (V$, 0 Vi,) 8 (Vi, 0 Oil) 0 co;, 0 v;,) The orthogonal given by 0 o;,,. complement of V?, in VI,, I is therefore 02, = (Vi, 0 04,) 0 co:, 0 v;,, 0 (Oi, 0 o;,,. (82) The family of functions (m&,(x - 2-.‘n))icz is an orthonormal basis of Vi, and ( fl+*, (X - 2-jn )),EZ is an orthonormal basis of O:, . Hence, (82) implies that (2-‘&,(x - 2-‘n) I&,(J - 2-/m), 2-‘&,(x - 2-/n) &,(y - 2:‘m), 2-‘$2~(x - - 2-‘n) 452iCY INTELLIGENCE. VOL. II, NO 7. JULY 1989 sum of the orthogonal spaces 02, L’(R’) = ii 02,. ,=-LX The family of functions (2-&(x 2-‘4!3,(x - 2-/n) il/z,(y - 2-Jm), - 2-ln) &,(y 2-‘ic/n(,x - 231) h(Y - 2-‘m), - 2--‘4)(n ,,,)tZ’ basis of L2( R*). ACKNOWLEDGMENT I would like to thank particularly R. Bajcsy for her advice throughout this research, and Y. Meyer for his help with some mathematical aspects of this paper. I am also grateful to J.-L. Vila for his comments. REFERENCES = (Oi, 0 Vi,) 0 (04, 8 Vi,). 8 (o;, MACHINE of is an orthonormal basis O?, . The vector space 02, is the orthogonal complement of V?, in V2, + I. Let 04, be the orthogonal complement of Vi, in Vl, I. Equation (81) yields V >,T’ = vi,+, AND is therefore an orthonormal 4 v,, = v;, 0 VA, ANALYSIS 2-‘f4)(,i,,)i)Ezz is an orthonormal basis of 02, The vector sDace L’C R’) can be decomnosed as a direct Ill E. Adelson and E. Simoncelli. “Orthogonal pyramid transform for image codinp.” ~roc. sPIE, Visucd Commun. Imuye Proc., 1987. I21 A. Arneodo: G. Grasseau, and H. Holachneider, “On the wavelet transform of multifractala,” Tech. Rep.. CPT, CNRS Luminy. Marseilles. France. 19X7. I31 G. 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Acoust., Speech, Signal Procew ing. vol. ASSP-32, pp. 645-648. June 1984. L411 A. Rosenfeld and M. Thurston, “Edge and curve detection for visual Scene analysis.” IEEE Trans. Comp;t.. vol. C-20. 1971. “Coarse-fine temolate matching,” IEEE Trans. Svst., Man, Cvf421 -. brm., vol. SMC-7, pp: 104-107, 1977. 1431 A. Watson, “Efficiency of a model human image code,” .I. Opt. Sot. Amer., vol. 4. pp. 2401-2417, Dec. 1987. I441 J. W. Woods and S. D. O’Neil, “Subband coding of images,” IEEE Trans. Aroust., Speech, Signal Processing. vol. ASSP-34, Oct. 1986. Stephane G. Mallat was born jn Paris, France. He graduated from Ecole Polytechnique, Paris, in 1984 and from Ecole Nationale Superieure des Telecommunications, Paris, in 1985. He received the Ph.D. degree in electrical engineering from the University of Pennsylvania, Philadelphia. PA, m 1988. Since September 1988. he has been an Asaistant Professor in the Department of Computer Science of the Courant Institute of Mathematical Sciences, New York University. New York. NY. His research interests include computer vision, signal processing, and apnlied mathematics.

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