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1 MORPHOLOGICAL PYRAMIDS AND WAVELETS BASED ON THE QUINCUNX LATTICE HENK J.A.M. HEIJMANS CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands and JOHN GOUTSIAS Center for Imaging Science, Dept. of Electrical and Comp. Eng. The Johns Hopkins University, Baltimore, MD 21218, USA Abstract. This paper is concerned with two types of multiresolution image decompositions, pyramids and wavelets. We present an axiomatic approach for both cases, encompassing linear as well as nonlinear decompositions. A wavelet decomposition is more specic in the sense that it always involves a pyramid transform. Both families will be illustrated by means of concrete examples using the quincunx scheme in two dimensions. One nonlinear wavelet transform will be discussed in more detail: it uses the lifting scheme and has the intriguing property that it preserves local maxima over a range of scales. Key words: multiresolution decomposition, pyramid, wavelet, lifting scheme, quincunx grid. 1. Introduction It is widely accepted that multiresolution approaches are extremely useful in various image processing applications. This is due to the fact that most images contain physically relevant features at dierent scales. For their proper understanding, multiresolution (or multiscale) techniques are indispensable. Another good reasons to take recourse to multiresolution approaches is that the corresponding algorithms oer various computational advantages. In this paper we present a brief overview of the axiomatic framework for the pyramid and the wavelet transform, which has been discussed in great detail in [5, 8], and we present worked-out examples for both cases based on the two-dimensional quincunx sampling scheme. An important feature of our framework is that it allows linear as well as nonlinear transforms. This is important to us, since our interest primarily goes to decompositions which are based on morphological operators. In Section 2 we introduce the pyramid transform and in Section 3 we discuss a particular example based on a morphological adunction. The general wavelet transform is introduced in Section 4. There we also explain how the lifting scheme can be used to design (linear and nonlinear) wavelet transforms. An interesting example, called max-lifting, based on morphological operators is discussed in Section 5. As we will show, the maor characteristic of this scheme is that it preserves local maxima. This work was supported in part by NATO Collaborative Research Grant CRG John Goutsias was also supported by the Oce of Naval Research (U.S.A.), Mathematical, Computer, and Information Sciences Division, under ONR Grant N

2 2 HENK J.A.M. HEIJMANS AND JOHN GOUTSIAS 2. The pyramid transform In this section we will formalize the concept of a pyramid, as rst introduced by Burt and Adelson [1], using general analysis and synthesis operators. For a comprehensive discussion we refer to [5]. Thus, consider a family of image spaces V ; 0. Assume that we can go from level to the next level +1 by means of an analysis operator : V! V +1. To goback from level + 1 to the lower level we need to dispose of a synthesis operator # : V +1! V. In this scheme, every analysis operator is designed to reduce the information contained in images at level, and as such, they are not invertible in general: the composition # (x) does only yield an approximation of the input image x 2 V. On the other hand, we demand that the synthesis operator does not cause a further reduction of the information content of an image. To achieve this, we will make the following assumption to which we refer as the pyramid condition: for every 0, the operators ; # satisfy # (x) =x for x 2 V +1: (1) This condition yields that the composition # is idempotent. The pyramid scheme introduced above can be used to obtain an alternative representation of an image x, provided that we have an addition _+ and a subtraction _, on V such that x 1 _+(x 2 _, x 1 )=x 2 for x 1 ;x 2 2 V. Given an input signal x 0 2 V 0,we consider the following recursive signal analysis scheme: x 0!fy 0 ;x 1 g!fy 0 ;y 1 ;x 2 g! where x +1 = (x ); y = x _, # (x +1) : We refer to this scheme as the pyramid transform. The original signal x 0 2 V 0 can be exactly reconstructed from x +1 and y 0 ;y 1 ;:::;y by means of the backward recursion x = # (x +1) _+ y ; 0; the inverse pyramid transform. For grey-level images, one can choose the standard addition and subtraction for _+ and _,, respectively. In the case of nitely many grey-levels, say n, one can also use the cyclic addition (i.e. addition modulus n). In the binary case, this corresponds with the `exclusive or'. 3. Morphological pyramids Here we investigate pyramid decompositions based on two basic morphological operators, erosion and dilation. Recall that for two complete lattices L and M, an operator pair (; ), with : L! M and : M! L, is called an adunction if (y) x if and only if y (x) for x 2Land y 2M. In that case is an erosion and a dilation; see [7]. We consider pyramids that satisfy the following constraints: (i) the image domains V are complete lattices; (ii) the analysis/synthesis pair ( ; # ) forms an adunction. Assume that there exist two sets S; Q and a binary relation on S Q denoted by s! q. Given a complete lattice T (later, T will have the interpretation of grey-level

3 MORPHOLOGICAL PYRAMIDS AND WAVELETS 3 set), we can dene an adunction ^ ( ; # )between T S and _ T Q in the following way: (x)(q) = x(s) and # (x 0 )(s) = x 0 (q) (2) s:s!q q:s!q V for x 2T S and x 0 2T Q. Here s:s!q x(s) stands for minfx(s) s 2 S and s! qg (or `inf' if the set is innite). Indeed, the pair ( ; # ) constitutes an adunction. In general it will not satisfy the pyramid condition, however. Thereto we need an additional assumption [5]. Proposition 1 The pair ( ; # ) given by (2) satises the pyramid condition (i.e. # = id on T Q ) if and only if for all q 2 Q there exists an s 2 S such that (i) s! q and (ii) s! q 0 where q 0 2 Q, implies q 0 = q. In the remainder of this section we are exclusively concerned with morphological adunction pyramids which correspond with a 2D quincunx sampling approach. Let S denote the square lattice comprising the integer points, i.e., S = f(s 1 ;s 2 ) s 1 ;s 2 2 Zg. Furthermore, let Q be the subset of S resulting from a quincunx sampling scheme, i.e., Q = f(q 1 ;q 2 ) q 1 ;q 2 2 Z and q 1 + q 2 even g. Finally, let S 0 Q be the set resulting after a second quincunx sampling step, i.e., S 0 = f(s 0 1 ;s0 2) s 0 1 ;s Zg. We dene the following two norms on S: Fig. 1. Arrows express the relations s! 0 q between s 2 S and q 2 Q (left), and q! 1 s 0 between q 2 Q and s 0 2 S 0 (right). The larger disks comprise the quincunx grid. ksk 1 = s 1 + s 2 and ksk 1 = maxfs 1 ; s 2 g; where s =(s 1 ;s 2 ) 2 S. Consider the binary relations s! 0 q i ks, qk 1 1 and q! 1 s 0 i kq, s 0 k 1 1 on S Q and Q S 0, respectively. These relations are illustrated in Fig. 1. Observe that both relations satisfy the conditions (i),(ii) of Proposition 1. Putting V 0 = T S and V 1 = T Q, the analysis and ^ synthesis operator given by _ (2), i.e., 0 (x)(q) = x(s) and 0 # (x0 )(s) = x 0 (q); s:s! 0q q:s! 0q

4 HENK J.A.M. HEIJMANS AND JOHN GOUTSIAS form an adunction between V 0 and V 1 and satisfy the pyramid condition.

1s 0 we obtain an analysis/synthesis pair ( 1 ; 1 # ) which is an adunction between V 1 and V 2 and satises the pyramid condition.

The odd levels in the pyramid are shown after a 45 clockwise rotation. We make two Fig. 2. Morphological adunction pyramid for the quincunx scheme.

important observations regarding this example: (1) since every analysis/synthesis pair constitutes a morphological adunction, the approximation operator # is a

4 4 HENK J.A.M. HEIJMANS AND JOHN GOUTSIAS form an adunction between V 0 and V 1 and satisfy the pyramid condition. Similarly, putting V 2 = T S0 and dening ^ _ 1 (x)(s0 )= x(q) and 1 # (x0 )(q) = x 0 (s 0 ); q:q! 1s 0 s 0 :q! 1s 0 we obtain an analysis/synthesis pair ( 1 ; 1 # ) which is an adunction between V 1 and V 2 and satises the pyramid condition. Now, since S 0 =2S, we can repeat the same procedure by putting Q 0 =2Q and S 00 =2S 0. In Fig. 2 we compute 2 levels of the corresponding pyramid transform. The odd levels in the pyramid are shown after a 45 clockwise rotation. We make two Fig. 2. Morphological adunction pyramid for the quincunx scheme. From left to right: image x # (with = 0 at the bottom level), approximation ^x = (x ), and detail y = x, ^x. important observations regarding this example: (1) since every analysis/synthesis pair constitutes a morphological adunction, the approximation operator # is a morphological opening [7]. This means in particular that the approximation image ^x = # (x ) is never larger than the original image x, and therefore the detail image y = x, ^x is nonnegative. (2) In the expressions for and # we only need to consider those points that lie in the domain of the image, i.e., a square window. This does not aect the validity of the pyramid condition, but it does destroy translation invariance.

5 4. Wavelets and lifting MORPHOLOGICAL PYRAMIDS AND WAVELETS 5 A serious drawback of the pyramid transform is that its output signal comprises more data than the input signal. The wavelet transform (see [9] for a comprehensive account in the linear case), to be discussed below, does not have this drawback. In this case the detail signal is generated by a second analysis operator!, mapping V into another space W +1, in such away that x 0 = (x) and y0 =! (x) ointly contain the same amount of data as x. Furthermore, there exists a synthesis operator such that the perfect reconstruction condition # holds, as well as # ( (x);! (x)) = x; x 2 V (3) ( # (x; y)) = x; and!( # (x; y)) = y; for x 2 V +1; y 2 W +1 : Refer to Fig. 3 for an illustration. Often, e.g. in the linear case, # is of the form # (x0 ;y 0 )= # (x0 ) _+! # (y0 ); (4) and in this case we say that the wavelet transform is uncoupled [8]. V +1 W +1 Synthesis Analysis ψ A Ψ B ω A Analysis V Fig. 3. The general wavelet transform. The lifting scheme introduced by Sweldens [10], provides a simple, exible, and ecient tool for the construction of linear as well as nonlinear wavelet transforms. A general lifting scheme starts with an invertible transformation of the input data x 0 into two or more channels (or bands). We restrict ourselves to the two-channel case for simplicity. Thus application of to x 0 yields two output signals, a coarse signal x 1 and a detail signal y 1. Application of,1 to x 1 ;y 1 returns the input signal: x 0 =,1 (x 1 ;y 1 ). In practice, will often be a known wavelet transform, for example the lazy wavelet which splits the input data into even and odd samples. By a concatenation of so-called prediction and update lifting steps (see Fig. 4) one arrives at a lifted wavelet. In Fig. 4 the prediction operators are denoted by i and the update operators by i. If the lifting scheme consists of one prediction step followed by one update step, and x 0 1 ;y0 1 are computed by the following scheme, (x 1 ;y 1 )=(x 0 ); y 0 1 = y 1, (x 1 ); x 0 1 = x 1 + (y 0 1);

6 6 HENK J.A.M. HEIJMANS AND JOHN GOUTSIAS x x 1 ' x 0 Σ π 1 λ 1 π i λ i + + y 1 y 1 ' + + Fig. 4. Lifting scheme. then we arrive at a lifted transform 0 (x 0 )=(x 0 1 ;y0 1). This can be inverted by x 1 = x 0 1, (y 0 1); y 1 = y (x 1 ); x 0 =,1 (x 1 ;y 1 ): Later we use this scheme to obtain a two-dimensional wavelet transform for the case that splits an input image according to the quincunx sampling lattice. In [8] we have shown that a linear transformation followed by one nonlinear prediction or update step yields an uncoupled wavelet transform. However, two nonlinear lifting steps result in a coupled wavelet transform, in general. 5. Max-lifting for the quincunx lattice In this section we present a particular example of a nonlinear wavelet scheme associated with the two-dimensional quincunx sampling lattice. Consider the partition of the square lattice S into two disoint subsets, Q and R = S n Q. We dene an adacency relation expressing when s is a neighbour of s 0, i.e., s s 0 if ks,s 0 k 1 =1. Thus is a symmetric relation on S S, and s s 0 can only hold if either s or s 0 (but not both) is an element ofq. Observe that r! 0 q if and only if r q for r 2 R and q 2 Q. Now let the operator govern the splitting of a signal x 0 on S into two subsampled signals, x 1 dened on Q and y 1 dened on R, i.e., x 1 (q) = x 0 (q) for q 2 Q and y 1 (r) = x 0 (r) for r 2 R. Consider the coupled wavelet transform obtained by applying rst a prediction lifting _ (x)(r) = x(q) (5) q:qr and then an update lifting (y)(q) = maxf0; _ r:rq y(r)g (6) This means that the prediction of the signal at a point r is given by the maximum of its 4 neighbours. The update operator is chosen so that local maxima of the input signal x 0 are mapped to the next level x 0 1. The next result, a proof of which can be found in [8], gives a formal statement. But rst we need two more denitions. We

7 MORPHOLOGICAL PYRAMIDS AND WAVELETS 7 Fig. 5. Wavelet decomposition of an image based on the max-lifting scheme. Bottom row: original image x 0. Middle row: wavelet transformed images x 0 1 and y0 1 (after rotation). Top row: images x 00 1 and y00 1 resulting from wavelet transform of x0 1. Note that the detail image may contain positive (bright) and negative (dark) greyvalues. write s s 0 if s = s 0 or ks, s 0 k 1 =2. Given a signal x on S and a point s 2 S we denote by A(x s) the set of neighbours s 0 of s such that x(s 0 ) x(s 00 ) for all neighbours s 00 of s. Observe that A(x s) contains at least one point. Proposition 2 Let x 0 be an input signal on S, let (x 1 ;y 1 ) be its splitting into subsignals on Q and R, respectively, and y 0 1(r) =y 1 (r), (x 1 )(r); x 0 1(q) =x 1 (q)+(y 0 1)(q); where and are given by (5)-(6). Then the following holds: (a) x 0 (q) x 0 1(q) maxfx 0 (s) s = q or s qg, for q 2 Q. (b) x 0 (r) maxfx 0 1(q) q rg, for r 2 R. (c) Assume that r 2 R is such that x 0 (r) x 0 (s) for s r and s r, then x 0 1(q) =x 0 (r) for every q 2 A(x 0 r). Thus the max-lifting scheme preserves local maxima. But we can also show that this scheme will never create new maxima.

8 8 HENK J.A.M. HEIJMANS AND JOHN GOUTSIAS Proposition 3 Suppose that the coarse signal x 0 1 hasalocal maximum at q 2 Q in the following sense: x 0 1(q 0 ) x 0 1(q) for q 0 2 Q with kq, q 0 k 1 2 (with every point q there correspond nine such points, including q itself ). Then x 0 hasalocal maximum at some s 2 S with s = q or s q and x 0 (s) =x 0 1(q). In Fig. 5 we apply the max-lifting scheme to a particular image. 6. Summary and Conclusions The pyramid condition given in (1), though seemingly straightforward, imposes relatively strong conditions on the analysis and synthesis operators. In the linear case, this condition is necessary and sucient in order that the pyramid transform can be extended to a wavelet transform. For the nonlinear case, this problem is open. Construction of nonlinear wavelets with prescribed properties (in the spirit of vanishing moments conditions in the linear case) is a research area which is almost entirely unexplored; some early work in this direction can be found in [2, 3, 4, 6]. The main tool, and to the best of our knowledge the only tool so far, to address this problem is the lifting scheme. A great deal of future research on nonlinear wavelets will have to face the question how to build schemes that yield decompositions which are useful in applications such as compression, denoising, image fusion, or image retrieval. We believe that the max-lifting scheme will turn out useful for some of these applications. We are currently exploring this issue. References 1. P. J. Burt and E. H. Adelson, The Laplacian pyramid as a compact image code. IEEE Transactions on Communications 31 (1983), 532{ R. Claypoole, R. G. Baraniuk, and R. D. Nowak, Lifting construction of non-linear wavelet transforms. In IEEE International Symposium on Time-Frequency and Time-Scale Analysis (Pittsburgh, 1998), pp. 49{ R. L. de Queiroz, D. A. Florencio and R. W. Schafer, Nonexpansive pyramid for image coding using a nonlinear lterbank. IEEE Transactions on Image Processing 7, 2 (1998), 246{ O. Egger and W. Li, Very low bit rate image coding using morphological operators and adaptive decompositions. In Proceedings of the IEEE International Conference on Image Processing, volume II (Austin, Texas, 1994), IEEE, pp. 326{ J. Goutsias and H. J. A. M. Heimans, Multiresolution signal decomposition schemes. Part 1: Morphological pyramids. To appear in IEEE Transactions on Image Processing. 6. F. J. Hampson and J.-C. Pesquet, M-band nonlinear subband decompositions with perfect reconstruction. IEEE Transactions on Image Processing 7, 11 (1998), 1547{ H. J. A. M. Heimans, Morphological Image Operators. Academic Press, Boston, H. J. A. M. Heimans and J. Goutsias, Multiresolution signal decomposition schemes. Part 2: Morphological wavelets. Research Report PNA-R9905, CWI, S. Mallat, A wavelet tour of signal processing. Academic Press, San Diego, W. Sweldens, The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3, 2 (1996), 186{200.

A new class of morphological pyramids for multiresolution image analysis

A new class of morphological pyramids for multiresolution image analysis new class of morphological pyramids for multiresolution image analysis Jos B.T.M. Roerdink Institute for Mathematics and Computing Science University of Groningen P.O. Box 800, 9700 V Groningen, The Netherlands