Regularity Analysis of Non Uniform Data

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1 Regularity Analysis of Non Uniform Data Christine Potier and Christine Vercken Abstract. A particular class of wavelet, derivatives of B-splines, leads to fast and ecient algorithms for contours detection of 1D and 2D signals by a discrete dyadic nite multiscale wavelet analysis (Mallat and Zhong). To detect higher order singularities, the multiscale analysis algorithm, with higher order spline wavelets, is very simply expressed in terms of B-splines subdivision and derivation formulas. This formulation allows us to generalize the algorithm for non uniformly sampled data. This algorithm is used to locate the singularities of a function. x1. Introduction The detection of singularities with multiscale transforms has been widely studied not only in mathematics but also in signal processing, computer vision,... For contours detection, the signal can be smoothed, at dierent scales and the edges are found as local extrema of the gradient modulus of the smoothed signal (Canny) or as zero-crossings of the Laplacian of the smoothed signal (Marr and Hilbreth). Mallat and al. [3] showed the equivalence between the Canny edge detector and the local maxima of a wavelet transform modulus detection. The signal regularity, at each point x, is measured by the decay, across scales, of its wavelet coecients centered at x. For this analysis, wavelet transforms with dyadic sampling, particularly orthonormal bases, may introduce distortion since they are not translation invariant. A particular class of wavelet, derivatives of B-splines, leads to fast and ecient algorithms for edges detection of 1D and 2D signals by a discrete dyadic nite scale analysis (Mallat and Zhong). For this class, the scaling function is a quadratic spline if the wavelet is the 1 st derivative of a cubic spline. More generally, a wavelet which is the p th derivative of a smoothing function is well tted to Curves and Surfaces II 1 P. J. Laurent, A. Le Mehaute, and L. L. Schumaker (eds.), pp. 1{4. Copyright oc 1991 by AKPeters, Boston. ISBN All rights of reproduction in any form reserved.

2 2 C. Potier and C. Vercken characterize singularities of (p? 1) th derivative through multiscale analysis using B-spline functions. The two scales dierence equation is a subdivision equation and the wavelet coecients are obtained, at each scale, from the derivation formulas. This formulation allows us to generalize the algorithm to non uniformly sampled data to locate singularities. x2. Dyadic Multiscale Wavelet Transform and Singularities analysis Let be a function 2 L 1 \ L 2 such as the R function x is integrable. The function can be considered as a wavelet if = 0. The dilation of at the scale s is dened by: s(x) = 1 s (x s ) (1) Let f be a function 2 L 1 \ L 2. The wavelet transform W s f(x) of f at a scale s and a position x is dened by the convolution product : W s f(x) = f s (x) (2) The dyadic wavelet transform is the sequence fw 2 j f(x) = f 2 j (x)g j2zz. Mallat[1] showed that the analysis is complete and redundant and f(x) can be recovered from its dyadic wavelet transform if there exist positive constants A and B such as 8! 2 C j A +1?1 j ^(2 j!)j 2 B (3) We now introduce a particular set S(; p) of wavelets for singularities analysis. S(; p) = f j 9 2 C 2p, compactly supported, R dx = 1, = (p) g The function is called the smoothing function. From the denition of S(; p), 2 C p (IR) and is a wavelet with p vanishing moments : Z +1?1 x i (x)dx = 0 0 i < p (4) The wavelet transform of f(x) at a scale s and a position x, veries : Ws p f(x) = f s(x) = f (s p dp (x) dp )(x) = dx p sp dx p (f s)(x) (5) and Ws pf(x) is proportional to the pth derivative of f smoothed by s (x). If f (p?1) has an isolated singularity at x 0, Ws pf has a local maximum at x 0. If 1 is the rst derivative of, 1 is well tted to localize a discontinuity of f (a sharp edge). The local extrema of Ws 1 f(x) correspond to the zero-crossings of Ws 2f(x) and to the inection points of f s(x). More precisely [3], the decay of wavelet moduli across scales depends on the Lipschitz regularity of f. If f 2 L 2 is uniformly Lipschitz < p over ]a; b[, there exists a positive constant K such that, for x 2]a; b[, s > 0 and 2 S(; p): jws p f(x)j Ks (6) To analyze singularities of f (p?1) we use a wavelet 2 S(; p) and study jws pf(x)j=s.

3 Regularity Analysis 3 x3. Discrete Dyadic Wavelet Transform In numerical applications, the input signal fx i ; y i g 0iN?1 is measured at a nite resolution that we call scale s = 1. The input signal S 1 f is a smoothed approximation of f at scale s = 1 and we can write: S 1 f(x) = f (x) (7) R where (x) is a smoothing function verifying (x) = 1 that we call the scaling function. The dilation of the scaling function is dened by equation (1). We suppose that ( x) and ( x ) can be expressed as linear combinations 2 2 of translated (x): ( x n=b 2 ) = n=a n (x? n) and ( x n=d 2 ) = n (x? n) (8) The scaling function can be used at scale 2 j to obtain a smoothed signal S 2 j f(x) = f 2 j (x). The smoothed signal represents the trend of the signal at the scale 2 j whereas W 2 j f(x) = f 2 j (x) represents the details of the signal at the scale 2 j. S 2 j f(x) may be viewed as a low-pass lter and W 2 j f(x) as a high-pass lter. The n discrete dyadicowavelet transform at scale 2 J is the sequence of details W 2 j f = f 2 j (x) and the tendency S 2 f. 1j<J J If fs 2 j f(n) = d j n g n2z are the coecients of the wavelet transform at the scale 2 j, S 2 j+1f(n) and W 2 j+1f(n) can be computed by using relations (8): S 2 j+1f(m) = n=b n=a n d j m?2 j n and W 2 j+1f(m) = n=c n=d n=c n d j m?2 j n (9) x4. Discrete Dyadic Spline Wavelet Transform It is well known that the polynomial splines spaces are well tted to multiscale wavelet transform [2] [5]. A spline function of order a is a piecewise polynomial function of degree a? 1 which is C a?2 for a 2. Its p th derivative may be used to detect singularities of f (p?1) if 2p = a? 2. If the wavelet (x) is the p th derivative of a spline of order 2p + 2, to satisfy (8) must be of the same order k = p + 2 than the scaling function. For example, let be a spline of order 6, and = ". Then is a cubic spline which is C 2 and is well tted to detect discontinuities of degree 1. The splines of order k with minimal support are called B-spline. The cardinal B-spline of order k is B k (x) = B k?1 (x) B 1 (x) where B 1 (x) is the B-spline of order 1 dened on [?1=2; 1=2]. It is dened on knot sequence [?k ; (?k+2) ; : : : ; k ]. Usually the cardinal B-spline B k(x) is dened on knot sequence [0; 1; : : : ; k]. To obtain functions centered on 0, we use translated cardinal B-splines.

4 4 C. Potier and C. Vercken The coecients f n g and f n g can be obtained by using the dilation properties of polynomial B-splines and derivation formulas. If B k denotes the k th order cardinal B-spline dened on knot sequence [?k ; (?k+2) ; : : : ; k ], B k ( x ) is the B-spline of order k dened on knot sequence [?k;?k + 2; : : : ; k]. 2 Then we can write: B k ( x n=k=2 2 ) = n=?k=2 a j B k (x? n) = n=k=2 n=?k=2 1 k B 2 k?1 k (x? n) (10) n + k=2 Similarly, the p th derivative of a (k + p) th order B-spline can be expressed: n=p=2 B (p) k+p (x) = n=?p=2 d j B k (x? n) = n=p n=0 (?1) n p n B k (x? n? p 2 ) (11) In case of regular knot sequence ft i = ig, the coecients fa n g and fd n g depend only on the order k of the spline since p = k? 2. The cardinal B-splines B k (x? i) currently denoted B i;k verify two normalizing relations P i=+1 i=?1 B i;k (x) = 1 and R B i;k (x)dx = 1. Hence, = B k (x) can be taken as scaling function. x5. Irregular Sampling In case of irregular knot sequence ft i g, the B-spline B i;k (x) dened on knots [t i?k=2 ; : : : ; t i+k=2 ] veries no more the relation R B i;k (x)dx = 1. Since we need as scaling function and smoothing function, functions verifying R f(x) = 1, we have to use normalized spline functions with compact support. The spline functions with minimal support verifying this normalizing equation are called the M-splines and denoted M i;k. The relation between B i;k and M i;k is: M i;k (x) = k (t i+k=2? t i?k=2 ) B i;k(x) (12) The relation corresponding to the relation (8) that expresses the scaling function 2s in terms of s, may be obtained by applying the subdivision formulae on B i;k dened on knot sequence [: : : ; t i?2 ; t i ; t i+2 ; : : :] if k is even and on knot sequence [: : : ; t i?1 ; t i+1 ; : : :] if k is odd. The subdivision is made by introducing one knot in each interval [1]. We obtain the coecients fa i;n g and then the coecients fh i;n g are obtained by using the relation (12): B i;k ( x n=k=2 2 ) = n=?k=2 a i;n B i+n;k (x) ; M i;k ( x n=k=2 2 ) = n=?k=2 h i;n M i+n;k (x) (13)

5 Regularity Analysis 5 Obviously, the coecients h i;n verify the fundamental relation: j=k=2 P j=?k=2 h i;j = 1 R R because M i;k (x)dx = 1 and M i;k ( x )dx = 1 for all i and k. 2 The subdivision is made at each scale s = 2 j and we obtain coecients that depend on the position i and on scale s = 2 j. For example, if k = 4 the relation between coecients a s i;n and hs i;n at scale s is: and the coecients h s i;n are: h s i;n = a s (t i+(2+n)s? t i?(2?n)s ) i;n (t i+4s? t i?4s ) (14) h s i;?2 = (t i?s? t i?4s )(t i?3s? t i?4s ) (t i+2s? t i?4s )(t i+4s? t i?4s ) h s i;?1 = (t i?s? t i?4s )(t i+s? t i?3s ) (t i+2s? t i?4s )(t i+4s? t i?4s ) h s i;0 = 1 - h s i;?2 - hs i;?1 - hs i;1 - hs i;2 h s i;1 = (t i+4s? t i+s )(t i+3s? t i?s ) (t i+4s? t i?2s )(t i+4s? t i?4s ) h s i;2 = (t i+4s? t i+s )(t i+4s? t i+3s ) (t i+4s? t i?2s )(t i+4s? t i?4s ) Coecients gi;n s at scale s are given by derivation formulae: g s = 30s 2 i;?1 (t i+3s? t i?3s )(t i+2s? t i?3s ) g s 30s 2 i;1 = (t i+3s? t i?3s )(t i+3s? t i?2s ) g s i;0 =?(g s i;?1 + gs i;1 ) Figure 1: (a) M 4 ( x) and h 2 im i;4 (x) dened on knots [1; 2:4; 3; 3:4; 4; 5; 7; 8; 9:5], (b) M 6 "(x) and g i M i;4 (x) dened on same knots.

6 6 C. Potier and C. Vercken Fig. 1(a) represents the cubic M-spline M 4 ( x ) dened on knot sequence 2 [1; 3; 4; 7; 9:5]. To obtain relation (13) we have introduced knots [2:4; 3:4; 5; 8]. We have drawn fh i M i (x)g i=1;4 where functions M i (x) are dened on ve successive knots. Fig. 1(b) represents the function M 6 " (x) where M 6 is dened on knots sequence [1; 2:4; 3; 3:4; 4; 5; 7] and the functions fg i M i (x)g i=1;3 Practically, if the subdivision schemes are applied from the coarse to the ne grid the discrete sequences converge - to a B-spline for an initial sequence yi 0 = i0 - to for an initial sequence yi 0 = d i At each step, we compute values y p+1 j by: y p+1 2i = n a 2(i?n);2n y p i?n y p+1 2i+1 = n a 2(i?n);n y p i?n coecients a i;j verifying n a 2(i?n);2n = 1 n a 2(i?n);n = 1 (15) Figure 2: Subdivision converging to x6. Discontinuity Detection In practice, the original discrete data have a nite number N of values. To solve the border problems, the signal is periodized by symmetry at each extremity as in a cosine transform. This periodization often introduce articial discontinuity at the borders and the periodization should depend on the shape of the signal at the border. We have compared the wavelet transform analysis on uniform sampling and on irregular sampling. Fig. 3(a) represents the discrete wavelet transform of the function f(x) = sin(2x) + jsin(8=3x)j for x 2 [0:2; 1:7] uniformly sampled on N = 256 points ft i g. To analyze the singularities of 1 st derivative of these data, we use the wavelet = M " 6 and the scaling function = M 4.

7 Regularity Analysis 7 s=1 s=1 s=2 s=2 s=4 s=4 (a) (b) Figure 3: (a) jw s f(x)j=s on regular sampling at scale s = 1; 2 and 4 and (b) on irregular sampling at same scale. Fig. 3(a) shows the wavelet transform computed at scale s = 1; 2 and 4. At each scale the local maxima have the same locations. On Fig. 3(b), the function is sampled on points t i verifying t i+1? t i = (1 + i ) h where i is a uniform random variable 2 [?0:2; 0:2]. At each scale, we compute: y 2s m = n=2 n=?2 h s i;n ys m?2 j n and v 2s m = n=1 n=?1 g s i;n ys m?2 j n The values fv 2s m g m=1;n are the wavelet coecients at scale s. Fig. 3(b) represents the wavelet transform computed at scale s = 1; 2 and 4. At scale 1, we can see a very oscillating function between two successive peaks corresponding to the maxima of Fig. 3(a). At scale 2, the function W s f is more smooth and at scale 4, the wavelet transform on irregular sampling looks like the wavelet transform obtained on regular sampling. Irregular sampling behaves as if it were noise. The number of maxima ndue to noise and their modulus decrease when scale increases. To distinguish a local maximum due to a singularity from a local maximum due to the noise, we compute the wavelet coecients at a few scales. The useful maxima remain stable in location and modulus. The values of the maxima are shown to be constant at each scale if the wavelet transform is computed exactly and not by discretization. To correct the error due to the discretization process [3][4], we have to normalize the

8 8 C. Potier and C. Vercken wavelet coecients at each scale s = 2 j by a factor that corresponds to the maximum modulus of the discrete wavelet transform of f(x) = x p?1 + if the analyzing wavelet is the p th derivative of a B-spline. Tab.1 gives the normalizing coecients for various wavelet analysis computed on regular sampling. On irregular sampling, the coecients which depend on the knot values, could be computed at each point. scale s=1 s=2 s=4 s=8 s=16 = B 0 4 = B = B " 6 = B Tab. 1: Normalizing coecients x7. Concluding remarks This multiscale wavelet analysis allows us to localize the singularities of irregularly sampled data. This algorithm is very fast and very ecient and depends only on the number N of data points. After the singularities localization, one can use appropriate functions to approximate the data. For approximation, the basis functions may be chosen as B-spline with multiple knots at each points where we have detected a singularity and simple knot elsewhere, the knot multiplicity corresponding to the degree of the singularity. References 1. Goldman R., Blossoming and Knot Insertion Algorithms for B-spline Curves, CAGD, 7 (1990), 69{ Mallat S., A Theory for Multiresolution Signal Decomposition : the Wavelet Representation, IEEE Trans. on PAMI, 11 (1989), 674{ Mallat S. and Hwang W., Singularity Detection and Processing with Wavelets, IEEE Trans. on Information Theory 38 (1992), 617{ Unser M., Aldroubi A. and Eden M., Fast B-spline Transforms for Continuous Image Representation and Interpolation, IEEE Trans. on PAMI, 13 (1991), 277{ Unser M. and Aldroubi A., Polynomial splines and Wavelets: A signal Processing Perspective, in Wavelets : A tutorial, C.K.Chui (eds.), Academic Press, (1991). Christine Potier and Christine Vercken Telecom Paris 46 rue Barrault 74634, Paris Cedex13, FRANCE Christine.Potier@ inf.enst.fr Christine.Vercken@ inf.enst.fr

X.-P. HANG ETAL, FROM THE WAVELET SERIES TO THE DISCRETE WAVELET TRANSFORM Abstract Discrete wavelet transform (DWT) is computed by subband lters bank

X.-P. HANG ETAL, FROM THE WAVELET SERIES TO THE DISCRETE WAVELET TRANSFORM 1 From the Wavelet Series to the Discrete Wavelet Transform the Initialization Xiao-Ping hang, Li-Sheng Tian and Ying-Ning Peng