ADDITIVE AND MULTIPLICATIVE ABRUPT JUMP ... - IEEE Xplore

ADDITIVE AND MULTIPLICATIVE ABRUPT JUMP DETECTION USING THE CONTINUOUS WAVELET TRANSFORM Marie CHABERT, Jean- Yves TO URNERET and Bancis CASTANIE ENSEEIHT/GAPSE, National Polytechnics Institute of Toulouse 2, rue Camichel, 31071 Toulouse France Tel: (33) 61 58 83 67 Fax: (33) 61 58 82 37 email: chabertCQlen7.enseeiht.fr ABSTRACT This paper addresses the problem of additive and multiplicative abrupt jump detection. Two detection algorithms based on the Continuous Wavelet " s f o r m are studied. Additive and multiplicative jumps are compared in the time-scale plane. Considering non-zero mean signals, we show that the same wavelet based algorithms can be used for the detection of additive and multiplicative jumps. However, an analysis of the variance in the time-scale plane allows us to distinguish these two kinds of jump. 1. INTRODUCTION This study aims at comparing additive and multiplicative jump detection. For the sake of simplicity, we consider the case of a step embedded in an additive or multiplicative Gaussian process. The addition of a step results in a mean value jump. In the multiplicative case, we consider a signal which is not zero mean, so that the multiplication by a step results in simultaneous mean value and variance jumps. Our jump date detection strategy is based on the Continuous Wavelet Transform (CWT) [l].Additive and multiplicativejumps are compared in the timescale plane. This study shows that additive jump detection algorithms can be applied to multiplicative jumps when the signature of the resulting mean value jump in the time-scale plane is considered. Once the jump date is estimated, variance estimation, either in the time-scale plane or on the original signal, allows us to distinguish between additive and multiplicative jumps. 2. ADDITIVE AND MULTIPLICATIVE

in the problem of line-by-line edge detection on digitalized images. Each line appears as a piecewise constant signal embedded in a zero mean white Gaussian noise. Abrupt transitions between two mean values correspond to object contours. The signal under study is:

+ s,d(t)

gad(t) = z ( t )

+

= z ( t ) Am.U(t - t o )

(1)

where U ( t ) is the unit step located at unknown date to, Am is the jump amplitude and z(t) is a Gaussian zero mean stationary random process with correlation function R,(r) and power spectral density Sz(f). In the case of a mean and variance jump (multiplication by a shifted step), the signal is:

+

y m ( t ) = ~ ' ( t ) . ~ , (= t )d ( t ) ( l Am.U(t - t o ) )

(2)

where x'(t) is a non-zero mean white Gaussian noise. The signal y,(t) can be seen as a Gaussian process with piecewise constant mean value and variance. This kind of signal is considered in many applications. These a p plications include speckle signal on piecewise constant backgrounds in radar images, mechanical vibrations, non linear time series and random communication models. 3. DETECTION IN THE TIME-SCALE

PLANE The detection consists in determining the jump instant t o . Our strategy uses the CWT defined by:

where $(t) is an

L2 normalized wavelet.

JUMPS In the case of a mean value jump, a step is added to a Gaussian process. The resulting signal is a Gaussian process with constant variance and piecewise constant mean value [2]. This kind of signal is often used

0-7803-3 192-3/96 $5.0091996IEEE

3.1. Additive and Multiplicative Jump Signatures

The signature of an additive or multiplicative jump is r ) ]. defined by E [Cy(a,

3002

For an additive jump, owing to the wavelet transform linearity, the conic jump signature can be easily computed [ 11:

with x" = x' - m,t. In both cases, the SNR is proportional to the scale a, highlighting the interest of working in the timescale plane. This property, not only valid for a step, depends on the Lipschitz exponent of the jump. Indeed, the Wavelet Transform evolution across scales depends on the local regularity of the function. This regularity can be measured by the Lipschitz exponent a. Let f ( t ) E L2 (W) with Lipschitz exponent a. There exits a constant K such that the CWT of f ( t ) satisfies [3]:

Cs,,(a,7)is the additive conic jump signature, I,,,(t) = s f , $(u)du is the wavelet integral and C,(a, T ) is the noise CWT. In the multiplicative case, a similar relation can be obtained:

p4G? [f(t)]l I Ka"+*

(13)

Consequently, the evolution across scale of the jump signature amplitude, and so of the SNR, depends on the local regularity of the jump. The SNR increases with scale only if a 2 0. This property holds for a step since its Lipschitz exponent is equal to zero.

+

3.3. Optimal wavelet;

(a,T ) is deterministic whereas CSmu, ( a ,T ) Note that CSod depends on the signal x'(t) and thus is stochastic. A linear relation between the additive and multiplicative jump signatures can be obtained:

Thus, additive and multiplicative jumps have similar signatures, provided m,l # 0.

An optimal wavelet, maximizing the SNR has been derived in the additive case [l]. Due to the above SNR formulation, the same holds for the multiplicative case. We consider lthe SNR for large scale a: the optimal wavelet maximizes 1I+(O)l 2 . For a real wavelet : of bounded support [-i?f,

+%]

The Cauchy-Schwartz inequality leads to:

11+(0)12 5

3.2. Signal to Noise Ratio in the Time-Scale

f

?)2(u)du/o

-At12

Plane We consider the CWT in the region of the time-scale plane defined by the conic jump signature. When the scale a tends to infinity, the variance of the CWT is asymptotically constant, whereas the signature amplitudes increase continuously with a. This can be summarized by expressing the signal to noise ratios: Additive case

when a

-

(15)

The Cauchy-Schwartz inequality becomes an equality when 1c, is a constant over the interval [-%, 01 . Consequently, the symmetrical Haar wavelet belongs to the class of optimal sol.utions for a bounded support wavelet and an abrupt inultiplicative jump. 3.4. Suboptimal Detectors

-+-CO

Multiplicative case

when a

du

-At12

+CO

Owing to these similarities between additive and multiplicative jumps, it is ]possible to use the same jump date detection algorithm for both cases. The matched filter, which is the optimal detector [4],requires the CWT noise power spectral density, which may be unknown. Even if it is known, the noise is not white in the time-scale plane. This complicates the computation of the filter. For these two reasons, the matched filter is,

3003

in general, ixnrealizable. Consequently, we study the performance of two sub-optimal 2-D detectors in the time-scale plane. The first approach consists in computing the CWT correlation with 2-D signature: rl(T)=

JJ C, (a,

7)C,

(a,7 - t ) o ~ t

(17)

IR2

The second uses a sum along n k e d scale slices of the CWT modulus: n

F2(7)

=~Icy(~z,T)I

(18)

0

i=l

For these two detectors, the jump date and amplitude estimation in the time-scale plane a m o y t s to a maximum research. A jump date estimator t o can be derived as follows:

ri(g)=

M ri(,-) ~ ~

i =i,2

(19)

In the next section, we show that, in the multiplicative case, the jump amplitude, Am, can be estimated by considering the variance in the time scale plane.

time Fig.1: The time-scale plane

Region 3 is the conic jump signature. In regions 1 and 2, the expression of variance can be simplified: Region 1:

Region 2:

4. VARIANCE IN THE TIMESCALE

war [Cy, ( a , , - ) ]= (1

PLANE The variance of Cv(a,T ) is quite different in the multiplicative and additive cases. In the multiplicative case, the variance depends on the jump date. Due to the contribution of Cs,,,(a,T ) , the expression of variance is different in the three domains bounded by the conic jump signature. This property allows us to distinguish additive and multiplicative ,jumps. In the additive case [l], the variance can be expressed as follows:

+ Am)2/w R,r

(w)

(f) dw

(25) The variances in these two regions of the time-scale plane are proportional to a factor (1 Am)2.The same relation holds for the signal z/(t). This allows us to estimate the jump amplitude:

+

Once the jump date has been computed, its amplitude can be estimated by relation (25) or (26).

5. SIMULATIONS where R, and 4 are the correlation functions of the noise and of the wavelet:

4 (U) =

s,

$ ( U ) $*

(U

+ U>dU

(21)

+

(22)

R, (U)= E (z ( U ) z (U w)] In the multiplicative case, we obtain:

I).

wz7- [Cym(a,41 = - IE [Cy, (a, JJRZ

(U)

sm('u

+

U)$*

l2 -t

(F) ( y d)d u Sm(U)$

(23) Let us denote by At the wavelet support for scale a = 1. The time-scale plane can be divided into three regions bounded by the conic jump signature (see Fig.1).

As an example, we consider N=2048 samples of a Gaussian distributed random sequence with m,! = 1 and a:, = 1. In the additive and multiplicative case, the jump date is to =lo00 and its amplitude Am = 0.5. The CWT is computed for scale varying from 3 to 7 with the symmetrical Haar wavelet of support 40. Figures 2 to 5 show the results of the 2-D detectors formed by the 2-D correlation with the signature and by the CWT modulus sum along fixed scale slices. It is easy to see that the qualitative behavior of the two approaches is very similar. Other detectors, using a fixed scale CWT slice or the CWT correlation with 1-D signature, can be considered. 2-D detectors should be prefered as they take advantage of information redundancy in the time-scale plane.

3004

5D

43 1)

a, 10

0

500

15co

1 0

0 0

am,

500

lax,

1m

am,

Fig.5: Multiplicative jump - ICWTl sum along fixed scale slices (maximum for T = 1007).

Fig. 2: Additive jump - 2-D correlation (maximum for T = 1003)

6. CONCLUSION

”1

I

t

43

0

503

1033

1503

am,

Fig.3: Additive jump - [CWTl sum along k e d scale slices (maximum abscissa in T = 999)

We have shown that the additive and mutiplicative jump signatures are very similar in the case of a nonzero mean signal. It allows us to apply additive jump detection algorithms to mmltiplicative jumps. The estimation of the jump date can then be implemented as in the additive case. Once tlhe jump date is estimated, we can distinguish additive From multiplicative jumps by estimating the variance in the time-scale plane before and after the conic signature, or on the initial signal before and after the jump date. The next step in our study will be to analyze multiplicative jump detection in the case of a zero mean signal (variance jumps). A non-linear transform on the resulting signal leads to simultaneous variance and mean jumps. This allows us to use the approaches detailed in this paper. 7. REFERENCES [l]A. Denjean and F. Castanik, “Mean Value Jump Detection: a Survey of Conventional and Wavelet

Based Methods,” in Wavelet Theory, Algorithms and Applications,C. I