Multiple Scale Correlation of Chirp Signal by ... - Semantic Scholar

MULTIPLE SCALE CORRELATION OF CHIRP SIGNAL BY DISCRETE WAVELET TRANSFORM Chinmoy Bhattacharya DEAL(DRDO), Dehradun, India [email protected]

Jibanananda Roy INTERRA SYSTEMS,Kolkata, India [email protected]

ABSTRACT Chirp Signal, a finite duration pulse with Linear Frequency Modulation (LFM) is the signal waveform widely used for coherent image formation. Coherent imagery is derived essentially as a 2-dimensional correlation of received, delayed chirp waveform with a 2dimensional matched filter. There is wide and active research interest in analyzing coherent imagery at multiple scales for multiresolution speckle reduction, clutter separation from desired targets and other low level vision requirements such as multiple scale segmentation, etc. This paper attempts to address these requirements during the image formation process; i.e., obtaining multiple resolution imagery by signal correlation at multiple scales. We derive a shift and scale invariant DWT algorithm, which functions as the cornerstone for such multiple scale correlation. We also briefly discuss implications of this image formation algorithm with particular attention to speckle reduction. 1. INTRODUCTION Coherent imaging sensors such as Synthetic Aperture Radar (SAR), Synthetic Aperture Sonar (SAS) frequently employ Linear FM (LFM) chirp signal as the transmitting waveform. The advantage in using LFM chirp waveform is that both the range and cross range matched filters can be approximated as quadratic functions in time whose scales are widely different in the two dimensions of range and cross range. Images formed by such coherent sensors as in SAR, in laser focusing [6] appear with a peculiar grainy appearance called speckle due to coherent averaging of Gaussian diffused background in the resolution cell. Traditional multilook processing techniques reduce speckle in SAR imagery at the cost of spatial resolution. Multiresolution techniques for speckle reduction have been attempted by decomposition of speckled imagery in discrete wavelet transform (DWT) domain [2]. Apart from this, it has been observed that the probability of detecting target signature against background clutter considerably improves in multiresolution setting [10]. Variation of target signature is distinctly different from random variation of clutter as a function of resolution. Such multiresolution DWT based techniques operate in image domain, which either are intensity images or

0-7803-7750-8/03/$17.00 ©2003 IEEE.

Avijit Kar Jadavpur University, Kolkata, India [email protected]

complex image data itself. Because of time-frequency interlocking in LFM chirp signal, the approximation (low frequency variation) and detail (high frequency variation) components of LFM chirp can be nicely separated in orthogonal DWT domain. This point to the possibility of achieving multiresolution in the signal domain itself by correlation of multiple scale chirp signals. One important shortcoming of DWT operation is the shift variance of the transform coefficients over the scales of transform making the proposition of multiple scale signal correlation difficult. Correlation at any scale of observation requires both even and odd circular shifts, thus the transform is oversampled and redundant. Shift invariant DWT (SIDWT) is attempted at multiple scales of analysis by searching for the best basis that best represents the transform coefficients in a shift invariant manner [7]. Such best basis selection necessarily requires a binary tree search algorithm and at best is an approximation for the orthogonal transform. In this paper we present an algorithm that circumvents shift variance by utilizing the orthogonal property of DWT in a novel way without taking the route of tree search algorithm [7]. Multiple scale correlation is performed over DWT coefficients of SAR signals and is the matter of Section 2. In Section 3 we briefly show the speckle model in signal domain and analyze how the present algorithm be utilized in reducing the effect of speckle in DWT domain. Section IV is a discussion on the implications of the algorithm in coherent image formation. 2. CHIRP CORRELATION IN DWT DOMAIN Chirp signal correlation in DWT domain was attempted in [8]; there the authors decomposed the chirp at multiple scales using orthogonal and bi-orthogonal wavelet kernels; but correlation at multiple scales was done in spectral domain using standard Fourier transform results. However the problem of shift variance of DWT coefficients is highlighted but not solved in DWT domain. Our approach starts with Beylkin’s algorithm [1] to produce SIDWT coefficients as shown in Fig.1. The Quadrature Mirror Filter (QMF) Bank produces the familiar 2-scale average and detail orthogonal DWT coefficients at the next higher scale. Stepping from scale

ICIP 2003

b e i1 (n)

xe (n)

2-channel QMF Bank 2

x (n) a ee i1 ( )

2-channel QMF Bank

z-1

z-1

z-1

2

z-1

a eoi1 (n)

xo (n)

b i1o(n) 2-channel QMF Bank 2

a oei1(n) z-1

z-1

z-1

a ooi1(n)

Scale 1

Scale 0

Scale 2

Fig.1 SIDWT decomposition upto three scales for signal x (n)

to scale we double the number of input vectors: i) coefficient produced from the original vector and ii) coefficients produced by the vector shifted by one sample. The length of each output vector of QMF bank gets halved because of dyadic downsampling. Thus for a signal vector of length N = 2n, there is an O (Nlog2 N) computation cost in stead of O (N2) for brute force SIDWT calculation at log2 N stages. Since we are interested only in approximation signal at the next higher scale the detail coefficients are not further analyzed as is done in wavelet packet transform. For two discrete signal vectors x (n) and y (n), the circular correlation is the left circulant matrix product,

x(n) o y(n) = X y ,

(1)

X is the left circulant matrix of the matching vector x (n). Orthogonal wavelet transform maintains the energy equation in the l 2 norm sense, i.e., if the two vectors are decomposed in their detail and approximate orthogonal components as,

x(n) = ∑ ∑ bij ψij (n) + ∑ aiJ φiJ ( n ) j i

(3)

j

the subscript j stands for 2 th scale approximation; WT is the symbol for orthogonal DWT operation on the approximation signal xj, yj. Equation (3) is a brute force method as at every scale one needs to synthesize the approximations xj(n), yj(n); recompute the DWT over the left circulant matrix Xj for the next higher scales. The transform vectors ψij(n) and ϕij (n) maintain orthogonality over even shifts at a scale 2j . For k = 2m, 〈 x j ( n + k ), y j ( n )〉 = 〈 b e i , ( j + 1) , d i , ( j + 1) 〉

+ ...

〈 a i , ( j + 1) , c i , ( j + 1) 〉 (4) where be(i+m), (j+1) = bi, (j+1) ; ae(i+m),(j+1) = ai,(j+1), are the 2scale DWT coefficients for even shifted vector xj(n+k) at 2j th scale of analysis. e

+ ...

〈 a o i ,( j +1) , ci ,( j +1) 〉 (5)

y( n ) = ∑ ∑ dij ψij ( n ) + ∑ ciJ φiJ ( n ), i

Then,

〈 x ( n ), y ( n )〉 = ∑ 〈bij ,d ij 〉 + 〈 aiJ , ciJ 〉

x j (n) o y j ( n ) = W T X j W T y j ,

〈 x o j (n + 2m ), y j ( n)〉 = 〈b o i ,( j +1) , d i ,( j +1) 〉

i

j i

scaling basis vector ϕ iJ (n) at the coarsest scale 2J. Hence from (1) and (2),

(2)

j

Here Σj stands for summation of orthogonal detail signals over dyadic scales 21 ≤ 2j ≤ 2J; Σi is the sum over shifts i for the wavelet basis vector ψij(n) at a scale 2j and for the

Equation (5) is for odd shifts in the circulant matrix at 2jth scale, with k = 2m +1, so that xoj(n+2m) = xj(n +k); bo(i+m),,(j+1) = boi,,(j+1) ; ao(i+m),,(j+1)= aoi,,(j+1) are the 2-scale DWT coefficients for odd shifted vector xoj(n) shifted 2m indices. From (4) and (5) it is seen that (3) can be implemented with even and odd indexed circulant shifts of DWT coefficients and interlacing the inner product operations

shown in (4) and (5). Shift invariance of correlation at 2jth scale of analysis is represented by (4) and (5) together. For scale invariance, we repeat the algorithm as shown in (4) and (5) with left circulant matrices of resultant DWT coefficients. For two dyadic, coarser scales of decomposition other than the original scale of input (j= 0) in Fig.1 the resultant DWT circulant matrices form a Kronecker product. Multiple scale correlation is, WT xe WT ai1ee ....    WT ai1eo....  (6) x(n) o y(n) =   WT y o WT aioe WT x 1 ....    WT aioo  1 ....   Here WT stands for left circulant matrix of DWT coefficients at each coarse scale; xe is the matching vector itself and xo is the odd shifted vector xe(n+1). It is to be noted that the scale and shift invariance of (6) is independent of signal waveform and orthogonal wavelet kernel size or shape. The simulation result for a real, delayed, chirp signal at three dyadic scales of analysis including the original scale is shown in Fig.3 (a)-(c).The target delay offset in Fig.2 is taken 83 time samples, the analyzing orthogonal wavelet is ‘daubchies5 (db5)’.

Transmitted LFM Chirp Waveform

1 0.5 0 -0.5 -1 -4

-3

-2

-1 0 1 2 3 time in sec. Delayed,Attenuated Return Chirp (Single Target)

1

4 -6

x 10

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-4

-2 0 2 Time in sec.(Range Direction)

4

6 -6

x 10

Fig.2. LFM Chirp (Upper), Delayed Return Chirp (lower plot) Multiple Scale Correlation for Return Chirp in Fig.2

80

(a) Scale 0 Matching('db5')

60 40 20 0 1.8

2

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40 (b) Scale 1 Matching('db5')

~ ~ ( n ), WT ~ x (n) = ~ x (n ) ⊗ ψ j −j/2 * ~ ψ (n ) = 2 ψ (−n / 2 j )

with

(7)

j

In (7) the ‘tilde’ symbols represent analytical form of real expressions, ⊗ stands for circular convolution. The transmitted analytical chirp waveform is given by, ~ x (n) = exp(iπα (nT f ) 2 )rect (nT f / T p ) (8)

In (8), α is the rate of frequency sweep in the pulse duration represented by the rect (.) function; Tf = (1/α Tp) is the sampling interval for the analytical signal. The target backscatter in the delayed chirp is the ensemble of all unresolved elements’ return, which here is represented as a point target of average reflectivity σ,

~ y (n) = σ exp(iπα (nTf −τ 0 ) 2 ) rect((nTf −τ 0 ) / Tp ) (9)

The delay, τ0 is determined with a resolution uncertainty of (1/αTp) in the sensor by matched filtering with the inverted replica of transmitted pulse; the complex response of matched filter in range direction is, ~ r (τ 0 ) = ~ x ( n) o ~ y ( n)

(10)

20 0 1.8 c o rre la t io n m a g n it u d e

2.1 Multiple scales correlation for complex radar signals The multiple scale correlation algorithm shown above in DWT domain is for real signals. SAR signals are complex in form necessitating complex DWT operation; complex DWT is the analytical form of the QMF bank of Fig.1.

2

2.2

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2.6

2.8

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3.2

20 (c) Scale 2 Matching('db5') 10 0

-10 1.8

2

2.2

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2.6 2.8 time in microsec.

3

3.2

Fig.3. Correlation at multiple scales by SIDWT for single target return shown in Fig.2 (lower plot)

The key point in SAR is its cross range resolution formation. If range migration in the phase history of backscatter in cross-range is neglected, the cross range response may also be represented as matched filter expression in (10). The 2-dimensional correlation result at original scale is given by [3], v ( mT s , nT f ) = σ exp( − i 4 π R 0 / λ ) ×

sin c ( π f R τ int ( mT s − s 0 )) × sin c ( πα T p ( nT f − τ 0 ))

(11 )

In (11) Ts is the pulse repetition interval of the radar; R0 is the shortest range to a target whose ground co-ordinates are proportional to cross-range and range delay (τ0, s0). Frequency sweep rate in cross-range is fR and coherent integration interval is τint; sin c(.) is the sinc function. The

cross-range time resolution (1/ fRτint) as like range resolution. Following (6) and (7) the complex correlation in (10) may be implemented in complex DWT domain with, ~ (n) = 2 ~ ~ ( n )), and ~ x (n) ⊗ ψ x ( n) ⊗ Re( ψ j j ~ ~ ~ ~ x ( n ) o y ( n) = W xW y (12 )

Scale 0('db3') 10 0 -10 0.25

0.3

0.35

T

0.4

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Scale 1('db3')

10 0

Correlation Mag.

T

Equation (12) represents the multiple scale correlation of analytical return chirp signal with the analytic transmitted signal in SIDWT domain. Fig.4 shows the real part of phase return in cross-range for 3 targets(upper plot) and the matching phase function (lower plot). Complex SIDWT correlation in cross-range with ‘db3’ wavelet is shown in the three plots of Fig.5. 3. SPECKLE REDUCTION IN SIGNAL DOMAIN

Multiple Scale Cross Range Correlation(3 Targets)

20

-10 0.25 10

0.3

0.35

0.4

Scale2 ('db3')

0 -10 0.3

0.35

0.4

0.45 0.5 0.55 Time in sec.(Cross- Range)

Fig.5. SIDWT Correlation . 3 Targets at Three scales value is the approximation of σs2for

CORRELATION

4. DISCUSSION

Speckle reduction techniques in image domain start by assuming speckle as multiplicative noise. In image domain homomorphic transform separates the multiplicative speckle noise into additive terms but such methods produce bias in the transform and speckle model is approximately Gaussian [9]. In complex signal domain speckle is considered additive, Gaussian, i.i.d. variable with zero mean [6]. The target backscatter with speckle from a single resolution cell from (9) may be modeled as, ~z ( n ) = ~ y ( n ) + σ s N [ 0 .1] (13 ) In orthogonal wavelet transform domain equation (13) is

Correlation of chirp signal at multiple scales as shown in Fig.3 demonstrates the multiresolution nature of the algorithm. At each higher scale the chirp rate becomes half of the preceding lower scale. Coarse resolution width gets doubled; correlation gain also is halved simultaneously. Formation of the image at multiple scales essentially is a transform operation; hence fine resolution imagery may be derived from coarse resolution ones which is reverse of conventional image formation methods. The merger of adjacent resolution cells at coarser scale is shown in Fig.5. Another implication of SIDWT is speckle reduction in fully coherent manner.

Real part of phase return

WT z = WT y + σ s ω (14) Here σs is the standard deviation of the speckle noise component; for convenience we show the real part only and ω is transform domain normal variable. To apply threshold limit in DWT domain estimate of σs is needed for which computation of the median of DWT coefficient vector is adopted in [5]. We take the conjecture that the average reflectivity of target σ is a slowly varying function and dominate the approximation coefficients of (14). In that case the norm of the difference of transform coefficient vector WTz of noisy backscatter from coefficient vector of reference chirp given in (8) produces an estimation vector for speckle variance [4]; the median Cross-range Phase Return from 3 Targets

4 2 0 -2 -4 0.4

0.6

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1

1.2

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Cross-Range Matching Function

1 0.5 0 -0.5 -1 -0.4

-0.3

-0.2

-0.1 0 0.1 Time in sec.(Cross-Range)

Fig.4. Cross –Range Phase Signal from 3 Targets (Real Part), Matching Phase Function

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