objective of image registration is to find a spatial transformation such that a .... Proof. The domain of JR!α is a convex polytope in which the ver- tices are ...
AN INFORMATION DIVERGENCE MEASURE FOR ISAR IMAGE REGISTRATION Yun He, A. Ben Hamza, Hamid Krim Department of Electrical and Computer Engineering North Carolina State University, Raleigh, NC 27695-7914, USA E-mails: {abhamza, yhe2, ahk}@eos.ncsu.edu ABSTRACT Entropy-based divergence measures have shown promising results in many areas of engineering and image processing. In this paper, a generalized information-theoretic measure called Jensen-R´enyi divergence is proposed. Some properties such as convexity and its upper bound are derived. Using the Jensen-R´enyi divergence, we propose a new approach to the problem of ISAR (Inverse Synthetic Aperture Radar) image registration. The goal is to estimate the target motion during the imaging time. Our approach applies JensenR´enyi divergence to measure the statistical dependence between consecutive ISAR image frames, which would be maximal if the images are geometrically aligned. Simulation results demonstrate a much improved performance of the proposed method in image registration. 1. INTRODUCTION Image registration is an important problem in computer vision, remote sensing, data processing and medical image analysis. The objective of image registration is to find a spatial transformation such that a dissimilarity metric achieves its minimum between two or more images taken at different times, from different sensors, or from different viewpoints. Inverse Synthetic Aperture Radar (ISAR) [1] is a microwave imaging system capable of producing high resolution imagery from data collected by a relatively small antenna. The ISAR imagery is induced by target motion, however, motion also blurs the resulting image. After conventional ISAR translational focusing process, image registration can be applied to estimate the target rotational motion parameter, then polar re-formating can be used to achieve a higher resolution image. During the last three decades, a wide range of registration techniques have been developed for various applications. These techniques can be classified [2] into correlation methods, Fourier methods, landmark mapping, and elastic model-based matching. In the work of Woods [3] and Viola [4], mutual information, a basic concept from information theory, is introduced as a measure for evaluating the similarity between images. When the two images are properly matched, corresponding areas overlap, and the resulting joint histogram contains high values for the pixel combinations of the corresponding regions. When the images are misregistered, non-corresponding areas also overlap and this will result in additional pixel combinations in the joint histogram. In case of misregistration, the joint histogram has less sharp peaks This work was supported by an AFOSR grant F49620-98-1-0190 and by ONR-MURI grant JHU-72798-S2 and by NCSU School of Engineering.
and is more dispersed than the correct alignment of the images. The registration criterion is then to find the transformation such that the mutual information of the corresponding pixel pair intensity values in the matching images is maximized. This approach is accepted by many [5] as one of the most accurate and robust registration measures. In this paper, a novel generalized information theoretic measure, called Jensen-R´enyi divergence and defined in terms of R´enyi entropy [6] is introduced. Jensen-R´enyi divergence is defined as the similarity measurement of any finite number of weighted probability distributions. Shannon mutual information is a special case of the Jensen-R´enyi divergence. This generalization endows us the ability to control the measurement sensitivity of the joint histogram, which would end up with a better registration result. In the section that follows, we give a brief statement of the problem. In section 3, we introduce the Jensen-R´enyi divergence and its properties. Section 4 is devoted to the application of the Jensen-R´enyi divergence in ISAR image registration. Finally, we provide some concluding remarks in the section 5. 2. PROBLEM STATEMENT ISAR imagery registration can be applied to estimate the target motion during the imaging time. Let T(l,θ,γ) be a Euclidean transformation with translational parameter l = (lx , ly ), rotational parameter θ and scaling parameter γ. Given two ISAR images f and r, the objective of image registration is to determine the spatial transformation parameters (l∗ , θ∗ , γ ∗ ) such that (l∗ , θ ∗ , γ ∗ )
=
! (p (f, T(l,θ,γ) r), arg max JRα 1 (l,θ,γ)
(1)
p2 (f, T(l,θ,γ) r), . . . , pn (f, T(l,θ,γ) r)) ! (·) is the Jensen-R´enyi divergence with order α and where JRα weight ! . Denote X = {x1 , x2 , . . . , xn } and Y = {y1 , y2 , . . . , yk } the sets of pixel intensity values of f and T(l,θ,γ) r respectively, then ωi = P (X = xi ) and pi (f, T(l,θ,γ) r) = (pij )1≤j≤k . pij = P (Y = yj |X = xi ), i = 1, 2, . . . , n, j = 1, 2, . . . , k is the conditional probability of Y = yj given X = xi for the corresponding pixel pairs in f and T(l,θ,γ) r. Here the Jensen-R´enyi divergence acts as a similarity measure between images, which will be explained further in the next section. ISAR imagery is induced by target motion, however, the target motion causes time-varying spectra of the received signals. Motion compensation has to be carried out to obtain a high resolution image. As the radar keeps tracking the target, the reflected signal is continuously recorded during the imaging time. By registration of a sequence of consecutive image frames, {fi }N i=0 , the target
motion during the imaging time can be estimated by interpolating {(li , θi , γi )}N i=1 . Then based on the trajectory of the target, translational motion compensation (TMC), and rotational motion compensation (RMC) [1] can be used to generate a clear image of the target. ´ 3. THE JENSEN-RENYI DIVERGENCE Let k ∈ N and X = {x1 , x2 , . . . , xk } be a finite set with a probPk ability distribution p = (p1 , p2 , . . . , pk ), i.e. j=1 pj = 1 and pj = P (X = xj ) ≥ 0, where P (·) denotes the probability. R´enyi entropy is a generalization of Shannon entropy, and is defined as [6] Rα (p) =
X α 1 log pj , 1−α j=1 k
α > 0 and α 6= 1.
(2)
For α > 1, the R´enyi entropy is neither concave nor convex. For α ∈ (0, 1), it is easy to see that R´enyi entropy is concave, and tends to Shannon entropy H(p) as α → 1. It can be easily verified that Rα is a non-increasing function of α, and hence Rα (p) ≥ H(p),
∀α ∈ (0, 1).
(3)
Definition 1 Let p1 , p2 , . . . , pn be n probability distributions of X Pnand ! = (ω1 , ω2 , . . . , ωn ) be a weight vector such that enyi divergence is defined i=1 ωi = 1 and ωi ≥ 0. The Jensen-R´ as
! (p , . . . , p ) = Rα JRα 1 n
n X i=1
!
ωi pi
−
n X
ωi Rα (pi ),
i=1
where Rα (p) is the R´enyi entropy, α > 0 and α 6= 1. Using the Jensen inequality, it is easy to check that the JensenR´enyi divergence is nonnegative for α ∈ (0, 1). It is also symmetric and vanishes if and only if the probability distributions p1 , p2 , . . . , pn are equal, for all α > 0. When α → 1, the Jensen-R´enyi divergence is exactly the generalized Jensen-Shannon divergence [7]. Unlike other entropy-based divergence measures such as the well-known Kullback Leibler divergence, the Jensen-R´enyi divergence has the advantage of being symmetric and generalizable to any finite number of probability distributions, with a possibility of assigning weights to these distributions. In the sequel, we will restrict α ∈ (0, 1), unless specified otherwise, and will use a base 2 for the logarithm, i.e., the measurement unit is in bits. The following result establishes the convexity of the JensenR´enyi divergence of a set of probability distributions.
! Proposition 1 For α ∈ (0, 1), the Jensen-R´enyi divergence JRα is a convex function of p1 , p2 , . . . , pn . Proof. Recall that the mutual information between two finite sets X = {x1 , x2 , . . . , xn } and Y = {y1 , y2 , . . . , yk } is given by [8] I(X; Y ) = H(Y ) − H(Y |X),
(4)
where H(Y ) is the Shannon entropy of Y and H(Y |X) is the conditional Shannon entropy of Y , given X.
Instead of using Shannon entropy in (4), the mutual information can be generalized using R´enyi entropy. Therefore, the αmutual information can be defined as Iα (X; Y ) = Rα (Y ) − Rα (Y |X), where Rα is the R´enyi entropy of order α ∈ (0, 1). Denote by P (X = xi ) = ωi , P (Y = yj |X = xi ) = pij and P (Y = yj ) = qj , then it is easy to check that
! (p , p , . . . , p ), Rα (Y ) − Rα (Y |X) = JRα 1 2 n
(5)
where pi = (pij )1≤j≤k , for all i = 1, . . . , n. For fixed ωi , the mutual information is a convex function of pij [8], then it can be verified that the α-mutual information is also a convex function of pij , leading to the Jensen-R´enyi divergence a convex function of p1 , p2 , . . . , pn . Proposition 2 The Jensen-R´enyi divergence achieves its maximum value when p1 , p2 , . . . , pn are degenerate distributions.
! is a convex polytope in which the verProof. The domain of JRα tices are degenerate probability distributions. That is, the maximum value of the Jensen-R´enyi divergence occurs at one of the degenerate distributions. Since the Jensen-R´enyi divergence is a convex function of
p1 , p2 , . . . , pn , then it achieves its maximum value when the R´enyi entropy function of the ! -weighted average of degenerate probability distributions, achieves its maximum value too. Next problem is to assign weights ωi to the degenerate distributions ∆1 , ∆2 , . . . , ∆n , that is to say, an assignment {ωi } −→ ∆i = {δij } must be found, where {δij } are probability mass functions, i.e. δij = 1 if i = j and 0 otherwise. The following upper bound thus holds n X
! ≤ Rα JRα
!
ωi ∆ i
.
(6)
i=1
Without loss of generality, consider the Jensen-R´enyi divergence with equal weights ωi = 1/n for all i, and denote it simply by JRα . Using (6), the following holds α JRα ≤ Rα (a) + log(n), (7) α−1 where
a = (a1 , a2 , . . . , ak )
such that
aj =
n X
δij .
(8)
i=1
Since Pk ∆1 , ∆2 , . . . , ∆n are degenerate distributions, then we have j=1 aj = n. From (7), it is clear that JRα achieves its maximum value when Rα (a) achieves its maximum too. In order to maximize Rα (a), the concept of majorization will be used [9]. Let (x[1] , x[2] , . . . , x[k] ) denote the non-increasing arrangement of the components of a vector x = (x1 , x2 , . . . , xk ). Definition 2 Let written a ≺ b, if
a and b
∈
Nk , a is said to be majorized by b,
( P P k a[j] = kj=1 b[j] Pj=1 P` ` j=1
a[j] ≤
j=1 b[j] ,
` = 1, 2, . . . , k − 1.
Since Rα is Schur-concave function, then Rα (a) ≥ Rα (b) whenever a ≺ b. The following result establishes the maximum value of the Jensen-R´enyi divergence. Proposition 3 Let with
p1 , p2 , . . . , pn
pi = (pi1 , pi2 , . . . , pik ),
Pk j=1
be n probability distributions pij = 1,
pij ≥ 0.
If n ≡ r (mod k), 0 ≤ r < k, then 1 JRα ≤ log 1−α
(k − r)q α + r(q + 1)α (qk + r)α
,
(9)
where q = (n − r)/k, and α ∈ (0, 1). Proof. It is clear that the vector z
r
}|
{ z k−r }| {
g = (q + 1, . . . , q + 1, q, . . . , q)
Transform processing of the polar formatted data would result in blurring at the edges of the target reflectivity image. Fig. 1 is a synthetic ISAR image of an aircraft MIG-25 [10]. The radar is assumed operating at 9GHz and transmits a stepped-frequency waveform. In each burst, 64 stepped frequency are used. The pulse repetition frequency is 15KHz. Basic motion compensation processing has been applied to the data. A total of 512 samples of the time history series are taken to reconstruct the image of this aircraft, which corresponds to 2.18s of integration time. As we can see, the resulting image is defocused due to the target rotation. In fact, the defocused image in Fig. 1 is formed by overlapping a series of MIG-25s at different viewing angles. By replacing the Fourier transform with the time varying spectral analysis techniques [11], we can take a sequence of snapshots of the target during the 2.18s of integration time. Fig. 2 shows the trajectory of the MIG-25, with 6 image frames taken at t = 0.1280s, 0.4693s, 0.8107s, 1.1520s , 1.4933s, 1.8347s respectively. (1)
(2)
(3)
(4)
(5)
(6)
is majorized by the vector a defined in (8). Therefore, Rα (a) ≤ Rα (g ). This completes the proof using (7). According to proposition 3, when n ≡ 0 (mod k) the following inequality holds JRα (p1 , p2 , . . . , pn ) ≤ log(k).
4. ISAR IMAGE REGISTRATION
Fig. 2. Trajectory of a sequence of MIG-25 image frames Then image registration can be applied to estimate the target motion in its trajectory. In this specific example, given a sequence of ISAR image frames {Ii }N i=0 which are observed in a time interval [0, T ], we search for the rotation angle {θi }N i=1 . Denote r = Ii−1 and f = Ii for i = 1, 2, . . . N , then by Equation (1), θi is given by
! (p (f, Tθ r), . . . , p (f, Tθ r)). θi∗ = arg max JRα 1 n i i θi
Fig. 1. ISAR image of moving target reconstructed by the Discrete Fourier Transform To form a radar image, N bursts of received signals are sampled and organized burst by burst into a M × N two-dimensional array. This sample matrix is not uniformly spaced in the spatial frequency, instead, it is polar formatted data. The Discrete Fourier
Fig. 3 shows the rotation angles {θi }N i=1 obtained by registering the 6 consecutive MIG-25 image frames. As we can see, α plays an important role in controlling the measurement sensitivity. When α < 1, the peak of the JR-divergence vs θ is much shaper than the traditional Shannon entropy (α = 1) based mutual information method. Obviously, a sharper peak would help to obtain a more accurate estimate of the true rotation angle. By interpolating {θi }N i=1 , we obtain a trajectory of the MIG25 rotational motion during the imaging time as shown in Fig. 3, then the polar re-formating [1] can be used to re-sample the received signal into rectangular format and generate a clear image of the MIG-25 based on all the received signals in the time interval [0, 2.18s], as demonstrated in Fig. 4.
(1)
(2)
0.8
0.8 α=0.3 α=1.0 α=2.0
0.7 0.6
0.6
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d(f2,f3)
d(f1,f2)
0.5
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α=0.3 α=1.0 α=2.0
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0.8 α=0.3 α=1.0 α=2.0
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α=0.3 α=1.0 α=2.0
0.7 0.6 d(f4,f5)
0.5
d(f3,f4)
0.5 0.4
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(5)
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(6)
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25 α=0.3 α=1.0 α=2.0
0.7 0.6
6. REFERENCES 20
[1] D.R. Wehner, High Resolution Radar, 2nd edition,Artech House Inc., Norwood, MA 02062, USA, 1995.
15 θ
d(f5,f6)
0.5 0.4
10
[2] L. Brown, “A Survey of Image registration Techniques,” ACM Computing Surveys, vol. 24, co. 4, pp. 325-376, 1992.
0.3 5 0.2 0.1
Fig. 4. Reconstructed MIG-25 by polar reformatting
0 0
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θ
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t
Fig. 3. Image registration of a MIG-25 Trajectory
5. CONCLUSIONS A generalized information-theoretic divergence measure based on the R´enyi entropy is proposed in this paper. We proved the convexity of this divergence measure and derived its maximum value. Using the Jensen-R´enyi divergence, we propose a new approach to the problem of ISAR image registration. The ISAR imagery is induced by target rotation, which in turn causes time varying spectra of the reflected signals and blurs the target image. The goal of ISAR image registration is to estimate the target motion for further motion compensation processing. Our approach applies JensenR´enyi divergence to measure the statistical dependence between consecutive ISAR image frames, which would be maximal if the images are geometrically aligned. Compared to the mutual information based registration techniques, the Jensen-R´enyi divergence endows us the ability to control the measurement sensitivity of the joint histogram. This flexibility would result in a better registration accuracy. Maximization of the Jensen-R´enyi divergence is a very general criterion, because no assumptions are made regarding the nature of this dependence and no limiting constraints are imposed on image contents. Simulation results demonstrate that our approach obtains an accurate estimation of target rotation automatically without any prior feature extraction.
[3] R.P. Woods, J.C. Mazziotta, S.R. Cherry, “MRI-PET registration with automated algorithm,” J. Comput. Assist. Tomogr., vol. 17, no. 4, pp. 536-546, 1993. [4] P. Viola and W. M. Wells, “Alignment by maximization of mutual information,” International Journal of Computer Vision, vol. 24, no. 2, pp. 173-154, 1997. [5] F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, P. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Trans. on Medical Imaging, vol. 16, no. 2, pp. 187-198, 1997. [6] A. R´enyi, On Measures of Entropy and Information, Selected Papers of Alfr´ed R´enyi, vol.2, pp. 525-580, 1961. [7] J. Lin, “Divergence Measures Based on the Shannon Entropy,” IEEE Trans. Information Theory, vol. 37, no. 1, pp. 145–151, 1991. [8] G. Gallager, Information Theory and Reliable Communications, John Willey Sons, 1968. [9] A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, 1979. [10] V.C. Chen and S. Qian, “Joint Time Frequency Transform for Radar Range-Doppler Imaging,” IEEE Trans. Aerospace and Electronic Systems, vol. 34, no. 2, pp. 486-499, 1998. [11] Y. He, A. Ben Hamza, H. Krim, V.C. Chen, “An information theoretic measure for ISAR imagery focusing,” Proc. SPIE, vol. 4116, 2000.
