A Bayesian approach to Fourier Synthesis inverse

Mono and Bistatic SAR Imaging geometries and the Fourier domain data. ..... which gives the possibility of jointly segmenting and reconstruction [18, 19, 20, 21].
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A Bayesian approach to Fourier Synthesis inverse problem with application in SAR imaging Sha ZHU and Ali Mohammad-Djafari Laboratoire des Signaux et Systèmes, Unité mixte de recherche 8506 (CNRS-SUPELEC-UNIV PARIS SUD) Supélec, Plateau de Moulon, 3 rue Juliot-Curie, 91192 Gif-sur-Yvette, France Abstract. In this paper we propose a Bayesian approach to the ill-posed inverse problem of Fourier synthesis (FS) which consists in reconstructing a function from partial knowledge of its Fourier Transform (FT) with application in SAR (Synthetic Aperture Radar) imaging. The function to be estimated represents an image of the observed scene. Considering this observed scene is mainly composed of point sources, we propose to use a Generalized Gaussian (GG) prior model, and then the Maximum A posterior (MAP) estimator as the desired solution. In particular, we are interested in bi-static case of spotlight-mode SAR data. In a first step, we consider real valued reflectivities but we account for the complex value of the measured data. The relation between the Fourier transform of the measured data and the unknown scene reflectivity is modeled by a 2D spatial FT. The inverse problem becomes then a FS and depending on the geometry of the data acquisition, only the set of locations in the Fourier space are different. We give a detailed modeling of the data acquisition process that we simulated, then apply the proposed method on those synthetic data to measure its performances compared to some other classical methods. Finally, we demonstrate the performance of the method on experimental SAR data obtained in a collaborative work by ONERA1 .

INVERSE PROBLEM OF FOURIER SYNTHESIS Fourier Synthesis in imaging As in many imaging systems, in SAR imaging, the main mathematical inverse problem becomes the Fourier Synthesis (FS) which consists in estimating an unknown function f (x, y) (scene) from some partial and truncated information of its Fourier Transform (FT). If we note by F(u, v) the FT of f (x, y), by G(u, v) the observed data in the Fourier domain (often obtained by a transformation of the observed signals) and by M(u, v) a binary function which is equal to one on the points where we have the data and zero elsewhere, then we can write the forward problems as

where F(u, v) =

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The French Aerospace Lab

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FIGURE 1. Mono and Bistatic SAR Imaging geometries and the Fourier domain data. Upper row shows the mono static case and lower row shows the bistatic case. In the equations k = 2π/ω represents wavenumber.

The points where the function M(u, v) is equal to one depends on the geometry of the SAR imaging. Two particular cases of mono and bistatic geometries are shown in Figure 1.For a given scene, considering 2D case, in mono-static case where the receiver and emitter antennas are colocated, for each angle θ of data, we obtain the line in the Fourier domain kx , ky . And the length of this line is proportional to the bandwidth of the emitted signal. In the bi- or multi-static case, we have to account for the time delay between the transmitter and the centre of the object, and the time delay between the object centre and the receiver. As shown in Figure 1 (lower part), that we have the same information in the Fourier domain but the shape of the points where we have information is more complexed. For further details see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] Evidently, if we could obtain M(u, v) = 1 everywhere in the Fourier domain, then the solution to the problem would be evident: fb(x, y) = IFT{G(u, v) M(u, v)} = IFT{G(u, v)}.

(3)

But, we know that this is not the case. However, the classical methods of inversion use this assumption. But, this means that all the non observed information (the values of G(u, v) on those points where M(u, v) = 0) are assumed to be zero. As we will see later, this is not the best we can do.

The main objective of this work is to propose effective Bayesian inversion method where we can include all the prior information we may know about the unknown scene in the form of a prior law. The rest of this paper is organized as follows: First we describe the basics of the Bayesian estimation framework and then we use the generalized Gaussian prior for the scene, and finally, we show some results.

INVERSION Upon to the validity of our forward models, the inverse problem becomes the Fourier synthesis, where G(u, v) and M(u, v) are given and we want to obtain fb(x, y) which b v) which represents the prediction of the data in the Fourier leads also to estimate G(u, domain and particularly on those points where we do not have any data. The inversion method in this case becomes: G(u, v), M(u, v) −→ Inversion −→ fb(x, y)

If we represent the target f (x, y) as a point in a vector space F (represented then by the vector f when the space is discrete in pixels) and the data by a point in the vector space G (represented by the vector g), the link between them is a projection H (or a matrix H), we can write g = H f + ε or g = H f + ε (4) where ε represents the errors. The operator H in our simple model is the Fourier transform and thus H is the matrix of 2D FT. If we represent by H t the adjoint operator of H , it is easy to show that classical IFT method corresponds to: bf = H t g. Evidently, in the case of complete data in the Fourier domain, we have H t H = I and this solution is perfect. In real case, the Fourier domain is never complete and the inversion becomes an ill posed problem. We need to include some prior information to obtain a unique and stable solution. There are great numbers of methods dealing with this inverse problem [11, 12, 6, 10]. One of the oldest and simplest one is the Gerchberg-Papoulis iterative method which imposes some constraint on the solution (band limited) in an iterative way. However, when the data are too incomplete, the results depend strongly on the type of constraints (prior knowledge of the band and positivity). Even if, this method may work in some cases, in general, we may need to include more prior information on the solution to obtain a satisfactory result. In this paper, we focus to the general regularization methods, and in particular the Bayesian estimation framework. The main steps are: i) assign the prior probability distributions p(g| f ) and p( f ) to translate our knowledge about the data g given f (forward model and the errors ε and the unknown image f ), ii) compute the posterior probability law: p( f |g) = p(g| f )p( f )/p(g) and, iii) use this posterior probability law to infer on the unknown f . Two main options here

are to compute the Maximum A Posterior (MAP) estimation: bf = arg max {p( f |g)} = arg min {J( f )} f f where J( f ) = − ln p(g| f ) − ln p( f ) or the Posterior Mean (PM): bf =

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f p( f |g) d f .

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The computation of the MAP solution needs an optimization algorithm and the PM solution needs an integration algorithm. When the error term ε in the model (4) is assumed to be centered, white, Gaussian with given variance σε 2 :   1 2 (7) kg − H f k p(g| f ) ∝ exp − 2σε 2 In this paper, we consider four prior laws: 1) Separable Generalized Gaussian (SGG): " # p( f ) ∝ exp −γ ∑ | f j | j

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h i β ∝ exp −γ | f |

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with 1 ≤ β ≤ 2. With these priors, we obtain: J( f ) = kg − H f k2 + λ ∑ | f j |β

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where Q( f ) = kg − H f k2 is recognized as the least squares (LS) criterion, Ω( f ) = ∑ j | f j |β as the regularization term and λ = σε 2 γ as the regularization parameter in the regularization theory literatures. (β = 2 corresponds to the Gaussian case ). This prior is appropriate for the scene with point sources (small size metallic reflectors). When there are a great number of these reflectors, β can be chosen near to 2 and when there are a few reflectors, we may choose β near to one. 2) Separable Cauchy (SC): # " 1 1 2 p( f ) ∝ ∏ p ∝ exp − ∑ ln(1 + | f j | ) 2 j 1 + | f j |2 j

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J( f ) = kg − H f k2 + λ ∑ ln(1 + | f j |2 )

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With this prior, we obtain:

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where λ = σε 2 This prior is also appropriate for the scene with point sources. It can model more appropriately when some of the reflectors may have very high amplitudes. 3) Generalized Gauss-Markov (GGM): "

p( f ) ∝ exp −γ1 ∑ | f j |β − γ2 ∑ | f j − f j−1 |β j

"

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with 1 ≤ β ≤ 2. With these priors, we obtain:

J( f ) = kg − H f k2 + λ1 k f kβ + λ2 kD f kβ

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where Q( f ) = kg−H f k2 is again the least squares criterion (LS), λ1 = σε 2 γ1 , λ2 = σε 2 γ2 and D = Toeplitz([−1, 1]) is a bi-diagonal Toeplitz matrix of first derivation. When λ1 = 0 we obtain the classical Ω( f ) = kD f k2 quadratic regularization term (Tikhonov). 4) General Markovian priors: "

p( f ) ∝ exp −γ ∑ φ( f j − f j−1 ) j

#

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where φ(.) is any potential function which can be chosen, for example, between the following: n o φ(t) = t 2, |t|β, ln(1 − |t|2) (15) and many other convex or non convex functions. With these priors, we obtain: J( f ) = kg − H f k2 + λ ∑ φ( f j − f j−1 ).

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The convex potentials are in general preferred because the resulted MAP criteria are convex and can be optimized more easily. While the non convex potentials are often used for their contour preserving properties [13, 14, 15, 16]. Two main difficulties are encountered when using these methods: • •

i) to have an appropriate optimization algorithm and ii) to be able to fix or to estimate the hyperparameters σε 2 , γ1 and γ2 or equivalently the regularization parameters λ1 and λ2 .

We use a gradient base algorithm and for the second we fix them experimentally. We are still working on these two points. We plan to use either Conjugate Gradient or a Preconditionned version of it to overcome the first difficulty and for the second problem we plan to use a full Bayesian approach where we obtain first the expression of the joint posterior law p(g| f , θ1 ) p( f |θ2 ) p(θ) (17) p( f , θ|g) = p(g)

where θ = (θ1 , θ2 ), θ1 = σε 2 , θ2 = γ1 or θ2 = (γ1 , γ2 ) depending on cases, and p(θ) is an appropriate prior law. Then, a joint estimation of f and θ can be obtained.

SIMULATION RESULTS Synthetic Fourier Synthesis data Here, we show a few results on two simulated targets f (x, y) for which we first computed F(u, v), defined a masque M(u, v) and simulated the data to be G(u, v) = M(u, v)F(u, v). We generated two sets of such data corresponding to two different cases with different bandwidth. Then, we used these data to test and compare different methods. The following figures show these data: orginal target f(x,y)

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FIGURE 2. Two simulated targets f (x, y) and the two data sets for each target

In Figure 3, we show the results obtained by classical Inverse FT, Quadratic regularization (QR) and the proposed MAP method with the three different prior laws: SGG (Eq. 8), SC (Eq. 10) and GGM (Eq. 12). Upper rows show the results with the data set 1 and lower rows with the data set 2. These reconstructed images have to be compared with the original targets in the Figure2. As we can see on these results, using appropriate priors for different scenes can give better results. For example, for a scene containing point sources (representing small metallic objects), it is more appropriate to use either separable generalized Gaussian(SGG) or separable Cauchy (SC). For a scene containing homogeneous regions (representing large homogeneous objects), it is more appropriate to use Generalized GaussMarkov prior.

CONCLUSIONS AND PERSPECTIVES In this paper, we proposed a Bayesian MAP method for the Fourier Synthesis inverse problem in SAR imaging. Different priors appropriate for different scenes have been

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FIGURE 3. Comparison of different inversion methods for the two targets 1 and 2. Upper rows show the results with data set 1 and lower rows with data set 2. These reconstruction results have to be compared with the original targets 1 and 2 of the Figure(2).

proposed: i) For a scene containing point sources, we use either separable generalized Gaussian(SGG) or separable Cauchy (SC); ii) For a scene containing homogeneous regions, we use Generalized Gauss-Markov. The proposed method with these different priors have been compared to the classical IFT and quadratic regularization.The proposed method has also been applied on a few set of real experimental data measured by ONERA. The details of this experimentation are reported in [17]. Some preliminary results can be seen in [10]. The next step will be using a more sophisticated prior model which gives the possibility of jointly segmenting and reconstruction [18, 19, 20, 21].

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

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