BI Hui，CHENG Yuan，ZHU Daiyin，and HONG Wen

1. College of Electronic and Information Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 211106，China;2. Aerospace Information Research Institute，Chinese Academy of Sciences，Beijing 100094，China

Abstract: Tomographic synthetic aperture radar (TomoSAR)imaging exploits the antenna array measurements taken at different elevation aperture to recover the reflectivity function along the elevation direction. In these years，for the sparse elevation distribution，compressive sensing (CS) is a developed favorable technique for the high-resolution elevation reconstruction in TomoSAR by solving an L1 regularization problem. However，because the elevation distribution in the forested area is nonsparse，if we want to use CS in the recovery，some basis，such as wavelet，should be exploited in the sparse representation of the elevation reflectivity function. This paper presents a novel wavelet-based L 1/2 regularization CS-TomoSAR imaging method of the forested area. In the proposed method，we first construct a wavelet basis，which can sparsely represent the elevation reflectivity function of the forested area，and then reconstruct the elevation distribution by using the L 1/2 regularization technique. Compared to the wavelet-based L1 regularization TomoSAR imaging，the proposed method can improve the elevation recovered quality efficiently.

Keywords: tomographic synthetic aperture radar (TomoSAR)，compressive sensing (CS)，L 1/2 regularization，wavelet basis.

1. Introduction

Tomographic synthetic aperture radar (TomoSAR) imaging is a three-dimensional (3D) radar imaging technique.It exploits the acquisitions of multi-baselines with different elevation aperture to recover the reflectively function in the elevation direction，and hence obtain the 3D image of the surveillance region by using spectral analysis (SA)or compressive sensing (CS) based methods [1-3].

CS as an important development in sparse signal processing was proposed by Donoho et al. in 2006 [4，5]，which can recover the sparse signal even from much fewer samples as required by the Shannon-Nyquist sampling theorem [6，7]. The optimal scheme to achieve CS reconstruction is a solution ofL0minimization，but it is intractable because of its NP-hard nature [8]. Thus，L1optimization，a relaxed estimation，is exploited as an alternative if several mathematical conditions are satisfied [9].L1regularization has been widely applied in CS reconstruction.Recently，Lq(0＜q＜1) regularization is proposed which is capable of obtaining more sparser solution thanL1in some situations [10]. In particular，it has been proven that，ifq∈ [1/2，1)，the smaller ofq，the sparser solution can be achieved，and ifq∈ (0，1/2]，the performance viaLqregularization with different values ofqhas no significant difference. This motivates us to chooseq=1/2 inLqregularization-based CS reconstruction [10].

In 2009，Zhu et al. first introduced the CS technique to TomoSAR [11，12]. Then，Zhu and Bamler applied CS successfully to reconstruct individual buildings from spaceborne TerraSAR-X data stacks by the scale-down byL1-norm minimization model selection estimation reconstruction (SL1MMER) algorithm in which they solved theL1-norm optimization problem [3]. In these years，Lq-norm (q∈ (0，1)) regularization has been also introduced to TomoSAR and achieved high-resolution reconstruction of artificial targets [13，14]. In urban environment，reflectivity profiles along elevation are always naturally sparse [3]，and hence CS with simple orthogonal bases，such as an identity matrix or a sinc basis [15]，is a favorable technique for the recovery of the complex reflectivity function. However，in the forested area，because the elevation distribution is usually non-sparse，CS cannot be directly used to the recovery. To solve this problem，in 2013，the wavelet basis was introduced to TomoSAR，which successfully realized the sparse representation of the elevation distribution，and reconstructed the reflectivity function by solving theL1optimization problem [16].

Inspired by the advantage ofL1/2regularization technique，in this paper，we present a novel wavelet-basedL1/2regularization CS-TomoSAR imaging method for the elevation recovery of the forested area. In the proposed method，we first construct a wavelet basis to sparsely represent the reflectivity function，then reconstruct the elevation distribution byL1/2regularization technique. Compared to beamforming，it can suppress elevation ambiguity well and achieve super-resolution imaging. While it shows better recovered accuracy of the elevation distribution，higher super-resolution ability，and much robust against noise than the correspondingL1based technique.

The rest of this paper is organized as follows. Section 2 introduces the TomoSAR imaging model briefly. Section 3 represents theL1regularization technique，and describes the proposed wavelet-basedL1/2regularization CS-Tomo-SAR imaging mechanism in detail. Section 4 shows the experimental results and the comparison between beamforming，wavelet-basedL1andL1/2regularization methods. Conclusions are given in Section 5.

2. TomoSAR imaging model

In TomoSAR，as shown in Fig.1，for an azimuth-range cell (x0，r0)，we can express the measurementgm(x0，r0) of themth acquisition as

wherebmdenotes the aperture of themth acquisition;Mdenotes the number of baselines; γ (s) denotes the reflectivity function; Δsdenotes the elevation scope;sdenotes the elevation direction; λ denotes the wavelength; andrdenotes the slant range.

Fig. 1 TomoSAR imaging geometry

After usingsl(l=1，2，···，L) to discretize γ (s)，(1) can be rewritten as

whereLis the number of elevation discrete indexes;rm，lis the slant range betweenbmand thelth elevation index，is a constant that denotes the elevation discretization interval. Letrepresent the data and the discrete reflectivity function vector，we can rewrite (2) as

In multiple baselines observations，after assuming the scattering elements located at different elevation positions are uncorrelated，we can express the covariance matrixCwith sizeM×Mas

where E (·) is the expectation operator; ( · )His the conjugate transpose operator; and diag(p)∈ RL×Lis a matrix whose main diagonal isand contains zeros in its off-diagonal elements [16]. The sparse elevation distribution is the precondition of CSTomoSAR imaging. However，in the forested area，γ is always non-sparse. Therefore，an orthogonal basis，or called sparse dictionary，Ψ ∈ RL×Lis required to projectγ as α which can be expressed as α =ΨHγ，where α is a sparse vector withKnon-zero elements (K?L).

3. Wavelet-based L1/2 regularization for CS-TomoSAR imaging

In the forested area，after choosing an appropriate sparse basis Ψ ，pcan be recovered [16]by

with ε being a constant decided by the noise level. Similar to the aboveL1regularization method，L1/2reconstruction can be achieved as

withL1/2-norm being denoted asThe alternative mechanism of (6) is

whereis the recovered elevation power distribution，and η is the regularization parameter.

With the help of the convergent iterative half soft thresholding (IHST) algorithm [10]，the solution ofL1/2the regularization problem in (7) can be expressed as

with μ and v ec(·) being a positive parameter and choosing the main diagonal elements of the matrix operator，respectively.is the complex half thresholding operator [10]. According to above thresholding operation，the iterative recovery algorithm for solving theL1/2regularization problem can be represented as

4. Experiment and discussion

In this section，we use simulated and real data to evaluate the performance of the proposed wavelet-basedL1/2regularization CS-TomoSAR imaging algorithm. For comparison，we also show the results of beamforming (L2regularization) [1]and wavelet-basedL1regularization approach [16].

4.1 Chosing of sparse dictionary

As seen from the imaging model in (5)，an appropriate choice of sparse dictionary is critical for CS based Tomo-SAR imaging in the forested area. According to the validation in [16，17]，in general，the elevation distribution of the forest can be regarded as two main parts，ground and canopy [16]. In this paper，Daubechies Symmlet wavelet is selected as the sparse dictionary Ψ to represent the elevation profile γ. The reason we select this wavelet is that，in a sense，it is optimal because it has minimal support for a given number of vanishing moments. In addition，previous studies on CS-TomoSAR imaging in the forested area have shown that Daubechies Symmlet wavelet has good performance in practical data processing [16].The sparse representation of the elevation profile in the wavelet domain is shown in Fig. 2.

Fig. 2 Sparse representation of the elevation profile in the wavelet domain

To support our choice，the first column in Fig. 2 shows the five kinds of common elevation distribution in the forested area. The second and third columns are the decomposition coefficients of elevation reflectivity function using Daubechies Symmlet wavelet [18]with five vanishing moments and three levels of decomposition and normalized sorted magnitudes of decomposition coefficients，respectively. Normalized recovered reflectivity functions by inverse discrete wavelet transformation using six largest wavelet coefficients are shown in the fourth column. In Fig. 2，we can see that for the five kinds of elevation distributions，Daubechies Symmlet wavelet can represent them sparsely，i.e.，the reflectivity functions along the elevation can be accurately reconstructed even by using only a small part of wavelet coefficients.

4.2 Experiment with simulated data

In this experiment，we collect the data with a 120 m elevation aperture based on uniform and random baseline distributions，separately. The wavelength and reference slant range are 0.80 m and 4 000 m，respectively. Then we can approximate the elevation Rayleigh resolution [2]as

In the experiments，the number of non-zero wavelet coefficientsKused in the elevation reconstruction is set as 10. The reason is that the signal is decreased by the noise，so more wavelet coefficients are needed to obtain a better elevation recovery. In addition，the smaller wavelet coefficients contain more details of the scattering which are also required for the high-quality reconstruction. Thus for above reasons，we setK=10. This not only reduces the required samples for CS reconstruction，but also ensures the recovery quality.

We first focus on the effectiveness of the waveletbasedL1/2regularization imaging method in the improvement of reconstructed image quality. Some noise with a signal-to-noise ratio (SNR) being equal to 10 dB is added to the simulated data. Fig. 3 shows the normalized profiles based on the beamforming，wavelet-basedL1，and the proposed method by the data of 13 uniform (see Fig. 3(a)) and 11 random baselines (see Fig. 3(e))，respectively. The left and right columns in Fig. 3 are the recovered results based on the uniform and random baseline distributions，respectively. From Fig. 3，we can see that not only uniform but also random baseline distribution，all three methods can find two scattering centers(canopy and ground) accurately. However，compared to beamforming and wavelet-basedL1regularization methods，the proposed method shows better recovered accuracy of scattering power both for canopy and ground areas.

Then the super-resolving ability of the wavelet-basedL1/2regularization imaging method will be also discussed in this section. Two scattering areas (representing ground and canopy) with the same amplitude and phase are set as the simulated scene along the elevation direction. Fig. 4 shows the comparison of the normalized elevation recovered profiles of beamforming，wavelet-basedL1/2regularization methods with a distanceDbetween two scattering centers of 6 m，8 m，10 m，and 14 m，respectively.

Fig. 3 Normalized elevation profiles reconstructed by beamforming，wavelet-based L1 and proposed wavelet-based L 1/2 method

Fig. 4 Normalized elevation profiles reconstructed by beamforming，wavelet-based L1 and proposed wavelet-based L 1/2 method based on the simulated data of random baseline distribution shown in Fig. 3(e)

The simulated data is generated according to the random baseline distribution in Fig. 3(e) with 10 dB Gaussian noise. From Fig. 4，we can see that when the distance is 14 m，all three methods can distinguish two scattering centers well. After the distance being equal to 10 m which is less than the Rayleigh resolution (ρs=13.33m)，the two scattering centers cannot be found in the image recovered by beamforming，while also can be found accurately in the images reconstructed by the wavelet-basedL1andL1/2regularization methods. However，once the distance further decreases to 8 m，L1is no longer able to distinguish two scattering areas，while the proposed wavelet-basedL1/2regularization method can still identify them well. This simulated results reveal that both wavelet-basedL1andL1/2methods can achieve super-resolution TomoSAR imaging in the forested area，whileL1/2has a better super-resolving ability thanL1.

4.3 Experiment with real data

For our purpose，BioSAR 2008 dataset (see Fig. 5) with sixL-band 2D complex images are used to further validate the proposed wavelet-basedL1/2regularization method. In this experiment，we select an azimuth slice indicated by the yellow rectangles and the red lines in Fig. 5 to show the experimental results.

Fig. 5 Polarimetric SAR image of the experimental area in Northern Sweden

Fig. 6 depicts the span of tomogram recovered by beamforming，wavelet-basedL1，and the proposed wavelet-basedL1/2regularization methods. Compared to the conventional beamforming method，we can see that both wavelet-basedL1andL1/2methods distinguish the canopy and ground in the forested area clearly with less ambiguity. While the presented method shows a better elevation recovered accuracy of scattering power thanL1especially in the area with the range from 5 000 m to 5 400 m.

Fig. 6 Span of tomogram obtained by different methods as a function of range and elevation using a 21-by-21 window with six L-band 2D focused SAR complex image data

5. Conclusions

In this paper，we introduce theL1/2regularization to CSTomoSAR imaging，and propose a novel wavelet-basedL1/2regularization CS-TomoSAR imaging method tailored to the forested area. For the non-sparse elevation distribution of the forest，it first constructs the wavelet basis which can represent the elevation reflectivity function sparsely，then reconstructs the elevation distribution by means of theL1/2regularization method，and hence obtains the high-resolution 3D image of the considered scene. Compared to the conventional beamforming technique，it suppresses elevation ambiguity well and shows better super-resolving ability. Compared to the waveletbasedL1regularization imaging technique，this method improves the recovered accuracy of the elevation profile，has a higher super-resolution ability，and is robust against Gaussian noise. The experimental results based on the simulated and real BioSAR 2008 airborne data validate the effectiveness of the proposed method.

Acknowledgment

We would like to thank Dragon 3 Project (ID10609) and Prof. Chen Erxue for providing the BioSAR 2008 dataset.