Improved ℓ₁-ℓ₂ Tomographic Reconstruction and Noise Estimation in Urban Areas

The application of sparse synthetic aperture radar tomographic (TomoSAR) reconstruction in urban areas faces several challenges, including deficiencies related to outliers and amplitude/position biases resulting from low SNR, small baseline aperture, and an insufficient number of acquisitions. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> norm-minimization for sparse reconstruction relies primarily on the Lagrange multiplier to promote sparsity in the solution. However, inaccurate estimation of the noise level and the Lagrange multiplier can significantly degrade reconstruction performance. To address these limitations, we propose an enhanced sparse reconstruction framework that integrates improved <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> norm minimization with refined noise estimation. We also introduce support shrinkage based on a modified singular value decomposition (M-SVD) to improve the localization of correct scatterers in elevation and reduce the outliers. We estimated the Lagrange multiplier for the shrunk system of equations, followed by a final reconstruction using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> norm-minimization. This study provides a comprehensive evaluation of the proposed method under various parameter settings and compares its performance against conventional approaches using simulated datasets to demonstrate the effectiveness of M-SVD in accurately localizing scatterers in elevation. Moreover, a practical application of a 3-D elevation point cloud of the Mapfre Tower in Barcelona using TerraSAR-X satellite data is generated.