We address the problem of compressed sensing (CS) with prior information: reconstruct a target CS signal with the aid of a similar signal that is known beforehand, our prior information. We integrate the additional knowledge of the similar signal into CS via ℓ1-ℓ1 and ℓ1-ℓ2 minimization. We then establish bounds on the number of measurements required by these problems to successfully reconstruct the original signal. Our bounds and geometrical interpretations reveal that if the prior information has good enough quality, ℓ1-ℓ1 minimization improves the performance of CS dramatically. In contrast, ℓ1-ℓ2 minimization has a performance very similar to classical CS and brings no significant benefits. All our findings are illustrated with experimental results.Index Terms-Compressed sensing, prior information, basis pursuit, ℓ1-ℓ1 and ℓ1-ℓ2 minimization, Gaussian width.i ∈ I c and β ≤ 1/|w i | .

Abstract:A new convolution neural network (CNN) architecture for semantic segmentation of high resolution aerial imagery is proposed in this paper. The proposed architecture follows an hourglass-shaped network (HSN) design being structured into encoding and decoding stages. By taking advantage of recent advances in CNN designs, we use the composed inception module to replace common convolutional layers, providing the network with multi-scale receptive areas with rich context. Additionally, in order to reduce spatial ambiguities in the up-sampling stage, skip connections with residual units are also employed to feed forward encoding-stage information directly to the decoder. Moreover, overlap inference is employed to alleviate boundary effects occurring when high resolution images are inferred from small-sized patches. Finally, we also propose a post-processing method based on weighted belief propagation to visually enhance the classification results. Extensive experiments based on the Vaihingen and Potsdam datasets demonstrate that the proposed architectures outperform three reference state-of-the-art network designs both numerically and visually.

We address the problem of Compressed Sensing (CS) with side information. Namely, when reconstructing a target CS signal, we assume access to a similar signal. This additional knowledge, the side information, is integrated into CS via ℓ1-ℓ1 and ℓ1-ℓ2 minimization. We then provide lower bounds on the number of measurements that these problems require for successful reconstruction of the target signal. If the side information has good quality, the number of measurements is significantly reduced via ℓ1-ℓ1 minimization, but not so much via ℓ1-ℓ2 minimization. We provide geometrical interpretations and experimental results illustrating our findings.Index Terms-Compressed sensing, basis pursuit, ℓ1-ℓ1 minimization, ℓ1-ℓ2 minimization, Gaussian width.