结合梯度投影稀疏重构和复数小波的图像重构

压缩感知主要包括随机投影和重构两部分。针对迭代收缩算法收敛速度较慢，普通二维小波变换缺少方向性表示的缺点，利用置乱离散余弦变换（PDCT）实现随机投影，重构时采用梯度投影算法，在简化计算的基础上，通过迭代的方式完善图像在双树复数小波域的变换系数，最后经反变换后得到重构图像。在同一重构算法下，比较了利用双树复数小波变换和双正交小波变换的重构结果，结果表明前者重构后的图像在细节和平滑度上优于后者，在峰值信噪比（PSNR）上平均高出约1.5 dB；同一稀疏域中，梯度投影算法的收敛速度优于迭代收缩算法；相同稀疏域和重构算法下，PDCT与结构随机矩阵相比在PSNR上略高。

Image reconstruction based on gradient projection for sparse representation and complex wavelet

Compressed sensing mainly contains random projection and reconstruction. Because of lower convergence speed of iterative shrinkage algorithm and the lacking of direction of traditional 2-dimensional wavelet transform， random projection was implemented by using Permute Discrete Cosine Transform （PDCT）， and the gradient projection was used for reconstruction. Based on the simplification of computation complexity， the transformation coefficients in the dual-tree complex wavelet domain were improved by iteration. Finally， the reconstructed image was obtained by the inverse transform. In the experiments， the reconstruction results of DT CWT （Dual-Tree Complex Wavelet Transform） and bi-orthogonal wavelet were compared with the same reconstruction algorithm， and the former is better than the latter in image detail and smoothness with higher Peak Signal-to-Noise Ratio （PSNR） of 1.5 dB. In the same sparse domain， gradient projection converges faster than iterative shrinkage algorithm. And in the same sparse domain and random projection， PDCT has a slightly higher PSNR than the structural random matrix.