In this effort, we propose a convex optimization approach based on weighted $\ell_1$-regularization for reconstructing objects of interest, such as signals or images, that are sparse or compressible in a wavelet basis. We recover the wavelet coefficients associated to the functional representation of the object of interest by solving our proposed optimization problem. We give a specific choice of weights and show numerically that the chosen weights admit efficient recovery of objects of interest from either a set of sub-samples or a noisy version. Our method not only exploits sparsity but also helps promote a particular kind of structured sparsity often exhibited by many signals and images. Furthermore, we illustrate the effectiveness of the proposed convex optimization problem by providing numerical examples using both orthonormal wavelets and a frame of wavelets. We also provide an adaptive choice of weights which is a modification of the iteratively reweighted $\ell_1$-minimization method.
翻译:在这项努力中,我们提出了一个基于加权$@ell_1$-常规化的组合优化法,用于重建在波盘中稀少或压缩的意向物体,例如信号或图像;我们通过解决拟议的优化问题,恢复与目标功能代表相关的波子系数;我们具体选择权重,并用数字显示所选择的权重允许从一组子标本或吵闹的版本中有效回收对象;我们的方法不仅利用了宽度,而且有助于促进通常由许多信号和图像展示的某种结构宽度;此外,我们通过提供数字例子,既使用极常波子又使用波子框架,说明拟议的二次曲线优化问题的有效性;我们还提供调整性重权重的选择,即修改迭性再加权的 $ell_1$-minmindict 方法。