关于带有各向异性总变差正则化的Cartesian线采样
On Cartesian line sampling with anisotropic total variation regularization
AI总结：
本文研究了利用各向异性总变差半范数从部分傅里叶系数恢复二维向量，证明了在满足特定稀疏性条件下，可通过采样特定方向的傅里叶系数精确恢复信号。
摘要：
本文考虑利用各向异性总变差半范数从部分傅里叶系数恢复二维向量$x\in \mathbb{C}^{N\times N}$。证明如果$(x_{k,j} - x_{k-1,j})_{k,j}$每列至多有$s_1$个非零系数，且$(x_{k,j} - x_{k,j-1})_{k,j}$每行至多有$s_2$个非零系数，则可通过采样$s_1$行和$s_2$列的傅里叶系数，乘以对数因子后精确恢复$x$。不同于标准压缩感知估计，所涉及的对数因子取决于梯度中非零元素的分离距离，而非$x$的环境维度$N^2$。
英文摘要：
This paper considers the use of the anisotropic total variation seminorm to recover a two dimensional vector $x\in \mathbb{C}^{N\times N}$ from its partial Fourier coefficients, sampled along Cartesian lines. We prove that if $(x_{k,j} - x_{k-1,j})_{k,j}$ has at most $s_1$ nonzero coefficients in each column and $(x_{k,j} - x_{k,j-1})_{k,j}$ has at most $s_2$ nonzero coefficients in each row, then, up to multiplication by $\log$ factors, one can exactly recover $x$ by sampling along $s_1$ horizontal lines of its Fourier coefficients and along $s_2$ vertical lines of its Fourier coefficients. Finally, unlike standard compressed sensing estimates, the $\log$ factors involved are dependent on the separation distance between the nonzero entries in each row/column of the gradient of $x$ and not on $N^2$, the ambient dimension of $x$.
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