Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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.  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 beten the nonzero entries in each row/column of the gradient of $x$ and not on $N^2$, the ambient dimension of $x$.
['Clarice Poon']
I am writing a research paper on a idea On Cartesian line sampling with anisotropic total variation regularization
[{"from": "human", "value": "I am writing a research paper on a idea On Cartesian line sampling with anisotropic total variation regularization"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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.  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 beten the nonzero entries in each row/column of the gradient of $x$ and not on $N^2$, the ambient dimension of $x$."}]