论文标题
关于解决方案的独特性，以追求连续性
On the Uniqueness of Solutions for the Basis Pursuit in the Continuum
论文作者
论文摘要
本文研究了从低频傅立叶系数中恢复对一维圆环的连续域逆问题，其中kc是截止频率。我们的方法包括最大程度地减少与观察结果一致的所有ra量度量中的总变化规范。我们将此问题称为连续体（BPC）中的基础追求。我们以独特性来表征（BPC）的解决方案集，并描述其稀疏的解决方案，该解决方案是很少有签名的狄拉克质量的总和。表征取决于仅取决于观测值的toeplitz和居比对称矩阵的光谱。更确切地说，我们证明（BPC）在且仅当此矩阵既不是正定的也不是负定确定性时具有独特的解决方案。如果它具有正征值和负特征值，则独特的解决方案是最多的2kc Dirac块的总和，至少一个正重量和一个负重量。如果此矩阵是正（分别为负）半准且等级不足的，则独特的解决方案由等于矩阵等级等级的多个狄拉克质量组成，所有矩阵的级别都具有非负（分别非阳性）权重。最后，如果（BPC）具有多个溶液，我们证明，如果矩阵分别为正（分别为负）确定，则有许多由KC+1个DIRAC质量组成的解决方案，具有非负（分别为非阳性）权重。
This paper studies the continuous-domain inverse problem of recovering Radon measures on the one-dimensional torus from low-frequency Fourier coefficients, where Kc is the cutoff frequency. Our approach consists in minimizing the total-variation norm among all Radon measures that are consistent with the observations. We call this problem the basis pursuit in the continuum (BPC). We characterize the solution set of (BPC) in terms of uniqueness and describe its sparse solutions which are sums of few signed Dirac masses. The characterization is determined by the spectrum of a Toeplitz and Hermitian-symmetric matrix that solely depends on the observations. More precisely, we prove that (BPC) has a unique solution if and only if this matrix is neither positive definite nor negative definite. If it has both a positive and negative eigenvalue, then the unique solution is the sum of at most 2Kc Dirac masses, with at least one positive and one negative weight. If this matrix is positive (respectively negative) semi-definite and rank deficient, then the unique solution is composed of a number of Dirac masses equal to the rank of the matrix, all of which have nonnegative (respectively nonpositive) weights. Finally, in cases where (BPC) has multiple solutions, we demonstrate that there are infinitely many solutions composed of Kc+1 Dirac masses, with nonnegative (respectively nonpositive) weights if the matrix is positive (respectively negative) definite.
