Problem statement

The interpolation problem in a CS framework can be summarized in the following model:

$$\mathbf{d}_{obs} = \mathbf{S}\mathbf{d}, \mathbf{m} = \mathbf{Ad}$$ (1)

where $\mathbf{d}_{obs}$ is the observed data, $\mathbf{S}$ is the sampling operator, $\mathbf{d}$ is the unknown data we would like to estimate, $\mathbf{A}$ is the sparsity-promoting transform, and $\mathbf{m}$ is the transform domain coefficients.

The synthesis based approach solves the following problem:

$$\min_{\mathbf{m}} \parallel\mathbf{d}_{obs}- \mathbf{SA}^{-1}\mathbf{m} \parallel_2^2 + \lambda \parallel \mathbf{m} \parallel_1,$$ (2)

where $\mathbf{A}^{-1}$ denotes the inverse sparsity-promoting transform.

The analysis-based approach solves the following problem:

$$\min_{\mathbf{d}} \parallel \mathbf{d}_{obs} - \mathbf{S}\mathbf{d} \parallel_2^2 + \lambda \parallel \mathbf{A} \mathbf{d} \parallel_1.$$ (3)

The analysis-based approach emphasizes the sparsity of the canonical transformed coefficients, so it tends to recover data with smooth regions; while the synthesis-based approach finds the sparsest approximation of the given data in the transformed domain (Liang et al., 2014).