## Problem statement

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

 (1)

where is the observed data, is the sampling operator, is the unknown data we would like to estimate, is the sparsity-promoting transform, and is the transform domain coefficients.

The synthesis based approach solves the following problem:

 (2)

where denotes the inverse sparsity-promoting transform.

The analysis-based approach solves the following problem:

 (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).

2020-04-11