Seislet-based morphological component analysis using scale-dependent exponential shrinkage

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Analysis-based iterative thresholding

A general inverse problem combined with a priori constraint can be written as an optimization problem

 (1)

where is the model to be inverted, and is the observations. To solve the problem with sparsity constraint , the iterative shrinkage-thresholding (IST) algorithm has been proposed (Daubechies et al., 2004), which can be generally formulated as

 (2)

where denotes the iteration number; and indicates the adjoint of . is an element-wise shrinkage operator with threshold :

 (3)

in which the soft thresholding function (Donoho, 1995) is

 (4)

Allowing for the missing elements in the data, the observations are connected to the complete data via the relation

 (5)

where is an acquisition mask indicating the observed and missing values. Assume is a tight frame such that , . It leads to

 (6)

in which we use and . Now we define a residual term as , thus Eq. (6) results in

 (7)

which is equivalent to solving

 (8)

Note that Eq. (8) analyzes the target unknown directly, without resort to and . Eq. (6) is referred to as the analysis formula (Elad et al., 2007). In this paper, we used the analysis formula because it directly addresses the problem in the data domain for the convenience of interpolation and signal separation.

 Seislet-based morphological component analysis using scale-dependent exponential shrinkage

Next: Understanding iterative thresholding as Up: MCA with scale-dependent shaping Previous: MCA with scale-dependent shaping

2021-08-31