Sparsity Constraint

In order to minimize the number of model space parameters necessary to represent the data in the Radon domain, I implemented the transform given by equations  28 and 29, with the Radon kernel given by equation  27 as a least squares problem with a sparsity constraint. As a linear transformation, the apex-shifted Radon transform can be represented simply as

$$\mathbf{d}=\mathbf{Lm} ,$$ (30)

where ${\bf d}$ is the (migrated) data in the angle domain, ${\bf m}$ is the model in the Radon domain and ${\bf L}$ is the forward apex-shifted Radon transform operator. To find the model ${\bf m}$ that best fits the data in a least-squares sense, I minimize the objective function:

$$f({\bf m}) = \Vert{\bf Lm-d}\Vert^2 + \epsilon^2b^2 \sum_{i=1}^n \ln\Big(1 +\frac{m_i^2}{b^2}\Big),$$ (31)

where the second term is a Cauchy regularization (, ) that enforces sparseness in the model space. Here $n$ is the size of the model space, and ${\epsilon}$ and $b$ are two constants chosen a-priori: ${\epsilon}$ which controls the amount of sparseness in the model space and $b$ which controls the minimum value below which everything in the Radon domain should be zeroed. The least-squares inverse of ${\bf m}$ is given by

$${\bf\hat{m}} = \left [ {\bf L'L}+\epsilon^2 {\bf Q}\right ]^{-1}{\bf L'd},$$ (32)

where ${\bf Q}$ is a diagonal matrix whose elements are given by (, ):

$$\left(1+\frac{m_i^2}{b^2}\right)^{-1}.$$ (33)

Because the model space can be large, I estimate ${\bf m}$ iteratively. Notice that the objective function in equation (31) is non-linear because the model appears in the definition of the regularization term. Therefore, I use a limited-memory quasi-Newton method (, ) to find the minimum of $f({\bf m})$.