L2L2SolverSPC

class colibri.recovery.solvers.spc.L2L2SolverSPC(y, acquisition_model)

Bases: Solver

Solver for the SPC acquisition model.

It describes the closed-form solution of the optimization problem.

\[\min_{\textbf{X}} \frac{1}{2}||\textbf{X} - \textbf{H}\textbf{X}||_2^2 + \rho||\textbf{X} - \tilde{\textbf{X}}||_2^2\]

where \(\textbf{X}\) is the tensor to be recovered, \(\textbf{Y}\) is the input tensor, \(\textbf{H}\) is the sensing matrix, and \(\rho\) is the regularization parameter.

in the case of the SPC acquisition model, the \(\textbf{X}\) is a matrix of size \((M\times N, L)\), where \(M\) and \(N\) are the height and width of the image, and \(L\) is the number of channels.

In this sense, \(\textbf{X}\) is the spatial vectorized form of the image. (since the SPC its broadcasting the sensing matrix over the channels)

The solution of the optimization problem is given by:

\[\hat{\textbf{X}} = (\textbf{H}^\top\textbf{H} + \rho \textbf{I})^{-1}(\textbf{H}^\top \textbf{Y} + \rho \tilde{\textbf{X}})\]

Parameters:

• y (torch.Tensor) – Input tensor with shape (B, L, M, N)

• acquisition_model (SPC) – Acquisition model