Matched Filter (Back-Projection)

The matched filter applies the adjoint:

$$\hat{\mathbf{x}}_\text{MF} = \mathbf{A}^H \mathbf{y}$$

This maximizes SNR for known targets but produces sidelobes for sparse scenes. It is the starting point for iterative methods.

ISTA solves the LASSO problem $\min_\mathbf{x} \frac{1}{2}\|\mathbf{y}-\mathbf{A}\mathbf{x}\|^2 + \lambda\|\mathbf{x}\|_1$:

$$\mathbf{x}^{(k+1)} = \mathcal{S}_\lambda\left(\mathbf{x}^{(k)} + \frac{1}{L}\mathbf{A}^H(\mathbf{y}-\mathbf{A}\mathbf{x}^{(k)})\right)$$

where $\mathcal{S}_\lambda$ is soft thresholding and $L = \|\mathbf{A}\|^2$.