Least-squares diffraction imaging using shaping regularization by anisotropic smoothing

Optimization

For inversion we adopt a conjugate gradients scheme (Fomel et al., 2007):

$$\mathbf{m}_d^i \leftarrow \mathbf{H}_{\epsilon,N,K} \mathbf{T_{... ...g], \mathbf{s}_{j}=-\nabla J(\mathbf{m}_{d}^{j}) + \beta_{j}\mathbf{s}_{j-1}$$ (5)

where $\mathbf{H}_{\epsilon,N,K}$ and $\mathbf {T_{\lambda }}$ are anisotropic-smoothing and thresholding operators, $-\nabla J(\mathbf{m}_{d}^{j})$ is the gradient at iteration $j$ , $\mathbf{s}_{j}$ is a conjugate direction, $\alpha_{j}$ is an update step length , and $\beta_j$ is designed to guarantee that $\mathbf{s}_j$ and $\mathbf{s}_{j-1}$ are conjugate. After several internal iterations $j$ of the conjugate gradient algorithm we generate $\mathbf{m}_d^j$ , to which we apply thresholding to drop samples corresponding to noise with values lower than the threshold $\lambda$ , and which we then smooth along edges by applying anisotropic smoothing operator $\mathbf{H}_{\epsilon,N,K}$ . Outer model shaping iterations are denoted by $i$ .

Inversion results also depend on the numbers of inner and outer iterations: their tradeoff determines how often shaping regularization is applied and therefore controls its strength. Regularization by early stopping can also be conducted. The optimization strategy with $\mathbf{H}_{\epsilon,N,K}$ removed corresponds to the iterative thresholding approach (Daubechies et al., 2004).

Least-squares diffraction imaging using shaping regularization by anisotropic smoothing