The restored image x is predicted from a corrupted observation y after training on a set of sample images S. A shrinkage (mapping) function {f}_{{\pi }_{i}}\left(v\right)={\sum }_{j=1}^{M}{\pi }_{i,j}\exp \left(-\frac{\gamma }{2}{\left(v-{\mu }_{j}\right)}^{2}\right) is directly modeled as a linear combination of
radial basis function kernels, where \gamma is the shared precision parameter, \mu denotes the (equidistant) kernel positions, and M is the number of Gaussian kernels. Because the shrinkage function is directly modeled, the optimization procedure is reduced to a single quadratic minimization per iteration, denoted as the prediction of a shrinkage field {g}_{\mathrm{\Theta }}\left(\text{x}\right)={\mathcal{F}}^{-1}\left\lbrack \frac{\mathcal{F}\left(\lambda {K}^{T}y+{\sum }_{i=1}^{N}{F}_{i}^{T}{f}_{{\pi }_{i}}\left({F}_{i}x\right)\right)}{\lambda {\check{K}}^{\text{*}}\circ \check{K}+{\sum }_{i=1}^{N}{\check{F}}_{i}^{\text{*}}\circ {\check{F}}_{i}}\right\rbrack ={\mathrm{\Omega }}^{-1}\eta where \mathcal{F} denotes the
discrete Fourier transform and F_x is the 2D convolution \text{f}\otimes \text{x} with
point spread function filter, \breve{F} is an optical transfer function defined as the discrete Fourier transform of \text{f}, and {\breve{F}}^{\text{*}} is the complex conjugate of \breve{F}. {\hat{x}}_{t} is learned as {\hat{x}}_{t}={g}_{{\mathrm{\Theta }}_{t}}\left({\hat{x}}_{t-1}\right) for each iteration t with the initial case {\hat{x}}_{0}=y, this forms a cascade of Gaussian
conditional random fields (or cascade of shrinkage fields (
CSF)). Loss-minimization is used to learn the model parameters {\mathrm{\Theta }}_{t}={\left\lbrace {\lambda }_{t},{\pi }_{\mathit{ti}},{f}_{\mathit{ti}}\right\rbrace }_{i=1}^{N}. The learning objective function is defined as J\left({\mathrm{\Theta }}_{t}\right)={\sum }_{s=1}^{S}l\left({\hat{x}}_{t}^{\left(s\right)};{x}_{gt}^{\left(s\right)}\right), where l is a
differentiable loss function which is greedily minimized using training data {\left\lbrace {x}_{gt}^{\left(s\right)},{y}^{\left(s\right)},{k}^{\left(s\right)}\right\rbrace }_{s=1}^{S} and {\hat{x}}_{t}^{\left(s\right)}. == Performance ==