Shrinkage fields is a random field-based machine learning technique that aims to perform high quality image restoration (denoising and deblurring) using low computational overhead.
The restored image x {\displaystyle x} is predicted from a corrupted observation y {\displaystyle y} after training on a set of sample images S {\displaystyle S} .
A shrinkage (mapping) function f π π --> i ( v ) = ∑ ∑ --> j = 1 M π π --> i , j exp --> ( − − --> γ γ --> 2 ( v − − --> μ μ --> j ) 2 ) {\displaystyle {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 γ γ --> {\displaystyle \gamma } is the shared precision parameter, μ μ --> {\displaystyle \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 Θ Θ --> ( x ) = F − − --> 1 [ F ( λ λ --> K T y + ∑ ∑ --> i = 1 N F i T f π π --> i ( F i x ) ) λ λ --> K ˇ ˇ --> * ∘ ∘ --> K ˇ ˇ --> + ∑ ∑ --> i = 1 N F ˇ ˇ --> i * ∘ ∘ --> F ˇ ˇ --> i ] = Ω Ω --> − − --> 1 η η --> {\displaystyle {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 F {\displaystyle {\mathcal {F}}} denotes the discrete Fourier transform and F x {\displaystyle F_{x}} is the 2D convolution f ⊗ ⊗ --> x {\displaystyle {\text{f}}\otimes {\text{x}}} with point spread function filter, F ˘ ˘ --> {\displaystyle {\breve {F}}} is an optical transfer function defined as the discrete Fourier transform of f {\displaystyle {\text{f}}} , and F ˘ ˘ --> * {\displaystyle {\breve {F}}^{\text{*}}} is the complex conjugate of F ˘ ˘ --> {\displaystyle {\breve {F}}} .
x ^ ^ --> t {\displaystyle {\hat {x}}_{t}} is learned as x ^ ^ --> t = g Θ Θ --> t ( x ^ ^ --> t − − --> 1 ) {\displaystyle {\hat {x}}_{t}={g}_{{\mathrm {\Theta } }_{t}}\left({\hat {x}}_{t-1}\right)} for each iteration t {\displaystyle t} with the initial case x ^ ^ --> 0 = y {\displaystyle {\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 Θ Θ --> t = { λ λ --> t , π π --> t i , f t i } i = 1 N {\displaystyle {\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 ( Θ Θ --> t ) = ∑ ∑ --> s = 1 S l ( x ^ ^ --> t ( s ) ; x g t ( s ) ) {\displaystyle 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 {\displaystyle l} is a differentiable loss function which is greedily minimized using training data { x g t ( s ) , y ( s ) , k ( s ) } s = 1 S {\displaystyle {\left\lbrace {x}_{gt}^{\left(s\right)},{y}^{\left(s\right)},{k}^{\left(s\right)}\right\rbrace }_{s=1}^{S}} and x ^ ^ --> t ( s ) {\displaystyle {\hat {x}}_{t}^{\left(s\right)}} .
Preliminary tests by the author suggest that RTF5[1] obtains slightly better denoising performance than CSF 7 × × --> 7 { 3 , 4 , 5 } {\displaystyle {\text{CSF}}_{7\times 7}^{\left\lbrace \mathrm {3,4,5} \right\rbrace }} , followed by CSF 5 × × --> 5 5 {\displaystyle {\text{CSF}}_{5\times 5}^{5}} , CSF 7 × × --> 7 2 {\displaystyle {\text{CSF}}_{7\times 7}^{2}} , CSF 5 × × --> 5 { 3 , 4 } {\displaystyle {\text{CSF}}_{5\times 5}^{\left\lbrace \mathrm {3,4} \right\rbrace }} , and BM3D.
BM3D denoising speed falls between that of CSF 5 × × --> 5 4 {\displaystyle {\text{CSF}}_{5\times 5}^{4}} and CSF 7 × × --> 7 4 {\displaystyle {\text{CSF}}_{7\times 7}^{4}} , RTF being an order of magnitude slower.
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