Regularization and denoising#

bartorch.priors provides the regularization terms of BART’s reconstructions. A term represents a functional \(g(Gx)\): BART builds the proximal operator of \(g\) and the linear transform \(G\), and the solvers of bartorch.optim apply them. An ImplicitPrior takes the place of a term with a denoiser; with spatial= it converts the complex image to the real planes an image-restoration network takes. Three functions apply BART’s denoisers to an image directly. Inverse problems and their solvers introduces functionals, proximal operators and the splitting that a nontrivial \(G\) requires.

Term	Functional \(g(Gx)\)	Transform \(G\)	Solvers
`L1`	\(\lambda\lVert x\rVert_1\)	\(I\)	IST, FISTA, ADMM, PRIDU
`Wavelet`	\(\lambda\lVert \Psi x\rVert_1\)	\(I\); \(\Psi\) is inside the proximal operator	IST, FISTA, ADMM, PRIDU
`LocallyLowRank`	\(\lambda\sum_b \lVert B_b x\rVert_*\)	\(I\); the blocks are inside the proximal operator	IST, FISTA, ADMM, PRIDU
`L2`	\(\tfrac{\lambda}{2}\lVert x\rVert_2^2\)	\(I\)	IST, FISTA, ADMM, PRIDU
`NonNegative`	Indicator of \(\operatorname{Re} x \ge 0\) and \(\operatorname{Im} x \ge 0\)	\(I\)	IST, FISTA, ADMM, PRIDU
`TotalVariation`	\(\lambda\sum_r \lVert (\nabla x)_r\rVert_2\) (isotropic)	Finite differences \(\nabla\)	ADMM, PRIDU
`FourierL1`	\(\lambda\lVert F x\rVert_1\)	Fourier transform \(F\)	ADMM, PRIDU
`Laplace`	\(\lambda\lVert L x\rVert_1\)	Laplacian \(L\)	ADMM, PRIDU
`ImaginaryL1`	\(\lambda\lVert \operatorname{Im} x\rVert_1\)	Imaginary part	ADMM, PRIDU
`ImaginaryL2`	\(\tfrac{\lambda}{2}\lVert \operatorname{Im} x\rVert_2^2\)	Imaginary part	ADMM, PRIDU
`TotalGeneralizedVariation`	Second-order TGV	Extends the variable	ADMM, PRIDU
`InfimalConvolutionTV`	Infimal convolution of two TV terms	Extends the variable	ADMM, PRIDU
`InfimalConvolutionTGV`	Infimal convolution of two TGV terms	Extends the variable	ADMM, PRIDU

\(\lambda\) is the term’s weight, relative to data divided by data_scaling(); \(\lVert\cdot\rVert_1\) of a complex array is the sum of the moduli, and a term’s joint_axes group entries into an \(\ell_2\) norm first. Every term is also accepted by bartorch.apps.pics(). Regularizer.detach() holds a term fixed in a differentiated solve.

Term classes#

Object	Description
`Regularizer`	Base class: the proximal operator and the transform of a BART term
`ImplicitPrior`	A denoiser or a real-valued image network in place of a proximal operator (plug-and-play), optionally through a transform

Sparsity-promoting terms#

Object	Description
`L1`	\(\ell_1\) norm of the image
`Wavelet`	\(\ell_1\) norm of the wavelet coefficients
`FourierL1`	\(\ell_1\) norm of the Fourier coefficients
`TotalVariation`	Total variation
`Laplace`	\(\ell_1\) norm of the Laplacian
`ImaginaryL1`	\(\ell_1\) norm of the imaginary part

Quadratic terms#

Object	Description
`L2`	Squared \(\ell_2\) norm of the image (Tikhonov)
`ImaginaryL2`	Squared \(\ell_2\) norm of the imaginary part

Low-rank terms#

Object	Description
`LocallyLowRank`	Nuclear norm of image blocks

Constraints#

Object	Description
`NonNegative`	Projection clamping the real and imaginary parts at zero

Terms with auxiliary variables#

The optimization variable is the image followed by auxiliary fields, which BART counts across the whole set of terms; these terms cannot be built alone, and the solvers return the image only.

Object	Description
`TotalGeneralizedVariation`	Second-order total generalized variation
`InfimalConvolutionTV`	Infimal convolution of total variation
`InfimalConvolutionTGV`	Infimal convolution of total generalized variation

Denoisers#

Object	Description
`rof`	Total-variation (Rudin-Osher-Fatemi) denoising
`tgv`	Second-order total generalized variation denoising
`nlmeans`	Non-local means filtering

Regularization and denoising

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