API for Regularizers

L1Regularization

Regularization term implementing the proximal map for the Lasso problem.

L2Regularization

Regularization term implementing the proximal map for Tikhonov regularization.

L21Regularization

Regularization term implementing the proximal map for group-soft-thresholding.

Arguments

• λ - regularization paramter

Keywords

• slices=1 - number of elements per group

LLRRegularization

Regularization term implementing the proximal map for locally low rank (LLR) regularization using singular-value-thresholding.

Arguments

• λ - regularization paramter

Keywords

• shape::Tuple{Int} - dimensions of the image
• blockSize::Tuple{Int}=(2,2) - size of patches to perform singular value thresholding on
• randshift::Bool=true - randomly shifts the patches to ensure translation invariance
• fullyOverlapping::Bool=false - choose between fully overlapping block or non-overlapping blocks

NuclearRegularization

Regularization term implementing the proximal map for singular value soft-thresholding.

Arguments:

• λ - regularization paramter

Keywords

• svtShape::NTuple - size of the underlying matrix

TVRegularization

Regularization term implementing the proximal map for TV regularization. Calculated with the Condat algorithm if the TV is calculated only along one real-valued dimension and with the Fast Gradient Projection algorithm otherwise.

Reference for the Condat algorithm: https://lcondat.github.io/publis/Condat-fast_TV-SPL-2013.pdf

Reference for the FGP algorithm: A. Beck and T. Teboulle, "Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems", IEEE Trans. Image Process. 18(11), 2009

Arguments

• λ::T - regularization parameter

Keywords

• shape::NTuple - size of the underlying image
• dims - Dimension to perform the TV along. If Integer, the Condat algorithm is called, and the FDG algorithm otherwise.
• iterationsTV=20 - number of FGP iterations

PositiveRegularization

Regularization term implementing a projection onto positive and real numbers.

RealRegularization

Regularization term implementing a projection onto real numbers.

innerreg(reg::AbstractNestedRegularization)

return the inner regularization term of reg. Nested regularization terms also implement the iteration interface.

sink(reg::AbstractNestedRegularization)

return the innermost regularization term.

sinktype(reg::AbstractNestedRegularization)

return the type of the innermost regularization term.

See also sink.

AbstractScaledRegularization

Nested regularization term that applies a scalefactor to the regularization parameter λ of its inner term.

See also scalefactor, λ, innerreg.

scalescalefactor(reg::AbstractScaledRegularization)

return the scaling scalefactor for λ

NormalizedRegularization

Nested regularization term that scales λ according to normalization scheme. This term is commonly applied by a solver based on a given normalization keyword

#See also NoNormalization, MeasurementBasedNormalization, SystemMatrixBasedNormalization.

NoNormalization

No normalization to λ is applied.

MeasurementBasedNormalization

λ is normalized by the 1-norm of b divided by its length.

SystemMatrixBasedNormalization

λ is normalized by the energy of the system matrix rows.

FixedParameterRegularization

Nested regularization term that discards any λ passed to it and instead uses λ from its inner regularization term. This can be used to selectively disallow normalization.

MaskedRegularization

Nested regularization term that only applies prox! and norm to elements of x for which the mask is true.

Examples

julia> positive = PositiveRegularization();

julia> masked = MaskedRegularization(reg, [true, false, true, false]);

julia> prox!(masked, fill(-1, 4))
4-element Vector{Float64}:
  0.0
 -1.0
  0.0
 -1.0

TransformedRegularization(reg, trafo)

Nested regularization term that applies prox! or norm on z = trafo * x and returns (inplace) x = adjoint(trafo) * z.

Example

julia> core = L1Regularization(0.8)
L1Regularization{Float64}(0.8)

julia> wop = WaveletOp(Float32, shape = (32,32));

julia> reg = TransformedRegularization(core, wop);

julia> prox!(reg, randn(32*32)); # Apply soft-thresholding in Wavelet domain

    PlugAndPlayRegularization

Regularization term implementing a given plug-and-play proximal mapping. The actual regularization term is indirectly defined by the learned proximal mapping and as such there is no norm implemented.

Arguments

• λ - regularization paramter

Keywords

• model - model applied to the image
• shape - dimensions of the image
• input_transform - transform of image before model

prox!(reg::AbstractParameterizedRegularization, x)

perform the proximal mapping defined by reg on x. Uses the regularization parameter defined for reg.

prox!(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)

construct a regularization term of type regType with given λ and kwargs and apply its prox! on x

norm(reg::AbstractParameterizedRegularization, x)

returns the value of the reg regularization term on x. Uses the regularization parameter defined for reg.

λ(reg::AbstractParameterizedRegularization)

return the regularization parameter λ of reg

norm(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)

construct a regularization term of type regType with given λ and kwargs and apply its norm on x