# FunctionOperators.jl

## Motivation

I wanted to write code for image reconstruction in Julia, which

• resambles the mathematical notation with abstract operators on multidimensional spaces,
• has minimal memory requirement and fast to run, and
• is easy to write and read.

FunctionOperator is an operator that maps from a multidimensional space to another multidimensional space. The mapping is defined by a function (forw), and optionally the reverse mapping can also be defined (backw). The input the mapping must be subtype of AbstractArray.

## Examples

### Create operator

The 2D Fourier transformation operator:

julia> using FFTW
julia> ๐ = FunctionOperator{Complex{Float64}}(
forw = x -> fft(x, (1,2)), backw = x -> ifft(x, (1,2)),
inDims = (128, 128), outDims = (128, 128))


Finite differences / Total Variance operator:

julia> โ = FunctionOperator{Complex{Float64}}(
forw = x -> (circ(x, (1,0)) - x).^2 + (circ(x, (0,1)) - x).^2,
inDims = (128, 128), outDims = (128, 128))


Or a sampling operator:

julia> mask = rand(128, 128) .< 0.3
julia> S = FunctionOperator{Complex{Float64}}(
inDims = (128, 128), outDims = (sum(mask),))


Then these operators can be combined (almost) arbitrarily:

julia> x = rand(128, 128);
julia> ๐ * โ * x == fft((circ(x, (1,0)) - x).^2 + (circ(x, (0,1)) - x).^2, (1,2))
true
julia> combined = S * (๐ + โ);
julia> combined * x == S * ๐ * x + S * โ * x
true


They can be combined with UniformScaling from LinearAlgebra:

julia> using LinearAlgebra
julia> 3I * โ * x == 3 * (โ * x)
true
julia> (๐ + (3+2im)I) * x == ๐ * x + (3+2im) * x
true


### Performance

With little effort we can achieve the same speed as we would have manually optimized functions. For example, consider the following function:

julia> using BenchmarkTools
julia> FFT_plan = plan_fft(x, (1,2));
julia> iFFT_plan = plan_ifft!(x, (1,2));
julia> function foo(output::Array{Complex{Float64},2}, x::Array{Complex{Float64},2},
mul!(output, FFT_plan, x)
mul!(output, iFFT_plan, output)
end;
julia> output = similar(x);
julia> @benchmark foo(output, x, FFT_plan, iFFT_plan, mask)
BenchmarkTools.Trial:
memory estimate:  0 bytes
allocs estimate:  0
--------------
minimum time:     390.961 ฮผs (0.00% GC)
median time:      418.149 ฮผs (0.00% GC)
mean time:        408.111 ฮผs (0.00% GC)
maximum time:     497.468 ฮผs (0.00% GC)
--------------
samples:          10000
evals/sample:     1


That function basically consist of three operations: A Fourier transform, a masking, and an inverse Fourier transform. Using FunctionOperators, we can write code that is more similar to the high-level description that has minimal run-time and memory overhead:

julia> ๐โ = FunctionOperator{Complex{Float64}}(
forw = (output, x) -> mul!(output, FFT_plan, x),
backw = (output, x) -> mul!(output, iFFT_plan, x),
inDims = (128, 128), outDims = (128, 128));
julia> Sโ = FunctionOperator{Complex{Float64}}(
forw = (output, x) -> output .= x .* mask,
inDims = (128, 128), outDims = (128, 128));
julia> combined = ๐โ' * Sโ * ๐โ;
julia> @benchmark mul!(output, combined, x)
BenchmarkTools.Trial:
memory estimate:  112 bytes
allocs estimate:  4
--------------
minimum time:     401.814 ฮผs (0.00% GC)
median time:      429.648 ฮผs (0.00% GC)
mean time:        427.211 ฮผs (0.00% GC)
maximum time:     681.116 ฮผs (0.00% GC)
--------------
samples:          10000
evals/sample:     1


For more detailed description, see tutorial.

Not a Julia package, but the main motivation behind creating this package is to have the same functionality as fatrix2 in the Matlab version of Michigan Image Reconstruction Toolbox (MIRT), (description).

The most similar Julia package is AbstractOperators.jl. The feature set of its MyLinOp type largely overlaps with FunctionOperator's features. The main difference is that composition in AbstractOperators is more intensive memory-wise as it allocates a buffer for each member of composition while FunctionOperators allocates a new buffer only when necessary. On the other hand, the difference between is significant only for memory-intensive applications.

FunctionOperators was also inspired by LinearMaps.jl The main difference is that LinearMaps support only mappings where the input and output are both vectors (which is often not the case in image reconstruction algorithms). LinearMapsAA.jl is an extension of LinearMaps.jl with getindex and setindex! functions making it conform to the requirements of an AbstractMatrix type. Additionally, a user can include a NamedTuple of properties with it, and then retrieve those later using the A.key syntax like one would do with a struct (composite type). From implementational point of view, both LinearMaps.jl and LinearMapsAA.jl uses more memory when LinearMaps with different input and output size are composed.

LinearOperators provides some similar features too, but it also requires the input and the output to be a vector.