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Merge pull request #26 from bionanoimaging/fractiona_fourier_transform
Fractiona fourier transform
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‎Project.toml‎

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@@ -15,6 +15,7 @@ Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
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ShiftedArrays = "1277b4bf-5013-50f5-be3d-901d8477a67a"
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[compat]
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ChainRulesCore = "1, 1.0, 1.1"
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FFTW = "1.5"
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ImageTransformations = "0.9"
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julia = "1, 1.6, 1.7, 1.8"
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[extras]
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FractionalTransforms = "e50ca838-b4f0-4a10-ad18-4b920bf1ae5c"
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ImageTransformations = "02fcd773-0e25-5acc-982a-7f6622650795"
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Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
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Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
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TestImages = "5e47fb64-e119-507b-a336-dd2b206d9990"
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Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
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[targets]
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test = ["Test", "Random", "ImageTransformations", "Zygote"]
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test = ["Test", "TestImages", "FractionalTransforms", "Random", "ImageTransformations", "Zygote"]

‎README.md‎

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@@ -29,10 +29,15 @@ The main features are:
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* array/image shearing
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* convenient wrappers of [NFFT.jl](https://github.com/JuliaMath/NFFT.jl)
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* several tools like `ffts`, `ft`, `fftshift_view` etc. allowing simpler use with Fourier transforms
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* Chirp Z-Transform
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* Fractional Fourier Transform
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* reexports [FFTW.jl](https://github.com/JuliaMath/FFTW.jl)
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Have a look in the [examples folder](examples/) for interactive examples. The [documentation](https://bionanoimaging.github.io/FourierTools.jl/dev/) offers a quick overview.
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## FFTW Threading
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By default we set 4 Threads. Use `FFTW.set_num_threads(N)` to set `N` threads.
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## Related Packages

‎examples/Project.toml‎

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Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
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FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
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FourierTools = "b18b359b-aebc-45ac-a139-9c0ccbb2871e"
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FractionalTransforms = "e50ca838-b4f0-4a10-ad18-4b920bf1ae5c"
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ImageShow = "4e3cecfd-b093-5904-9786-8bbb286a6a31"
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Images = "916415d5-f1e6-5110-898d-aaa5f9f070e0"
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IndexFunArrays = "613c443e-d742-454e-bfc6-1d7f8dd76566"
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NDTools = "98581153-e998-4eef-8d0d-5ec2c052313d"
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Napari = "ed46c112-542a-4e7e-ab0f-e39321fab6f0"
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Noise = "81d43f40-5267-43b7-ae1c-8b967f377efa"
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Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
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Pluto = "c3e4b0f8-55cb-11ea-2926-15256bba5781"

‎examples/fractional_fourier_transform.jl‎

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### A Pluto.jl notebook ###
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# v0.19.13
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using Markdown
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using InteractiveUtils
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# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
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macro bind(def, element)
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quote
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local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
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local el = $(esc(element))
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global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
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el
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end
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end
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# ╔═╡ a696290a-0122-11ed-01e5-a39256aed683
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begin
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using Pkg;
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Pkg.activate(".")
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using Revise
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end
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# ╔═╡ 459649ed-ca70-426e-8273-97b146b5bcd5
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using FourierTools, FFTW, NDTools, TestImages, Colors, ImageShow, PlutoUI, IndexFunArrays, Plots
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# ╔═╡ 55894157-a2d1-4567-99a8-a052d5335dd1
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begin
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"""
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simshow(arr; set_one=false, set_zero=false,
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f=nothing, γ=1)
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Displays a real valued array . Brightness encodes magnitude.
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Works within Jupyter and Pluto.
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# Keyword args
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The transforms are applied in that order.
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* `set_zero=false` subtracts the minimum to set minimum to 1
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* `set_one=false` divides by the maximum to set maximum to 1
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* `f` applies an arbitrary function to the abs array
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* `γ` applies a gamma correction to the abs
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* `cmap=:gray` applies a colormap provided by ColorSchemes.jl. If `cmap=:gray` simply `Colors.Gray` is used
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and with different colormaps the result is an `Colors.RGB` element type
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"""
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function simshow(arr::AbstractArray{T};
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set_one=true, set_zero=false,
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f = nothing,
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γ = one(T),
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cmap=:gray) where {T<:Real}
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arr = set_zero ? arr .- minimum(arr) : arr
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if set_one
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m = maximum(arr)
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if !iszero(m)
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arr = arr ./ maximum(arr)
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end
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end
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arr = isnothing(f) ? arr : f(arr)
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if !isone(γ)
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arr = arr .^ γ
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end
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if cmap == :gray
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Gray.(arr)
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else
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get(colorschemes[cmap], arr)
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end
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end
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"""
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simshow(arr)
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Displays a complex array. Color encodes phase, brightness encodes magnitude.
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Works within Jupyter and Pluto.
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# Keyword args
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The transforms are applied in that order.
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* `f` applies a function `f` to the array.
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* `absf` applies a function `absf` to the absolute of the array
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* `absγ` applies a gamma correction to the abs
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"""
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function simshow(arr::AbstractArray{T};
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f=nothing,
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absγ=one(T),
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absf=nothing) where (T<:Complex)
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if !isnothing(f)
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arr = f(arr)
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end
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Tr = real(T)
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# scale abs to 1
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absarr = abs.(arr)
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absarr ./= maximum(absarr)
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if !isnothing(absf)
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absarr .= absf(absarr)
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end
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if !isone(absγ)
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absarr .= absarr .^ absγ
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end
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angarr = angle.(arr) ./ Tr(2pi) * Tr(360)
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HSV.(angarr, one(Tr), absarr)
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end
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end
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# ╔═╡ 18b0700c-20fc-4e58-8950-ca09fe34ea19
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import FractionalTransforms
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# ╔═╡ 4371cfbf-a3b3-45dc-847b-019994fbb234
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md"## Fractional Fourier Transform on a Image"
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# ╔═╡ d90b7f67-4166-44fa-aab7-de2c4f38fc00
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img = Float32.(testimage("resolution_test_512"));
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# ╔═╡ 24901666-4cc4-497f-a6ff-68c3e7ead629
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md"
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Fractional order
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$(@bind s Slider(-3:0.001:3, show_value=true))
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"
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# ╔═╡ 08d95a42-24fe-4995-98d6-1f415de71538
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function frft_1D(x, order)
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x_ft = similar(x, complex(eltype(x)), size(x))
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for i = 1:size(x, 1)
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x_ft[i, :] = FractionalTransforms.frft(x[i, :], order)
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end
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for j = 1:size(x, 2)
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x_ft[:, j] = FractionalTransforms.frft(x_ft[:, j], order)
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end
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return x_ft
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end
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# ╔═╡ 7c445baa-d970-4954-a3dc-df828971bfd7
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[simshow(abs.(ft(img)), γ=0.2) simshow(sqrt(length(img)) .* abs.(frfft(img, s, shift=true)), γ=0.2) simshow(sqrt(length(img)) .* abs.(frft_1D(img, s)), γ=0.2)]
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# ╔═╡ 1915c023-69cf-4d18-90cb-b47465dbef69
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begin
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plot(log1p.(abs.(ft(img)[(end+begin)÷2+1,:] ./ sqrt(length(img)))))
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plot!(log1p.(abs.(frfft(img, s)[(end+begin)÷2+1,:])))
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end
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# ╔═╡ 3109fc21-50c6-46e6-850d-add6f54872d7
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begin
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plot((imag.(ft(img)[(end+begin)÷2+1,:] ./ sqrt(length(img)))))
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plot!((imag.(frfft(img, 0.99999999999)[(end+begin)÷2+1,:])))
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end
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# ╔═╡ 284cd6f2-1ee3-4923-afa6-ea57e93b28a7
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begin
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plot((angle.(ft(img)[(end+begin)÷2+1,:] ./ sqrt(length(img)))))
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plot!((angle.(frfft(img, 0.99999999999)[(end+begin)÷2+1,:])))
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end
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# ╔═╡ 227ae9a3-9387-4ac3-b391-e2a78ce40d49
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begin
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plot((real.(ft(img)[(end+begin)÷2+1,200:300] ./ sqrt(length(img)))))
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plot!((real.(frfft(img, s)[(end+begin)÷2+1,200:300])))
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end
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# ╔═╡ abff911a-e10d-4311-955a-7afc4e0d344c
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md"## Fractional Fourier Transform on Vector
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Comparison with [FractionalTransforms.jl](https://github.com/SciFracX/FractionalTransforms.jl) roughly matches.
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"
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# ╔═╡ bae3c5b7-8964-493b-9e7b-d343e092219c
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r = box(Float64, (301,), (201,))#.+ 0.4 .* randn((301,))
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# ╔═╡ 5655dc10-f4e9-4765-9a89-ac9702864de1
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plot(abs.(ft(r)))
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# ╔═╡ 07d2b3b6-3584-4c64-9c4a-138beb3d6b88
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@bind s2 Slider(-5:0.001:5, show_value=true)
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# ╔═╡ 1839f03e-6add-4c85-b6fd-9035656ed86c
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begin
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plot(real.(frfft(r, s2, shift=true)), label="FourierTools")
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#plot!(imag.(frfft(frfft(r, s2/2, shift=true), s2/2, shift=true)), label="FourierTools 2 Step")
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plot!(real.(FractionalTransforms.frft(r, s2)), label="FractionalTransforms")
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#plot!(imag.(FractionalTransforms.frft(ft(r) ./ sqrt(length(r)), -1+s2)), label="FractionalTransforms")
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#plot!(imag.(FractionalTransforms.frft(r, s2)))
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#plot!(real.(FractionalTransforms.frft(r, s2)))
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end
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# ╔═╡ f3cb2153-a7b3-46ed-adbb-038a812b6a81
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begin
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#plot(abs.(frfft(r, s2, shift=true, p_change=false)))
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plot(abs.(frfft(r, s2, shift=true, p_change=true)))
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#plot!(imag.(FractionalTransforms.frft(r, s2)), label="FractionalTransforms")
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#plot!(imag.(FractionalTransforms.frft(ft(r) ./ sqrt(length(r)), -1+s2)), label="FractionalTransforms")
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plot!(abs.(FractionalTransforms.frft(r, s2)))
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#plot!(real.(FractionalTransforms.frft(r, s2)))
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end
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# ╔═╡ 37ebf4d8-28fa-4d0b-929c-5df4c9f418e0
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md"## Gaussian Propagation"
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# ╔═╡ fab2b38f-7a93-438e-a1f9-9e58709aec2e
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x = -256:256
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# ╔═╡ 02708a88-14ce-45cc-8d40-71a74bc5a56d
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amp = exp.(-(x.^2 .+ x'.^2) ./ 3000);
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# ╔═╡ 77807bbe-a33a-4d65-8e06-446ad368784f
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phase_term = exp.(1im .* x .* 2π ./ 5 .+ 1im .* x'.^2);
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# ╔═╡ 696a77b2-a904-4cf8-805e-b66621dbbb8f
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field = amp .* phase_term;
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# ╔═╡ 4e53efc4-de25-4b97-8dc8-985d56b8bc67
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simshow(field)
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# ╔═╡ 3fa82b96-e701-40d1-89c2-8f71038b6d05
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simshow(ft(field))
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# ╔═╡ 4dcf3db5-6d37-4a09-a161-4af53ffc91ec
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@bind f2 Slider(-4:0.001:4, show_value=true)
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# ╔═╡ e4db42df-cfe9-4ae0-91f6-5672707d87d5
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simshow(frfft(field, f2))
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# ╔═╡ 67e0e7dc-4692-451c-97d0-742ab5df3853
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rev2(x) = ifftshift(reverse(fftshift(x)))
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# ╔═╡ a52deb8f-64e6-46ab-b77e-13b35a20f17c
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rev2([simshow(frfft(field, f2));;; simshow(frfft(ift(field), f2-1))][:, :, 2])
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# ╔═╡ ce65aa2c-e558-434b-aa8a-2268c47f5684
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md"### Comparison with two step FRFT with half the order"
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# ╔═╡ e0529213-f2c9-49e4-b2fe-96bbd16a77b7
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@bind f3 Slider(-4:0.001:4, show_value=true)
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# ╔═╡ 1fe0d80f-664b-4b9f-9ff3-95f0d00e32d5
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simshow(frfft(field, f3, p_change=false))
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# ╔═╡ e5f32874-5f98-4825-824a-780764e8ef91
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simshow(frfft(frfft(field, f3/2),f3/2))
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# ╔═╡ Cell order:
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# ╠═a696290a-0122-11ed-01e5-a39256aed683
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# ╟─55894157-a2d1-4567-99a8-a052d5335dd1
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# ╟─4371cfbf-a3b3-45dc-847b-019994fbb234
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‎src/FourierTools.jl‎

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include("correlations.jl")
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include("damping.jl")
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include("czt.jl")
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include("fractional_fourier_transform.jl")
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end # module

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