HW05_in_class.jl

### A Pluto.jl notebook ###
# v0.19.40

using Markdown
using InteractiveUtils

# 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).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ d1980bd2-babf-11ee-1dbb-dfefbdbdb36d
using PlutoUI, Plots, ImageShow, TestImages, FFTW, NDTools, IndexFunArrays, FileIO, FourierTools, SpecialFunctions, UrlDownload, ImageMagick

# ╔═╡ 17adc747-9822-42be-91c1-ad8a2e1532bc
md"# 0. Load packages"

# ╔═╡ d2c6102c-7a62-4899-9c5c-b08ce9a5baa8
FFTW.set_num_threads(4)

# ╔═╡ 8e6ec43c-9135-4829-8146-a56b8624750a
const TODO = nothing

# ╔═╡ 7331d6a5-dd03-42c7-9ab3-c84a641296bc
TableOfContents()

# ╔═╡ f326cd36-9c05-4b1c-81c0-56c954cd51f3
gauss_R(z::T, z_R) where T = iszero(z) ? T(Inf) : (1 + (z_R / z)^2)

# ╔═╡ 7c6b4ddd-ca5a-4b23-96f7-e896fb1cda6d
gauss_ψ(z, z_R) = atan(z, z_R)

# ╔═╡ dd16735b-c1d6-4977-a396-a3e23823ee68
gauss_w(z, z_R, w_0) = w_0 * sqrt(1 + (z / z_R)^2)

# ╔═╡ 8e04ae26-5474-4c2c-9fda-dbb7efd77acd
"""
	gauss_beam(y, x, z, λ, w_0)


Returns the eletrical field of a Gaussian beam at position `(y, x)` at optical axis position `z` with respect to the beam waist `w_0`.
Wavelength is `λ`.
"""
function gauss_beam(y, x, z, λ, w_0)
    k = π / λ * 2
    z_R = π * w_0^2 / λ
	r² = x ^ 2 + y ^ 2
    # don't put exp(i * k * z) into the same exp, it causes some strange wraps
	return w_0 / gauss_w(z, z_R, w_0) * exp(-r² / gauss_w(z, z_R, w_0)^2) * 
            exp(1im * k * z) * 
            exp(1im * (k * r² / 2 / gauss_R(z, z_R) - gauss_ψ(z, z_R)))
end

# ╔═╡ 5f21a849-0272-4bca-9659-54cd38068e09
"""
	bpm(field, λ0, Lx, Ly, z; window=true, paraxial=true, amplitude_array)

Propagates the array `field` with wavelength `λ0` and the filed size in meter size
`(Lx, Ly)`. The propagation distance `z` should be a vector of distances.

.
The returned array is a three dimensional array where `size(arr, 3) == size(z, 1)`.


If `window=true` we apply a Hann window function to dampen the boundaries.
A keyword `amplitude_array` can be provided, which multiplies with the field at each point. This allows to include obstacles.

If `paraxial=true` the Fresnel approximation is applied.
"""
function bpm(field, λ0, Lx, Ly, z, n=1; window=true, amplitude_array=ones(size(field)..., length(z)), paraxial=true)
	# free space wavenumber in m-1
	k0 = 2 * π / λ0
	# medium wavenumber m-1
	k = n * k0
	# medium in m
	dz = z[2] - z[1]

	# field parameters
	Nx = size(field, 1)
	dx = Lx / Nx
	x  = range(-Lx/2, Lx/2, Nx)
	fx = reshape(fftfreq(Nx, 1 / dx), (1, Nx))

	Ny = size(field, 1)
	dy = Ly / Ny
	y  = range(-Ly/2, Ly/2, Ny)
	fy = fftfreq(Ny, 1 / dy)

	if paraxial
		# important step, this calculates the Fourier space kernel
		H = exp.(-1im .* k .* λ0^2 .* (fx.^2 .+ fy.^2) ./ (2) * dz)
	else
		H = exp.(1im .* sqrt.(1 .+ 0im .- λ0^2 .* fx.^2 .- λ0^2 .* fy.^2) .* k .* dz) .* ((λ0^2 .* fx.^2 .+ λ0^2 .* fy.^2) .< 1)
	end
	# 3d output fields we save
	# third dimensions stores the different z propagation distances
	out_field = zeros(ComplexF64, (Ny, Nx, size(z, 1)))
	# first entry corresponds to z[1] = 0
	out_field[:, :, 1] = field
	
	# FFT plan for calculating FFTs
	# It's a more efficient syntax: p * x == fft(x)
	p = plan_fft(field, (1,2))
	window_f = window ? IndexFunArrays.window_hanning(size(out_field)[1:2], border_in=0.8) : 1
	# inverse FFT
	invp = inv(p)
	for z_index in 2:size(out_field, 3)
		u0 = out_field[:, :, z_index - 1] .* window_f .* amplitude_array[:, :, z_index - 1]
    	u1 = invp * ((p * u0) .* H)
		out_field[:, :, z_index] .= u1
	end
	return out_field
end

# ╔═╡ 1938a0dd-ffec-478e-870c-18d14f458f6e
md"# 1. Coherent Imaging with a 4f system

As seen in the image below, a 4f system consists of two lenses.
The object is placed 1f in front of the first lens and the image is located 1f behind the second lens.
In the middle, the Fourier plane is located.

Given the electrical field at the input plane, we can perform a Fourier transform to obtain the Fourier plane. At the Fourier plane we can apply the aperture function (pointwise multiplication `.*`).
A final Fourier transform provides us the electrical field at the exit.
At the image  plane we take `abs` to obtain the amplitude.



If the fast discrete Fourier transform (FFT) is applied, there happens a scaling between the fields:

$$U'(x', y') = \mathcal{F}[U(x,y)](x', y')$$

where $x' = \frac{\lambda \cdot f \cdot N}{L}$
$f$ is the focal length of the lens, $\lambda$ the wavelength, $L$ the field size and $N$ the discretization along one axis.
"

# ╔═╡ 8812ba45-091d-4b07-b941-0c51a6f8d198
urldownload("https://d3i71xaburhd42.cloudfront.net/9d8bdf497732e41e31f871b438d58263ab5a4b54/2-Figure2-1.png")

# ╔═╡ 21b57718-25bc-4143-a383-90124f561eea
md"## Question

Fill in the missing gaps for the circular aperture in the Fourier plane.
You can use `rr` and the `scale` keyword in the `rr` function.

To verify your solution, the cut-off radius of the aperture is $\approx 2\mathrm{mm}$
"

# ╔═╡ 2386c0bf-30d2-41e2-a085-6c422baf16a2
function prop_4f(field, λ, f, L, radius_aperture)
	field_padded = select_region(field, M = 2)
	L_padded = 2 * L
	field_ft = ft(field_padded)

	L_ft = λ * size(field_ft, 1) * f / L_padded
	
	aperture = rr(size(field_ft), scale=L_ft/size(field_ft, 1)) .<= radius_aperture
	field_ft .*= aperture 

	
	field_out = select_region(ift(field_ft), new_size=size(field))
	return (field_out)
end

# ╔═╡ d2f60c2f-9745-4fe0-bf9d-8f7d5877ea39
rr((5,5), scale=0.1)

# ╔═╡ 50358b73-9736-4195-98a0-b5f79b8cd150
begin
	img = sqrt.(Float32.(testimage("resolution_test_512")));
	img[490:512, :] .= 0.5 .* (1 .+ sin.((fftpos(512, 512, CenterFT)) .* π .* 2 / 8))'
	#img[:, :] .= 0.5 .* (1 .+ sin.((fftpos(512, 512, CenterFT)) .* π .* 2 / 8))'

end;

# ╔═╡ da84b8e5-374c-4970-990f-a99d70632f24
size(img, 1)

# ╔═╡ 64fba384-d96e-4895-9e2a-1bc03fb839ef
simshow([img img])

# ╔═╡ c7bdd3a3-59f2-451f-8ccb-0fdff26abc53
L = 2e-3

# ╔═╡ 0c9fcd30-c779-46ba-a231-0b421e2ccbdd
y = fftpos(L, size(img, 1), CenterFT)

# ╔═╡ 6feb5de4-dbf9-45cf-beda-28adee3564c5
x = y'

# ╔═╡ c5093acf-bb4e-402c-a1fa-430cafd9b523
λ = 633e-9

# ╔═╡ 0c936df2-e9bf-48e0-bb9a-42728b14af43
f = 100e-3

# ╔═╡ ab3a0a7d-9f4c-4bbf-b2f3-61e0e1f3a970
@bind radius Slider(range(0, 10e-3, 100), show_value=true, default=0.001)

# ╔═╡ 9019367d-2592-4401-9f38-217451c58492
img_prop_coherent = prop_4f(img, λ, f, L, radius);

# ╔═╡ 4572348d-d9d1-448d-9121-617e74de3a0e
heatmap(y, y, simshow(abs2.(img_prop_coherent)), xlim=(y[begin], y[end]))

# ╔═╡ b2e93c0b-9413-4cc2-a917-ae51963bd100
md"# 2. Incoherent Imaging
Incoherent imaging means that two neighbouring points are not coherent and hence do not interfere with each other.
A similar effect can be achieved by adding random phase shifts to the field.

Use the code below and add at the right position random phase shifts.
Which random values of the phase you have to choose?
See `randn` and the `rand` functions. Which is the correct one?

Use the code below and fix it at the right position.
"

# ╔═╡ 24c9fcf8-b13e-44e9-9982-c61e73a34240
function prop_4f_incoherent(field, λ, f, L, radius_aperture, N_iter=50)
	intensity = zeros((size(field)..., Threads.nthreads()))
	
	Threads.@threads for k in 1:N_iter
		random_phase = exp.(1im .* 2π .* rand(size(field)...))
		
		field_padded = select_region(field .* random_phase, M = 2)
		L_padded = 2 * L
		field_ft = ft(field_padded)
	
		L_ft = λ * f / L_padded * size(field_padded, 1)
		aperture = rr(size(field_ft), scale=L_ft / size(field_padded, 1)) .<= radius_aperture
		aperture = aperture
		field_ft .*= aperture 
	
		field_out = select_region(ift(field_ft), new_size=size(field))

		intensity[:, :, Threads.threadid()] .+= abs2.(field_out)
	end
	return sum(intensity, dims=(3,))[:, :, 1] ./ sqrt(N_iter)
end

# ╔═╡ b5a50850-ce01-49db-bb12-55adba69fc9c
img_prop_incoherent = prop_4f_incoherent(img, λ, f, L, radius);

# ╔═╡ f1341836-0297-49e4-a21d-7b8cbcc8fd59
heatmap(y,y, simshow(img_prop_incoherent), xlim=(y[begin], y[end]))

# ╔═╡ a72ff4fa-70a8-4088-b393-17daaabd6c19
md"## Question
Is the incoherent imaging better than the coherent one?
Explain what is the resolution limit between the two?
What is the resolution limit for the coherent one and what for the incoherent one?
"

# ╔═╡ 2f32c085-1d6b-4858-be6f-9bfc280c2695
md"## Answer

$$x_c = \frac{\lambda}{\text{NA}}$$

$$x_i = \frac{\lambda}{2\text{NA}}$$
"

# ╔═╡ 447ab89c-6fec-4dad-8cd5-7ccc3ef3c275
md"# 3. True Incoherent Imaging - Point Spread Function

True incoherent imaging can be described more efficiently than just adding randomly phased shifted fields.
To describe the imaging process, we can use the point spread function (PSF).

The PSF is the image of a point source and is the response of the imaging system to a point source.
It can be calculated by coherently propagating a point source through the imaging system. And then taking the `abs2` of the field at the image plane.

* propagate point source with `ft`
* apply circular aperture. Use again the `rr` function to create a circular aperture. Use the `scale=L_ft / N` to have the correct scaling of the aperture. `L_ft = λ * f / L * N` is the scaling factor for the aperture.
* propagate with another `ft`
* take `abs2.`

"

# ╔═╡ 002c1f76-ca75-43d5-a294-8f1c3cd5da56
@bind radius_psf Slider(range(0, 10e-3, 100), default=0.001, show_value=true)

# ╔═╡ 41f3b42a-49f5-485e-b1ef-fa8f24e85a42
"""
    calc_psf(N, λ, f, L, radius_aperture)

`N` is the pixel number of the field.
`λ` is the wavelength.
`f` is the focal length of the lens.
`L` is the field size.
`radius_aperture` is the radius of the aperture in meter.
"""
function calc_psf(N, λ, f, L, radius_aperture)

	field = delta((N, N))
	field_ft = ft(field)
	
	
	L_ft = λ * f / L * N
	aperture = rr((N, N), scale=L_ft / N) .< radius_aperture
	aperture = aperture ./ length(aperture)

	field_ft = field_ft .* aperture

	intensity_detector = abs2.(ft(field_ft))

	psf = intensity_detector ./ sum(intensity_detector)

	return psf
end

# ╔═╡ 41e06229-d84c-4d5e-a3de-57da92e93bf4
psf = calc_psf(size(img, 1), λ, f, L, radius);

# ╔═╡ a4284aa6-097d-4ed4-a8aa-2c6dada33e7d
simshow(psf)

# ╔═╡ 0535e742-de35-40e9-b072-c60710959bdf
simshow(abs.(jinc.(rr(size(psf), scale=0.2)).^2))

# ╔═╡ 90837c30-4886-4701-9c60-da945fe9ff14
heatmap(y,y, simshow(psf), xlim=(y[begin], y[end]))

# ╔═╡ b5f3df13-2aa2-4c0c-bb9a-b07bb0190e91
heatmap(y,y, simshow(img_prop_incoherent), xlim=(y[begin], y[end]))

# ╔═╡ 66ea19ac-e8c1-4b62-bc05-9386a44e0d85
md"## Question
Convolve the `img` with `psf` and compare again with `img_prop_incoherent` and `img_prop_coherent`.

Can you observe better resolution again?
"

# ╔═╡ cbfa1584-312c-4e43-b949-2c5139c59d6a
md"## Answer

Yes we can.
"

# ╔═╡ 6e44b629-f5a9-4724-98f3-b864659d35a5
function my_conv(img, psf)
	real.(ift(ft(img) .* ft(psf)))
end

# ╔═╡ 7f0ff049-bfbc-45e4-8ae0-e8cb4c2274eb
real.(ift(ft(img) .* ft(psf))) ≈ abs.(ift(ft(img) .* ft(psf)))

# ╔═╡ 9c626ddd-2b39-427b-a598-46db2b619af9
@time ift(ft(img) .* ft(psf));

# ╔═╡ 0b2e60c1-f11f-4197-bd7d-84902c8ec5fb
@time irft(rft(img) .* rft(psf), size(psf, 1));

# ╔═╡ 25e6b2b1-164c-426f-9ffa-3b26e4cc315f
ift(ft(img) .* ft(psf));

# ╔═╡ 57b8cfb2-a26d-4b06-9320-21ffb69bbb55
img_incoherent_psf = my_conv(abs2.(img), psf);

# ╔═╡ 5bc460dd-68d2-4326-b6b5-993efd9486e6
heatmap(y,y, simshow(img_incoherent_psf), xlim=(y[begin], y[end]))

# ╔═╡ 0ef096a9-3979-461e-8642-5e362c8cfc65
md"# 4. Imaging two Gaussians

Now we want to image two Gaussians blobs.
First use the `prop_4f` for coherent propagation.
Then use `prop_4f_incoherent`.

Additionally, apply a global phase shift of `π` to `blob2`.
And then image again with coherent and incoherent.


Provide a plot of the cross section of all the four images. 
Normalize them to 1 with `maxnormf`.
"

# ╔═╡ 74f0d4f6-22e3-4628-88b8-e450e4c82a1b
sz4 = (64, 64)

# ╔═╡ b87cceb8-dc09-44e9-92ee-74c5c329e83b
λ4 = 633e-9

# ╔═╡ bc124403-07dc-47a9-8557-d83ec4096ed9
L4 = sz4[1] * λ4 / 2

# ╔═╡ 658f936c-ce5e-460c-a8e7-d6b40a6dad73
f4 = 100f-3

# ╔═╡ 33d74a95-7144-4654-b841-a3b735fa85ee
blob1 = disc(sz4, 6, offset=(20, 30));

# ╔═╡ b18ebc68-9ee3-4585-bd0a-647589e9f08f
blob2 = disc(sz4, 6, offset=(37, 30));

# ╔═╡ 376cc9a8-8c02-4d9b-846a-f3e31e69a5d9
blobs_coherent = abs2.(prop_4f(blob1 .+ blob2, λ4, f4, L4, 8e-3));

# ╔═╡ 6089a4ae-ea45-4283-9f16-0bd87b77f514
blobs_phase =  abs2.(prop_4f(blob1 .+ blob2 .* exp(1im * π), λ4, f4, L4, 8e-3));

# ╔═╡ 27137726-297d-4771-a8a1-752f91f4f4f2
blobs_incoherent = abs2.(prop_4f_incoherent(blob1 .+ blob2, λ4, f4, L4, 8e-3));

# ╔═╡ c460493c-79b9-44f0-983f-ab659d08568b
blobs_incoherent_phase = abs2.(prop_4f_incoherent(blob1 .+ blob2 .* exp(1im * π), λ4, f4, L4, 8e-3));

# ╔═╡ 7c32e38c-9eea-4940-a9d5-cfbffeeb291b
[simshow(blobs_coherent) simshow(blobs_incoherent) simshow(blobs_phase)]

# ╔═╡ 0402636e-e4eb-48c1-a1c6-9625aeee5d93
maxnormf(x) = x ./ maximum(x)

# ╔═╡ e9670f97-99b4-423c-861b-d3b460af7c6c
md"## Question
What differences do you observe between all the images? Which has the best resolution?

"

# ╔═╡ 97bd88c4-18db-40e1-a116-081d31ca07b4
md"## Answer
See [Horstmeyer, Roarke, et al. \"Standardizing the resolution claims for coherent microscopy.\" Nature Photonics 10.2 (2016): 68-71.](https://rdcu.be/dBL9X):

_Consider two point sources separated by a small distance and emitting mutually incoherent light, which are resolved at the Rayleigh limit (that is, just separated by a clear dip in intensity) when imaged by a specific microscope. The same point sources in the same locations will not be resolved by the same microscope (that is, will appear as only one large spot) if the sources are instead emitting light that is coherent and in-phase. However, the two sources become fully 'resolved' when their emissions are in anti-phase (that is, shifted by π radians) to each other._

_This means that the intensity profile of an image rendered by a coherent imaging system is phase dependent. As such, simply measuring and reporting its intensity response is an unsuitable means to characterize resolution in an unambiguous manner. Further compounding the issue, many recent coherent imaging methods rely on computational post-processing, with a digitally manipulated image formation pipeline. Another factor that needs to be taken into account is the presence of noise, which can impact image quality and resolution limits4, but is challenging to encompass within a single measurement or scalar performance metric._
"

# ╔═╡ ac39b3f4-7cd8-4c57-a6c2-c4bddd958181
md"# 5. Pinhole Imaging

Let's do imaging through a pinhole.

For that we propagate the object a certain distance before a pinhole.
Then we multiply with a small pinhole.

Finally we perform a far field propagation, hence a `ft` operation to a screen.


## Question
* Find a good value for the aperture radius such that you see an image of Fabio. Can you quantitatively argue why this value works?
* Why do we need an incoherent propagation and why does the coherent one fail?

"

# ╔═╡ dad298c5-e25b-4326-a2a6-14daa15767af
"""
	angular_spectrum(field, z, λ, L)

Returns the the electrical field with physical length `L` and wavelength `λ` propagated with the angular spectrum method of plane waves (AS) by the propagation distance `z`.
"""
function angular_spectrum(field::Matrix{T}, z, λ, L; pad_factor = 2) where T
	@assert size(field, 1) == size(field, 2) "Restricted to quadratic fields."
	# we need to apply padding to prevent circular convolution
	L_new = pad_factor .* L
	
	# applies zero padding
	field_new = select_region(field, new_size=size(field) .* pad_factor)
	
	# helpful propagation variables
	k = 2π / λ
	N = size(field_new, 1)
	f_y = similar(field, real(eltype(field)), (N,1))
	f_y .= fftfreq(N, N / L_new)
	f_x = f_y'
	
	# transfer function kernel of angular spectrum
	H = exp.(1im .* k .* z .* sqrt.(0im .+ 1 .- abs2.(f_x .* λ) .- abs2.(f_y .* λ)))
	
	# bandlimit according to Matsushima
	# and fuzzy logic 
	Δu =   1 / L_new
	u_limit = 1 / (sqrt((2 * Δu * z)^2 + 1) * λ)
	
	# bandlimit filter
	W = ((abs2.(f_y) ./ u_limit^2 .+ abs2.(f_x) * λ^2) .< 1) .*
		 ((abs2.(f_x) ./ u_limit^2 .+ abs2.(f_y) * λ^2) .< 1)

	# propagate field
	field_out = fftshift(ifft(fft(ifftshift(field_new)) .* H .* W))
	# take center part because of circular convolution
	field_out_cropped = select_region(field_out, new_size=size(field))
	
	# return final field and some other variables
	return field_out_cropped, (;H, W)
end

# ╔═╡ 897a0965-f495-4c82-aee8-60f5599778cb
fabio = sqrt.(Float32.(testimage("fabio_gray_256")))[begin:2:end, begin:2:end] .+ 0im;

# ╔═╡ 37ad4839-dcce-40db-b890-258613cc98f8
heatmap(y,y, simshow(fabio), xlim=(y[begin], y[end]))

# ╔═╡ a1b76d0d-deed-401b-a452-6c3f46e9654c
"""
	coherent_pinhole_camera(img, z, λ, L; radius=1e-3)

`img` is the input field. `z` is the propagation distance between object an pinhole.
`λ` is the wavelength. `L` is the size of the input image.
`radius` is the aperture radius in meters.
"""
function coherent_pinhole_camera(img, z, λ, L; radius=1e-3)
	dx = L / size(img, 1)
	pinhole = rr(size(img), scale=dx) .<= radius;
	part1, t = angular_spectrum(img, z, λ, L);
	part2 = ft(part1 .* pinhole)
	out = abs2.(part2)
	return out
end

# ╔═╡ 26acacc8-c70b-4e87-9336-241712503762
"""
	incoherent_pinhole_camera(img, z, λ, L; radius=1e-3, N=200)


`img` is the input field. `z` is the propagation distance between object an pinhole.
`λ` is the wavelength. `L` is the size of the input image.
`radius` is the aperture radius in meters.

This function averages over `N` images with random phase, this simulates incoherent light.
"""
function incoherent_pinhole_camera(img, z, λ, L; radius=1e-3, N=200)
	dx = L / size(img, 1)
	out = zeros((size(img)..., Threads.nthreads()))
	pinhole = rr(size(img), scale=dx) .<= radius;
	
	Threads.@threads for _ in 1:N
		part1, t = angular_spectrum(img .* cis.(2π .* rand(size(img)...)), z, λ, L);
		part2 = ft(part1 .* pinhole)
		out[:, :, Threads.threadid()] .+= abs2.(part2)
	end
	
	return sum(out, dims=(3,))[:, :, 1]
end

# ╔═╡ 6fc457d8-6e9b-45e1-9435-898aebbd2108
z5 = 0.2

# ╔═╡ d0485b3b-7ef6-43d1-a347-4164c626d10e
L5 = 0.004

# ╔═╡ d5f96173-4027-411a-b2aa-ab24c29dbe9b
y4 = fftpos(L5, sz4[1], CenterFT);

# ╔═╡ 71356c5e-6d9a-4227-bba9-0cbb527fcf48
begin
	plot(y4, maxnormf(blobs_coherent)[:,30], label="coherent blobs")
	plot!(y4, maxnormf(blobs_phase)[:,30], label="coherent but phase")
	plot!(y4, maxnormf(blobs_incoherent)[:,30], label="incoherent")
	plot!(y4, maxnormf(blobs_incoherent_phase)[:,30], label="incoherent phase")
end

# ╔═╡ f5664869-f2a2-42de-9467-5bc6c1857ec1
radius_pinhole = 0.27e-3

# ╔═╡ a21bdc4f-c712-4bc0-a64b-ca76b31ca1ee
λ5 = λ

# ╔═╡ 329226da-4794-48c0-b143-799b2c666378
fabio_pinhole_c = coherent_pinhole_camera(fabio, z5, λ5, L5, radius=radius_pinhole);

# ╔═╡ a096cd69-9a00-4b28-8d7f-1aa8c7b95a00
fabio_pinhole = incoherent_pinhole_camera(fabio, z5, λ5, L5, radius=radius_pinhole, N=1000);

# ╔═╡ 1863f777-ed76-4e28-a82c-c01e57888331
L5^2 / λ5 / size(fabio, 1)

# ╔═╡ f7c5163f-e364-4af1-bc92-4583b01d0de8
[simshow(fabio_pinhole_c) simshow(fabio_pinhole |> reverse) simshow(abs2.(fabio))]

# ╔═╡ 1ccea702-b279-434f-8631-c02c2528b6d3
√(λ5 * z5 * 2) / 2

# ╔═╡ a0456ef6-fce0-4906-ba81-fec7b6f3d334
md"## Answer
The pinhole camera can be easily understood from geometrical optics perspective.

If we would have a coherent signal only, most of the light will be simply blocked by the aperture since the light travels mostly straight.


To find the best radius $r = D / 2$ we have to find the optimum regime between geometrical and wave optics.
From geometrical optics, the smaller $D$ is, the better the resolution (small $B_G$).

However, the smaller $D$, wave optics will produce a large diffraction spot (Fraunhofer Diffraction).
The optimum is, when both are equal

$$B_G = B_W$$

$$2D = 2.44 λ z / D $$

$$D ≈ \sqrt{\lambda z}$$
"

# ╔═╡ 123fcda7-e0be-4d91-9ed7-d6bebc2840c2
urldownload("https://felix.sumpi.org/pinhole_camera.png")

# ╔═╡ d70e8383-813b-4802-9c76-3328b25e131e
mm = randn((256,256))

# ╔═╡ 06ea5301-843a-4069-8a76-2981457dc22d
@time ft(mm);

# ╔═╡ 96eeab44-8a57-4391-999c-e2e219d86bc7
@time fftshift(fft(ifftshift(mm)));

# ╔═╡ 00000000-0000-0000-0000-000000000001
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[[deps.GLFW_jll]]
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[[deps.GR_jll]]
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[[deps.Gettext_jll]]
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[[deps.Ghostscript_jll]]
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[[deps.Glib_jll]]
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[[deps.Graphite2_jll]]
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[[deps.HTTP]]
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[[deps.HarfBuzz_jll]]
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[[deps.HypertextLiteral]]
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[[deps.ImageIO]]
deps = ["FileIO", "IndirectArrays", "JpegTurbo", "LazyModules", "Netpbm", "OpenEXR", "PNGFiles", "QOI", "Sixel", "TiffImages", "UUIDs"]
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[[deps.ImageMagick]]
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[[deps.ImageMagick_jll]]
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[[deps.ImageMetadata]]
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[[deps.ImageShow]]
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[[deps.Imath_jll]]
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    [deps.IntervalSets.extensions]
    IntervalSetsRandomExt = "Random"
    IntervalSetsRecipesBaseExt = "RecipesBase"
    IntervalSetsStatisticsExt = "Statistics"

[[deps.IrrationalConstants]]
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[[deps.JLFzf]]
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[[deps.LAME_jll]]
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[[deps.LERC_jll]]
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[[deps.LLVMOpenMP_jll]]
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[[deps.LZO_jll]]
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[[deps.LibCURL]]
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[[deps.LibCURL_jll]]
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[[deps.LibGit2]]
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[[deps.LibGit2_jll]]
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[[deps.Libffi_jll]]
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[[deps.Libgcrypt_jll]]
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[[deps.Libglvnd_jll]]
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[[deps.Libgpg_error_jll]]
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[[deps.Libiconv_jll]]
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[[deps.Libmount_jll]]
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[[deps.Libtiff_jll]]
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[[deps.Libuuid_jll]]
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[[deps.LittleCMS_jll]]
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    LogExpFunctionsChangesOfVariablesExt = "ChangesOfVariables"
    LogExpFunctionsInverseFunctionsExt = "InverseFunctions"

    [deps.LogExpFunctions.weakdeps]
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[[deps.Logging]]
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[[deps.LoggingExtras]]
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[[deps.MIMEs]]
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[[deps.MKL_jll]]
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[[deps.MacroTools]]
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[[deps.MappedArrays]]
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[[deps.Markdown]]
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[[deps.MbedTLS]]
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uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.TestImages]]
deps = ["AxisArrays", "ColorTypes", "FileIO", "ImageIO", "ImageMagick", "OffsetArrays", "Pkg", "StringDistances"]
git-tree-sha1 = "0567860ec35a94c087bd98f35de1dddf482d7c67"
uuid = "5e47fb64-e119-507b-a336-dd2b206d9990"
version = "1.8.0"

[[deps.TiffImages]]
deps = ["ColorTypes", "DataStructures", "DocStringExtensions", "FileIO", "FixedPointNumbers", "IndirectArrays", "Inflate", "Mmap", "OffsetArrays", "PkgVersion", "ProgressMeter", "UUIDs"]
git-tree-sha1 = "34cc045dd0aaa59b8bbe86c644679bc57f1d5bd0"
uuid = "731e570b-9d59-4bfa-96dc-6df516fadf69"
version = "0.6.8"

[[deps.TranscodingStreams]]
git-tree-sha1 = "54194d92959d8ebaa8e26227dbe3cdefcdcd594f"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.10.3"
weakdeps = ["Random", "Test"]

    [deps.TranscodingStreams.extensions]
    TestExt = ["Test", "Random"]

[[deps.Transducers]]
deps = ["Adapt", "ArgCheck", "BangBang", "Baselet", "CompositionsBase", "ConstructionBase", "DefineSingletons", "Distributed", "InitialValues", "Logging", "Markdown", "MicroCollections", "Requires", "Setfield", "SplittablesBase", "Tables"]
git-tree-sha1 = "3064e780dbb8a9296ebb3af8f440f787bb5332af"
uuid = "28d57a85-8fef-5791-bfe6-a80928e7c999"
version = "0.4.80"

    [deps.Transducers.extensions]
    TransducersBlockArraysExt = "BlockArrays"
    TransducersDataFramesExt = "DataFrames"
    TransducersLazyArraysExt = "LazyArrays"
    TransducersOnlineStatsBaseExt = "OnlineStatsBase"
    TransducersReferenceablesExt = "Referenceables"

    [deps.Transducers.weakdeps]
    BlockArrays = "8e7c35d0-a365-5155-bbbb-fb81a777f24e"
    DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
    LazyArrays = "5078a376-72f3-5289-bfd5-ec5146d43c02"
    OnlineStatsBase = "925886fa-5bf2-5e8e-b522-a9147a512338"
    Referenceables = "42d2dcc6-99eb-4e98-b66c-637b7d73030e"

[[deps.Tricks]]
git-tree-sha1 = "eae1bb484cd63b36999ee58be2de6c178105112f"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.8"

[[deps.URIs]]
git-tree-sha1 = "67db6cc7b3821e19ebe75791a9dd19c9b1188f2b"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.5.1"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unitful]]
deps = ["Dates", "LinearAlgebra", "Random"]
git-tree-sha1 = "3c793be6df9dd77a0cf49d80984ef9ff996948fa"
uuid = "1986cc42-f94f-5a68-af5c-568840ba703d"
version = "1.19.0"

    [deps.Unitful.extensions]
    ConstructionBaseUnitfulExt = "ConstructionBase"
    InverseFunctionsUnitfulExt = "InverseFunctions"

    [deps.Unitful.weakdeps]
    ConstructionBase = "187b0558-2788-49d3-abe0-74a17ed4e7c9"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.UnitfulLatexify]]
deps = ["LaTeXStrings", "Latexify", "Unitful"]
git-tree-sha1 = "e2d817cc500e960fdbafcf988ac8436ba3208bfd"
uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
version = "1.6.3"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.UrlDownload]]
deps = ["HTTP", "ProgressMeter"]
git-tree-sha1 = "758aeefcf0bfdd04cd88e6e3bc9feb9e8c7d6d70"
uuid = "856ac37a-3032-4c1c-9122-f86d88358c8b"
version = "1.0.1"

[[deps.Wayland_jll]]
deps = ["Artifacts", "EpollShim_jll", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "7558e29847e99bc3f04d6569e82d0f5c54460703"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+1"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "93f43ab61b16ddfb2fd3bb13b3ce241cafb0e6c9"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.31.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Zlib_jll"]
git-tree-sha1 = "07e470dabc5a6a4254ffebc29a1b3fc01464e105"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.12.5+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "afead5aba5aa507ad5a3bf01f58f82c8d1403495"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6035850dcc70518ca32f012e46015b9beeda49d8"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "34d526d318358a859d7de23da945578e8e8727b7"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.15.0+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.43.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+1"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "1ea2ebe8ffa31f9c324e8c1d6e86b4165b84a024"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.43+0"

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"]
git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.3+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

# ╔═╡ Cell order:
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