pluto_WF_square.jl

### A Pluto.jl notebook ###
# v0.19.32

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

# ╔═╡ 8f3a2089-c44c-450d-8d10-d355c9503ed7
begin
	using PlutoUI
    using LaTeXStrings
	using Plots
	using Colors 
	using ImageFiltering
	using Images
	using ImageMagick
	using TestImages
	using ImageView
	using ImageTransformations
	using LinearAlgebra
	using LowRankApprox
end

# ╔═╡ 6b1ce739-aca6-47d6-842a-701712b2cc61
md"""
#### FBI Transform and Wave Front Set
"""

# ╔═╡ 33e36dbc-6682-4de6-a911-7eb2a0ec4f05
md"""
 $1/h$ = $(@bind λ1 Slider(100:100:5000, show_value=true, default=2000)) 
"""

# ╔═╡ 97846a02-a4e2-4220-b585-0ebed4e52a46
begin
    # f(x,y)
    A=zeros(200,200);
    A[51:150,51:150].=1;
    AA=A[26:175,26:175];

    # FBI transform of f
    F0=zeros(150,150);
    F1=zeros(150,150);
    F2=zeros(150,150);

    for j=1:150
        for k=1:150
            for q=-25:25
                for p=-25:25
                    E3=exp(-λ1*(-p/50)^2/2);
                    E4=exp(-λ1*(q/50)^2/2);
                    F0[j,k]=F0[j,k]+p*q*A[j+25+p,k+25+q]*E3*E4;
                    F1[j,k]=F1[j,k]+p*A[j+25+p,k+25+q]*E3*E4;
                    F2[j,k]=F2[j,k]+q*A[j+25+p,k+25+q]*E3*E4;
                end
            end
        end
    end
	
end

# ╔═╡ d9475fc3-39be-4adc-b112-0a0b98c0e045
begin
    MF0=abs.(F0);
    M0=maximum(MF0);
    MF1=abs.(F1);
    M1=maximum(MF1);
    MF2=abs.(F2);
    M2=maximum(MF2);

	Z10=zeros(2,1);
    Z11=zeros(2,1);
    Z12=zeros(2,1);

    for j=1:150
        for k=1:150
            if (MF0[j,k]>50*M0/100)
               Z10=[Z10 [j;k]];
            end
        end
    end

    for j=1:150
        for k=1:150
            if (MF1[j,k]>80*M1/100)
               Z11=[Z11 [j;k]];
            end
        end
    end

    for j=1:150
        for k=1:150
            if (MF2[j,k]>80*M2/100)
               Z12=[Z12 [j;k]];
            end
        end
    end

	(L00,L0)=size(Z10);
    Z0=Int64.(Z10[:,2:L0]);
    (L11,L1)=size(Z11);
    Z1=Int64.(Z11[:,2:L1]);
    (L22,L2)=size(Z12);
    Z2=Int64.(Z12[:,2:L2]);

	Z=[Z0 Z1 Z2];
	
	for j=1:1
		a=j
	end
end

# ╔═╡ 01408c00-677e-4527-8111-5130f157ae08
begin

	p1=plot(Gray.(A),
    title="Original Grayscale Image", 
    #    right_margin=Plots.Measures.Length(:mm, 10.0),
    size=(400,500), 
    xticks = ([21 floor(89/2)  69;], [-1,0,1]),
	xaxis=false,
    xlabel="x",
    yticks = ([21 floor(89/2)  69;], [1,0,-1]),
	yaxis=false,
    ylabel="y")
    savefig("./gray_square_1.png") 
 
    p2=scatter(Z[1,:], Z[2,:], 
               title="The singular support of a square \n detected by the FBI transform", 
		       aspect_ratio=1.0,
               color="magenta",
               size=(400,500),
               xaxis="x",
               xticks = ([26 75 126;], [-1,0,1]),
               yaxis="y", 
		       yticks = ([26 75 126;], [-1,0,1]),
               legend = :false)
               savefig("./ss_square.png")    

    plot(p1, p2, 
     layout=2, 
     size=(1100,400), 
     left_margin=Plots.Measures.Length(:mm, 5.0),
     right_margin=Plots.Measures.Length(:mm, 15.0),
     top_margin=Plots.Measures.Length(:mm, 5.0),
     bottom_margin=Plots.Measures.Length(:mm, 5.0))
     #savefig("./fbi_square.png") 

end

# ╔═╡ 2e7b3ce5-4388-4cc8-a1eb-20587b891702
md"""
 $1/h$ = $(@bind λ2 Slider(10:10:2000, show_value=true, default=100)) 

 $|ξ|$ = $(@bind ρ Slider(1:1:100, show_value=true, default=10)) 
"""

# ╔═╡ 39606315-c436-467b-ae47-4bad4d385975
begin
	G0=ComplexF64.(zeros(L0-1,32));
    G1=ComplexF64.(zeros(L1-1,32));
    G2=ComplexF64.(zeros(L2-1,32));
	
	for l=1:32
        for n=1:L0-1
            for q=-25:25
                for p=-25:25
                    E1=exp(im*1000*ρ*λ2*(p/50)*sin(pi*(l-1)/16));
                    E2=exp(-im*1000*ρ*λ2*(q/50)*cos(pi*(l-1)/16));
                    E3=exp(-(-p/50)^2/2);
                    E4=exp(-(q/50)^2/2);
                    #E5=p/50-im*100*ρ*λ2*sin(pi*(l-1)/16);
                    #E6=q/50+im*100*ρ*λ2*cos(pi*(l-1)/16);
                    G0[n,l]=G0[n,l]+A[Z0[1,n]+25+p,Z0[2,n]+25+q]*E1*E2*E3*E4;#*E5*E6;
                end
            end
        end
    end

	for l=1:32
        for n=1:L1-1
            for q=-25:25
                for p=-25:25
                    E1=exp(im*ρ*λ2*(p/50)*sin(pi*(l-1)/16));
                    E2=exp(-im*ρ*λ2*(q/50)*cos(pi*(l-1)/16));
                    E3=exp(-(-p/50)^2/2);
                    E4=exp(-(q/50)^2/2);
                    E5=p/20-im*ρ*λ2*sin(pi*(l-1)/16);
                    G1[n,l]=G1[n,l]+A[Z1[1,n]+25+p,Z1[2,n]+25+q]*E1*E2*E3*E4*E5;
                end
            end
        end
    end

    for l=1:32
        for n=1:L2-1
            for q=-25:25
                for p=-25:25
                    E1=exp(im*ρ*λ2*(p/50)*sin(pi*(l-1)/16));
                    E2=exp(-im*ρ*λ2*(q/50)*cos(pi*(l-1)/16));
                    E3=exp(-(p/50)^2/2);
                    E4=exp(-(q/50)^2/2);
                    E6=q/20+im*ρ*λ2*cos(pi*(l-1)/16);
                    G2[n,l]=G2[n,l]+A[Z2[1,n]+25+p,Z2[2,n]+25+q]*E1*E2*E3*E4*E6;
                end
            end
        end
    end
	
end

# ╔═╡ 0b9683ae-e548-4541-9cf7-c3cc2d13266f
begin
    MG0=abs.(G0);
    m0=maximum(MG0);
    MG1=abs.(G1);
    m1=maximum(MG1);
    MG2=abs.(G2);
    m2=maximum(MG2);

    U1=zeros(1,1);
    V1=zeros(1,1);
    W1=zeros(1,1);

	for l=1:32
        for n=1:L0-1
            if (MG0[n,l]>1*m0/1000)
                U1=[U1 (Z0[2,n]-75.5)/50];
                V1=[V1 (75.5-Z0[1,n])/50];
                W1=[W1 pi*(l-1)/16];
            end
        end
    end

    for l=1:32
        for n=1:L1-1
            if (MG1[n,l]>65*m1/100)
                U1=[U1 (Z1[2,n]-75.5)/50];
                V1=[V1 (75.5-Z1[1,n])/50];
                W1=[W1 pi*(l-1)/16];
            end
        end
    end

    for l=1:32
        for n=1:L2-1
            if (MG2[n,l]>65*m2/100)
                U1=[U1 (Z2[2,n]-75.5)/50];
                V1=[V1 (75.5-Z2[1,n])/50];
                W1=[W1 pi*(l-1)/16];
            end
        end
    end

	(L33,L3)=size(U1);
    U=U1[1,2:L3];
    V=V1[1,2:L3];
    W=W1[1,2:L3];

	for j=1:1
		b=j;
	end
end

# ╔═╡ 399b12c4-3545-4a84-8834-5ccec168d1fc
begin

	q1=plot(Gray.(A),
    title="Original Grayscale Image", 
    #    right_margin=Plots.Measures.Length(:mm, 10.0),
     size=(400,500), 
     xticks = ([21 floor(89/2)  69;], [-1 0 1]),
	 xaxis=false,
     xlabel="x",
     yticks = ([21 floor(89/2)  69;], [1 0 -1]),
	 yaxis=false,
     ylabel="y")
     #savefig("./gray_square_1.png") 
	
    q2=scatter3d(U, V, W, 
    title="The wave front set of a square \n detected by the FBI transform \n \$(x,y,|ξ|cosθ,|ξ|sinθ)\$", 
    color="cyan",
    size=(600,500),
    xaxis="x",
	xticks=([-1 0 1;],[-1,0,1]),
    yaxis="y", 
	yticks=([-1 0 1;],[-1,0,1]),
    zaxis="θ",
    zticks=([0 pi/2 pi 3*pi/2;],[0,"π/2","π","3π/2"]),
    legend = :false)
    #savefig("./wf_square.png") 

    #plot(q1, q2,
      #layout=2, 
      #size=(1100,800), 
      #left_margin=Plots.Measures.Length(:mm, 5.0),
      #right_margin=Plots.Measures.Length(:mm, 15.0),
      #top_margin=Plots.Measures.Length(:mm, 5.0),
      #bottom_margin=Plots.Measures.Length(:mm, 5.0))
     #savefig("./fbi_square.png") 

end

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version = "0.9.10+4"

[[deps.Xorg_libXtst_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll", "Xorg_libXi_jll"]
git-tree-sha1 = "0c0a60851f44add2a64069ddf213e941c30ed93c"
uuid = "b6f176f1-7aea-5357-ad67-1d3e565ea1c6"
version = "1.2.3+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+0"

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

[[deps.adwaita_icon_theme_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "hicolor_icon_theme_jll"]
git-tree-sha1 = "37c9a36ccb876e02876c8a654f1b2e8c1b443a78"
uuid = "b437f822-2cd6-5e08-a15c-8bac984d38ee"
version = "3.33.92+5"

[[deps.at_spi2_atk_jll]]
deps = ["ATK_jll", "Artifacts", "JLLWrappers", "Libdl", "Pkg", "XML2_jll", "Xorg_libX11_jll", "at_spi2_core_jll"]
git-tree-sha1 = "f16ae690aca4761f33d2cb338ee9899e541f5eae"
uuid = "de012916-1e3f-58c2-8f29-df3ef51d412d"
version = "2.34.1+4"

[[deps.at_spi2_core_jll]]
deps = ["Artifacts", "Dbus_jll", "Glib_jll", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXtst_jll"]
git-tree-sha1 = "d2d540cd145f2b2933614649c029d222fe125188"
uuid = "0fc3237b-ac94-5853-b45c-d43d59a06200"
version = "2.34.0+4"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.gdk_pixbuf_jll]]
deps = ["Artifacts", "Glib_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pkg", "Xorg_libX11_jll", "libpng_jll"]
git-tree-sha1 = "e9190f9fb03f9c3b15b9fb0c380b0d57a3c8ea39"
uuid = "da03df04-f53b-5353-a52f-6a8b0620ced0"
version = "2.42.8+0"

[[deps.hicolor_icon_theme_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b458a6f6fc2b1a8ca74ed63852e4eaf43fb9f5ea"
uuid = "059c91fe-1bad-52ad-bddd-f7b78713c282"
version = "0.17.0+3"

[[deps.iso_codes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51559b9071db7e363047a34f658d495843ccd35c"
uuid = "bf975903-5238-5d20-8243-bc370bc1e7e5"
version = "4.11.0+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+0"

[[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", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+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+0"

[[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 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
# ╟─8f3a2089-c44c-450d-8d10-d355c9503ed7
# ╟─6b1ce739-aca6-47d6-842a-701712b2cc61
# ╟─33e36dbc-6682-4de6-a911-7eb2a0ec4f05
# ╟─01408c00-677e-4527-8111-5130f157ae08
# ╟─97846a02-a4e2-4220-b585-0ebed4e52a46
# ╟─d9475fc3-39be-4adc-b112-0a0b98c0e045
# ╟─2e7b3ce5-4388-4cc8-a1eb-20587b891702
# ╟─399b12c4-3545-4a84-8834-5ccec168d1fc
# ╟─39606315-c436-467b-ae47-4bad4d385975
# ╠═0b9683ae-e548-4541-9cf7-c3cc2d13266f
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002