Re: 2-D Butterworth lowpass filter?

• To: mathgroup at smc.vnet.net
• Subject: [mg113829] Re: 2-D Butterworth lowpass filter?
• From: hadi motamedi <motamedi24 at gmail.com>
• Date: Sun, 14 Nov 2010 06:11:29 -0500 (EST)

```On 11/12/10, Patrick Scheibe <pscheibe at trm.uni-leipzig.de> wrote:
> Hi,
>
> using Gonzalez/Woods again (page 273, 3rd ed.) the transfer function is
> given by
>
> d[u_, v_, p_, q_] := Sqrt[(u - p/2)^2 + (v - q/2)^2]
> h[u_, v_, p_, q_, n_, d0_] := 1/(1 + (d[u, v, p, q]/d0)^(2 n));
>
> I'm using a compiled version of it
>
> hc = Compile[{u, v, p, q, n, d0},
>    1/(1 + (Sqrt[(u - p/2)^2 + (v - q/2)^2]/d0)^(2 n))];
> hcompiled[u_?NumericQ, v_?NumericQ, p_?NumericQ, q_?NumericQ,
>   n_?NumericQ, d0_?NumericQ] := hc[u, v, p, q, n, d0]
>
> With this the plots on page 274 are
>
> Plot3D[hcompiled[u, v, 0, 0, 3, 3], {u, -15, 15}, {v, -15, 15},
>  PlotRange -> All, PlotPoints -> 30]
> Plot[Evaluate[h[u, 0, 0, 0, #, 5] & /@ Range[4]], {u, 0, 15},
>  PlotStyle -> {Thick}, PlotRange -> All]
>
> And now (taking the CenteredFourier from Matthias)
>
> CenteredFourier[img_] :=
>   Module[{data = ImageData[img], dim}, dim = Dimensions[data];
>    Fourier[data*(-1)^Table[i + j, {i, First[dim]}, {j, Last[dim]}]]];
>
> CenteredInverseFourier[F_] := Module[{dim}, dim = Dimensions[F];
>    Image[Re[
>      InverseFourier[F]*(-1)^
>        Table[i + j, {i, First[dim]}, {j, Last[dim]}]]]];
>
> ButterWorthFilter[img_, n_, d0_] := Module[{nx, ny, bw},
>   {nx, ny} = ImageDimensions[img];
>   bw = Table[hcompiled[u, v, nx, ny, n, d0], {v, 1, ny}, {u, 1, nx}];
>   CenteredInverseFourier[bw*CenteredFourier[img]]
>   ]
>
>
> And to test it you could try to create figure 4.45 of the book (please
> check that there are no unwanted newlines in the link-strings when you
> copy it to the notebook!)
>
> img = Import[
>    "http://www.imageprocessingplace.com/downloads_V3/dip3e_downloads/\
> dip3e_book_images/DIP3E_CH04_Original_Images.zip",
>    "DIP3E_Original_Images_CH04/Fig0445(a)(characters_test_pattern).\
> tif"];
> GraphicsGrid@
>  Partition[
>   Flatten[{img,
>     ButterWorthFilter[img, 2, #] & /@ {10, 30, 60, 160, 460}}], 2]
>
>
> I must admit that the images in the book are more blurred with the same
> setting.
>
> Cheers
> Patrick
>
Thank you very much for your help. I tried for your code but I didn't
see the resulted filtered image. Can you please let me know how to
display the resulted filtered image?

```

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