       Re: 2-D Butterworth lowpass filter?

• To: mathgroup at smc.vnet.net
• Subject: [mg113791] Re: 2-D Butterworth lowpass filter?
• From: Patrick Scheibe <pscheibe at trm.uni-leipzig.de>
• Date: Sat, 13 Nov 2010 00:58:18 -0500 (EST)

```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], {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[
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

On Fri, 2010-11-12 at 05:28 -0500, hadi motamedi wrote:
> Dear All
> Can you please let me know how can I design and apply 2-D Butterworth
> lowpass filter to my image in Mathematica? Please be informed that I
> can obtain its Fourier transform but I need to design and apply
> lowpass Butterworth filtering to it.
> Thank you
>

```

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