Help needed - compute convolution kernel from a frequency filter

Hi,
My problem is the following:
Given a integer N, I need to generate a NxN kernel that approximates an
arbitrary frequency filter. So I can use the (small) kernel to filter the
image in spatial domain with convolution operations. 
For example I use the lowpass filter:

  (lowLv-hiLv) * e^ [-(f/Ft)^2] + hiLv

where f is the euclidean distance between center and a "pixel" in
frequency domain, Ft is the cutoff freq and lowLv and hiLv are
multiplicative coefficients for low and hi frequencies. 

I hope that this filter is correct. Despite this fact, my teacher suggest
me to take the inverse FFT of the filter to find out the kernel... but
how?

I try to find answers searching with Google and reading the book "Digital
Image Processing" but I don't find what I need.

Please help me. Best regards.

Hi,

If you have a matrix representing the response of the filter in the
frequency domain,
then you can invoke the command ifft2 (2D inverse FFT) in matlab to
get the time domain kernel.
Now the important thing is to verify that your frequency filter has
the conjugate symmetry,
so that the ifft2 will give you a real kernel.


On 24 mar, 20:39, "boiacan" <boia...@gmail.com> wrote:
> Hi,
> My problem is the following:
> Given a integer N, I need to generate a NxN kernel that approximates an
> arbitrary frequency filter. So I can use the (small) kernel to filter the
> image in spatial domain with convolution operations.
> For example I use the lowpass filter:
>
>   (lowLv-hiLv) * e^ [-(f/Ft)^2] + hiLv
>
> where f is the euclidean distance between center and a "pixel" in
> frequency domain, Ft is the cutoff freq and lowLv and hiLv are
> multiplicative coefficients for low and hi frequencies.
>
> I hope that this filter is correct. Despite this fact, my teacher suggest
> me to take the inverse FFT of the filter to find out the kernel... but
> how?
>
> I try to find answers searching with Google and reading the book "Digital
> Image Processing" but I don't find what I need.
>
> Please help me. Best regards.

>Hi,
>
>If you have a matrix representing the response of the filter in the
>frequency domain,
>then you can invoke the command ifft2 (2D inverse FFT) in matlab to
>get the time domain kernel.
>Now the important thing is to verify that your frequency filter has
>the conjugate symmetry,
>so that the ifft2 will give you a real kernel.


Thanks for your answer but I'm not sure to understand.
I'll try to give you more details.

I've to do the whole thing in C++. I've no matrix representing the
response of the filter (I've just the filter function) and I can't use
external libraries for FFT (for example FFTW) so give a simple 1D FFT
algorithm I can compute the inverse (or forward) transform for rows and
cols in the usual way.

But the tricky part (for me :-D) is to set up an array of data (that is a
matrix in 2D) that represents the frequency filter and then to do reverse
Fourier transform this data and scale up the  transform result to an NxN
kernel (of course I'm not even sure if this procedure is the correct one).

On Mar 24, 2:39 pm, "boiacan" <boia...@gmail.com> wrote:
> Given a integer N, I need to generate a NxN kernel that approximates an
> arbitrary frequency filter. So I can use the (small) kernel to filter the
> image in spatial domain with convolution operations.

The problem you're trying to solve is called "FIR filter design", as
you're trying to find a finite convolution kernel (a finite impulse
response, or FIR, filter) that best approximates an given frequency
response.

FIR filter design is covered in many textbooks on DSP, and many sites
online, and there are several free software packages to help you.
Google it (knowing the correct name will help)!

(You're trying to solve the 2d version of the FIR problem, which is
slightly more complicated than most of what you'll find online.
However, there are many papers on multidimensional filter design which
may help you.  Alternatively, as a simple approximation you can simply
apply 1d filters along each dimension, especially if the frequence
response you're trying to apply is separable.)

> I hope that this filter is correct. Despite this fact, my teacher suggest
> me to take the inverse FFT of the filter to find out the kernel... but
> how?

Your teacher's suggestion is a reference to what is called the "window
method" of FIR filter design.  Look it up.  (However, there may be
other methods that produce better approximations, see e.g.
http://gmeteor.sourceforge.net/ for references and code for one such
method.)

Regards,
Steven G. Johnson