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This module extends the ideas of the Discrete Fourier Transform (DFT) into two-dimensions, which is necessary for any image processing.

2d dft

To perform image restoration (and many other useful image processing algorithms) in a computer, we need a FourierTransform (FT) that is discrete and two-dimensional.

F k l u 2 k N v 2 l N F u v
for k 0 N 1 and l 0 N 1 .
F u v m n f m n u m v m
F k l m N 1 0 n N 1 0 f m n 2 k m N 2 l n N
where the above equation ( ) has finite support for an N x N image.

Inverse 2d dft

As with our regular fourier transforms, the 2D DFT also has an inverse transform that allows us to reconstruct an imageas a weighted combination of complex sinusoidal basis functions.

f m n 1 N 2 k N 1 0 l N 1 0 F k l 2 k m N 2 l n N

Periodic extensions

Illustrate the periodic extension of images.
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2d dft and convolution

The regular 2D convolution equation is

g m n k N 1 0 l N 1 0 f k l h m k n l

Below we go through the steps of convolving two two-dimensional arrays. You can think of f as representing an image and h represents a PSF, where h m n 0 for m n 1 and m n 0 . h h 0 0 h 0 1 h 1 0 h 1 1 f f 0 0 f 0 N 1 f N 1 0 f N 1 N 1 Step 1 (Flip h ):

h m n h 1 1 h 1 0 0 h 0 1 h 0 0 0 0 0 0
Step 2 (Convolve):
g 0 0 h 0 0 f 0 0
We use the standard 2D convolution equation ( ) to find the first element of our convolved image. In order to better understand what ishappening, we can think of this visually. The basic idea is to take h m n and place it "on top" of f k l , so that just the bottom-right element, h 0 0 of h m n overlaps with the top-left element, f 0 0 , of f k l . Then, to get the next element of our convolved image, we slide the flipped matrix, h m n , over one element to the right and get the following result: g 0 1 h 0 0 f 0 1 h 0 1 f 0 0 We continue in this fashion to find all of the elements ofour convolved image, g m n . Using the above method we define the general formula to find a particular element of g m n as:
g m n h 0 0 f m n h 0 1 f m n 1 h 1 0 f m 1 n h 1 1 f m 1 n 1

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Circular convolution

What does H k l F k l produce?

2D Circular Convolution

g ~ m n IDFT H k l F k l circularconvolutionin2D

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Due to periodic extension by DFT ( ):

Based on the above solution, we will let

g ~ m n IDFT H k l F k l
Using this equation, we can calculate the value for each position on our final image, g ~ m n . For example, due to the periodic extension of the images, when circular convolution is applied we willobserve a wrap-around effect.
g ~ 0 0 h 0 0 f 0 0 h 1 0 f N 1 0 h 0 1 f 0 N 1 h 1 1 f N 1 N 1
Where the last three terms in are a result of the wrap-around effect caused by the presence of the images copies located all around it.

Zero padding

If the support of h is M x M and f is N x N , then we zero pad f and h to M N 1 x M N 1 (see ).

Circular Convolution = Regular Convolution

Computing the 2d dft

F k l m N 1 0 n N 1 0 f m n 2 k m N 2 l n N
where in the above equation, n N 1 0 f m n 2 l n N is simply a 1D DFT over n . This means that we will take the 1D FFT of each row; if wehave N rows, then it will require N N operations per row. We can rewrite this as
F k l m N 1 0 f m l 2 k m N
where now we take the 1D FFT of each column, which means that if we have N columns, then it requires N N operations per column.
Therefore the overall complexity of a 2D FFT is O N 2 N where N 2 equals the number of pixels in the image.

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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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