Digital images, Fourier Transformation |

In order to be analysed, a signal need to be digitized, i.e. represented as a sequence of numbers (real, integer... values) as a function of time, position etc.
In the present illustration, the reference signal is a digital photograph of statue from the 'Avenue of the Chinese Musicians, Allerton Park, University of Illinois at Urbana-Champaign, IL, USA (2004) :
This image can be seen as an
array of 256*256 cells (pixels), each containing an integer number
comprised between 0 (black) and 255 (white) coding for the
corresponding gray level :
In a first
step, for simplification consider a second image also in 8 bits gray
scale but composed only of 4*4 pixels, each identified by their row m and column n numbers:
Gray scale image
Pixels values
Then, when digitized, this image is the array I(n,m), with n the column number and m
the row number (e.g. I(2,0)=255).
The Fourier Transform (noted FT) of this N*N pixels images (here N=4) is: where h and k are two integers from -N/2 à N/2-1.
The inverse tranformation (Inverse Fourier Transform)
allowing to form back the original image is:
Often the original signal I(n,m)
is real (and we will limit ourself to such signals), and this implies a symmetry in its Fourier Transform F(h,k).
Indeed:
The Fourier Transform F(h,k) of the image I(n,m) is usually displayed as an array F(h,k) of N*N
pixels containing complex numbers, named the Fourier Coefficients of the image I(n,m)
:
Each of these cartesian (x+iy) complex numbers can be written on polar form |F(h,k)|.e
^{i.φ(h,k)}, where |F(h,k)| is the modulus and φ(h,k) the phase of the Fourier coefficient F(h,k), leading to a graphical representation of the Fourier Transform through gray scale maps of moduli and phases:
The diplay of the Fourier Transform is centered around F(h=0,k=0) (due to the symmetry F(-h,-k)=F
^{*}(h,k)). The central coefficent F(h=0,k=0)
is particular since it is always a real value and is equal to the sum of the pixel values of I(n,m) image: Fourier coefficients in first
column and row k=-2 and h=-2 are also particuliar since they correspond
to the highest possible absolute values of the h and k indices. In
order to work with arrays of same sizes for the image and its Fourier
Transform only the coefficients with k and K negative are displayed
(here column h=-2 and row k=-2).
Note : with the definition of the Fourier Transform we used here N should be a even number (h have to be an integer ranging from N/2 to N/2-1), and in order to use the software employed to make the illustrations in these pages (DigitalMicrograph @ Gatan; Fast Fourier Transform) N should be equal to 2 ^{j} with j another integer. |