11.00D.1
Chapter 11.00D
Physical Problem for Fast Fourier Transform
Computer Engineering
Problem Statement
Steganography is the art and science of hiding information in images. The basic idea
is not to let the enemy know that message is being transmitted. This is different from
message encryption, which is designed to make it hard to understand the message once a
message is found. Steganography makes it hard to find the message itself. For instance, in
Figure 1 one of the images has the preamble of the US constitution embedded in it. Can you
guess? Looking at the images it is impossible to figure it out. There are various ways for
performing steganography and there are various way to figure out if there is a hidden
message in an image. Detecting steganography is another problem, which we will leave for
another day. In this problem, we are going to look at one method for doing steganography
using the Discrete Fourier Transform, computed using the Fast Fourier Transform.
Figure 1
Which of these two images has a hidden message embedded in it? One of them has
the amendments of the US constitution in it.
Problem Formulation
Image is a collection of gray level values at set of predetermined sites known as
pixels, arranged in an array. These gray level values are also known as image intensities. For
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Chapter 11.00D
this problem, an image is nothing but a two dimensional array of numbers,
)
,
(
n
m
I
. The size
of the array is the size of the image. One place to hide information in images is in the Fourier
or frequency domain of images. Yes, we can define “frequency” of images too.
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 Spring '08
 Kaw,A
 DFT

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