Transcript Fourier transform

Dr. Abdul Basit Siddiqui
FUIEMS
Quiz
Time 30 min.
 How the coefficents of Laplacian Filter are generated.
Show your complete work. Also discuss different
versions of Laplacian Filter.
 Applay the 3x3 laplacian filter on the following piece of
an Image f(x,y). What will be its effect.
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Introduction
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Background (Fourier Series)
 Any function that periodically repeats itself can be
expressed as the sum of sines and cosines of
different frequencies each multiplied by a different
coefficient
 This sum is known as Fourier Series
 It does not matter how complicated the function is;
as long as it is periodic and meet some mild
conditions it can be represented by such as a sum
 It was a revolutionary discovery
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Introduction to DFT
• The Fourier Transform is an important image
processing tool which is used to decompose an
image into its sine and cosine components.
• The output of the transformation represents the
image in the Fourier or frequency domain, while
the input image is the spatial domain equivalent.
• In the Fourier domain image, each point represents
a particular frequency contained in the spatial
domain image.
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Introduction to DFT
• The Fourier Transform is used in a wide range of
applications, such as image analysis, image filtering, image
reconstruction and image compression.
• The DFT is the sampled Fourier Transform and therefore
does not contain all frequencies forming an image, but
only a set of samples which is large enough to fully
describe the spatial domain image
• The number of frequencies corresponds to the number of
pixels in the spatial domain image, i.e. the image in the
spatial and Fourier domain are of the same size.
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Background (Fourier Transform)
 Even functions that are not periodic (but whose area under the
curve is finite) can be expressed as the integrals of sines and
cosines multiplied by a weighing function
 This is known as Fourier Transform
 A function expressed in either a Fourier Series or transform can be
reconstructed completely via an inverse process with no loss of
information
 This is one of the important characteristics of these representations
because they allow us to work in the Fourier Domain and then
return to the original domain of the function
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Fourier Transform
• ‘Fourier Transform’ transforms one function into
another domain , which is called the frequency
domain representation of the original function
• The original function is often a function in the
Time domain
• In image Processing the original function is in the
Spatial Domain
• The term Fourier transform can refer to either the
Frequency domain representation of a function or
to the process/formula that "transforms" one
function into the other.
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Our Interest in Fourier Transform
• We will be dealing only with functions (images) of
finite duration so we will be interested only in Fourier
Transform
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Applications of Fourier Transforms
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1-D Fourier transforms are used in Signal Processing
2-D Fourier transforms are used in Image Processing
3-D Fourier transforms are used in Computer Vision
Applications of Fourier transforms in Image processing: –
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Image enhancement,
Image restoration,
Image encoding / decoding,
Image description
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One Dimensional Fourier Transform
and its Inverse
 The Fourier transform F (u) of a single variable, continuous
function f (x) is
 Given F(u) we can obtain f (x) by means of the Inverse
Fourier Transform
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One Dimensional Fourier Transform
and its Inverse
 The Fourier transform F (u) of a single variable, continuous
function f (x) is
 Given F(u) we can obtain f (x) by means of the Inverse
Fourier Transform
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Discrete Fourier Transforms (DFT)
1-D DFT for M samples is given as
The Inverse Fourier transform in 1-D is given as
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Discrete Fourier Transforms (DFT)
1-D DFT for M samples is given as
The inverse Fourier transform in 1-D is given as
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Two Dimensional Fourier Transform
and its Inverse
 The Fourier transform F (u,v) of a two variable, continuous
function f (x,y) is
 Given F(u,v) we can obtain f (x,y) by means of the Inverse
Fourier Transform
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2-D DFT
where f(a,b) is the image in the spatial domain and the
exponential term is the basis function corresponding to each
point F(k,l) in the Fourier space
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Fourier Transform
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2-D DFT
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Image Restoration
• In many applications (e.g., satellite imaging, medical
imaging, astronomical imaging, poor-quality family
portraits) the imaging system introduces a slight
distortion
• Image Restoration attempts to reconstruct or recover
an image that has been degraded by using a priori
knowledge of the degradation phenomenon.
• Restoration techniques try to model the degradation
and then apply the inverse process in order to
recover the original image.
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Image Restoration
• Image restoration attempts to restore images that
have been degraded
– Identify the degradation process and attempt to reverse it
– Similar to image enhancement, but more objective
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A Model of the Image Degradation/
Restoration Process
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A Model of the Image Degradation/
Restoration Process
• The degradation process can be modeled as a degradation function
H that, together with an additive noise term η(x,y) operates on an
input image f(x,y) to produce a degraded image g(x,y)
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