6.frequency domain image_processing
TRANSCRIPT
Image processing in frequency Domain
Department of Computer Science And EngineeringShahjalal University of Science and Technology
Nashid AlamRegistration No: [email protected]
Masters -2 Presentation(Backup Slides# 6)
Introduction to Frequency domain
Deal with images in:-Spatial domain-Frequency domain
Difference between spatial domain and frequency domain
Spatial domain : - Deal with images as it is. - The value of the pixels of the image change with respect to scene.
Frequency domain : -Deal with the rate at which the pixel values are changing in spatial domain.
For simplicity , Let’s put it this way.
DIFFERENCE BETWEEN SPATIAL DOMAIN AND FREQUENCY DOMAIN
Directly deal with the image matrix.
For simplicity , Let’s put it this way.
Image Processing in Frequency Domain
o Transform the image to its frequency distribution. o Black box system perform what ever processing it has to perform oThe output of the black box is not an image , - The output it is a transformation.
o After performing inverse transformation - The output is converted into an image which is then viewed in spatial domain.
It can be pictorially viewed
Frequency Components
Any image in spatial domain can be represented in a frequency domain.
But what do this frequencies actually mean?
Frequency components are divided into two major components.
1. HIGH FREQUENCY COMPONENTSHigh frequency components correspond to edges in an image.
2. LOW FREQUENCY COMPONENTSLow frequency components in an image correspond to smooth regions.
TRANSFORMATION
Transformation: A signal can be converted from spatial domain into frequency domain using mathematical operators called transformation.
kind of transformation:
Fourier Series Fourier transformation Laplace transform Z transform
Fourier Transform
Fourier Transform
f(m, n) is a function of two discrete spatial variables m and n,
Two-dimensional Fourier transform of f(m, n) :
The variables ω1 and ω2 are frequency variablesω1 and ω2 both are periodic with period 2πWhere, -2π<= ω1 and 2π<= ω2
F(ω1,ω2 ) is often called the frequency-domain representation of f(m, n)
Fourier Transform
F(0,0 ) is the sum of all the values of f(m, n)
F(0,0 ) is often called the constant component or DC component of the Fourier transform.
(DC stands for direct current:It is an electrical engineering term that refers to a constant-voltage power source, as opposed to a power source whose voltage varies sinusoidally.)
frequency-domain representation of f(m, n):
Fourier Transform
The inverse two-dimensional Fourier transform is given by:
This equation means that f(m, n) can be represented as:A sum of an infinite number of complex exponentials (sinusoids) with different frequencies.
Example2D Fourier Transform
Consider a function f(m, n) : 1 within a rectangular region
0 everywhere else.
To simplify the diagram:
f(m, n) is shown as a continuous function, eventhough the variables m and n are discrete
Example2D Fourier Transform
The magnitude of the Fourier transform, |F(ω1,ω2 )| is shown in mesh plot.
This reflects the fact:-Horizontal cross sections of are narrow pulses,vertical cross sections are broad pulses.
-Narrow pulses have more high-frequency content than broad pulses.
Example2D Fourier Transform
The plot also shows :
More energy at high horizontal frequencies Less energy at high vertical frequencies.
Example Of How To Do This Is In Next Slide
Implemented example
ExampleRelationship to 2D Fourier transform with spatial image
Construct a matrix f that is similar to the function f(m, n) f = zeros(30,30);f(5:24,13:17) = 1;imshow(f,'notruesize')
(b) Main Image (Toy image Constriction)
in spatial domain
(c)Fourier transform of the image(Frequency Domain)
(a) Toy Image
Example2D Fourier Transform
Main Image in spatial domain
Orthogonal
Compute and visualize the 30-by-30 DFT of f(m,n)
Target
Main Image in spatial domain
Fourier transform of the image
(Frequency Domain) Inverse Fourier transform
(Going back to Spatial domain)
Example2D Fourier Transform
Compute and visualize the 30-by-30 DFT of f(m,n)
F = fft2(f);F2 = log(abs(F));imshow(F2,[-1 5],'notruesize'); %% colormap(jet); colorbar
Example2D Fourier Transform
Doesn’t match the target Fourier Transformed Image
The resolution is much lower
DC coefficient of F(0,0) is displayed in the upper-left corner instead of the traditional location in the center
(b) Target Fourier Transformed Image
(b) Visualize the 30-by-30 DFT
F(0,0) value is called DC coefficients
DC=Direct Current
Example2D Fourier Transform
The resolution is increased by zero-padding f when computing its DFT.
Need more computation
%resolution is increased by zero-padding f%zero-pads f to be 256-by-256 before %computing the DFTF = fft2(f,256,256);
Dealing with Resolution
Example2D Fourier Transform
DCT coefficients of Fourier transform are displayed much more finelyThe result is :
(a) DCT result Before zero-padding (b) DCT result After zero-padding(fine result)
Need more computation Dealing with ResolutionF = fft2(f,256,256);
Example2D Fourier Transform
Need more computation Dealing with displaying DC coefficient of F(0,0)
fix this problem by using the function fftshift,which swaps the quadrants of F so that the DC coefficient is in the center.
imshow(log(abs(F)),[-1 5]);
(b) Target Fourier Transformed Image
Example2D Fourier Transform
compute the inverse DFT
(a) Fourier transform of the image
(Frequency Domain)
(b) Inverse Fourier transform
(Going back to Spatial domain)