Image processing in frequency Domain

Transcription

1 Image processing in frequency Domain

2 Introduction to Frequency Domain Deal with images in: -Spatial domain -Frequency domain

3 Frequency Domain In the frequency or Fourier domain, the value and location are represented by sinusoidal relationships that depend upon the frequency of a pixel occurring within an image. In this domain, pixel location is represented by its x- and y-frequencies and its value is represented by an amplitude. Images can be transformed into the frequency domain to determine which pixels contain more important information and whether repeating patterns occur.

4 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.

5 DIFFERENCE BETWEEN SPATIAL DOMAIN AND FREQUENCY DOMAIN For simplicity, Let s put it this way. Directly deal with the image matrix.

6 Image Processing in Frequency Domain o o o For simplicity, Let s put it this way. Transform the image to its frequency distribution. Black box system perform what ever processing it has to perform The output of the black box is not an image, - The output it is a transformation. After performing inverse transformation - The output is converted into an image which is then viewed in spatial domain. o It can be pictorially viewed

7 Frequency Components Any image in spatial domain can be represented in a frequency domain. But what do this frequencies actually mean? v Frequency components are divided into two major components. 1. HIGH FREQUENCY COMPONENTS High frequency components correspond to edges in an image. 2. LOW FREQUENCY COMPONENTS Low frequency components in an image correspond to smooth regions.

8 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

9 Fourier Transform

10 Fourier Transform f(m, n) is a function of two discrete spatial variables m and n, Two-dimensional Fourier transform of f(m, n) : F(ω1,ω2 ) is often called the frequency-domain representation of f(m, n) The variables ω1 and ω2 are frequency variables ω1 and ω2 both are periodic with period 2π Where, -2π<= ω1 and 2π<= ω2

11 Fourier Transform frequency-domain representation of f(m, n): 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.)

12 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.

13 2D Fourier Transform Example 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, even though the variables m and n are discrete

14 2D Fourier Transform Example

15 2D Fourier Transform Example The plot also shows : More energy at high horizontal frequencies Less energy at high vertical frequencies. 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. Example Of How To Do This Is In Next Slide

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32 Implemented example

33 Relationship to 2D Fourier transform with spatial image Construct a matrix f that is similar to the function f(m, n) (a) Toy Image (b) Main Image (Toy image Constriction) in spatial domain Example 1 f = zeros(30,30); f(5:24,13:17) = 1; imshow(f,'notruesize') (c)fourier transform of the image(frequency Domain)

34 Target Main Image in spatial domain Fourier transform of the image (Frequency Domain) Inverse Fourier transform (Going back to Spatial domain)

35 Relationship to 2D Fourier transform with spatial image Example 2 2D FFT

36 Filtering an image within frequency domain Within the frequency domain, convolution can be performed by multiplying the FFT (Fast Fourier Transform) of the image by the FFT of the kernel, and then transforming back into the spatial domain. The kernel is padded with zero values to enlarge it to the same size as the image before the forward FFT is applied. These types of filters are usually specified within the frequency domain and do not need to be transformed.

37 Cont... The following examples in this section will focus on some of the basic filters applied within the spatial domain using the CONVOL function: Low Pass Filtering High Pass Filtering Directional Filtering Laplacian Filtering Since filters are the building blocks of many image processing methods, these examples merely show how to apply filters, as opposed to showing how a specific filter may be used to enhance a specific image or extract a specific shape. This basic introduction provides the information necessary to accomplish more advanced image-specific processing.

38 Low Pass Filter Example Low past filter can be used to smooth an image Eg. Gaussian Low Pass Filter (others Butterworth LPF)

39 High Pass Filter Example High past filter can be used to sharpening an image Eg. Butterworth High Pass Filter