Digital Image Processing btech third year

Digital Image Processing

1. ‹#› IMAGE ENHANCEMENT IN FREQUENCY DOMAIN

2. ‹#› Image Enhancement Purposes: To make an image better appealing and easier to deal with than the original image Three categories: 1. Spatial domain methods : operate on the images itself, Point processing, e.g., image averaging; logic operation; contrast stretching ... Mask processing, e.g., filtering or mask operation, (blurring, median

3. ‹#› 2. Frequency domain methods : work on the Fourier transformed output of the image, examples: from the convolution theory g(x,y) = f(x,y) ⊗ h(x,y) => G(u,v) = F(u,v) • H(u,v) => certain properties of F(u,v) can be emphasized into G(u,v) => spatial domain g(x,y) = F -1 {G(u,v)} 3. Combination of the above two categories

4. ‹#› Recap Filtering Filtering is a technique for modifying or enhancing an image. For example, you can filter an image to emphasize certain features or remove other features. In principle, filtering is the term used for any operation that is applied to pixels in an image. This also includes smoothing , edge enhancement and resolution recovery . Generally, the aim of filtering is to allow improved extraction of relevant information .

5. ‹#› So far we processed the image ‘directly’, i.e. the transformation was a function of the image itself. We called this the SPATIAL domain. So what’s the FREQUENCY domain ?

6. ‹#› Spatial & frequency domain approaches Two different approaches in image processing Spatial Domain Approaches Involve more computation Frequency Domain Approaches More flexible & involve less computation

7. ‹#› Let’s first forget about images, and look at SOUND. SOUND: 1 dimensional function of changing (air-pressure in time) Pressure Time t If the function is periodic, we perceive it as sound with a certain frequency (else it’s noise). The frequency defines the pitch . The amplitude of the curve defines the volume.

8. ‹#› The SHAPE of the curve defines the sound character Flute String Brass

9. ‹#› Listening to an orchestra, you can distinguish between different instruments, although the sound is a SINGLE FUNCTION ! Flute String Brass

10. ‹#› If the sound produced by an orchestra is the sum of different instruments, could it be possible that there are BASIC SOUNDS, that can be combined to produce every single sound ?

11. ‹#› The answer (Charles Fourier, 1822): Any function that periodically repeats itself can be expressed as the sum of sines/cosines of different frequencies, each multiplied by a different coefficient

12. Jean Baptiste Joseph Fourier (1768-1830) Had crazy idea (1807): Any periodic function can be rewritten as a weighted sum of Sines and Cosines of different frequencies. Don’t believe it? Neither did Lagrange, Laplace, Poisson and other big wigs Not translated into English until 1878! But it’s true! called Fourier Series Possibly the greatest tool used in Engineering ‹#›

13. ‹#› Periodic Signals A continuous-time signal x(t) is periodic if: x(t + T) = x(t) Fundamental period T , of x(t) is smallest T satisfying above equation. Fundamental frequency : f = 1/T Fundamental angular frequency : ω = 2π/T = 2πf

14. ‹#› y(x) = A sin(fx + p) Any mathematical function that periodically repeats itself can be expressed as a sum of sines &/or cosines with different amplitudes A , frequencies f , and phases p . Fourier Series It does not matter how complicated the function is – as long as it is periodic (& meets certain mathematical conditions), it can be represented by such a sum.

15. ‹#› Fourier Transform Even functions that are not periodic (but whose area under the curve is finite) can be expressed as the integral of sines &/or cosines multiplied by a weighting function.

16. ‹#› Images are functions of finite duration Hence we will be dealing with Fourier Transform tool

17. ‹#› Fourier Transform & The frequency Domain

18. ‹#› The One-Dimensional Discrete Fourier Transform (DFT) & its Inverse The Fourier transform of a discrete function of one variable, f(x), x = 0, 1, …, M-1, is given by the following equation where

19. ‹#› Conversely, given F(u), f(x) can be obtained by means of the inverse Fourier transform , i.e. we can obtain the original function back using the inverse DFT

20. ‹#› To compute F(u): Substitute u = 0 in the exponential term & then sum for all values of x. Next substitute u = 1 in the exponential & repeat the summation over all values of x. Repeat this process for all M values of u in order to obtain the complete Fourier transform.

21. ‹#› Each term of the Fourier transform is composed of the sum of all values of the function f(x) The domain (values of u) over which the values of F(u) range is called the Frequency domain because u determines the frequency components of the transform

22. ‹#› Each of the M terms of F(u) are known as Frequency Component of the transform

23. ‹#› Fourier Transform – A “mathematical prism”

24. A prism which splits the white light into different colours (components with a different wavelength / frequency). The prism acts like a Fourier transform operator. The Fourier transform is a mathematical procedure which decomposes a signal into its sinusoid components with different frequencies. Inverse procedure - the Inverse Fourier transform - can be used to recombine the components into the composed signal. One can manipulate the frequency components in the frequency space as needed for different purposes. Jean Baptiste Joseph Fourier (1768-1830) - the concept of the transform first suggested in 1810 in the paper on heat conduction ‹#›

25. ‹#› Example: A simple one-dimensional DFT Discrete Function & it Fourier Spectrum M = 1024 A = 1 K = 8

26. ‹#› The function f(x) for x = 0, 1, …, M-1 represents M samples These samples are not necessarily always taken at integer values of x. They are taken at equally spaced, but otherwise arbitrary points. x represents the I point in the sequence. The I value of the sampled function is then f(x ). The next sample has been taken a fixed interval ∆x units away to give f(x + ∆x)

27. ‹#› The k th sample gives us f(x + k ∆x) The final sample is ? f(x + [M-1] ∆x)

28. ‹#› Variable u has similar interpretation But … The sequence always starts at true zero frequency . Thus, the sequence for the values of u is 0, ∆u, 2∆u, …, [M-1]∆u Inverse relationship exits between a function & its transform. Hence

29. ‹#› The Two-Dimensional Discrete Fourier Transform (DFT) & its Inverse The discrete Fourier transform of a function (image) f(x, y) of size M x N is given by following equation

30. ‹#› Similarly, given F(u, v), we can obtain f(x, y) via inverse Fourier transform

31. ‹#› Two Dimensional Discrete Fourier Transform (DFT) Pair u & v Transform or Frequency variable x & y Spatial or image variable

32. ‹#› A common practice … Multiply the i/p image function by (-1) x + y prior to computing the Fourier transform This shifts the origin of F(u, v) to frequency coordinates (M/2, N/2),which is the centre of the M x N area occupied by the 2-DFT

33. ‹#› The value of transform at (u, v) = (0, 0) Which is the average of f(x, y) f(x, y) is an image The value of the Fourier transform at origin is equal to the average gray-level of the image