Transforms and Frequency Filtering

Transcription

1 Transforms and Frequency Filtering Khalid Niazi Centre for Image Analysis Swedish University of Agricultural Sciences Uppsala University

2 2 Reading Instructions Chapter 4: Image Enhancement in the Frequency Domain Book: Digital Image Processing by Gonzalez & Woods, Third Edition, 2007

3 3 Image processing g(x, y) = T[f(x, y)] Original image f. Result image g after transformation T. T can be performed point-wise (gray-level transformation) 1 x 1 locally (in a small neighborhood) m x n globally (the whole image) M X N

4 4 Image representations An image is a function of x and y. f(x, y) One possible way to investigate its properties is to display the function values as grey-level intensities; this is the normal representation. Another possibility is to transform the function values, e.g., to spatial frequencies through the Fourier transform. It is still the same image, but in a different representation. This gives a partitioning of the frequencies in the image.

5 5 Frequency domain representation We need to know Frequency Sinusoids Euler s formula

6 6 Frequency domain representation Frequency: Cycles per second Sinusoids: Common name for Cosine and Sine Euler's formula

7 7 Frequency domain representation Using Euler s formula one can write a Sinusoid as, 0

8 8 Frequency domain representation x(t) = x1(t) + x2(t) + x3(t) + x4(t) x1(t) = 10 cos(2*pi*1*t+pi/32) x2(t) = 6 cos(2*pi*1.5*t+pi/10) x3(t) = 4 cos(2*pi*2.7*t+pi/7) x4(t) = 2 cos(2*pi*3.4*t+pi/13) time

9 9 Frequency domain representation frequency

10 10 Frequency domain representation Conjugate symmetric pair of points in the frequency domain of a signal corresponds to a sinusoid in the spatial domain A one-dimensional sinusoid has frequency, phase, and amplitude No time information in the frequency domain

11 11 Frequency domain representation Y[x, y] = sin[2π(ux+ Vy)] Where x and y are horizontal and vertical dimensions of the image Y. U and V represents the frequencies along these dimensions. Here the image has two frequencies, i.e., horizontal and vertical frequencies MATLAB demonstration

12 12 Frequency domain representation Conjugate symmetric pair of points in the frequency domain of an image corresponds to a two-dimensional sinusoid in the spatial domain A one-dimensional sinusoid has frequency, phase, and amplitude and two dimensional sinusoid also has a direction Two-dimensional sinusoids are directional in nature

13 13 Fourier transform F(u, v) is generally complex: F(u, v) = R(u, v) + I(u, v) = F(u, v) exp ( φ(u, v)) R(u, v) is the real component of F(u, v). I(u, v) is the imaginary component of F(u, v). F(u, v) is the magnitude function, also called the Fourier spectrum. φ(u, v) is the phase angle. F(u, v) is composed of an infinite sum of sine and cosine terms, where u and v determines the frequency of its corresponding sine-cosine pair.

14 Summation of Fourier frequency 14 terms to fit a step function

15 15 Display of Fourier images The magnitude function, or Fourier spectrum, F(u, v) can be displayed as an intensity function, where the brightness is proportional to the amplitude. The spectrum most often has a large dynamic range, e.g., Only the brightest parts of the spectrum are visible. By a logarithmic transform an increase in visible detail is possible: D(u, v) = c log(1 + F(u, v) )

16 16 Display of Fourier images

17 17 The Fourier spectrum F(u, v) F(0, 0) is the mean grey-level in the image, i.e., the lowest frequency. The farther away from (u, v) = (0, 0) we get, the higher the frequencies represented by F(u, v) are. F(u, v) contains information about low frequencies (areas with slowly changing grey-level) if (u, v) is close to (0, 0). F(u, v) contains information about high frequencies (abrupt changes in grey-level, such as edges and noise) if (u, v) is far away from (0, 0). F(0, 0) is usually centered in the image showing the Fourier spectrum.

18 18 Properties of the Fourier transform Separability Translation Periodicity Conjugate Symmetry Rotation Convolution

19 19 Separability The separability property is a great advantage for images with M = N. F(u, v) or f(x, y) can be obtained by two successive applications of the simple 1D Fourier transform or its inverse, instead of by one application of the more complex 2D Fourier transform. That is, first transform along each row, and then transform along each column, or vice versa.

20 20 Translation A shift in f(x, y) = f(x x0, y y0) does not affect the magnitude function F(u, v) : It only affects the phase angle φ(u, v). Keep in mind when displaying the magnitude function! MATLAB demonstration

21 21 Periodicity The discrete Fourier transform is periodic: F(u, v) = F(u +M, v) = F(u, v + N) = F(u +M, v + N) Why Periodic? Only one period is necessary to reconstruct f(x, y)

22 22 Conjugate Symmetry

23 23 Rotation Dependency Rotating f(x, y) by an angle α rotates F(u, v) by the same angle. Similarly, rotating F(u, v) rotates f(x, y) by the same angle.

24 24 Convolution theorem Convolution in the spatial domain multiplication in frequency domain f(x, y) g(x, y) = F 1{F(u, v) G(u, v)} and vice versa F(u, v) G(u, v) = F{f(x, y) g(x, y)}

25 25 Examples

26 26 Examples

27 27 Enhancement in frequency domain Steps in filtering in the frequency domain (simplified): 1. Compute the Fourier-transform of the image to be enhanced. 2. Multiply the result by a frequency filter. 3. Compute the inverse Fourier-transform to produce the enhanced image.

28 28 Enhancement in frequency domain When filtering images by a mask, convolution in the spatial domain is used. We can get the frequency filter by computing the Fourier transform of the spatial filter (the mask). Then, filtering by multiplying the Fourier transformed image and the frequency filter is equal to filtering by convolution in spatial domain.

29 29 Smoothing frequency-domain filters High frequencies are attenuated; noise and edges are blurred. Ideal lowpass filter: ILPF Butterworth lowpass filter: BLPF Gaussian lowpass filter: GLPF

30 30 Ideal lowpass filtering Only values of F(u, v) near (u, v) = (0, 0) remains after filtering Only low frequencies remains after filtering

31 31 Gaussian lowpass filtering The frequency filter must have the same size as the original image, this is achieved by filling up the frequency filter with zeros.

32 32 Gaussian lowpass filtering

33 Sharpening frequency-domain 33 filters Low frequencies are attenuated; noise and edges are enhanced. The reverse operation of lowpass filters. Ideal highpass filter: IHPF Butterworth highpass filter: BHPF Gaussian highpass filter: GHPF The Laplacian in the frequency domain

34 34 Ideal highpass filtering Only values of F(u, v) far from (u, v) = (0, 0) remains after filtering Only high frequencies remains after filtering.

35 35 Gaussian highpass filter By taking one minus the Gaussian lowpass filter (GLPF), the Gaussian highpass filter (GHPF) is achieved.

36 36 Gaussian highpass filter

37 37 Laplacian in frequency domain

38 38 Laplacian in frequency domain

39 Decimation-free directional filter 39 bank(ddfb)

40 40 DDFB

41 41 DDFB

42 42 DDFB

43 43 Directional analysis

44 44 Directional analysis

45 45 Directional analysis

46 46 DDFB

47 47 Directional analysis

48 48 Directional analysis

49 49 Directional analysis

50 50 Directional analysis

51 51 Directional analysis

52 52 Directional analysis

53 53 Directional analysis

54 54 Directional analysis

55 55 Directional analysis

56 56 Directional analysis

57 57 Design of DDFB

58 58 DDFB Divides an Image into its directional components It is common practice to divide an image into eight directional components. But it is mostly dependent on the image information Noise is omni-directional

59 59 Image enhancement with DDFB

60 60 Image enhancement with DDFB

61 61 Image enhancement with DDFB

62 62 Next lecture Image Restoration Time1315hrs Date: 28/10/09