ENG4BF3 Medical Image Processing Image Enhancement in Frequency Domain

2 Image Enhancement Original imageEnhanced image Enhancement: to process an image for more suitable output for a specific application.

3 Image Enhancement Image enhancement techniques:  Spatial domain methods  Frequency domain methods Spatial (time) domain techniques are techniques that operate directly on pixels. Frequency domain techniques are based on modifying the Fourier transform of an image.

4 Fourier Transform: a review Basic ideas:  A periodic function can be represented by the sum of sines/cosines functions of different frequencies, multiplied by a different coefficient.  Non-periodic functions can also be represented as the integral of sines/cosines multiplied by weighing function.

5 Joseph Fourier ( ) Fourier was obsessed with the physics of heat and developed the Fourier transform theory to model heat- flow problems.

6 Fourier transform basis functions Approximating a square wave as the sum of sine waves.

7 Any function can be written as the sum of an even and an odd function E(-x) = E(x) O(-x) = -O(x)

8 Fourier Cosine Series Because cos(mt) is an even function, we can write an even function, f(t), as: where series F m is computed as Here we suppose f(t) is over the interval (–π,π).

9 Fourier Sine Series Because sin(mt) is an odd function, we can write any odd function, f(t), as: where the series F’ m is computed as

10 Fourier Series So if f(t) is a general function, neither even nor odd, it can be written: Even componentOdd component where the Fourier series is

11 The Fourier Transform Let F(m) incorporates both cosine and sine series coefficients, with the sine series distinguished by making it the imaginary component: Let’s now allow f(t) range from –  to , we rewrite: F(u) is called the Fourier Transform of f(t). We say that f(t) lives in the “time domain,” and F(u) lives in the “frequency domain.” u is called the frequency variable.

12 The Inverse Fourier Transform Given F(u), f(t) can be obtained by the inverse Fourier transform Inverse Fourier Transform We go from f(t) to F(u) by Fourier Transform

13 2-D Fourier Transform and the inverse Fourier transform Fourier transform for f(x,y) with two variables

14 Discrete Fourier Transform (DFT) A continuous function f(x) is discretized as:

15 Discrete Fourier Transform (DFT) Let x denote the discrete values (x=0,1,2,…,M-1), i.e. then

16 Discrete Fourier Transform (DFT) The discrete Fourier transform pair that applies to sampled functions is given by: u=0,1,2,…,M-1 x=0,1,2,…,M-1 and

17 2-D Discrete Fourier Transform In 2-D case, the DFT pair is: u=0,1,2,…,M-1 and v=0,1,2,…,N-1 x=0,1,2,…,M-1 and y=0,1,2,…,N-1 and:

18 Polar Coordinate Representation of FT The Fourier transform of a real function is generally complex and we use polar coordinates: Magnitude: Phase: Polar coordinate

19 Fourier Transform: shift It is common to multiply input image by (-1) x+y prior to computing the FT. This shift the center of the FT to (M/2,N/2). Shift

20 Symmetry of FT For real image f(x,y), FT is conjugate symmetric: The magnitude of FT is symmetric:

21 FT IFT

22 IFT

23 The central part of FT, i.e. the low frequency components are responsible for the general gray-level appearance of an image. The high frequency components of FT are responsible for the detail information of an image.

24 u v Image Detail Generalappearance Frequency Domain (log magnitude)

25 10 %5 %20 %50 %

26 Frequency Domain Filtering

27 Edges and sharp transitions (e.g., noise) in an image contribute significantly to high-frequency content of FT. Low frequency contents in the FT are responsible to the general appearance of the image over smooth areas. Blurring (smoothing) is achieved by attenuating range of high frequency components of FT. Frequency Domain Filtering

28 –f(x,y) is the input image –g(x,y) is the filtered –h(x,y): impulse response Convolution Theorem G(u,v)=F(u,v) ● H(u,v) g(x,y)=h(x,y)*f(x,y) Filtering in Frequency Domain with H(u,v) is equivalent to filtering in Spatial Domain with f(x,y). Multiplication in Frequency Domain Convolution in Time Domain

29 Spatial domain Gaussian lowpass filterGaussian highpass filter Frequency domain Examples of Filters

30 Ideal low-pass filter (ILPF) D 0 is called the cutoff frequency. (M/2,N/2): center in frequency domain

31 Shape of ILPF Spatial domain Frequency domain

32 ringing and blurring Ideal in frequency domain means non-ideal in spatial domain, vice versa. FT

33 Butterworth Lowpass Filters (BLPF) Smooth transfer function, no sharp discontinuity, no clear cutoff frequency.

34 Butterworth Lowpass Filters (BLPF) n=1n=2n=5n=20

35 No serious ringing artifacts

36 Smooth transfer function, smooth impulse response, no ringing Gaussian Lowpass Filters (GLPF)

37 Spatial domain Gaussian lowpass filter Frequency domain GLPF

38 No ringing artifacts

39 Examples of Lowpass Filtering

40 Examples of Lowpass Filtering Original image and its FTFiltered image and its FT Low-pass filter H(u,v)

41 Sharpening High-pass Filters H hp (u,v)=1-H lp (u,v) Ideal: Butterworth: Gaussian:

42

43 High-pass Filters

44 Ideal High-pass Filtering ringing artifacts

45 Butterworth High-pass Filtering

46 Gaussian High-pass Filtering

47 Gaussian High-pass Filtering Original image Gaussian filter H(u,v) Filtered image and its FT

48 Laplacian in Frequency Domain )(),( 22 1 vuvuH  ),()(] ),(),( [ vuFvu y yxf x yxf        Laplacian operator Spatial domain Frequency domain

49

50 Subtract Laplacian from the Original Image to Enhance It ),()(),(),( 22 vuFvuvuFvuG  ),(),(),( 2 yxfyxfyxg  new operator Spatial domain Original image enhanced image Laplacian output Frequency domain )(1),( 22 2 vuvuH  ),( 1 vuH 1  Laplacian

51

52 Unsharp Masking, High-boost Filtering Unsharp masking: f hp (x,y)=f(x,y)-f lp (x,y) H hp (u,v)=1-H lp (u,v) High-boost filtering: f hb (x,y)=Af(x,y)-f lp (x,y) f hb (x,y)=(A-1)f(x,y)+f hp (x,y) H hb (u,v)=(A-1)+H hp (u,v) One more parameter to adjust the enhancement

53

54 An image formation model We can view an image f(x,y) as a product of two components: i(x,y): illumination. It is determined by the illumination source. r(x,y): reflectance (or transmissivity). It is determined by the characteristics of imaged objects.

55 Homomorphic Filtering In some images, the quality of the image has reduced because of non-uniform illumination. Homomorphic filtering can be used to perform illumination correction. The above equation cannot be used directly in order to operate separately on the frequency components of illumination and reflectance.

56 Homomorphic Filtering ln : DFT : H(u,v) : (DFT) -1 : exp :

57 By separating the illumination and reflectance components, homomorphic filter can then operate on them separately. Illumination component of an image generally has slow variations, while the reflectance component vary abruptly. By removing the low frequencies (highpass filtering) the effects of illumination can be removed. Homomorphic Filtering

58 Homomorphic Filtering

59 Homomorphic Filtering: Example 1

60 Homomorphic Filtering: Example 2 Original imageFiltered image

61 End of Lecture