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## Digital Image Processing Week V

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**Frequency Domain Processing**Digital Image Processing Week V Thurdsak LEAUHATONG**Background: Fourier Series**Fourier series: Anyperiodic signals can be viewed as weighted sum of sinusoidal signals with different frequencies Frequency Domain: view frequency as an independent variable**Fourier Tr. and Frequency Domain**Fourier Tr. Time, spatial Domain Signals Frequency Domain Signals Inv Fourier Tr. 1-D, Continuous case Fourier Tr.: Inv. Fourier Tr.:**Example of 1-D Fourier Transforms**Notice that the longer the time domain signal, The shorter its Fourier transform (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.**Fourier Tr. and Frequency Domain (cont.)**1-D, Discrete case Fourier Tr.: u = 0,…,M-1 Inv. Fourier Tr.: x = 0,…,M-1 F(u) can be written as or where**Relation Between Dx and Du**For a signal f(x) withMpoints, let spatial resolution Dx be space between samples in f(x) and let frequency resolutionDu be space between frequencies components in F(u), we have Example: for a signal f(x) with sampling period 0.5 sec, 100 point, we will get frequency resolution equal to This means that in F(u) we can distinguish 2 frequencies that are apart by 0.02 Hertz or more.**2-Dimensional Discrete Fourier Transform**For an image of size MxN pixels 2-D DFT u = frequency in x direction, u = 0 ,…, M-1 v = frequency in y direction, v = 0 ,…, N-1 2-D IDFT x = 0 ,…, M-1 y = 0 ,…, N-1**2-Dimensional Discrete Fourier Transform (cont.)**F(u,v) can be written as or where For the purpose of viewing, we usually display only the Magnitude part ofF(u,v)**Relation Between Spatial and Frequency Resolutions**where Dx = spatial resolution in x direction Dy = spatial resolution in y direction Du = frequency resolution in x direction Dv = frequency resolution in y direction N,M = image width and height ( Dxand Dyare pixel width and height.)**f(x,y)**How to Perform 2-D DFT by Using 1-D DFT 1-D DFT by row F(u,y) 1-D DFT by column F(u,v)**f(x,y)**How to Perform 2-D DFT by Using 1-D DFT (cont.) Alternative method 1-D DFT by column 1-D DFT by row F(x,v) F(u,v)**1-D DFT**From DFT: -N 2N N 0 We display only in this range DFT repeats itself every N points (Period = N) but we usually display it for n = 0 ,…, N-1**Conventional Display for 1-D DFT**f(x) DFT N-1 N-1 0 0 Time Domain Signal High frequency area Low frequency area The graph F(u) is not easy to understand !**Conventional Display for DFT : FFT Shift**FFT Shift: Shift center of the graph F(u) to 0 to get better Display which is easier to understand. N-1 0 High frequency area Low frequency area -N/2 N/2-1 0**2-D DFT**g(x,y) 2-D DFT: -M For an image of size NxM pixels, its 2-D DFT repeats itself every N points in x-direction and every M points in y-direction. 0 M We display only in this range 2M -N 0 N 2N**Conventional Display for 2-D DFT**F(u,v) has low frequency areas at corners of the image while high frequency areas are at the center of the image which is inconvenient to interpret. High frequency area Low frequency area**2-D FFT Shift : Better Display of 2-D DFT**2-D FFT Shift is a MATLAB function: Shift the zero frequency of F(u,v) to the center of an image. 2D FFTSHIFT High frequency area Low frequency area**2-D FFT Shift (cont.) : How it works**-M 0 Display of 2D DFT After FFT Shift M Original display of 2D DFT 2M -N 0 N 2N**Example of 2-D DFT**Notice that the longer the time domain signal, The shorter its Fourier transform**Example of 2-D DFT**Notice that direction of an object in spatial image and Its Fourier transform are orthogonal to each other.**Example of 2-D DFT**2D DFT F=fft2(f); S=abs(F); imshow(S,[ ]); Original image Fc=fftshift(F); Sc=abs(Fc); imshow(Sc,[ ]); 2D FFT Shift**Example of 2-D DFT**2D DFT F=fft2(f); S=abs(F); imshow(S,[ ]); Original image Fc=fftshift(F); Sc=abs(Fc); imshow(Sc,[ ]); 2D FFT Shift**Basic Concept of Filtering in the Frequency Domain**From Fourier Transform Property: We cam perform filtering process by using Multiplication in the frequency domain is easier than convolution in the spatial Domain.**Filtering in the Frequency Domain with FFT shift**F(u,v) H(u,v) (User defined) g(x,y) FFT shift X 2D IFFT 2D FFT FFT shift f(x,y) G(u,v) In this case, F(u,v) and H(u,v) must have the same size and have the zero frequency at the center.**Multiplication in Freq. Domain = Circular Convolution**f(x) DFT F(u) G(u) = F(u)H(u) IDFT g(x) h(x) DFT H(u) Multiplication of DFTs of 2 signals is equivalent to perform circular convolution in the spatial domain. f(x) h(x) “Wrap around” effect g(x)**Multiplication in Freq. Domain = Circular Convolution**Original image H(u,v) Gaussian Lowpass Filter Filtered image (obtained using circular convolution) Incorrect areas at image rims**Linear Convolution by using Circular Convolution and Zero**Padding f(x) Zero padding DFT F(u) G(u) = F(u)H(u) h(x) Zero padding DFT H(u) IDFT Concatenation g(x) Padding zeros Before DFT Keep only this part**Linear Convolution by using Circular Convolution and Zero**Padding**Filtering in the Frequency Domain : Example**Lowpass Filter Highpass Filter (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.**Ideal Lowpass Filter**Ideal LPF Filter Transfer function where D(u,v) = Distance from (u,v) to the center of the mask. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.**Examples of Ideal Lowpass Filters**The smaller D0, the more high frequency components are removed.**Results of Ideal Lowpass Filters**Ringing effect can be obviously seen!**How ringing effect happens**Surface Plot Ideal Lowpass Filter with D0= 5 Abrupt change in the amplitude**How ringing effect happens (cont.)**Surface Plot Spatial Response of Ideal Lowpass Filter with D0= 5 Ripples that cause ringing effect**Butterworth Lowpass Filter**Transfer function Where D0= Cut off frequency, N = filter order. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.**Results of Butterworth Lowpass Filters**There is less ringing effect compared to those of ideal lowpass filters!**Gaussian Lowpass Filter**Transfer function Where D0= spread factor. (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition. Note: the Gaussian filter is the only filter that has no ripple and hence no ringing effect.**Results of Gaussian Lowpass Filters**No ringing effect!**Application of Gaussian Lowpass Filters**Original image Better Looking The GLPF can be used to remove jagged edges and “repair” broken characters.**Highpass Filters**Hhp = 1 - Hlp**Ideal Highpass Filters**Ideal LPF Filter Transfer function where D(u,v) = Distance from (u,v) to the center of the mask.**Butterworth Highpass Filters**Transfer function Where D0= Cut off frequency, N = filter order.**Gaussian Highpass Filters**Transfer function Where D0= spread factor.**Laplacian Filter in the Frequency Domain**From Fourier Tr. Property: Then for Laplacian operator We get Image of –(u2+v2) Surface plot**Sharpening Filtering in the Frequency Domain**Spatial Domain Frequency Domain Filter