Image Enhancement in the frequency domain. Inel 5046 Prof. Vidya Manian

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1 Image Enhancement in the frequency domain Inel 5046 Prof. Vidya Manian

2 Introduction 2D Fourier transform Basics of filtering in frequency domain Ideal low pass filter Gaussian low pass filter Ideal high pass filter Gaussian high pass filter FT properties and theorems

3 2D DFT and its inverse M 1N 1 1 Fuv (, ) = f( xye, ) MN M 1N 1 u= 0 v= 0 x= 0 y= 0 f( x, y) = F( u, v) e j2 π ( ux / M + vy / N ) j2 π ( ux/ M+ vy/ N) Fourier spectrum, phase angle and power spectrum Fuv (, ) = [ R( uv, ) + I( uv, )] φ(,) uv tan 2 2 1/2 1 = Puv (,) = Fuv (,) 2 2 R u v I u v Iuv (,) Ruv (,) = (,) + (,) 2

4 Associations in frequency domain Each term of F(u,v) contains all values of f(x,y) modified by the values of the exponential terms Hard to associate specific components of image and transform Frequencies in FT with intensity variations in an image Low frequency-smooth regions, high frequencyfaster grey level changes (edges and noise)

5 Basics of filtering in the frequency domain 1. Multiply the input image by (-1) x+y to center the transform 2. Compute F(u,v), the DFT of the image 3. Multiply F(u,v) by a filter function H(u,v) 4. Compute the inverse DFT of (3) 5. Obtain the real part of the result Multiply the result in (5) by (-1) x+y

6 Preprocessing-cropping image to closest even dimension, grey-level scaling, converison to floating point

7 Some basic filters F(0,0) is the dc component the average value of the image Notch filter sets F(0,0) to zero and leaves all other frequency components of the FT untouched. 0 if (u,v)=(m/2,n/2) Huv (, ) = 1 otherwise

8 Low pass and high pass filters

10 Filters based on Gaussian functions H(u) frequency domain Gaussian filter function Hu ( ) Ae 2 2 /2σ Corresponding filter in the spatial domain = hx ( ) = 2π (1) constitute a Fourier transform pair, both components of which are Gaussian and real (2) These functions behave reciprocally. When H(u) has a broad profile (large value of σ), h(x) has a narrow profile, and vice versa. u x Ae π

12 Smoothing frequency domain filters Ideal low pass filter Cuts off all high frequency components of the Fourier transform that are at a distance greater than a specified distance D 0 from the origin of the (centered) transform 1 if D(u,v) D Huv (, ) = 0 0 if D(u,v)>0 If image is of size MxN, center of the frequency rectangle is at (u,v)=(m/2,n/2) D(u,v)=[(u-M/2) 2 +(v-n/2) 2 ] 1/2

16 Gaussian low pass filtering Huv (, ) = Let σ = D Huv (, ) = Ae 0 Ae D 2 2 ( u, v)/2σ 2 2 (, )/2 0 D u v D

17 GLPF

18 Sharpening frequency domain filters Ideal high pass filter Edges and abrupt changes in gray levels are associated with high-frequency components Perform reverse operation of IDLPF H ( u, v) = 1 H ( u, v) hp 0 if D(u,v) D Huv (, ) = 1 if D(u,v)>D0 lp 0

19 IDHPF

20 Gaussian high pass filter (GHPF) Huv (, ) = 1 e 2 2 D ( uv, )/2D0

21 Laplacian in the frequency domain n d f( x) n I = ( ju) F( u) n dx 2 2 f( x, y) f( x, y) I + = x y 2 2 ( ) (, ) = u + v F u v 2 2 (, ) ( ) (, ) 2 2 ( ju) F( u, v) ( jv) F( u, v) I f( x, y) = = ( u + v ) F( u, v) F( u, v) Laplacian can be implemented using the filter Huv = u + v Fuv

22 Laplacian FT pair Origin of F(u,v) is centered using f(x,y)(-1)x+y, before taking the transform.center of filter function is shifted: H(u,v)= -[(u-m/2) 2 +(v-n/2) 2 ] 1/2 Laplacian filtered image in the spatial domain is obtained by computed the IFT of H(u,v)F(u,v) { Fuv} 2 f ( xy, ) =I 1 -[(u-m/2) 2 +(v-n/2) 2 ] 1/2 (, ) Computing Laplacian in spatial domain and computing FT of the result is equivalent to multiplying F(u,v) by H(u,v) { Fuv} f ( xy, ) -[(u-m/2) +(v-n/2) ] (, ) /2

24 FT is separable

Digital Image Processing. Chapter 4: Image Enhancement in the Frequency Domain

Digital Image Processing. Chapter 4: Image Enhancement in the Frequency Domain Digital Image Processing Chapter 4: Image Enhancement in the Frequency Domain Image Enhancement in Frequency Domain Objective: To understand the Fourier Transform and frequency domain and how to apply