Image Processing for Mechatronics Engineering For senior undergraduate students Academic Year 2017/2018, Winter Semester
Lecture 14: Filtering in Frequency Domain 09.12.2017
Dr. Mohammed Abdel-Megeed Salem Media Engineering Technology, German University in Cairo
Course Info - Contents 1. 2. 3. 4. 5. 6. 7.
Introduction Elementary Image Information and Operations Fundamentals of Signal and Image Processing Image Acqusition and Digitization Sensing and Perception (HVS) Color Image Processing Image Processing Operations 1. 2. 3. 4.
Appendix 1: Random Process & Probability Distribution Function Elementary and Point Image Operations Local Image Operations and Filters Global Image Operation and Transforms
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Outline Global Image Operation and Transforms 1. Fourier Transform 1. 2. 3. 4.
Signal Approximation Fourier Transform for 1D Signals Fourier Transform for 2D Signals Image Filtering in Frequency Domain
2. Discrete Cosine Transform 1. 2.
DCT-based Compression Hough Transform
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Image Filtering in Frequency Domain • if we modify one of the Fourier coefficients before applying the inverse 2D DFT, then we obtain a modified function I . • We modify the 2D DFT I of I by multiplying the values in I, position by position, with the corresponding complex numbers in G [i.e., I(u, v) ·G(u, v)]. – We denote this operation by I ◦ G.
• The resulting complex array is transformed by the inverse 2D DFT into the modified image J . Salem, Image Processing for Mechatronics Engineering, Winter Semester 2017
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Image Filtering in Frequency Domain • Steps of Fourier Filtering Given is an image I and a complex-valued filter function G in the frequency domain. Input Image
Input Filter
Fourier Transform
Fourier Transform
Point -wise multi plicat ion
Salem, Image Processing for Mechatronics Engineering, Winter Semester 2017
Inverse Fourier Transform
Lecture 14
Output Processed Image
5
Image Filtering in Frequency Domain
• 1D profiles of rotation-symmetric filter functions. Top: A linear high-pass filter and an ideal low-pass filter. Bottom: An exponential high-emphasis filter and a linear band-pass filter
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Ideal Low Pass Filter The ideal low-pass filter H(u,v) is described as H(u,v) =
1
radius(u,v) d0
0
radius(u,v) > d0
radius(u,v) = u2 + v2 ,
d0 = cut off frequency H(u,v)
H(u,v) v u
1 0
d0 radius(u,v)
http://www1.idc.ac.il/toky/imageproc-10/
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Ideal Low Pass Filter
99.7%
99.37%
98.65%
http://www1.idc.ac.il/toky/imageproc-10/
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The Ringing Problem
• 1D signal describing a step (in bold grey) and frequency components (in shades of brown to orange) whose addition defines the blue signal, which approximates the ideal step. http://www1.idc.ac.il/toky/imageproc-10/
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The Ringing Problem H(u,v)
sinc(x)
Inverse Fourier Transform
h(x,y)
As d0 increases, the ringing and the bluering decreases. 250 200 150 100 50 0
Freq. domain
http://www1.idc.ac.il/toky/imageproc-10/
0
50
100
150
200
250
Spatial domain
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The Gaussian Filter H(u,v)
H(u,v)
1
Softer Blurring + no Ringing
v
u
H(u,v) =
e
0
d0
radius(u,v)
-d2(u,v)/(2d02)
raduis(u,v) = u2 + v2 http://www1.idc.ac.il/toky/imageproc-10/
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The Gaussian Filter
99.11%
98.74%
96.44%
http://www1.idc.ac.il/toky/imageproc-10/
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The Gaussian Filter H(u,v)
Inverse Fourier Transform
h(x,y)
300 250 200 150 100 50 0
Freq. domain
http://www1.idc.ac.il/toky/imageproc-10/
0
50
100
150
200
250
300
Spatial domain
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In Spatial Domain? Averaging = convolution with
111 111 111
= point multiplication of the transform with sinc: 0.15 1 0.1
0.8 0.6
0.05
0.4 0.2
0 0
50
Image Domain
100
0 -50
0
50
Frequency Domain
http://www1.idc.ac.il/toky/imageproc-10/
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In Spatial Domain? 1 2 1 2 4 2 1 2 1
Gaussian Averaging = convolution with
= point multiplication of the transform with a gaussian. 0.15 1 0.1
0.8 0.6
0.05
0.4 0.2
0 0
50
Image Domain
100
0 -50
0
50
Frequency Domain
http://www1.idc.ac.il/toky/imageproc-10/
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Ideal High Pass Filter The ideal high-pass filter H(u,v) is described as H(u,v) =
0
radius(u,v) d0
1
radius(u,v) > d0
radius(u,v) = u2 + v2 ,
d0 = cut off frequency
H(u,v)
H(u,v)
v u
1 0
d0
radius(u,v)
http://www1.idc.ac.il/toky/imageproc-10/
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The High-pass Gaussian Filter H(u,v)
H(u,v) 1
1 1/ e v
u
H(u,v) = 1- e
0
d0
radius(u,v)
-d2(u,v)/(2d02)
raduis(u,v) = u2 + v2 http://www1.idc.ac.il/toky/imageproc-10/
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The High-pass Gaussian Filter Original
High Pass Filtered
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Emphasize High Frequency Filter The ideal high-pass filter H(u,v) is described as H(u,v) =
0
radius(u,v) d0
1
radius(u,v) > d0
radius(u,v) = u2 + v2 , H'(u,v) = K0 + H(u,v)
d0 = cut off frequency H'(u,v)
(Typically K0 =1)
1 0
d0
radius(u,v)
http://www1.idc.ac.il/toky/imageproc-10/
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Emphasize High Frequency Filter
Original
High Frequency Emphasis
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Emphasize High Frequency Filter
Original
High Frequency Emphasis
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Emphasize High Frequency Filter
Original
High pass Emphasis
Salem, Image Processing for Mechatronics Engineering, Winter Semester 2017
High Frequency Emphasis + Histogram Equalization
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Band-pass Filtering H(u,v) =
0
raduis(u,v) d0 -
1
d0-
0
radius(u,v) > d0 +
w 2
w 2
raduis(u,v) d0 +
w 2
w 2
radius(u,v) = u2 + v2 d0 = cut off frequency w = band width H(u,v)
H(u,v)
v u
1 radius(u,v)
0
Salem, Image Processing for Mechatronics Engineering, Winter Semester 2017
d0- w 2
d0
d0+ w 2
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Local Frequency Filtering H(u,v) =
1
radius1(u,v) d0 or radius2(u,v) d0
0
otherwise
H(u,v) v
radius1(u,v) = (u-u0)2 + (v-v0)2 radius2(u,v) = (u+u0)2 + (v+v0)2
u H(u,v)
d0 = local frequency radius
1
u0,v0 = local frequency coordinates
0 -u0,-v0
Salem, Image Processing for Mechatronics Engineering, Winter Semester 2017
d0
radius(u,v) u0,v0
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Band Rejection Filtering H(u,v) =
0
radius1(u,v) d0 or radius2(u,v) d0
1
otherwise H(u,v)
radius1(u,v) = (u-u0)2 + (v-v0)2
v
radius2(u,v) = (u+u0)2 + (v+v0)2 u
d0 = local frequency radius
H(u,v)
u0,v0 = local frequency coordinates
1
0 -u0,-v0 Salem, Image Processing for Mechatronics Engineering, Winter Semester 2017
d0
radius(u,v) u0,v0
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Readings • Rafael G. Gonzalaz and Richard E. Woods, Digital Image Processing, 3rd Edition, Pearson Edu., 2008. – [Section 4.3: Smoothing Frequency-Domain Filters] – [Section 4.4: Sharpening Frequency-Domain Filters]
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Contacts Image Processing for Mechatronics Engineering, for senior students, Winter Semester 2017 Dr. Mohammed Abdel-Megeed M. Salem Media Engineering Technology, German University in Cairo Office: C7.311 Ext. 3580 Tel.: +2 011 1727 1050 Email:
[email protected]
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