04-2 - Using frequency filters

04-2 - Using frequency filters - 4/28/2008 Using frequency...

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4/28/2008 1 Using frequency domain filters Spring 2008 ELEN 4304/5365 DIP 1 by Gleb V. Tcheslavski: gleb@ee.lamar.edu http://ee.lamar.edu/gleb/dip/index.htm Image smoothing - ILPF 0 1 (,) Duv D An ideal low-pass filter (ILPF) is specified as 0 ( , ) 0( , ) Huv D = > Here D 0 is a positive constant and D ( u,v ) is the distance between the origin and the point ( u,v ) in the frequency domain: ( ) ( ) 22 2 2 u P v Q = + Spring 2008 ELEN 4304/5365 DIP 2 P and Q are sizes of the zero-padded image.
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4/28/2008 2 Image smoothing - ILPF D 0 is also called a cut-off frequency . One way to specify a cut-off frequency is by circles enclosing specific amount of total power P T Spring 2008 ELEN 4304/5365 DIP 3 frequency is by circles enclosing specific amount of total power in an image, where 11 2 00 (,) QQ PP T uv u v F u v −− == ∑∑ Image smoothing - ILPF If the DFT is centered, a circle of radius D 0 will enclose α percent of power: 100 ( ) Puv P ( , ) T =⋅ for the summation over the values being inside the circle or on its boundary. Test image and its DFT Spring 2008 ELEN 4304/5365 DIP 4 Test circles show ILPFs
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4/28/2008 3 Image smoothing - ILPF The original image and results of filtering with circular ILPF of radii in pixels (enclosing % of total power): 10 (87), 30 (93.1), 60 (95.7), 160 (97.8), and 460 (99.2). Spring 2008 ELEN 4304/5365 DIP 5 The filters removed 13, 6.9, 4.3, 2.2, and 0.8 % of total power. Observe blurring and ringing. Image smoothing - ILPF Spring 2008 ELEN 4304/5365 DIP 6 A spatial representation of ILPF of radius 10. the center lobe causes blurring, the outer lobes introduces ringing…
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4/28/2008 4 Butterworth LPF A transfer function of a Butterworth LPF (BLPF) of order n and with cutoff frequency at the distance D 0 from the origin: [] 2 0 1 (,) 1( , ) n Huv Duv D = + ( ) ( ) 22 ( , ) Duv u P v Q = −+ Where: Spring 2008 ELEN 4304/5365 DIP 7 P and Q are sizes of the zero-padded image. Butterworth LPF D ( u,v ) Spring 2008 ELEN 4304/5365 DIP 8 Unlike the ILPF, the BLPF transfer function does not have sharp discontinuities. No visible ringing and smooth transition between low and high frequencies.
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4/28/2008 5 Butterworth LPF The original image and results of filtering with circular BLPF of order 2 and of radii in pixels (enclosing % of total power): 10 (87), 30 (93.1), 60 (95.7), 160 (97.8), and 460 (99.2). The filters removed 13 6 9 Spring 2008 ELEN 4304/5365 DIP 9 The filters removed 13, 6.9, 4.3, 2.2, and 0.8 % of total power. Observe no ringing. Butterworth LPF n = 1 n = 2 n = 5 n = 20 Spring 2008 ELEN 4304/5365 DIP 10 BLPF of order 1 has neither ringing nor negative values; BLPF of order 1 has mid ringing and small negative values; while the order increases, ringing becomes more pronounces. For n = 20, BLPF is similar to ILPF. The second order is a good compromise between effective LP filtering and acceptable ringing.
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4/28/2008 6 Gaussian LPF (GLPF) A transfer function of a Gaussian LPF (GLPF) with cutoff frequency at the distance D 0 from the origin: 22 0 (,) 2 () Du v D Huv e ( , ) = (,) 2 2 Duv u P v Q =− + Where: P and Q are sizes of the zero-padded image.
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04-2 - Using frequency filters - 4/28/2008 Using frequency...

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