Lecture #7. EECS490: Digital Image Processing. Image Processing Example Fuzzy logic. Fourier Transform. Basics Image processing examples
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1 Lecure #7 Image Processing Example Fuzzy logic Basics Image processing examples Fourier Transorm Inner produc, basis uncions Fourier series
2 Image Processing Example original image Laplacian o image (c) Sharpened by adding Laplacian Sobel gradien image 00 R. C. Gonzalez & R. E. Woods
3 Image Processing Example (e) Blurred Sobel gradien image Mask = (c) x (e) Add Mask o original Power Law Inensiy Transorm 00 R. C. Gonzalez & R. E. Woods
4 Basic Fuzzy Logic crisp membership uncion uzzy membership uncion Probabiliy: here is a 50% chance ha a paricular person is young Fuzzy logic: a person s membership wih he se o young people is 0.5 Basically everyone is young o some degree. A membership uncion represens ha degree. 00 R. C. Gonzalez & R. E. Woods
![Basic Fuzzy Logic =max[ A (z), B (z)] OR =min[](/docs-images/77/76462136/images/5-3.jpg "A (z), B (z)] AND 00 R. C. Gonzalez & R. E.")
5 Basic Fuzzy Logic =max[ A (z), B (z)] OR =min[ A (z), B (z)] AND 00 R. C. Gonzalez & R. E. Woods
6 Basic Fuzzy Logic Commonly used membership uncions used o describe inpus and oupus. 00 R. C. Gonzalez & R. E. Woods
7 Fuzzy Inpu Variables We will use a single color o describe a rui wih a color ha changes rom green o yellow o red as i ripens. A paricular color z o has a membership value green (z o ), yellow (z o ), and red (z o ) in all hree inpu membership uncions. 00 R. C. Gonzalez & R. E. Woods
8 Fuzzy Oupu Variables The rui can be verdan(uni o ea), halmaure (ripening), and maure (ripe) The oupu variable is mauriy which is hard o quaniy 00 R. C. Gonzalez & R. E. Woods
9 Fuzzy Sysem We now need o relae he inpu membership uncions o he oupu membership uncions his is called implicaion inpu membership This is a simple plo o he relaionship beween he wo membership uncions oupu membership This is simply he membership uncion or red AND maure or red maure redaure (z,v)=min[ red (z), maure (v)] 00 R. C. Gonzalez & R. E. Woods
![), redaure (z o,v)] red (z o ) is a](/docs-images/77/76462136/images/10-5.jpg "consan c which clips he oupu membership")
10 A uzzy oupu or a given inpu We now need o evaluae each oupu membership uncion or he given inpu value Q 3 (v)= red (z o ) AND redaure (z o,v) =min[[ red (z o ), redaure (z o,v)] red (z o ) is a consan c which clips he oupu membership uncion as shown above Q 3 is sill a membership uncion! 00 R. C. Gonzalez & R. E. Woods
11 All uzzy oupus or a given inpu There are 3 dieren inpu membership uncions each o which can be ANDed wih maure. This gives hree dieren oupu membership uncions. The sysem oupu is he maximum value a each poin or Q=Q 1 OR Q OR Q 3. This is he maure membership uncion or a speciic color z o 00 R. C. Gonzalez & R. E. Woods
12 Deuzziicaion The oupu is sill a se. The acual membership value is he cener o graviy o he oupu se. v 0 = K v=1 K v=1 vq( v) Qv ( ) 00 R. C. Gonzalez & R. E. Woods
13 The Enire Process 00 R. C. Gonzalez & R. E. Woods
14 Fuzzy Conras Enhancemen IF a pixel is dark THEN make i darker IF a pixel is gray THEN make i gray IF a pixel is brigh THEN make i brigher The oupu memberships are only hree values. 00 R. C. Gonzalez & R. E. Woods
15 Conras Enhancemen 1. Compue he inpu membership uncion AND he oupu membership uncion. For a speciic value o inpu gray level we map ono a single oupu plane. The membership is 1 or deep blacks and gradually decreases o zero. Do his or each oupu. 3. Deermine he oal membership uncion and compue he cener o graviy o he oupu v 0 = μ dark ( z 0 ) v dark + μ gray ( z 0 ) v gray + μ brigh ( z 0 ) v brigh μ dark ( z 0 )+ μ gray ( z 0 )+ μ brigh ( z 0 )
16 Fuzzy Conras Enhancemen 00 R. C. Gonzalez & R. E. Woods
17 Fuzzy Conras Enhancemen Original hisogram Equalized hisogram Fuzzy membership uncions Fuzzy conras enhanced hisogram 00 R. C. Gonzalez & R. E. Woods
18 Fuzzy Boundary Exracion IF a pixel belongs o a uniorm region THEN make i whie ELSE make i black IF d is zero AND d 6 is zero THEN z 5 is whie IF d 6 is zero AND d 8 is zero THEN z 5 is whie IF d 8 is zero AND d 4 is zero THEN z 5 is whie IF d 4 is zero AND d is zero THEN z 5 is whie ELSE z 5 is black 00 R. C. Gonzalez & R. E. Woods
19 Fuzzy Boundary Exracion Rules 00 R. C. Gonzalez & R. E. Woods
20 Fuzzy Boundary Exracion Inpu membership uncion or ZERO inensiy dierences Oupu membership uncion or black and whie 00 R. C. Gonzalez & R. E. Woods
21 Fuzzy Sysem inpu membership oupu membership This is a plo o he relaionship beween he wo membership uncions This is he membership uncion or dierence AND whie
22 A uzzy oupu or a given inpu We now need o evaluae each oupu membership uncion or he given inpu value This would be he oupu membership or a speciic inensiy dierence inpu AND whie
23 Fuzzy Boundary Exracion 00 R. C. Gonzalez & R. E. Woods
24 Sum o Funcions
25 Fac: Any Real Signal has a Frequency- Domain Represenaion Odd-order harmonics sq = 1 n= n + 1 () sin ( n + 1) The erms shown (blue) sum o he rippling square wave (black). As he number o erms in he sum becomes large, i approaches a square wave (red) by Richard Alan Peers II
26 Frequency-Domain Represenaion Any periodic signal can be described by a sum o sinusoids. sq = 1 n= n + 1 () sin ( n + 1) The sinusoids are called basis uncions. The mulipliers are called Fourier coeiciens by Richard Alan Peers II
27 Frequency-Domain Represenaion Any periodic signal can be described by a sum o sinusoids. sq = 1 n= n + 1 () sin ( n + 1) The sinusoids are called basis uncions. The mulipliers are called Fourier coeiciens. Basis uncions by Richard Alan Peers II
28 Frequency-Domain Represenaion Any periodic signal can be described by a sum o sinusoids. sq = 1 n= n + 1 () sin ( n + 1) The sinusoids are called basis uncions. The mulipliers are called Fourier coeiciens. The Fourier coeiciens (o a square wave) by Richard Alan Peers II
29 The similariy beween uncions and g on he inerval (-, ) can be deined by, g = ()g * () d where g * () is he complex conjugae o g This number, called he inner produc hough o The Inner Produc: a Measure o Similariy as he amoun o g in or as o (). and g, can also be he projecion o ono g. I and g have he same energy, hen heir inner produc is maximal i = g. On he oher hand i, g = 0, hen and have nohing in common. g by Richard Alan Peers II
30 Inner Produc o a Periodic Funcion and a Sinusoid, g = () ( sin )d () (, g = cos )d, g = () cos = = ()e j ()e j ( ) j sin d d ( ) d e j = cos( ) j sin( ) = 3 dieren represenaions by Richard Alan Peers II
![() ( )d g = sin, () ( ) ( ) [ ] () () d e d e d i g i i = = = sin cos, () ( )d g = cos, real number resuls yield he](/docs-images/77/76462136/images/31-4.jpg "ampliude o ha sinusoid in he uncion.")
31 () ( )d g = sin, () ( ) ( ) [ ] () () d e d e d i g i i = = = sin cos, () ( )d g = cos, real number resuls yield he ampliude o ha sinusoid in he uncion. Inner Produc o a Periodic Funcion and a Sinusoid by Richard Alan Peers II
![() ( )d g = sin, () ( ) ( ) [ ] () () d e](/docs-images/77/76462136/images/32-3.jpg "d e d i g i i = = = sin cos, () ( )d g =")
32 () ( )d g = sin, () ( ) ( ) [ ] () () d e d e d i g i i = = = sin cos, () ( )d g = cos, Complex number resul yields he ampliude and phase o ha sinusoid in he uncion. Inner Produc o a Periodic Funcion and a Sinusoid by Richard Alan Peers II
33 The Fourier Series is he decomposiion o a -periodic signal ino a sum o sinusoids. () = A0 + = n 1 A n n cos + B n n sin periodic : such ha ( ± n) = ( ). The represenaion o a uncion by is Fourier Series is he sum o sinusoidal basis uncions muliplied by coeiciens. A B n n = = () cos n n n d or n 0 () sin d or n 0 n Fourier coeiciens are generaed by aking he inner produc o he uncion wih he basis by Richard Alan Peers II
34 The Fourier Series can also be wrien in erms o complex exponenials e ()= C n n= n= n= + j n = C n + j n e + n n = C n cos + n + j C n sin n + n j = 1 C n = C n e + j n Cn = Cn e + j n = 1 = 1 j ()e n d n () cos n j sin n n d e ± jx = cos x ± j sin x ( + n) = ( ) or all inergers n by Richard Alan Peers II
35 Why are Fourier Coeiciens Complex Numbers? () = C n represens he ampliude, A= C n, and relaive phase,, o ha par o he original signal, (), ha is a sinusoid o requency n = n. e n= C n + j n where C n = C n e + j n by Richard Alan Peers II
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