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# Class Note for ECE 482 at UA-Comp Visn Dig Image Proc(4)

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This 12 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Alabama - Tuscaloosa taught by a professor in Fall. Since its upload, it has received 13 views.

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Date Created: 02/06/15
Computer Vision amp Digital Image Processing Fourier Transform Properties the Laplacian Convolution and Correlation Electrical amp Computer Engineering Dr D J Jackson Lecture 91 Periodicity of the Fourier transform The discrete Fourier transform and its inverse are periodic with period N Fuv FuNv FuvN FuNvN Although Fuv repeats itself infinitely for many values of u and v only Nvalues of each variable are required to obtain fXy from Fuv ie Only one period ofthe transform is necessary to specify Fuv in the frequency domain Similar comments may be made for fXy in the spatial domain Electrical amp Computer Engineering Dr D J Jackson Lecture 92 Conjugate symmetry of the Fourier transform If fXy is real true for all of our cases the Fourier transform exhibits conjugate symmetry FuvFuv or the more interesting FUVl FUlVl where Fuv is the complex conjugate of Fuv Electrical amp Computer Engineering Dr D J Jackson Lecture 93 Implications of periodicity amp symmetry Consider a 1D case Fu FuN indicates Fu has a period oflength N Fu Fu shows the magnitude is centered about the origin Because the Fourier transform is formulated for values in the range from 0N1 the result is two backtoback half periods in this range To display one full period in the range move shift the origin ofthe transform to the point uN2 Electrical amp Computer Engineering Dr D J Jackson Lecture 94 Periodicity properties H i ll Fourier spectrum With back to back half periods in the range N n M V N A i i 011 1 L E Om period 4 a Shifted spectrum EFMI with a full period in the A 0 M2 N I L One period 4 1 7 same rang C Electrical amp computer Engineering Dr D J Jackson Lecture 95 Periodicity properties 2D Example Erwin I am arm quot12 5 2 Edit Mndaw he p El Edit widow Eda Electrical amp computer Engineering Dr D J Jackson Lecture 95 Distributivity amp Scaling The Fouriertransform and its inverse are distributive over addition but not over multiplication So 5f1xay f2xy 3f1xy 5f2xy 5f1xay X f2xy i 5f1xy X 5f2xy For two scalars a and b afxyltgt aFuv fwcby ltgt iFuavb labl Electrical amp Computer Engineering Dr D J Jackson Lecture 97 Average Value A widely used expression for the average value of a 2D discrete function is 1171er my ZZfxy From the definition of Fuv for uv0 N71 N71 F0022fxy x0 y0 Therefore 1 fxy F00 Electrical amp Computer Engineering Dr D J Jackson Lecture 98 The Laplacian The Laplacian of a two variable function fxy is given as 62 62 WW ax 3 From the definition ofthe 2D Fourier transform Sszx yltgt 42702012 vZFuv The Laplacian operator is useful for outlining edges in an image Electrrczu a Cnmrluler Enurneerrnu Dr D l Jacksnn lecture 979 The Laplacian Matlab example Given Fuv use the Laplaclan to construct an edge outllned representatlon of the fxy ffmapbmpread39lena128bmp39 Ffft2f Fedgezeros128 for u1128 for v11ZE Fedgeuvc 2tp1AzftuA2vA2Fuv end end fedge1fft2Fedge 1magerealfedgecolormapgray256 Electrrczu a Cnmrluler Enurneerrnu Dr D l Jacksnn lecture 94H Convolution amp Correlation The convolution of two functions fx and gX is denoted fxgx and is given by f x goc Iforgxa am Where 0c is a dummy variable of integration Example Consider the following functions f0c and 0c g f06 g0 12 10c 10c Electrical amp Computer Engineering Dr D J Jackson Lecture 911 1D convolution example Compute g0c by folding g0c about the origin gm g0c 12 12 X X 1 1 Compute gX0c by displacing g0c by the value X g0c gov0c 12 12 1 0L 1 x X Electrical amp Computer Engineering Dr D J Jackson Lecture 912 1D convolution example continued Then for any value X we multiply gXOc and f0c and integrate from 00 to 00 For Osx 31 we have For 1 s X s 2 we have f0cggtlt Oc f0cgX 0c 1 1 l 1 0L 1 0L 59m amp CommerEngmeemg Dr D J Jackson Lama 1D convolution example continued Thus we have xZ 03x31 fxgx lax2 ISxSZ 0 elsewhere Graphically fXgX 12 Electrical amp Computer Engineering Dr D J Jackson Lecture 914 Convolution and impulse functions Of particular interest will be the convolution of a function fX with an impulse function 5XX0 l fx5xxOdx fxo The function 5XX0 may be viewed as having an area of unity in an infinitesimal neighborhood around X0 and O elsewhere That is T6xixodxif6xixodxl Electrical amp Computer Engineering Dr D J Jackson Lecture 915 Convolution and impulse functions continued We usually say that 6X X0 is located at xx0 and the strength of the impulse is given by the value of fX at xx0 If fXA then A6X X0 is impulse of strength A at XX0 Graphically this is xu x A5x x0 Electrical amp Computer Engineering Dr D J Jackson Lecture 916 Convolution with an impulse function Given fX is fx A L CL X o and gx8XT 6X 6XT gm Electrical amp Computer Engineering Dr D J Jackson Lecture 917 Convolution with an impulse function continued fxgx is Electrical amp Computer Engineering Dr D J Jackson Lecture 918 Convolution and the Fourier transform fX gx and FuGu form a Fourier transform pair If fX has transform Fu and gx has transform Gu then fxgx has transform FuGu f xgx lt3 F uGu f xgx lt3 F u Gu These two results are commonly referred to as the convolution theorem Electrical amp Computer Engineering Dr D J Jackson Lecture 919 Frequency domain filtering Enhancement in the frequency domain is straightforward Compute the Fourier transform Multiply the result by a filter transform function Take the inverse transform to produce the enhanced image In practice small spatial masks are used considerably more than the Fourier transform because oftheir simplicity of implementation and speed of operation However some problems are not easily addressable by spatial techniques Such as homomorphic filtering and some image restoration techniques Electrical amp Computer Engineering Dr D J Jackson Lecture 920 Lowpass frequency domain filtering Given the following relationship Guv H uvF w where Fuv is the Fourier transform of an image to be smoothed The problem is to select an Huv that yields an appropriate Guv We will consider zerophaseshift filters that do not alter the phase of the transform ie they affect the real and imaginary parts of Fuv in exactly the same manner Electrical amp Computer Engineering Dr D J Jackson Lecture 921 Ideal lowpass filter lLPF A transfer function for a 2D ideal lowpass filter lLPF is given as 1 if Duv D0 Huv 0 1f DuvgtD0 where D0 is a stated nonnegative quantity the cutoff frequency and Duv is the distance from the point uv to the center of the frequency plane Duv Vuz v2 Huv V H Electrical amp Computer Engineering Dr D J Jackson Lecture 922 Ideal lowpass filter ILPF continued The point D0 traces a circle from the frequency origin giving a locus of cutoff frequencies all are at distance D0 from the origin One way to establish a set of standard loci is to compute circles that encompass various amounts of the total signal power PT PT is given by WW4 PT ZZPuv u0 v0 where Puv is given as Pu v Fu v2 R2 u v 12 u v o For the centered transform a circle of radius r encompasses 3 percent of the power where 8 100Z ZPW v P the summationis over allpoints it Vcncompasscd by thecircle Electrical amp Computer Engineering Dr D J Jackson Lecture 923

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