linearsystems 2 prelim

linearsystems 2 prelim - 2.0 Basic Imaging...

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2.0 Basic Imaging Principles—Signals and Systems Signals model physical processes; systems model how edical imaging systems create new signals (images) medical imaging systems create new signals (images) from those original signals •Signals •Point impulse and its use to characterize the impulse response pp p •Comb and sampling functions, rect and sinc, complex exponential and sinusoids •Linear systems hift i i •Shift invariance •Separable signals •Fourier transform •Transfer function and relation to impulse response •Relation of the output of a linear, shift-invarient system to the input •Sampling •Nyquist theorem and aliasing
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Summary of Last Lecture: We can characterize a linear system’s output, g, in terms of its input, f, and the impulse response or point spread function, h as a superposition: In the case of a linear, shift invariant system, d we have the convolution g=h*g ∫∫ = η ξ d d y x h f y x g ) , ; , ( ) , ( ) , ( ) , ( ) , ; , ( = y x h y x h and we have the convolution g=h*g For well behaved PSF the output resembles the input. = d d y x h f y x g ) , ( ) , ( ) , ( If we use an input of a given frequency, The system response is 1 )} ( { = x FT δ ) ( )} 2 {exp( 0 0 u u x u j FT = δ π ) ( sinc )} ( { u x rect FT =
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2.5
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Conjugate symmetry If f(x,y) is a complex-valued function with 2D FT F(u,v), then the 2d FT of f*(x,y) (complex conjugate of f) is simply related to the FT of f(x,y): v F 2D (f*)(u,v)=F*(-u,-v) If f(x,y) is real-valued, then f=f* and we have conjugate u symmetry, often called Hermitian symmetry: F(u,v)=F*(-u,-v) or F(-u,-v)=F*(u,v) this case if F= jF then we have: in this case, if F=F R +jF I , then we have: F R (u,v)=F R (-u,-v) and | F R (u,v)|=|F R (-u,-v)| ,ie, are even functions. nd And: F I (u,v)=-F I (-u,-v) and ,ie, are odd functions. ) , ( ) , ( v u F v u F −∠ =
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Conveniently, the basis functions of the 2DFT are separable into separate x and y integrations: dx x p( = dy vy j y u r dydx vy ux j y x f v u F ) 2 exp( ) , ( )) ( 2 exp( ) , ( ) , ( π ∫∫ = + x x p( = Where: dx ux j y x f y u r ) 2 exp( ) , ( ) , (
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Properties of the 2-D Fourier Transform
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This note was uploaded on 10/30/2010 for the course MP 230 taught by Professor Macfall during the Fall '10 term at Duke.

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linearsystems 2 prelim - 2.0 Basic Imaging...

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