linearsystems 1

# linearsystems 1 - 2.0 Basic Imaging PrinciplesSignals and...

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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-invariant system to the input •Sampling •Nyquist theorem and aliasing

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Read Prince, Chapter 1 for a quick overview of Medical Imaging. You should already be familiar with: i D lt f ti ) d fi iti ifti li Dirac Delta function, δ (x), definition, sifting, scaling Comb function (array of delta functions) Rect and Sinc functions, scaling and shifting Exponential functions: exp(j2 π x) est to review Prince, Chapter 2, sections 1, 2 Best to review Prince, Chapter 2, sections 1, 2
Example of systems and signals: Projection X-ray The system is the setup yp that provides I 0 and measures I(y) y () ) x μ (x,y) g(y) I 0 I(y) I(y)/I 0 =exp(-g(y))

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2.3 Systems η , ξ x,y

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In the general case, applying linearity = y x f H y x g )] , ( [ ) , ( ∫∫ = η ξ δ d d y x f H ] ) , ( ) , ( [ linearity = d d y x H f )] , ( [ ) , ( = d d y x h f ) , ; , ( ) , ( hus, in a linear system, it is completely described by Thus, in a linear system, it is completely described by the PSF and the superposition integral se of the PSF in practice is awkward since it is a 4D Use of the PSF in practice is awkward since it is a 4D function—in general

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Example: PSF varies with position, Finite hole pinhole camera: PSF(B) A B PSF(A) Input “f( ξ )” System “H” Output “g(x; ξ )”
if

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Example: pinhole camera: Either the pinhole is “infinitesimal” or the source tent is restricted so that all angles are very small
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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 1 - 2.0 Basic Imaging PrinciplesSignals and...

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