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 medical 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 •Comb and sampling functions, rect and sinc, complex exponential and sinusoids •Linear systems Shift 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, and we have the convolution g=h*g ∫∫ = η ξ η ξ η ξ d d y x h f y x g ) , ; , ( ) , ( ) , ( ) , ( ) , ; , ( η ξ η ξ = y x h y x h ∫∫ ξ ξ ξ d d h f ) ( ) ( ) ( For well behaved PSF the output resembles the input. = η η η y x 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 =
2.5

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