## ESE589HW1-18

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## ESE589HW1-18

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- School:
- Washington University in St. Louis
- Course:
- Ese 589 - Biological Imaging Technology

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ESE 589 Spring 2018 Homework 1 Assigned February 6 2018 Due February 12 2018 Read Part I Basic Imaging Principles Chapters 1 2 and 3 in Medical Imaging Signals and Systems by Prince and Links Part II Radiography Chapters 4 5 and 6 in Medical Imaging Signals and Systems by Prince and Links Recommended Problems Do not hand in 1 Consider an unusual imaging system whose aperture consists of a set of pinholes on a rectangular grid spaced every 1 m in the x direction and every 2 m in the y direction There are 15 holes in each row x direction and 7 holes in each column y direction A mathematical model for the point spread function can be written in at least two ways as h x y 7 3 x k y 2l k 7 l 3 7 3 h x y x k y 2l k 7 l 3 a Compute the transfer function H u v for this aperture b Find an approximate expression for the bandwidths in each of the frequencies u and v conjugate to x and y Suppose this aperture is used in place of a pinhole in a pinhole camera so that the image on a focal plane at distance d from the aperture For an object at distance z in front of the aperture with source image s x y an approximate equation for the image i x y ignoring several geometric effects but accounting for image inversion is yz xz i x y h s d d d d Suppose that the source image is zero outside of some region s x y 0 for x 4 mm or y 8 mm In terms of the distance of the focal plane from the aperture d find the set of distances z such that the image in the focal plane i x y consists of 105 7 15 non overlapping copies of the source image j 2 u x v y 2 Let f x y e 0 0 and let s x y f x y cos 2 u0 x v0 y For each of these functions a state whether it is separable b find its periods and c find its Fourier transform x 2 y 2 Let h x y e be the point spread function for a linear shift invariant system d find the output of this system with input f x y and with input s x y 3 Consider a linear shift invariant imaging system with point spread function PSF 1 x2 y 2 h x y x2 y 2 0 a Find the FWHM of the PSF b Find the line spread function LSF c Find the FWHM of the LSF d Are the answers in parts a and c different Explain e Find the modulation transfer function f Sketch the output of this system if the input is f x y x y x 3 y x 3 y 3 2 2 g Sometimes the usual intuition for understanding PSFs and their FWHM does not apply directly For example if the peak of the PSF is not at the origin and is in fact not at one point then the usual definition of the FWHM breaks down somewhat As an example consider a circularly symmetric nonnegative PSF whose LSF which equals the parallel beam projection is 1 l g l 0 l Without necessarily finding the PSF compare the FWHM of the LSF and the PSF Hint The point spread function is confined to a circle of radius and all integrals through the PSF equal 1 Where must the peak values of the PSF be Note that g l can be considered to be the Radon transform of the PSF at angle 4 Please submit all problem sets electronically on BlackBoard Solve the following problems and submit complete problem solutions show all steps required to solve the problem and include a problem statement 1 Note that the sum ux vy can be written as an inner product between two vectors x ux vy u v Let F u v be the Fourier transform of f x y y T F u v f x y e j 2 ux vy dxdy or in vector notation F u f x e j 2 u x dx Define the rotation matrix cos sin R sin cos and let g x f Rx the function f x y rotated by an angle of around the origin Using the vector notation show that the Fourier transform of g x satisfies G u F Ru That is a rotation in the spatial domain by an angle yields a rotation in the Fourier frequency domain by the same angle 2 Let the input to a linear shift invariant system be f x y x a line impulse along the y axis and let l x h f x y h x d d h x d be the line spread function corresponding to the point spread function PSF h x y a Find the Fourier transform of the line spread function in terms of the transfer function H u v Now consider a rotated line impulse g x f Rx using the notation from recommended problem 1 above b Find the Fourier transform of the convolution of g x f Rx with h x y c Argue that if all such integrals against rotated line impulses were known then h x y could be found 3 Consider a linear shift invariant imaging system with PSF 1 1 x 2 h x y rect x rect y where rect x 5 7 0 x 1 2 Let the input be a contrast phantom represented as f x y A B cos 2 ax by Find the output of the linear shift invariant imaging system 4 Problem 2 22 in Prince and Links 5 Problem 3 6 Anisotropic PSF 6 Problem 3 10 in Prince and Links 7 Problem 3 16 in Prince and Links

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