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 μm in the x direction and every 2 μm 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 73737373, , 2,2klklh x y x k y lh x y x k y l (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 , , ,xz yzi x y h s d ddd Suppose that the source image is zero outside of some region, , 0, for 4 mm or 8 mm.s x y x y 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. 2. Let 002( , )j u x v yf x y e and let 00( , ) ( , ) cos 2s x y f x y u x v y . For each of these functions, (a) state whether it is separable, (b) find its periods, and (c) find its Fourier transform. Let 22( , )xyh 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 yand with input( , )s x y. 3. Consider a linear shift invariant imaging system with point spread function (PSF)22221,( , )0,xyh x yxy (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 33( , ) 322f x y x y x y x y (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, ,()0, .lgll 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 α and all integrals through the PSF equal 1. Where must the peak values of the PSF be? Note that ()glcan be considered to be the Radon transform of the PSF at angle θ. 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 xux vy u vy. Let ( , )F u v be the Fourier transform of( , )f x y, 2( , ) ( , )j ux vyF u v f x y e dxdy, or in vector notation,2( ) ( )TjF f e duxu x x. Define the rotation matrix cos sinsin cosR and let ()gfx Rx, the function ( , )f x y rotated by an angle of θ around the origin. Using the vector notation, show that the Fourier transform of ()g x satisfies ( ) ( )GFu Ru. That is, a rotation in the spatial domain by an angle θ yields a rotation in the Fourier (frequency) domain by the same angle θ.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 ()gfx Rx, using the notation from recommended problem 1 above. (b) Find the Fourier transform of the convolution of ()gfx Rx with ( , )h x y. (c) Argue that if all such integrals against rotated line impulses were known, then ( , )h x ycould be found. 3. Consider a linear shift invariant imaging system with PSF 12121, |x|( , ) , where 570, |x|>yxh x y rect rect rect x Let the input be a contrast phantom represented as))(2cos(),( byaxBAyxf . 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
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