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Exam #2

Name: ____________________________________________________________

13 pages, 64 questions, 75 minutes.

True or False (T/F) ?

1._______Photoelectric effect is the primary mechanism for contrast in projection radiography and CT.

2. _______Like the Fourier Transform, the Radon Transform has an exact inverse.

3._______In CT and projection radiography, higher frequencies penetrate the body more, and in ultrasound, higher

frequencies penetrate the body less.

4._______Undersampling the projection data in CT results in aliasing that manifests as streaking artifacts around

small bright objects.

5. _______Nuclear medicine utilizes ionizing electromagnetic radiation, like CT and x-ray.

6. _______The direction of photon propagation distinguishes the x-rays detected in projection radiography and CT

from the gamma rays detected in nuclear medicine.

7. _______Planar scintigraphy is the only direct nuclear medicine imaging technique.

8._______All nuclear medicine techniques detect the distribution of radiotracers introduced into the body.

9._______Nuclear medicine techniques form an image by detecting the varying intensity of gamma rays.

10._______Nuclear medicine techniques depend on the attenuation of gamma rays to create contrast in the image.

11.______Collimation is used to prevent blurring from Compton Scatter when PET imaging.

12.______The projection slice theorem provided the basis for us to develop reconstruction algorithms given

projection data.

13.______The acoustical impedances of most soft tissues in the body match each other very closely.

14. ______Ultrasound is a functional imaging modality because it can form an image quickly.

15. ______Ultrasound is characterized by sound wave frequencies greater than 20kHz.

16.______In order to image deeper in the body, it is sometimes necessary to increase the frequency of theultrasound machine.

17.______SPECT has no corresponding projection mode imaging technique (as CT has projection radiography).

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Identify the modalities from the following choices:X-ray (plane film)CTNuclear MedicineUltrasound

24. ___________________

20. ____________________

21. __________________

19. ___________________

22. ________________________

18. ________________________

23. ____________________

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Fill in the blank / Short answer.

25. How would you categorize a wave in which the particle motion is parallel to the direction of

propagation? ______________________________

26. How would you categorize a wave in which the particle motion is perpendicular to the direction

of propagation? ______________________________

27. How would you categorize a wave which cannot propagate in a vacuum?_____________________

28. How would you categorize a wave which can propagate in a vacuum?________________________

29. With increasing frequency, the ability to resolve objects with ultrasound improves or decreases?

______________________

30. Assuming integer increments, how many unique “views” (projections) of your object f(x,y) do you

have when projection imaging? __________

31. Our formulation of reconstruction methods was based on 1st generation CT technology. What

physical assumption did this enforce?______________________________________________________

32. This represents a “standard attenuation equation” through a homogenous slab:

ି

a) With regard to CT imaging, what is

B ? ___________________________________________________________________________

A ?___________________________________________________________________________

μ ?___________________________________________________________________________

d ? ___________________________________________________________________________

b) With regard to Ultrasound imaging, what is

B ? ___________________________________________________________________________

A ?___________________________________________________________________________

μ ?___________________________________________________________________________

d ? ___________________________________________________________________________

33. In CT, what are we (ideally) mapping? __________________________________________________

34. In Ultrasound, what are we (ideally) mapping?__________________________________________

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35. Given f(x,y): and the following standard frame of reference:

A black circle with intensity = 0 with 2 identical squares centered along a 45 degree line.

Sketch the following: Include (approximate) relative intensities between the three sketches and keepconsistent axis limits in order to show the relative locations of features.

l

a) g(l,0)

l

b) g(l,45)

l

c) g(l,90)

x [space units]y[s

pace

units

]

θ=0o

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36. Given g(l,θ)=cos(√ω0l )δ(θ-45o), what is f(x,y) ? (Show any necessary sketches or work)

37. Given ( , ) ( , ) ( cos sin )g f x y x y dxdy

For all l and θ constant; g(l,θ) is a______________________________________

For all l and all θ ; g(l,θ) is ___________________________________________

38. What is (,) = +) (, called ? _______________________________________

39. Sketch b45(x,y) given

f(x,y) = b45(x,y) =

40. Write the equation for straight backprojection summation:

f1(x,y) =

41. In the above equation, label the “backprojection” step and the “summation” step.

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42. Given G(q,θ) is the Fourier transform of g(l,θ), what does the following equation tell us? Give itsformal name as well as a brief description.

(,) = (,)

43. This leads straightforwardly to a reconstruction method for f(x,y) since

f2(x,y)=F2D-1{G(q,θ)}

What is this reconstruction method called? ____________________________________________

44. Given G(q,θ) is the 1D Fourier transform of g(l,θ), what is

න (,)ஶ

ିஶ

= ______________________________________________________________

45. Write the equation for filtered backprojection summation:

f3(x,y) =

46. In the above equation, label the filter step, the backprojection step and the summation step.

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47. The method of reconstructing f(x,y) from g(l,θ) in the following manner

ସ݂(ݕ,ݔ) = න [ (ܿ )݈ ∗ ݃( ߠୀ௫௦ఏା௬௦ఏ݀[(ߠ݈,

గ

is called _______________________________________________

48. Of the 4 methods to reconstruct f(x,y) from g(l,θ) given in problems 40, 43, 45, and 47, which is themathematically precise formulation for f(x,y)? (f1(x,y), f2(x,y), f3(x,y), or f4(x,y))

_________________________

49. The precise inverse Radon transform is not usually practically implementable.

a) Why not ?

b) What is added to the expression/formula to make it implementable?

50. If there are two primary means of blurring the acquired projections in CT, and they can beexpressed as s(l) and w(l), rewrite the expression for f4(x,y) shown in problem 47 to include them.

51. What are the two sources of blurring in CT?

1)_________________________________________________________________

2)_________________________________________________________________

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52. The simplified imaging equation for planar scintigraphy can be developed as

where Φ(x,y) is _________________________________________________

Az(x,y) is _____________________________________________________

μ(x,y,z’;E) is ___________________________________________________

53. Write the simplified imaging equation for planar scintigraphy including the blurring due to acollimator with PSF hC(x,y;|z0|) and the blurring due to intrinsic camera effects, expressed with PSF hI(x,y)

_________________________________________________________________________________

54. We can write an equation for the data collected during a SPECT exam to be in the same format asthe equation that expresses the data collected during a CT exam (equation found in problem 37) :

dxdyyxyxA )sincos(),(),(

When relating the two imaging modalities in this way, one can view the desired objects to reconstructas having the following relationship:

f(x,y) in CT (map of ________________________) ↔ A(x,y) in SPECT (map of __________________)

55. The reconstruction technique used in PET and SPECT imaging is _____________________________

56. What sharply distinguishes PET from SPECT?

0

0

( , , '; ) '

20

1( , ) ( , )

4z

x y z E dz

zx y A x y ez

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Fill in the blanks with appropriate titles and axes labels.

57.

58.

59.

a) ______ mode ultrasound

b) ______________________ (axis label)

a) ______ mode ultrasound

c) ______________________ (axis label)b)

___

___

___

___

___

___

___

(axi

sla

be

l)

a) ______ mode ultrasound

c) ______________________ (axis label)b)

___

___

___

___

___

___

___

(axi

sla

bel

)

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60. Consider a an ultrasound transducer with aperture S(r)=rect(r/D), where r = 22 yx , designed

to propagate waves in the z direction.

a) What shape is this transducer in the x-y plane?_______________________________________

b) The region where the z distance from the transducer is less than D2/λ is called the

__________________________________________________________

c) The region considered the “far field” of the transducer, where the distance from thetransducer is greater than D2/λ, is called the

___________________________________________________________

d) In this (far field) region, the more accurate field model, or diffraction formulation, of thetransducer is related to S(r) in what way?

_____________________________________________________________________

e) If the transducer aperture is changed from the original geometry specified above to

D

rrectrS

2)(

What happens to the beamwidth in the region where z< D2/λ? ________________________

What happens to the beamwidth in the region where z> D2/λ? ________________________

Given the following standard projection angle frame of reference

Match the following objects to their corresponding Radon Transform (sinogram) on thelast page.

61. _______________ 62. _______________63. _______________

x [space units]

y[s

pace

units]

-10 -5 0 5 10

-10

-8

-6

-4

-2

0

2

4

6

8

10x [space units]

y[s

pace

units]

-10 -5 0 5 10

-10

-8

-6

-4

-2

0

2

4

6

8

10

x [space units]y[s

pace

units

]

θ=0o

11

A B C

D E F

G HI

50 100 150 200 250 300 350

20

40

60

80

100

120

140

160

180

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64. Suppose a 4 MHz acoustic pulse travels from a transducer through 1.3 cm of fat, then encounters

an interface with kidney at normal incidence.

Given:

R = reflectivity =

ti

ti

i

r

ZZ

ZZ

I

I

coscos

coscos

12

12

μa= attenuation factor [cm-1] =67.8

fa [dBcm-1MHz-1]Characteristic impedance, Z

[kg m-2s-1] · 106 Velocity [m/s]

Fat 0.63 1.35 1450

Kidney 1.0 1.62 1560

a) At what time interval after the transmitted pulse will the first reflected pulse (the echo) arrive

back at the transducer?

b) Taking only attenuation into account, what is the fraction (decimal or percentage) of the incident

waveform that encounters the fat/kidney interface? *please read carefully*

fat kidney

1.3cm

fat

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c) Taking only reflection characteristics at the fat/kidney interface into account, what is the fraction

(decimal or percentage) of the incident waveform that is transmitted into the kidney?

d) Taking both attenuation and reflection losses into account, what will be the amplitude loss in

decibels of the returning waveform?

e) What change in your answer would you see if the frequency of the transducer was changed to

10MHz and why ? (no numbers necessary – just words)