BME 470/570 Project

Category-selection deadline: November 8, 2024 11:59 am

Preference-presentation deadline: November 11, 2024, 11:59 am

Report due date: December 16, 2024 11:59 pm

Instructions

• InHW1,welearnedconceptsinfunctionalanalysisforimagingthroughsolvingmathematical

problems. InHW2, welearnedaboutsomeofthecomputationaltoolsforanalysisofimaging

systems such as singular-value decomposition and eigen analysis. Having learned these theo-

retical and computational tools, in this assignment, our objective will be to use these tools to

analyzetwoimagingsystems! Further, wewillusethesetoolstoconductartificial-intelligence

(AI)-based operations on the data obtained with these systems. We will conduct this analysis

through a step-wise procedure.

• The assignment will be graded based on a presentation and a report.

• Foreachofthetwoimagingsystemsmentionedbelow,thereareeightsub-parts. Thelastthree

sub-parts involve concepts from artificial intelligence (AI). However, these three questions are

more challenging than the other questions due to multiple reasons such as this not being

a course on AI, the AI-related questions being more involved and longer and that question

wouldrequiremorecomputations. Toensurefairnesswhileprovidingstudentswithflexibility,

we offer the following option to choose the questions a student wants to attempt:

– The eight sub-parts are divided into two categories, sub-parts (1)-(5) and sub-parts 6-8.

You are required to choose between one of these two categories by the deadline stated

above.

– If you have taken BME 570/ESE 5981, you have the following options:

∗ If you choose sub-parts (1)-(5), in your presentation, you would be randomly as-

signed to present one of these sub-parts for one system. You would have to then

presentthatsub-partindividually. Inyourwrittenreport,youwouldhavetoprovide

solutions to all 5 sub-parts and for both the systems.

∗ If you choose sub-part 6-8, you will be assigned one of these sub-parts for one of the

two systems. You can collaborate and present that sub-part as a team of 3 people.

In your written report, you would need to provide solution only to sub-part and

only for the system that you present.

– If you have taken BME 470, you have the following options:

∗ If you choose sub-parts (1)-(5), you can pair up with another student who has

taken BME 470 to work on those sub-parts. You both can then present a combined

presentation and submit a combined report.

∗ If you choose sub-parts (6)-(8), you can work in a team of 4. The other team

members could have taken BME 470/BME 570/ESE 5981.

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– You need to inform the instructor and the TAs if you want to work on sub-parts (1)-

(5), or on sub-part (6)-(8). If you choose to work on sub-part (6)-(8), you would need

to inform the composition of your team and if you have a preference for System 1 vs.

System 2.

– In case we do not hear from a student, we will assume that the student has chosen

subparts (1)-(5) for their presentation and written report.

• The presentation and report components of this assignment are described below.

• Presentation:

– The objective of the presentation is to learn how to communicate mathematically com-

plex ideas. This is a very important skill while presenting work on theoretical and

computational imaging at venues such as conferences.

– Thepresentationswilloccuroverthelastfourdaysoftheclass(Nov.20,Nov.25,Dec.2,

and Dec. 4).

– In several of the questions, we ask that the student display the results for five objects.

Therequirementoffiveobjectfunctionsisonlyforthepresentation. Also, one of these

objects should be the Shepp-Logan phantom. The other four object functions can

be your choice.

– If the student chooses to attempt sub-parts (1)-(5), they will be assigned one sub-part

within this category to present. The student will be assigned 8 minutes, of which 5-

6 minutes are assigned to presentation and the remaining time for a question-answer

session. The questions will be asked by the students, TAs, and the instructor.

– For sub-part (6)-(8), the team of students will get a total of 15/20 minutes, with 12

or 15 minutes assigned for presentation, depending on whether there are three or four

members in the team, respectively. All members of the team need to present. Next, 5

minutes will be assigned for questions.

– In the presentation, the student should focus on outlining the concept they learned in

the sub-part they are presenting. A guideline to organize the presentation is as follows:

∗ Introduce and very briefly go through how the problem was solved.

∗ Discuss the results.

∗ Describe the concept that was learned.

More detailed guidelines have been provided in presentation format on Canvas.

– We have tried to ensure that sub-parts (1)-(5) have equal levels of difficulty. In case

you choose this category of question to attempt, the students can mail their preferences

on which of these sub-parts they would like to present to the instructor and teaching

assistants by the deadline specified above. To ensure that different students present

different sub-parts, the sub-parts will be assigned to the students through a random-

selection procedure that will account for the student preferences. The sub-parts will be

assigned by around Nov. 14. Students who choose their own imaging system will need

to mention this in their email and those students will be assigned one sub-part for that

system.

– The question-answer session at the end of each presentation is highly encouraged and

will be part of the grading (both for the presenter and the rest of the students).

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Figure 1: Demonstrating the procedure to display a sinogram

• Report:

– The report for this assignment is due at the deadline specified above.

– In the report, first answer all the sub-parts for one system before proceeding to the next

system.

– Start each sub-part on a new page.

– In several of the questions below, we ask that the student display the results for five

object functions. For the report, the results need to be displayed only for one object

function: the Shepp-Logan phantom.

– Collaborationontheassignmentisfine, butanywrittenmaterial, includingcode, should

be your own.

– We recommend for the report to be typed.

– Organizeyoursolutionssothattheyareeasytofollow. Somesuggestionsincludeprovid-

ing titles for plots, plotting legible figures, and referring to the plots by figure numbers

in your solution. For example, in questions that require plotting and/or providing com-

ments on the plots, the template below could be used as a guideline:

∗ Provide theoretical details of the solution and conceptual understanding.

∗ Provide any implementation details. This is a great place to provide the code.

∗ Plot results. For each plot, provide title and caption.

∗ Providecommentsontheresults. Ineachcomment,refertotheplotbyFig. number

(e.g. Fig. 3a)

A general thought to keep in mind while organizing your solution is putting yourself in

the shoes of the grader.

• In some of the questions below, you are asked to display the data vectors as a sinogram. In

this sinogram image, the pixels for each detector position are stacked in a vertical strip, with

agreylevelineachpixelcorrespondingtothedataforthatpixel, andthestripsforsuccessive

angles are placed next to each other. An example is shown in Fig. 1.

• Your entire submission should be a single file containing your answers to all the questions in

PDF format. Do not submit multiple files or zip files.

• All the code used for this assignment must be submitted. If you prefer to not include the

code along with your solution, then include the code as an Appendix in your submitted file.

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The key purpose of this project is to become familiar with analyzing imaging systems using the

tools learned in class. We will consider two imaging systems in this project.

(A) System 1: 2-D tomographic system: Considerasimplified2-Dtomographicsystem. The

field of view (FOV) is a disk of unit radius in the plane. The kernel for the imaging system

is given by

(cid:40)

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for(cid:0)

−1+

m−1(cid:1)

< y <=

(cid:0)

−1+

m(cid:1)

h (x,y) = 16 16 (1)

m

0,otherwise

In other words, each sensitivity function is equal to 1 in a horizontal strip of width 1/16, and

zero outside this strip. The strips are non-overlapping and stack up to cover the whole unit

disk. This corresponds to a 32 pixel detector oriented vertically with perfect collimation. To

get the next 32 sensitivity functions we rotate these by π/16. We continue rotating by π/16

to generate a total of M = 32×16 sensitivity functions.

(B) System 2: 2-D radial MRI: Consider a highly simplified 2D radial MRI system. The

kernel for the imaging system is given by:

h (r) = exp(−2πik ·r) (2)

m m

and the FOV is the disk centered at the origin with radius 1. The first 32 k are (m−16,0)

m

for m = 1,2,...,32. To get the next 32 k we rotate the first 32 by π/16 . We continue

m

rotating by π/16 to generate a total of M = 32×16 sensitivity functions.

For each of the systems above, or for a system of your choice, perform the following:

1. Systemmodeling: Simulatetheforwardmodelforthissystemusingthefollowingexpansion

functions for representing the object:

(a) 64×64 pixels

(b) 32×32 pixels

(c) Fourier-basis functions for the square of side-length 2 units for system A and system B.

These would be of dimensions 64×64.

In the process, you will obtain the H matrices for each of the expansion functions. For 5

object functions in the unit disk and using these H matrices, display and compare the data

vectors that you get from the three different discretizations.

2. SVD analysis of DD operator: For each of the H matrices generated in part 1, perform

the following:

(a) Compute the SVD and plot the singular value spectra.

(b) Plot some of the singular vectors in object space as images.

(c) For each singular vector plotted above, plot the corresponding singular vectors in data

space in the sinogram format.

(d) For five object functions on the unit disk, use the SVD analysis to determine and then

display the measured and null components of the object.

(e) Comment on all the results.

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3. Reconstruction with DD operator: For five object functions of your choice on the unit

disk and each H matrix obtained in 1, generate the data vector and perform the following:

(a) UsetheSVDtoperformapseudoinversereconstructionforeachH matrixforfiveobject

functions of your choice.

(b) Repeat with Poisson noise added to the data. To add Poisson noise in Matlab, you can

use the poissrnd command. Remember that the Poisson noise must be added to the

data, not to the object. Also, ensure that the sinogram pixels have values higher than

unity. Note that Poisson distribution outputs only integer values.

(c) You will find that adding noise leads to poor-quality reconstruction when all the SVD

values are used. Explain why.

(d) Reduce the number of SVD values by discarding the values with smaller magnitude and

repeat the reconstruction. This process is referred to as regularization.

4. SVD analysis of CD operator: In this sub-part, we work directly with the original

continuous-to-discrete (CD) operator H, the kernel of which is given by Eq. (1) and Eq. (2)

for the two systems. Note that the operator cannot be discretized in the problems below.

(a) Describe the procedure to compute the SVD. Hint: Use similar approach as problem 3

in Homework 2.

(b) Plot the singular value spectra.

(c) Plot the some of the singular vectors in object space as images. Note that you will need

to discretize the singular vectors.

(d) Plot some of the singular vectors in data space in the sinogram format.

(e) For five object functions on the unit disk, use the SVD analysis to determine and then

displaythemeasuredandnullcomponentsoftheobject. Againyouwillneedtodiscretize

the object functions.

5. Reconstruction with CD operator: Herewewillperformreconstructionwiththeoriginal

CD operator. Again note that the operator cannot be discretized in the problems below.

(a) Use the SVD to perform a pseudoinverse reconstruction of the CD H operator for five

object functions on the unit disk.

(b) Repeat with Poisson noise added to the data. Remember that the Poisson noise must be

added to the data, not to the object. Also, ensure that the sinogram pixels have values

higher than unity. Note that Poisson distribution outputs only integer values.

(c) The results from part (b) above will likely be poor-quality reconstruction. Explain why

that is the case.

(d) Implement the regularization-based strategy described above to address the issue of

poor-quality reconstruction.

In the next question, we will explore the application of the concepts we learned in class in the

emerging era of AI, and more specifically, deep learning. We will use the tool of convolutional

neural network (CNN) in these questions, and some working knowledge of how to train and

test CNNs will be assumed. There are multiple references to this though, and several open-

source codes. The emphasis will not be on AI, but instead on the tools we learned in the

class.

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