Reference no: EM132266680 , Length: 3190 words

This case study will involve reconstruction of medical images from the Positron Emission Tomography (PET) described in Section 1.7.5 of the text.

You are required to submit a complete optimization study, including modelling, developing an appropriate optimization algorithm, implementing the algorithm using MATLAB, and interpreting the optimization result.

Answer the questions posed and follow the hints and suggestions given. This project is to be completed individually; you may consult each other during the conceptualization and implementation stage (but the report is individual).

If you want to learn more about the PET imaging modality, you can easily conduct a quick online search.

Requirements

- Submit a complete optimization study, including modelling, developing an appropriate optimization algorithm, implementing the algorithm using MATLAB, and interpreting the optimization result by the due date.

- Answer the questions posed and follow the hints ansd suggestions given.

PROCEDURE

(1) Introduction: (Background, problem statement, study objective)
(2) Method of Analysis and Key concepts used
(3) Results
(4) Conclusion and Recommendation

Case Study - Positron Emission Tomography Image Reconstruction

Tasks to be accomplished in the case study

a) Show that the likelihood that the total emissions received by all detector pairs is y given that the expected number of emissions from all voxels is x is given by:

P(y|x) = Π_{j ((}e^{-y^j}(y^_j)^yj)/y_j!) = Π_{j ((}e^{-Σ_ic_i,jx_i}(Σ_ic_i,jx_i)^yj)/y_j!), where yˆ_j = Σ_iC_i,jx_i

(Use hints given at the end of this document.)

Prove the final expression for the maximum likelihood estimation objective function f_ML• Hint: Take the logarithm of the likelihood and omit the constant term that does not depend on x.

b) Derive the formulas for the gradient and Hessian matrix of f_ML.

c) Show that the Hessian of f_ML may be dense, even when its matrix factors are sparse. Do this by considering C = (IIe_n) and y = yˆ = e_2n+1 where I is the identity matrix, and ek is a vector of ones of size k. Show that every element of the Hessian is nonzero.

d) The purpose of this problem is to write a program in MATLAB to solve a PET image reconstruction problem. Choose the algorithm that you think will be most efficient to solve this case, based on the information that you have up to this point.

(i) Write a MATLAB script of your chosen solution method and test it on a problem with n = 9 variables corresponding to a 3 x 3 grid, and with N = 33 detector pairs. The data are C = (B B B), where B is a sparse nx(n + 2) matrix with the following nonzero entries:

B_i,i = a, B_i,i+1 = b, B_i,i+2 = a, i = 1, ... , n,  where a = 0.18, b = 0.017, and

y^T =( 0 0 1 19 27 30 40 50 35 15 1 0 0 1 7 20 38 56 55 38 20 7 1 0 1 3 17 38 40 20 7 1 0).

(ii) Test your software on a problem with n = 1080 variables corresponding to a 36 x 30 grid, and with N = 2164 detector pairs. The data are C = (B 2B), where B is defined as in part (i) with the parameter values a = 0.15 and b = 0.05. The vector y can be downloaded in text format from the Web page for this book. Display the values of the first row of the reconstructed image.

(iii) Identify the image you obtained in (ii). You will need software for displaying intensity images.

Attachment:- Case study.rar