CS 736 : Assignment 3 solution

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5 marks are reserved for submission in the described format. 1. (65 marks) Reconstructing a Phantom Magnetic Resonance Image. Downloadthe2Dnoiselessimageandthesparsely-acquiredkspace-data(i.e.,frequency-domain data) image available at http://www.cse.iitb.ac.in/~suyash/cs736/assignmentImageReconstructionPhantom.mat. zip In the kspace-data image, the center of kspace (i.e., low frequencies) lies at the corners of the image. Pixels where the data isn’t acquired have value zero in the kspace-data image. The kspace-mask image indicates the pixels where the data wasn’t acquired. Implement a maximum-a-posteriori Bayesian image-reconstruction algorithm that uses a sparseacquisition transformation model, a noise model, and a MRF-prior model that uses a 4-neighbor neighborhood system (each pixel has 4 neighbors: left, right, up, down; the neighborhood wraps around at image boundaries) that has cliques of size no more than 2. Usegradientascent(ordescent)optimizationwithdynamicstepsize. Ensurethatthevalueofthe objective function (i.e., the log posterior or its negative) at each iteration increases (or decreases if using gradient descent). Use the inverse Fourier transform of the kspace data as the initial solution. Use 3 different MRF priors where the potential functions V (xi,xj) := g(xi −xj) underlying the MRF penalize the difference between the neighboring pixel values xi,xj as follows (see class Notes at http://www.cse.iitb.ac.in/~suyash/cs736/Notes_AlgoMIP_ImagePrior.pdf for details). You may rely on the circshift() function in Matlab when computing differences between every pixel in the image and its neighbor. Introduce a parameter α ∈ [0,1] to control the weighting between the prior (weight α) and the likelihood (weight 1−α). Specifically, implement the following functionality as part of the reconstruction algorithm: (a) (1 marks) A Complex-Gaussian noise model. You don’t need the noise level because that parameter can be absorbed in 1−α that you’ll tune manually (Tuning α essentially manipulates the noise level, in case of the likelihood. So we can ignore the noise level σ when tuning α manually. Use σ = 1).
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(b) (1 marks) MRF prior: Quadratic function: g1(u) := |u|2. (c) (1 marks) MRF prior: Discontinuity-adaptive Huber function: g2(u) := 0.5|u|2, when|u|≤ γ and g(u) := γ|u|−0.5γ2, when|u| γ, where 0 < γ < ∞is a constant. (d) (1 marks) MRF prior: Discontinuity-adaptive function: g3(u) := γ|u|− γ2 log(1 + |u|/γ), where 0 < γ < ∞is a constant. (e) (16 marks) A sparse-acquisition transformation model that is able to simulate a general sparse MRI acquisition strategy that acquires any subset of pixels in the Fourier domain. For each MRF prior, manually tune the parameters α and γ (where applicable) to reconstruct the noisy image in order to achieve the least possible relative root-mean-squared error (RRMSE). The RRMSE for 2 complex images A and B is defined as : RRMSE(A,B) =qPp(|A(p)|−|B(p)|)2/qPp|A(p)|2, where the summation is over all pixels p. Always use the noiseless image as A. Report the following: (a) (0mark)ReporttheRRMSEbetweentheinverseFourier-kspaceDataandnoiselessimages. (b) (18 marks) Report the optimal values of the parameters and the corresponding RRMSEs for each of the 3 reconstruction algorithms. For each optimal parameter value reported (for each of the 3 reconstruction algorithms), give evidence of the optimality of the reported valuesbyreportingtheRRMSEvaluesfortwonearbyparametervalues(aroundtheoptimal) at plus/minus, say, 20% of the optimal value. That is, if a∗,b∗ are the optimal parameter values, then report: a∗,b∗,RRMSE(a∗,b∗), RRMSE(1.2a∗,b∗),RRMSE(0.8a∗,b∗), RRMSE(a∗,1.2b∗),RRMSE(a∗,0.8b∗). (Tip: the optimal values for α might be very close to extreme marks of the allowed range. Be aware of that possibility.) . (c) (18 marks) Show the following 5 images (at each pixel, show the magnitude of the pixel value) in the report using exactly the same colormap (i) Noiseless image, (ii) inverseFourierkspaceData image, (iii) Image reconstructed using quadratic prior g1(·) and optimal parameter tuning, (iv) Image reconstructed using Huber prior g1(·) and optimal parameter tuning, (v) Image reconstructed using discontinuity-adaptive prior g3(·) and optimal parameter tuning. (d) (9 marks) Show the plots of the objective-function values (vertical axis) versus iteration (horizontal axis) corresponding to each of the 3 reconstructed results in (iii), (iv), and (v) above. 2. (30 marks) Reconstructing a Magnetic Resonance Image of the Brain. Download the kspace-data image available at http://www.cse.iitb.ac.in/~suyash/cs736/ assignmentImageReconstructionBrain.mat.zip In the kspace-data image, the center of kspace (i.e., low frequencies) lies at the corners of the image. Pixels where the data isn’t acquired have value zero in the kspace-data image. The kspace-mask image indicates the pixels where the data wasn’t acquired. 2 Use all 3 maximum-a-posteriori Bayesian reconstruction algorithms implemented to reconstruct the noisy brain image. Manuallytunetheparameterstogivethebestreconstructedimagethat,basedonyourjudgment, gives the right tradeoff between noise/artifact removal and edge preservation. Report the following: (a) (20 marks) Show the following 4 images (at each pixel, show the magnitude of the pixel value) in the report using exactly the same colormap (i) InverseFourier-KspaceData image, (ii) Image reconstructed using quadratic prior g1(·) and manual parameter tuning, (iii) Image reconstructed using Huber prior g1(·) and manual parameter tuning, and (iv) Image reconstructed using discontinuity-adaptive prior g3(·) and manual parameter tuning. (b) (10 marks) Show the plots of the objective-function values (vertical axis) versus iteration (horizontal axis) corresponding to each of the 3 reconstructed results in (ii), (iii), and (iv) above.