Notes

On Spherical Harmonics

1. Spherical Harmonics Fitting

   • Developed a 7-page note which serves as an accompanying note for the MATLAB toolbox Spherical Harmonics Fitting.
   • The note covers both the technical and practical aspects of spherical harmonics fitting.
     • The technical aspect includes three different formulations of the least squares problem associated with spherical harmonics fitting and the derivation of a regularizer.
     • The practical aspect includes line-by-line explanation of the usage of the toolbox using a simple example of reconstructing a tensor-valued orientation distribution function (ODF).
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On Statistical Estimation

1. Score Matching

   • Developed a 6-page note that provides an in-depth explanation of score matching.
   • The note provides more detailed explanations of the theorems presented in the paper with elementary but more rigorous proofs, which include:
     • Theorem 1: Simplification of the objective function.
     • Theorem 2: Well-definedness of the estimator obtained by minimization of the exact objective function.
     • Corollary 3: Consistency of the estimator obtained by minimization of the approximated objective function based on sampling.
   • The note also includes explanation of the motivation behind score matching and clarification of some of the mild yet essential regularity assumptions omitted by the author.
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On Ill-posed Inverse Problems

1. Plug-and-Play ADMM using MATLAB and PyTorch

   • Developed a 6-page tutorial on how to implement the Plug-and-Play ADMM algorithm using MATLAB and PyTorch.
   • MATLAB is used to for solving the regularized least-squares problem associated with the first step.
   • PyTorch is used to export a pre-trained denoiser model, which is plugged into the ADMM algorithm.
   • Besides providing a step-by-step guide on how to implement the algorithm, the aim of this tutorial is three-fold:
     1. To explain one of the caveats of running python scripts in MATLAB.
     2. To compare the performance and convergence property of lsqr for the original problem and pcg for the corresponding normal problem.
     3. To provide a template for handling variable passing between MATLAB and PyTorch via ProtoBuf.
   • In addition, the tutorial also provides a brief introduction and problem formulation of the Plug-and-Play ADMM algorithm from a mathematical perspective.
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On Medical Image Registration

1. Converting ANTs affine matrix to a $4 \times 4$ homogeneous matrix

   • Developed a 4-page note that demonstrates two methods of converting an affine matrix obtained from ANTs to a $4 \times 4$ homogeneous matrix, which can be subsequently incorporated into a pipeline dealing with both image volumes and surfaces.
   • The aim of this note is to provide a clear and concise explanation of the mathematical concepts behind this seemingly simple yet sometimes frustrating procedure, to facilitate the understanding of the relationship between LPS and RAS spaces, and to provide a practical example of how to perform the conversion.
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2. Deformable Image Registration

   • Developed a 2-page note that clarifies notations and concepts that are frequently abused and confused in the field of deformable image registration.
   • The note also provide definitions of important terms and their mathematical formulations.
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On Real Analysis

1. Review Notes for MATH 522: Advanced Calculus II

   • Developed a 40-page note that summarizes the materials covered in MATH 522: Advanced Calculus II.
   • Topics covered in this note include:
     • Metric spaces
     • Jordan content and Riemann integration
     • Differential forms
     • Surfaces
     • Functional analysis
     • ODE theory
     • Lebesgue measure and integration
   • The aim of this note is to organize the materials in a way that is easy to understand and to provide a quick reference when studying for midterms and finals.
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