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Introduction to MATLAB and MATLAB Programming

Introduction to MATLAB and MATLAB Programming. #1. Outline. Who are we and why are we here? (to paraphrase James Stockdale, 1992) Who am I and how did I get here? Who are you and why are you here? MATLAB. Started programming (BASIC) in 1981

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Introduction to MATLAB and MATLAB Programming

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  1. Introduction to MATLAB and MATLAB Programming #1

  2. Outline • Who are we and why are we here? (to paraphrase James Stockdale, 1992) • Who am I and how did I get here? • Who are you and why are you here? • MATLAB

  3. Started programming (BASIC) in 1981 • Started imaging algorithm and software development (Pascal) in 1983 • Cambridge Instruments Quantimet 900 • 1 MB memory • Booted from 8” floppy disk • $100,000

  4. BCC ASET (1983), AA (1986) • BASIC, Pascal, FORTRAN, x86 Assembler • TSU BSEE (1986) • NCSU MSEE (1988) • MATLAB • C/C++ • APL, ProLog • NCSU PhD (1996)

  5. 3D ICUS • 3D Intracoronary Ultrasound • Lumen and medial-adventitial boundary estimation

  6. 3D Imaging Geometry

  7. TIPS

  8. Surgical Planning

  9. MRA Vessel Extraction

  10. Objectives • Develop a basic working knowledge of MATLAB (notation, syntax, built-in functions) • Learn to use the command line (MATLAB as a calculator) • Learn to develop scripts and functions • Learn basic programming skills • Learn to use the debugger to troubleshoot scripts and functions

  11. Objectives • Generate 2D and 3D data plots • Read, write, and manipulate numeric data (e.g. text files, binary files, Excel) • Read, write, manipulate, and display image data • Learn the basics of linear algebra • Learn about the FFT and complex variables

  12. What is MATLAB? • MATLAB - Matrix Laboratory • What is a matrix? • Fundamental data type in MATLAB • An image • What is an image? • A rectangular array of numbers • A point in a high-dimensional space • A representation of a system of equations

  13. What is MATLAB? • What is a matrix? • A linear mapping from Rnto Rm (or Cnto Cm)** • What’s a linear mapping? • Lines map to lines. Origin maps to the origin. • A collection of vectors • What is a vector? • Magnitude and direction (velocity, force, etc.) • An n-tuple of numbers

  14. What is MATLAB? • History • Netlib(netlib.org) • Large software library written in FORTRAN (FORmulaTRANslator) • Cleve Moler (EISPACK, LINPACK, cofounder of MathWorks) • Computing environment for technical computation, visualization, design, simulation, and implementation in a wide range of application areas.

  15. Common MATLAB Applications • Numerical Methods • Linear algebraic equations • Roots and optimization • Curve fitting • Integration and differentiation • Differential equations • MATLAB toolboxes • Statistics, optimization, image processing, computer vision, bioinformatics, …

  16. Why MATLAB? • Pros: • Easy to start using MATLAB with little or no prior programming experience • Relatively easy to learn • Can be used as a calculator or as a programming language • Can be used to test simple algorithms (command line or scripts) prior to coding in a higher-level language

  17. Why MATLAB? • Pros: • MATLAB has a large library of optimized, robust mathematical functions • MATLAB has a large collection of very powerful toolboxes (e.g. image processing, statistics) • Easy to produce 2D and 3D graphs

  18. Why Not MATLAB? • Cons: • Cost associated with ease of use • Interpreted language • Slow (10-100x) compared to some other programming languages (e.g. C, C++, FORTRAN, Pascal) • May be too slow to solve large problems • Fairly expensive (free alternatives available) • Designed for scientific computing • Not always the best solution for other applications

  19. 2-by-2 System of Equations • Can solve graphically • Can add/subtract equations

  20. 2-by-2 System of Equations • Can solve with the cross or outer product • cross ()

  21. n-by-n System of Equations • What about higher dimensions? • Cross product only works in 3D. • Harder (impossible?) to visualize. • Circuit analysis example • Graph theory (non-planar) and linear algebra • 6-by-6 system of equations

  22. Matrix Operations • The entry (or element) in row i and row j of a matrix, A, is denoted by aij • The entry in row i of a vector (i.e. matrix with only one column) vis denoted by vi.

  23. Matrix Operations • Two matrices are equal if: • They are the same size • All corresponding entries are equal • Two matrices can be added together if they have the same size (i.e. the same number of rows and columns)

  24. Matrix Operations • Multiplication by a scalar • Matrix-vector multiplication • If A is m-by-n and x is n-by-1, b is m-by-1.

  25. Matrix Operations • Matrix-vector multiplication

  26. Matrix Operations • Matrix-matrix multiplication • If A is m-by-r and B is r-by-n, C is m-by-n.

  27. Matrix Operations • Matrix-matrix multiplication • To find cij (i.e. the element of C in row i and column j) multiply corresponding entries in row i of A and column j of B, and add them together.

  28. Matrix Operations

  29. Matrix Operations • Zero matrix • A matrix where each element is 0 • zeros()

  30. Matrix Operations • Identity matrix • Square (i.e.m=n) • Diagonal entries equal to 1; all others are 0. • Multiplication by the identity matrix doesn’t change the matrix • eye()

  31. Row Reduction • Elementary row operations • Multiply a row by a scalar. • Add a scalar multiple of one row to another. • Exchange two rows. • Solve Ax=b • Row reduction with augmented matrix

  32. 2-by-2 cont’d • Back to 2-by-2 example • Rewrite as a matrix equation • Form an augmented matrix

  33. 2-by-2 cont’d • Solve by row reduction • Make left side look like an identity matrix using only elementary row operations

  34. Rank • r=rank(A) • Number of leading 1’s in the row-reduced matrix • From previous example, rank(A)=2

  35. Matrix Inverse • Matrix inverse • A must be square and full rank • If a matrix can be found such that AB=In, then B is called the inverse of A.

  36. 2-by-2 cont’d • Computing the inverse

  37. Matrix Inverse • If A is n-by-n and rank(A)=n, then A-1exists. • Implies there is a solution for every b • b is in the range of A • inv() • If A and B are both invertible,

  38. 2-by-2 cont’d • Inconsistent (no solution) • n=2, Rank=1 • Inverse doesn’t exist

  39. 2-by-2 cont’d • Consistent (infinite solutions) • n=2, Rank=1 • Inverse doesn’t exist

  40. Matrix Transpose • Transpose (‘ in MATLAB) • Swap rows and columns

  41. Scalar Product • Also called inner or dot product • Vectors must have the same length (size) • Multiply corresponding elements and add

  42. Scalar Product • Inner (dot) product • Also written as:

  43. Scalar Product • Properties

  44. Norms • Vector norms • 1-norm • 2-norm (Euclidean norm or length) • Infinity-norm

  45. Norms • Matrix Norm • 1-norm (column sum) • 2-norm – later • Infinity norm (row sum) • Frobenius norm

  46. Norms • Vector norm properties • Matrix norm properties

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