Announcements Problem Set 2, handed out today, due next Tuesday. Late Homework should be turned into my office with date and time written on it. Mail problem sets in one zipped file to farrell (at) cs.umd.edu. When ing matlab code to Ryan, use subject: CMSC 426 Matlab code.

Matlab tutorial and Linear Algebra Review Today’s goals: Learn enough matlab to get started. Review some basics of Linear Algebra Essential for geometry of points and lines. But also, all math is linear algebra. (ok slight exaggeration). Many slides today adapted from Octavia Camps, Penn State.

Starting Matlab For PCs, Matlab should be a program. For Sun’s: Numerical Analysis and Visualization Matlab 6.1

Help help help command Eg., help plus Help on toolbar demo Tutorial: /matlab/

Matlab interpreter Many common functions: see help ops

Vectors Ordered set of numbers: (1,2,3,4) Example: (x,y,z) coordinates of pt in space.

Indexing into vectors

Vector Addition v w V+w

Scalar Product v av

Operations on vectors sum max, min, mean, sort, … Pointwise:.^

Inner (dot) Product v w  The inner product is a SCALAR!

How do we prove these properties of the inner product? Let’s start with the fact that orthogonal vectors have 0 inner product. Suppose one vector is (x,y), and WLOG x,y>0. Then, if we rotate that by 90 degrees counterclockwise, we’ll get (y, -x). Rotating the vector is just like rotating the coordinate system in the opposite direction. And (x,y)*(y,-x) = xy – yx = 0. Next, note that v*w = (v*w)/(||v||||w||) * ||v||||w|| This means that if we can show that when v and w are unit vectors v*w = cos alpha, then it will follow that in general v*w = ||v|| ||w|| cos alpha. So suppose v and w are unit vectors. Next, note that if w1 + w2 = w, then v*w = v*(w1+w2) = v*w1 + v*w2. For any w, we can write it as the sum of w1+w2, where w1 is perpendicular to v, and w2 is in the same direction as v. So v*w1 = 0. v*w2 = ||w2||, since v*w2/||w2|| = 1. Then, if we just draw a picture, we can see that cos alpha = ||w2|| = v*w2 = v*w.

Matrices Sum: A and B must have the same dimensions

Matrices Product: A and B must have compatible dimensions Identity Matrix:

Matrices Transpose: If A is symmetric

Matrices Determinant: A must be square

Matrices Inverse: A must be square

Indexing into matrices

Euclidean transformations

2D Translation t P P’

2D Translation Equation P x y txtxtxtx tytytyty P’ t

2D Translation using Matrices P x y txtxtxtx tytytyty P’ t tP

Scaling P P’

Scaling Equation P x y s.x P’ s.y

Rotation P P’

Rotation Equations Counter-clockwise rotation by an angle  P x Y’ P’  X’ y

Why is this true? (cos theta, - sin theta) is what we get if we rotate (1,0) clockwise by theta. So it is just the x axis, rotated by theta. Similarly, (sin theta, cos theta) is the y axis rotated clockwise by theta. Then, x’ is the inner product of the rotated x axis with (x,y). So it is just the x coordinate of the point according to the new x axis. And similarly, y’ is the coordinate of (x,y) relative to the new y axis.

Degrees of Freedom R is 2x2 4 elements BUT! There is only 1 degree of freedom:  The 4 elements must satisfy the following constraints:

Stretching Equation P x y S x.x P’ S y.y

Stretching = tilting and projecting (with weak perspective)

Linear Transformation SVD

Affine Transformation

Files Matlab

Functions Format: function o = test(x,y) Name function and file the same. Only first function in file is visible outside the file. Look at sample function

Images Black and white image is a 2D matrix. Intensities represented as pixels. Color images are 3D matrix, RBG. Matlab

Debugging Add print statements to function by leaving off ; keyboard debug and breakpoint

Conclusions Quick tour of matlab, you should teach yourself the rest. We’ll give hints in problem sets. The inner product will be very important.