cs249 lecture notes, Fall 2001
Week 10, Monday

Assignment for next time: draw a sketch of a blueprint
for a self-similar figure (as in Figure 5.22) and compute
the values of e, f, phi, psi, r and s for each lens system.


Dot products
------------

In linear algebra, the dot product of two vectors is
a scalar, which is the sum of the products of the
corresponding elements of the vectors.

The operation is only defined if the vectors have
the same length.

MATLAB complicates this picture a litte by distinguishing
between row vectors and column vectors.

The vectors we have seen so far are row vectors:

>> a = [1 2]

a =

     1     2

>> b = [3 4]'

b =

     3
     4

>> a * b

ans =

    11

The apostrophe ' is the transpose operator.  It converts
a row vector to a column vector or back.

Why doesn't this work?

>> a * a
??? Error using ==> *
Inner matrix dimensions must agree.


Why DOES this work?

>> b * a

ans =

     3     6
     4     8



Matrices
--------

A matrix is a bunch of numbers in a box.

In MATLAB, you can create a matrix the same way
you create vectors, with semi-colons between the
rows:

>> a = [1 2; 3 4]

a =

     1     2
     3     4

Also, the ones command creates a matrix with the
given number of rows and columns

>> ones(2, 3)

ans =

     1     1     1
     1     1     1

I used matrices to write drawLine and triangle 

    function drawLine (v1, v2)
    z = [v1;v2];
    line (z(:,1), z(:,2))

    function triangle (v1, v2, v3, color)
    z = [v1;v2;v3];
    fill (z(:,1), z(:,2), color)


Matrix-vector multiplication
----------------------------

Multiplying a matrix by a vector yields a vector:

>> a = [1 2; 3 4]

a =

     1     2
     3     4

>> a * [1 1]'

ans =

     3
     7

Again, the dimensions have to agree and the orientation of
the vector has to be right.

>> a * [1 1]
??? Error using ==> *
Inner matrix dimensions must agree.

You can keep all this stright by drawing a picture.


Matrix operators
----------------

You can think of a matrix as an operator on vectors.

Multiplying by a matrix transforms one vector into
another.

In particular, there are special matrices that have
an identifiable effect on vectors.

Example, the rotation matrix R(theta) rotates a vector
counterclockwise by theta degrees

                cos (theta)    -sin (theta)
   R(theta) =   sin(theta)    cos (theta)

In MATLAB, we can write

function fern ()
v = [1 1]';
r = rotate (pi/4)
r * v

function r = rotate (theta)
r = [cos(theta) -sin(theta); sin(theta) cos(theta)];

And get:

r =

    0.7071   -0.7071
    0.7071    0.7071


ans =

    0.0000
    1.4142

Draw a picture and convince yourself that it is doing the
right thing.

What is the visual effect of multiplying a vector by a scalar?

What is the visual effect of adding two vectors?


Affine transformation
---------------------

Affine transformation is a generalization of rotation than
includes the possibility of flipping and scaling.

                r cos (phi)   s -sin (psi)
   R(theta) =   r sin (phi)   s  cos (psi)

r controls the scaling in the x direction
s controls the scaling in the y direction

if psi = phi, then it rotates angle phi counterclockwise

if phi = pi and psi = 0, it reflects around the y axis


Write a MATLAB function that computes the matrix for
an affine transformation described by parameters r, s, 
phi and psi.


Barnsley's fern
---------------

Barnsley described a fern in terms of a set of transformations
(see handout)

The idea is that if something is self similar, we can describe
it entirely just by enumerating the _ways_ in which it is
self-similar!

Potentially a very dense form of image encoding.

Blueprints: diagram showing the initial image and each transformation
	    of it

Lens: a transformation defined by translation (e, f),
      rotation (phi, psi) and scaling (r, s).

MRCM: multiple reduction copy machine

Next time: draw that fern!