===== Image coordinate system =====

The choice of an image coordinate system is a matter of convention in most cases. As introduced in the lectures, we will use the left handed image coordinate system, with the first axis (//u//) pointing right and the second axis (//v//) pointing down. Coordinates of a point are then written as an algebraic **column** vector (euclidean) <latex>(u, v)^\top</latex>. In Matlab, we will write e.g.

  [u;v]

Since //1-based// indexing is used in Matlab, the **centre** of the top left pixel has coordinates ''[1;1]'' and the centre of the bottom right pixel has coordinates ''[count_u; count_v ]'', where ''count_u'' is number of image columns and ''count_v'' is number of image rows (both in pixels). See <imgref image1>.

<imgcaption image1| Image coordinate system>{{:courses:a4m33tdv:cviceni:image_coords.png}}</imgcaption>

When drawing points or lines in Matlab, correct orientation of ordinate must be set.

  plot( u, v );
  set( gca, 'ydir', 'reverse' );
  
It is also possible to draw a bitmap image first, and then draw points or lines over the bitmap. Correct orientation is then set automatically by imaging the bitmap.

  im = imread( 'image.jpg' );
  image( im );
  hold on
  plot( u, v );
  hold off

Matlab considers images as ordinary matrices, so the standard matrix indexation (row, column) is used. Let ''[u;v]'' be the integer coordinates in the grayscale image (centre of a pixel). Then the intensity of the pixel is obtained as

  im(v,u)

===== Matlab Help for Typical Situations in Homeworks ======

=== Some recommendations for the fastest progress ===
  - Do not trust any results unless you verified them.
  - <fc #FF0000>Always verify your results</fc>.
  - Try to visualize your results
  - Each Matlab command has help. E.g. for ''plot''<code>help plot % text version
doc plot % html version</code>
  - Try to encapsulate general tasks into a functions and reuse them.
  - Remind the rule 1.

== Debugging ==

Matlab has a powerful debugger. Either put breakpoints everywhere you want, or use the command ''keyboard'' in your code to stop processing at the place. Exit the debugging mode:

  dbquit
  
You can also start debugging automatically in case of error by turning on
  
  dbstop error

=== Vectors and its transformations ===

<fc #FF0000>Vectors are stored as column matrices.</fc>
  u = [1;2];
  X = [1;2;3];
  Xmore = [1 2 3; 4 5 6]';  % transpose at the end

Vectors are multiplied by a matrices on the left:
  P = [1 0 0 -5; 0 1 0 -6; 0 0 1 1];  % a camera
  ux_homog = P * [X;1]; % projection

=== Special matrices: Identity, Zeros, Ones ===
  I = eye(3); % 3x3 identity
  I2 = diag( [1 1 1] ); % same result
  Z = zeros(3); % 3x3 matrix of zeroes
  O = ones(3); % 3x3 matrix of ones

=== Euclidean distance of two 3D points ===
  X1 = [1;2;3];
  X2 = [4;5;6];
  d2 = norm(X1-X2); % cannot be used for more vectors
  d2 = sqrt(sum((X1-X2).^2)); % for matrices with points stored in columns

=== Dot (scalar) product ===
  u1 = [4;5;1];
  u2 = [7;8;1];
  dot12 = u1' * u2;
  
 === Cross product ===
  u1 = [4;5;1];
  u2 = [7;8;1];
  u12x = cross(u1,u2);
  % verification of orthogonality by dot product 
  % (remember the recommendations 1 and 2)
  u12x' * u1 % should be zero
  u12x' * u2 % should be zero