~~NOTOC~~

====== Homework 01 - Image Coordinate System, Affine Transformation ======

==== Input Data ====

Find your data in the assignment '00data: Your data' in the submission system; the image ''daliborka_01.jpg'' and the set of seven 2D points ''U2''.

{{http://cw.felk.cvut.cz/courses/GVG/2014/HomeWorks/daliborka/daliborka_01.jpg?direct&200|daliborka_01.jpg}}

==== Task ====

Make Matlab script ''hw01.m'' to do the following. 

  - Load ''daliborka_01.jpg'' image into your Matlab workspace and display it. Use ''subfig.m'' utility in [[http://cw.felk.cvut.cz/courses/GVG/2017/HomeWorks/Tools|Tools repository]]. <code>img = imread( 'daliborka_01.jpg' ); % load the image
subfig(2,2,1); % create a new figure
image( img ); % display the image, keep axes visible
axis image % display it with square pixels</code> 
  - Manually acquire the set of seven points in the image corresponding to the tops of the five chimneys and the two towers. The points must be ordered **from left to right**. Use <code>[x,y] = ginput(7)</code>
  - Store the point coordinates in a 2×7 matrix ''u''. Every point is represented by a **column** vector of coordinates, i.e. ''u(:,1)'' are coordinates of the first point.
  - In the bitmap image, colorize the pixels that are nearest to the acquired points (use the ''round'' function to convert fractional coordinates into integer coordinates). Use the following colors in the following order: **<fc #FF0000>red = [255 0 0]</fc>, <fc #00FF00>green = [0 255 0]</fc>, <fc #0000FF>blue = [0 0 255]</fc>, <fc #FF00FF>magenta = [255 0 255]</fc>, <fc #00FFFF>cyan = [0 255 255]</fc>, <fc #FFDD00>yellow = [255 255 0]</fc>, white = [255 255 255]** to colorize the respective seven points. The colors are defined by their R, G, B values: **color = [R G B]**. The order of colors must correspond to the order of points in the matrix ''u''. Store the modified bitmap image as a file ''01_daliborka_points.png''.<code>imwrite(img, '01_daliborka_points.png');</code>
  - Implement estimation of the affine transformation ''A'' (2×3 matrix) from n given points ''u2'' to points ''u'' as a function <code>A = estimate_A( u2, u ); % u2 and u are 2xn matrices</code>
    - Perform all possible selections of 3 point correspondences from all n correspondences. Use e.g.<code>nchoosek(1:n, 3 )</code>
    - For each triple compute the affine transformation ''Ai'' (exactly).
    - Compute transfer errors of the particular transformation for every correspondence, i.e., the euclidean distances between points ''U'' and points ''U2'' transferred by ''A''. Find the maximum error over the correspondences.
    - From the all computed transformations ''Ai'' select the one ''A'' that has the maximum transfer error minimal.
    - Assume general number of points, though we have only 7 points and 35 possibilities<code>n = size(u,2); % number of columns</code>
  - Display the modified image, the set of points ''u'' using the same colors as above, and 100× magnified transfer errors for the best ''A'' as red lines.<code>% assume we have points in ''u'' and points ''u2'' transferred by ''A'' in ''ux''
e = 100 * ( ux - u ); % magnified error displacements
...
hold on % to plot over the image
...
plot( u(1,4), u(2,4), 'o', 'linewidth', 2, 'color', 'magenta' ) % the 4-th point 
plot( [ u(1,4) u(1,4)+e(1,4) ], [ u(2,4) u(2,4)+e(2,4) ], 'r-', 'linewidth', 2 ); % the 4-th error
...
hold off</code> (If the errors are too large, it is optional to plot them magnified only 10×, as blue lines).
  - Export (print) the plot as a pdf file ''01_daliborka_errs.pdf''. Use ''fig2pdf.m'' utility in [[http://cw.felk.cvut.cz/courses/GVG/2017/Tools|Tools repository]].<code>fig2pdf( gcf, '01_daliborka_errs.pdf' )</code>
  - Store the points and the matrix in a file ''01_points.mat''.<code>save( '01_points.mat', 'u', 'A' )</code>

^  Example of a points and transfer errors  ^
|  {{courses:gvg:labs:01_daliborka_points.png|}}  |


==== Upload ====

Upload an archive containing the following files:
  - ''01_points.mat''
  - ''01_daliborka_points.png''
  - ''01_daliborka_errs.pdf''
  - ''estimate_A.m''
  - ''hw01.m'' - your Matlab implementation. It makes all required figures, output files and prints. 
  - any other your files required by hw01.m.  

Note: <fc #FF0000>All files must be in the same directory. Do not use any subdirectories.</fc>