~~NOTOC~~
====== Homework 03 - Perspective Camera Pose and Internal Calibration  ======

==== Implementation ====

  [ K R C ] = Q2KRC( Q );

Create a function ''Q2KRC'' for decomposing a camera matrix ''Q'' (3×4) into into the projection centre ''C'' (3×1), rotation matrix ''R'' (3×3) and upper triangular matrix ''K'' (3×3) such that

''Q'' = λ ( ''K'' ''R'' |  - ''K'' ''R'' ''C'' )

where ''K(3,3) = 1'', ''K(1,1) > 0'', and ''det(R) = 1''.

  plot_csystem( Base, b, color, name );
  
Create a function for drawing a coordinate system base ''Base'' located in the system origin ''b''. The base and origin is expressed in the world coordinate system δ. The base consists of a three or two three-dimensional column vectors. E.g.<code>plot_csystem( eye(3), zeros(3,1), 'k', '\\delta'' )</code> should plot the δ system. The function should label each base vector (e.g. δ_x, δ_y, δ_z). The ''text'' function can be used, e.g.

  text( x, y, z, [ name '_x' ], 'color', color )
  
(here ''x'',''y'',''z'' are coordinates of the end of the first base vector (base x-axis), ''name'' and ''color'' are the input arguments). Note that Matlab can plot greek numbers using TeX sequences ( '''\\alpha''', '''\\beta''', etc.).

==== Steps ====
  - Decompose the optimal camera matrix ''Q'' you have recovered in HW-02. Let the horizontal pixel size be 5 μm. Compute ''f'' (in metres) and compose matrix ''Pb'' (Pβ) using ''K'', ''R'', ''C'', and ''f''.
  - For the camera, compute bases and centres of coordinate systems α, β, γ, δ, ε, κ, υ. Express all bases and centres in the world coordinate system δ. The bases should be stored in matrices ''Alpha'', ''Beta'', ''Gamma'', ''Delta'', ''Epsilon'', ''Kappa'', ''Nu'', respectively, the coordinate system centres should be stored in matrices ''a'', ''b'', ''g'', ''d'', ''e'', ''k'', ''n'', respectively. 
  - Save ''Pb'', ''f'', all bases and coordinate system centres into ''03_bases.mat''.<code>save( 'bases.mat', 'Pb', 'f', 'Alpha', 'a', 'Beta', 'b', 'Gamma', 'g', ...
      'Delta', 'd', 'Epsilon', 'e', 'Kappa', 'k', 'Nu', 'n', '-v6' );</code>
  - For following plots, multiply the first vectors of bases α, β  by image width (1100) and the second vectors of bases α, β by image height (850).
  - Draw the coordinate systems δ (black), ε (<fc #FF00FF>magenta</fc>), κ (<fc #800000>brown</fc>), υ (<fc #00FFFF>cyan</fc>), draw the system β (<fc #FF0000>red</fc>) with its base scaled-up 50 times additionally , and draw the 3D scene points (109 points, <fc #0000FF>blue</fc>). Label each base vector (e.g. δ_x, δ_y, δ_z). Export as ''03_figure1.pdf''.
  - Draw the coordinate systems  α (<fc #00FF00>green</fc>), β (<fc #FF0000>red</fc>), γ (<fc #0000FF>blue</fc>), draw the image points (109 points, <fc #0000FF>blue</fc>). Label each base vector. Export as ''03_figure2.pdf''.
  - Draw the coordinate systems δ (black), ε (<fc #FF00FF>magenta</fc>), plot the 3D scene points (<fc #0000FF>blue</fc>), and plot centers (<fc #FF0000>red</fc>) of all cameras you have tested in HW-02 (using the decomposition). Zoom-in such that the coordinate systems are clearly visible. Export as ''03_figure3.pdf''. Note that the coordinate system ε is for the optimal camera only.

==== Upload ====

Upload an archive containing the following files:
  - ''03_bases.mat''
  - ''03_figure1.pdf'', ''03_figure2.pdf'', ''03_figure3.pdf''
  - ''Q2KRC.m'', ''plot_csystem.m''
  - ''hw03.m'' – your Matlab implementation entry point; it makes all required figures, output files and prints. 
  - any other files required by hw03.m. 

Note: <fc #FF0000>The required files must be in the root directory of the archive.</fc>