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

==== Implementation ====

^ matlab ^ python ^
| ''[ K, R, C ] = Q2KRC( Q )'' |  ''K, R, C = hw03.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.

Create a function 'plot_csystem' for drawing a coordinate system with base ''Base'' located in the origin ''b'' with a given ''name'' and ''color''. The base and origin are expressed in the world coordinate system δ. The base consists of a two or three three-dimensional column vectors of coordinates. E.g.

| ''plot_csystem(eye(3),zeros(3,1),'k','\\delta');''  | ''hw03.plot_csystem(ax,np.eye(3),np.zeros([3,1]),'k','d')''  |

should plot the $\delta$ system. The function should label each base vector (e.g. $\delta_x$, $\delta_y$, $\delta_z$).

==== 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''.
  - 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.

|  ''save( '03_bases.mat', 'Pb', 'f', ...''\\ '''Alpha', 'a', ...''\\ '''Beta', 'b',...''\\ '''Gamma', 'g', ...''\\ '''Delta', 'd', ...''\\ '''Epsilon', 'e',...''\\ '''Kappa', 'k', ...''\\ '''Nu', 'n' );'' |  ''sio.savemat( '03_bases.mat', { 'Pb':Pb, 'f':f,''\\ '''Alpha':Alpha, 'a':a,''\\ '''Beta':Beta, 'b':b,''\\ '''Gamma':Gamma, 'g':g,''\\ '''Delta':Delta, 'd':d,''\\ '''Epsilon':Epsilon, 'e':e,''\\ '''Kappa':Kappa, 'k':k,''\\ '''Nu':Nu, 'n':n } )'' |

==== Upload ====

Upload an archive containing the following files:
^ matlab ^ python ^
|  ''03_bases.mat''  ||
|  ''03_figure1.pdf'', ''03_figure2.pdf'', ''03_figure3.pdf''  ||
| ''Q2KRC.m'', ''plot_csystem.m'' | |
| ''hw03.m''  | ''hw03.py'' containing the required functions |
|  any other file used by your solution  ||

The input entry point script ''hw03'' should make all required figures, output files and prints without manual intervention. 

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