~~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 non-zero scale of the image projection matrix ''Q'' (3×4) into into the camera projection center ''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 $\delta$. 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(np.eye(3),np.zeros([3,1]),'k','d')''  |

should plot the $\delta$ system. The function should label each base vector (e.g. $\delta_x$, $\delta_y$, $\delta_z$). The automatic evaluation just verifies if the function with this name is present in your code.

==== Steps ====
  - Decompose the optimal scaled image projection matrix ''Q'' you have recovered in HW-02. Let the horizontal pixel size be 5 μm. Compute ''f'' (in metres) and compose the image projection matrix ''Pb'' ($\mathtt{P}_\beta$) using ''K'', ''R'', ''C'', and ''f''.
  - For the camera, compute bases and centres of coordinate systems $(o, \alpha)$, $(C, \beta)$, $(C, \gamma)$, $(O, \delta)$, $(C, \epsilon)$, $(O, \kappa)$, $(C, \nu)$ (see Figure 7.2 in {{https://cw.fel.cvut.cz/wiki/_media/courses/gvg/pajdla-gvg-lecture-2021.pdf|GVG Materials}}). Express all bases in basis $\delta$ and centres in the world coordinate system $(O, \delta)$. 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 the following plots, multiply the first vectors of bases $\alpha$, $\beta$  by image width (1100) and the second vectors of bases $\alpha$, $\beta$ by image height (850).
  - Draw the coordinate systems with bases $\delta$ (black), $\epsilon$ (<fc #FF00FF>magenta</fc>), $\kappa$ (<fc #800000>brown</fc>), $\nu$ (<fc #00FFFF>cyan</fc>), draw the system with basis $\beta$ (<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. $\delta_x$, $\delta_y$, $\delta_z$). Export as ''03_figure1.pdf''.
  - Draw the coordinate systems with bases $\alpha$ (<fc #00FF00>green</fc>), $\beta$ (<fc #FF0000>red</fc>), $\gamma$ (<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 with bases $\delta$ (black), $\epsilon$ (<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 with basis $\epsilon$ 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>