Knowledge and Thinking
Homogenous coordinates
We can use a function ( ax+by+c = 0 ) to describe a line and ((x,y)) is a point in the line. so the three parameters a,b,c are enough to describe a line. ( l = left(egin{array}{l} a \ b \ c end{array} ight) ) but the scalar fator play a not important role in the function. It means ( l^prime = fl ) . If we ignore the scalar, we can see that ( l = left(egin{array}{l} x \ y \ 1 end{array} ight) ), which mean that the line will be not changed by multiplying a factor except 0 . We can see that ( x = lambda y ), x and y are equivilent as the script. We can easliy add an array to construct a homogeneous vector. ( left(egin{array}{l} x \ yend{array} ight) Rightarrow left(egin{array}{l} x \ y \ 1 end{array} ight) )
A significant advantage of homogenous coordinates is that we cannot operate translation by a matrix multiplication without homogenous coordinates. e.g. ( egin{pmatrix} 1 & 0 & 0\ 0 & 2 & 0\ 0 & 0 &3end{pmatrix} egin{pmatrix}x \ y\ zend{pmatrix} = egin{pmatrix}x \ 2y \ 3z end{pmatrix} ) The result can never be such as ( egin{pmatrix} x \ y+2 \ z end{pmatrix} ). If we use homogenous coordinates ( mathbb {R}^3 Rightarrow mathbb{R}^4 ) the scalar and translation transform can be realized by using matrix multiplication. (egin{pmatrix}1 & 0 & 0 & 0 \0 & 1 & 0 & 2 \0 & 0 & 0 & 1 \ 0 & 0&0&1end{pmatrix} egin{pmatrix} x\y\z \ 1 end{pmatrix} = egin{pmatrix}x \ y+2 \ z \ 1 end{pmatrix} )
Euclidean Movement
Any movement of a point in the euclidean space can be expressed by a rotation and a translation as in figure1.
(a) (b)
(c)
Figure 1 (a) pure translation of a polygon (b) pure Rotation of a polygon.
(c) an Affine transformation contains a translation and a rotation
The rotation matrix R should be an orthogonal matrix and ( R^TR = I ) ( det(R) = 1 ) For example a commonly used rotation in ( mathbb{R^2} ) is ( egin{pmatrix} cos heta & sin heta \ -sin heta & cos heta end{pmatrix} ) and any rotation in ( mathbb{R^2} ) can be realized by modifying this matrix. such as multiply a scalar.
An Euclidean Movement in homogeous coordinates is ( P = RP+T ). It can be expressed by a matrix ( egin{bmatrix} R & t \ 1 & 0 end{bmatrix} ) and a movement in euclidean space is (P_2 = MP_1)。
Cross product is a widely used calculation in computer vision. The difference from dot product is that, cross product brings a new vector which perpendicular to the original vectors. A 3D corrdinates can be produced by cross product.
calibration matrix of the pinhole camera
The procedure of conversion from a 3D point to a 2D point can be divided into 3 steps.
First of all, in homogenous coordinates, the point is from ( mathbb P^4 ) to( mathbb P^3 ) (in euclidean space is ( mathbb R^3 ) to ( mathbb R^2 ) ), it is a linear transform. we need a matrix like ( egin{pmatrix} 1 & 0 & 0 & 0\ 0 &1 & 0 & 0\ 0 & 0 & 1 &0 end{pmatrix} )
Second step, the 3D point can be transformed into 2D point by using the pinhole camera model regard to (f) focal length. As the previous script, ( egin{bmatrix} x\ y end{bmatrix} = frac{f}{Z}egin{bmatrix}X \ Y end{bmatrix} ) ( Zegin{bmatrix} x\ y\1 end{bmatrix} = egin{bmatrix} X\Y\Z end{bmatrix} )The matrix is ( egin{pmatrix}f & 0 & 0\0 & f & 0 \0 & 0 & 1end{pmatrix} )
Third step, In the 3D space the unit of a coordinates is mm. butthe coordinates of a point in a image is in pixel . We should multiply a scalar in pixels/mm and a skew parameter because there exist a angel between thef two coordinates, the correction matrix is ( egin{pmatrix} s_{x} & s_{ heta} & o_{x} \ 0 & s_{y} & o_{y} \ 0 & 0 & 1 end{pmatrix} )
The final projection should be ( underbrace{left[egin{array}{ccc} s_{x} & s_{ heta} & o_{x} \ 0 & s_{y} & o_{y} \ 0 & 0 & 1 end{array} ight]left[egin{array}{ccc} f & 0 & 0 \ 0 & f & 0 \ 0 & 0 & 1 end{array} ight]}_{:=K} underbrace{left[egin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 end{array} ight]}_{Pi_{0}}left[egin{array}{c} X \ Y \ Z \ 1 end{array} ight] )