courses:tdv:labs:toolbox [CourseWare Wiki]

Toolbox of Elementary and Helper Functions

Following functions should be implemented. It is important to keep the specified calling convention (order of arguments, shape of matrices, etc).

Note: vectors of coordinates (e.g., points) should be always columns . Thus a set of 5 euclidean points in a plane is written as a matrix with 2 rows and 5 columns (i.e., 2 x 5).

e2p — Transformation of euclidean to projective coordinates

Synopsis	`u_p = e2p( u_e )`
Inputs	`u_e`	d by n matrix; n euclidean vectors of dimension d
Outputs	`u_p`	d+1 by n matrix; n homogeneous vectors of dimension d+1

p2e — Transformation of projective to euclidean coordinates

Synopsis	`u_e = p2e( u_p )`
Inputs	`u_p`	d+1 by n matrix; n homogeneous vectors of dimension d+1
Outputs	`u_e`	d by n matrix; n euclidean vectors of dimension d

>> u = e2p( [[1;2] [3;4] [5;6]] )

u =
     1     3     5
     2     4     6
     1     1     1

>> p2e( u )

ans =
     1     3     5
     2     4     6

vlen — Column vectors length

Synopsis	`l = vlen( x )`
Inputs	`x`	d by n matrix; n vectors of dimension d
Outputs	`l`	1 by n row vector; euclidean lengths of the vectors

>> vlen( [ [0.5;0.5;1] [3;4;0]])

ans =
    1.2247    5.0000

sqc — Skew-symmetric matrix for cross-product

Synopsis	`S = sqc(x)`
Inputs	`x`	vector 3×1
Outputs	`S`	skew symmetric matrix (3×3) for cross product with `x`

Notes: The vector x characterises the left as well as the right null space of the matrix S. For any vector y, cross product x×y equals to S y.

>> sqc( [1;2;3] )

ans =
   0    -3     2
   3     0    -1
  -2     1     0

EutoRb — Essential matrix decomposition with cheirality

This is obsolete, create EutoRt instead.

Synopsis	`[R, b] = EutoRb( E, u1, u2 )`
Inputs	`E`	essential matrix (3×3)
	`u1`, `u2`	corresponding image points in homogeneous coordinates (3×n), used for cheirality test
Outputs	`R`	relative rotation (3×3) or [] if cheirality fails
	`b`	relative translation, euclidean (3×1), unit length

Notes: The sessential matrix E is decomposed such that E = R * sqc( b ).

EutoRt — Essential matrix decomposition with cheirality

Synopsis	`[R, t] = EutoRt( E, u1, u2 )`
Inputs	`E`	essential matrix (3×3)
	`u1`, `u2`	corresponding image points in homogeneous coordinates (3×n), used for cheirality test
Outputs	`R`	relative rotation (3×3) or [] if cheirality fails
	`t`	relative translation, euclidean (3×1), unit length

Notes: The sessential matrix E is decomposed such that E ~ sqc(t) * R. Note that t = -R*b.

Pu2X — Binocular reconstruction by DLT triangulation

Synopsis	`X = Pu2X( P1, P2, u1, u2 )`
Inputs	`P1, P2`	projective camera matrices (3×4)
	`u1`, `u2`	corresponding image points in homogeneous coordinates (3×n)
Outputs	`X`	reconstructed 3D points, homogeneous (4×n)

err_F_sampson — Sampson error on epipolar geometry

Synopsis	`err = err_F_sampson( F, u1, u2 )`
Inputs	`F`	fundamental matrix (3×3)
	`u1`, `u2`	corresponding image points in homogeneous coordinates (3×n)
Outputs	`e`	Squared Sampson error for each correspondence (1×n).

u_correct_sampson — Sampson correction of correspondences

Synopsis	`[nu1, nu2] = u_correct_sampson( F, u1, u2 )`
Inputs	`F`	fundamental matrix (3×3)
	`u1`, `u2`	corresponding image points in homogeneous coordinates (3×n)
Outputs	`nu1,nu2`	corrected corresponding points, homog. (3×n).