Images rotation and standards (Perspective)

The original pose of a camera (without any translation or rotation) is given in the PATB convention. In this configuration the image point of view (optical center) is located on the scene origin, the view direction is oriented toward -Z axis, the top direction of the image is given by the Y axis and the right direction of the image given by the X axis.

This basic orientation is different from the image inner pixel axis regarding the Y axis. Other basic orientations exist and each image acquisition device or software giving external parameter is susceptible to use a different basic orientation. When applying to the image a rotation given within a particular basic orientation, a first rotation must be applied to switch from the PATB convention basic orientation to the considered basic orientation.

Photogrammetry angles (ω φ κ)

The main image orientation process consists in providing for each image to texture, three photogrammetric angles that will define the image orientation in the scene. The supported rotation convention is the Standard Omega Phi Kappa (Euler Angles)

In this rotation configuration the rotation are performed in the Order X => Y => Z.

The rotations values are given by omega, phi and kappa Ox(ω), Oy(φ), Oz(κ).

The rotation matrix corresponding to this rotation is computed as follows:

Valid camera internal parameters

When loading camera internal parameters in the command, a check is done to ensure that the parameters match the current image definition. It means the following assertion must be satisfied:

• Image Width = Sensor Width x pixel size

• Image Height = Sensor Height x pixel size

Perspective projection model

Classical perspective projection using a standard pinhole camera model coupled with lens radial and tangential distortions are expressed as follows:

Knowing a 3D Point P located at the coordinates [X Y Z] in the world coordinate system, the first step is to compute the point P coordinates in the camera coordinate system. This point P in the camera coordinates system can be expressed as P' of coordinates [X' Y' Z'] such as P' = R P + t. With R the rotation matrix and t the camera position in the world coordinates as expressed in the camera external parameters.

The homogeneous coordinates of the point P' are computed to obtain the point p = [X' / Z' Y' / Z' Z' / Z'] = [x y 1]

This point p is then shifted according to the principal point of symmetry (PPS [x_p y_p]) location such as x' = x - x_p and y' = y-y_p

Using these centered coordinates the computation of the distortions is possible.

Radial distortion:

Δx_r = x' (1.0 - k₀ + k₁r² + k ₂ r⁴ + k₃r⁶ + k₄r⁸ + k₅r¹⁰ + k₆r¹²)

Δy_r = y' (1.0 - k₀ + k₁r² + k₂r⁴ + k₃r⁶ + k₄r⁸ + k₅r¹⁰ + k₆r¹²)

Tangential distortion:

Δx_d = p₁ (r² + 2x'²) + 2 p₂ x' y'

Δy_d = p₂ (r² + 2y'²) + 2 p₁ x' y'

The distorted point coordinates are simply:

x_c = Δx_r + Δx_d

y_c = Δy_r + Δy_d

From these distorted coordinates, the final UV coordinates are obtained as:

U = f x_c + ( PPA _x / S_W )

V = f y_c + ( PPA_y / S_H )

Images rotation and standards (Spherical)

The original pose of a spherical image (without any translation or rotation) is the following. The spherical image center is located on the scene origin, the center of the spherical image is corresponds to the X axis, the top direction of the image is given by the -Z axis and the right direction of the image given by the Y axis.

Photogrammetry angles (ω φ κ)

The main image orientation process consists in providing for each image to texture, three photogrammetric angles that will define the image orientation in the scene. The supported rotation convention is the Standard Omega Phi Kappa (Euler Angles)

In this rotation configuration the rotation are performed in the Order X => Y => Z.

The rotations values are given by omega, phi and kappa Ox(ω), Oy(φ), Oz(κ).

The rotation matrix corresponding to this rotation is computed as follows: