pycvcam.ZernikeDistortion._transform#

ZernikeDistortion._transform(normalized_points, *, dx=False, dp=False)[source]#

Compute the transformation from the normalized_points to the distorted_points.

Lets consider normalized_points in the camera normalized coordinate system \(\vec{x}_n = (x_n, y_n)\), the corresponding distorted_points in the camera normalized coordinate system are given \(\vec{x}_d\) can be obtained by :

\[x_{d} = x_{n} + \sum_{n=0}^{N_{zer}} \sum_{m=-n}^{n} C^{x}_{n,m} Z_{nm}(\rho, \theta)\]

\[y_{d} = y_{n} + \sum_{n=0}^{N_{zer}} \sum_{m=-n}^{n} C^{y}_{n,m} Z_{nm}(\rho, \theta)\]

The jacobians with respect to the distortion parameters is an array with shape (n_points, 2, n_params), where the last dimension represents the parameters in the order of the class attributes (Cx[0,0], Cy[0,0], Cx[1,-1], Cy[1,-1], Cx[1,1], Cy[1,1], …). The jacobian with respect to the normalized points is an array with shape (n_points, 2, 2).

The derivative of the distorted points with respect to the normalized points is given by:

\[\frac{\partial x_{d}}{\partial x_{n}} = 1 + \sum_{n=0}^{N_{zer}} \sum_{m=-n}^{n} C^{x}_{n,m} \frac{\partial Z_{nm}}{\partial x_{n}}\]

\[\frac{\partial x_{d}}{\partial y_{n}} = \sum_{n=0}^{N_{zer}} \sum_{m=-n}^{n} C^{x}_{n,m} \frac{\partial Z_{nm}}{\partial y_{n}}\]

Where:

\[\frac{\partial Z_{nm}}{\partial x_{n}} = \frac{\partial Z_{nm}}{\partial \rho} \cdot \frac{\partial \rho}{\partial x_{n}} + \frac{\partial Z_{nm}}{\partial \theta} \cdot \frac{\partial \theta}{\partial x_{n}}\]

\[\frac{\partial Z_{nm}}{\partial y_{n}} = \frac{\partial Z_{nm}}{\partial \rho} \cdot \frac{\partial \rho}{\partial y_{n}} + \frac{\partial Z_{nm}}{\partial \theta} \cdot \frac{\partial \theta}{\partial y_{n}}\]

pycvcam.ZernikeDistortion._transform#

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