GeneralizedGaussian.compute_jacobian¶
giant.point_spread_functions.gaussians
:
- GeneralizedGaussian.compute_jacobian(x, y, computed)[source]¶
This method computes the Jacobian of the PSF with respect to a change in the state.
This is used internally for computing the covariance. It returns a 5xn matrix defined as
\[\begin{split}\mathbf{J} = \left[\begin{array}{cccccc} \frac{\partial f}{\partial x_0} & \frac{\partial f}{\partial y_0} & \frac{\partial f}{\partial a} & \frac{\partial f}{\partial b} & \frac{\partial f}{\partial c} & \frac{\partial f}{\partial A} \end{array}\right]=\left[\begin{array}{c} \left(2a(x-x_0)+2b(y-y_0)\right)f(x, y) \\ \left(2c(y-y_0)+2b(x-x_0)\right)f(x, y) \\ -(x-x_0)^2f(x, y) \\ -2(x-x_0)(y-y_0)f(x, y) \\ -(y-y_0)^2f(x, y) \\ \frac{f(x, y)}{A} \end{array}\right]^T\end{split}\]- Parameters:
x (ndarray) – The x values to evaluate the Jacobian at as a length m array
y (ndarray) – The y values to evaluate the Jacobian at as a length m array
computed (ndarray) – The PSF evaluated at x and y as a length m array
- Returns:
The Jacobian matrix as a 5xm numpy array