Gaussian2D¶
-
class
astropy.modeling.functional_models.
Gaussian2D
(amplitude=1, x_mean=0, y_mean=0, x_stddev=None, y_stddev=None, theta=None, cov_matrix=None, **kwargs)[source] [edit on github]¶ Bases:
astropy.modeling.Fittable2DModel
Two dimensional Gaussian model.
Parameters: - amplitude : float
Amplitude of the Gaussian.
- x_mean : float
Mean of the Gaussian in x.
- y_mean : float
Mean of the Gaussian in y.
- x_stddev : float or None
Standard deviation of the Gaussian in x before rotating by theta. Must be None if a covariance matrix (
cov_matrix
) is provided. If nocov_matrix
is given,None
means the default value (1).- y_stddev : float or None
Standard deviation of the Gaussian in y before rotating by theta. Must be None if a covariance matrix (
cov_matrix
) is provided. If nocov_matrix
is given,None
means the default value (1).- theta : float, optional
Rotation angle in radians. The rotation angle increases counterclockwise. Must be None if a covariance matrix (
cov_matrix
) is provided. If nocov_matrix
is given,None
means the default value (0).- cov_matrix : ndarray, optional
A 2x2 covariance matrix. If specified, overrides the
x_stddev
,y_stddev
, andtheta
defaults.
Other Parameters: - fixed : a dict, optional
A dictionary
{parameter_name: boolean}
of parameters to not be varied during fitting. True means the parameter is held fixed. Alternatively thefixed
property of a parameter may be used.- tied : dict, optional
A dictionary
{parameter_name: callable}
of parameters which are linked to some other parameter. The dictionary values are callables providing the linking relationship. Alternatively thetied
property of a parameter may be used.- bounds : dict, optional
A dictionary
{parameter_name: value}
of lower and upper bounds of parameters. Keys are parameter names. Values are a list or a tuple of length 2 giving the desired range for the parameter. Alternatively, themin
andmax
properties of a parameter may be used.- eqcons : list, optional
A list of functions of length
n
such thateqcons[j](x0,*args) == 0.0
in a successfully optimized problem.- ineqcons : list, optional
A list of functions of length
n
such thatieqcons[j](x0,*args) >= 0.0
is a successfully optimized problem.
See also
Notes
Model formula:
\[f(x, y) = A e^{-a\left(x - x_{0}\right)^{2} -b\left(x - x_{0}\right) \left(y - y_{0}\right) -c\left(y - y_{0}\right)^{2}}\]Using the following definitions:
\[ \begin{align}\begin{aligned}a = \left(\frac{\cos^{2}{\left (\theta \right )}}{2 \sigma_{x}^{2}} + \frac{\sin^{2}{\left (\theta \right )}}{2 \sigma_{y}^{2}}\right)\\b = \left(\frac{\sin{\left (2 \theta \right )}}{2 \sigma_{x}^{2}} - \frac{\sin{\left (2 \theta \right )}}{2 \sigma_{y}^{2}}\right)\\c = \left(\frac{\sin^{2}{\left (\theta \right )}}{2 \sigma_{x}^{2}} + \frac{\cos^{2}{\left (\theta \right )}}{2 \sigma_{y}^{2}}\right)\end{aligned}\end{align} \]- If using a
cov_matrix
, the model is of the form: - \[f(x, y) = A e^{-0.5 \left(\vec{x} - \vec{x}_{0}\right)^{T} \Sigma^{-1} \left(\vec{x} - \vec{x}_{0}\right)}\]
where \(\vec{x} = [x, y]\), \(\vec{x}_{0} = [x_{0}, y_{0}]\), and \(\Sigma\) is the covariance matrix:
\[\begin{split}\Sigma = \left(\begin{array}{ccc} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \end{array}\right)\end{split}\]\(\rho\) is the correlation between
x
andy
, which should be between -1 and +1. Positive correlation corresponds to atheta
in the range 0 to 90 degrees. Negative correlation corresponds to atheta
in the range of 0 to -90 degrees.See [1] for more details about the 2D Gaussian function.
References
[1] (1, 2) https://en.wikipedia.org/wiki/Gaussian_function Attributes Summary
amplitude
input_units
This property is used to indicate what units or sets of units the evaluate method expects, and returns a dictionary mapping inputs to units (or None
if any units are accepted).param_names
theta
x_fwhm
Gaussian full width at half maximum in X. x_mean
x_stddev
y_fwhm
Gaussian full width at half maximum in Y. y_mean
y_stddev
Methods Summary
evaluate
(x, y, amplitude, x_mean, y_mean, …)Two dimensional Gaussian function fit_deriv
(x, y, amplitude, x_mean, y_mean, …)Two dimensional Gaussian function derivative with respect to parameters Attributes Documentation
-
amplitude
¶
-
input_units
¶ This property is used to indicate what units or sets of units the evaluate method expects, and returns a dictionary mapping inputs to units (or
None
if any units are accepted).Model sub-classes can also use function annotations in evaluate to indicate valid input units, in which case this property should not be overridden since it will return the input units based on the annotations.
-
param_names
= ('amplitude', 'x_mean', 'y_mean', 'x_stddev', 'y_stddev', 'theta')¶
-
theta
¶
-
x_fwhm
¶ Gaussian full width at half maximum in X.
-
x_mean
¶
-
x_stddev
¶
-
y_fwhm
¶ Gaussian full width at half maximum in Y.
-
y_mean
¶
-
y_stddev
¶
Methods Documentation
-
static
evaluate
(x, y, amplitude, x_mean, y_mean, x_stddev, y_stddev, theta)[source] [edit on github]¶ Two dimensional Gaussian function
-
static
fit_deriv
(x, y, amplitude, x_mean, y_mean, x_stddev, y_stddev, theta)[source] [edit on github]¶ Two dimensional Gaussian function derivative with respect to parameters