model_fitting.py

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# -*- coding: utf-8 -*-
 
# Copyright © 2019 Université de Genève, LMU Munich - Faculty of Physics, IAP-CNRS/Sorbonne Université
#
# This library is free software; you can redistribute it and/or modify it under
# the terms of the GNU Lesser General Public License as published by the Free
# Software Foundation; either version 3.0 of the License, or (at your option)
# any later version.
#
# This library is distributed in the hope that it will be useful, but WITHOUT
# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
# FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
# details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this library; if not, write to the Free Software Foundation, Inc.,
# 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
from __future__ import division, print_function
 
import math
import sys
import warnings
from enum import Enum
 
import _SourceXtractorPy as cpp
 
try:
    import pyston
except ImportError:
    warnings.warn('Could not import pyston: running outside sourcextractor?', ImportWarning)
from astropy import units as u
from astropy.coordinates import SkyCoord
 
from .measurement_images import MeasurementGroup
 
 
class RangeType(Enum):
    LINEAR = 1
    EXPONENTIAL = 2
 
 
class Range(object):
    r"""
    Limit, and normalize, the range of values for a model fitting parameter.
 
 
    Parameters
    ----------
    limits : a tuple (min, max), or a callable that receives a source, and returns a tuple (min, max)
    type : RangeType
 
    Notes
    -----
    RangeType.LINEAR
        Normalized to engine space using a sigmoid function
 
        .. math::
 
            engine = \ln \frac{world - min}{max-world} \\
            world = min + \frac{max - min}{1 + e^{engine}}
 
    RangeType.EXPONENTIAL
        Normalized to engine space using an exponential sigmoid function
 
        .. math::
 
            engine = \ln \left( \frac{\ln(world/min)}{\ln(max /world)} \right) \\
            world = min * e^\frac{ \ln(max / min) }{ (1 + e^{-engine}) }
 
    """
 
    def __init__(self, limits, type):
        """
        Constructor.
        """
        self.__limits = limits
        self.__callable = limits if hasattr(limits, '__call__') else lambda v, o: limits
        self.__type = type
 
    def get_min(self, v, o):
        """
        Parameters
        ----------
        v : initial value
        o : object being fitted
 
        Returns
        -------
        The minimum acceptable value for the range
        """
        return self.__callable(v, o)[0]
 
    def get_max(self, v, o):
        """
        Parameters
        ----------
        v : initial value
        o : object being fitted
 
        Returns
        -------
        The maximum acceptable value for the range
        """
        return self.__callable(v, o)[1]
 
    def get_type(self):
        """
        Returns
        -------
        RangeType
        """
        return self.__type
 
    def __str__(self):
        """
        Returns
        -------
        str
            Human readable representation for the object
        """
        res = '['
        if hasattr(self.__limits, '__call__'):
            res += 'func'
        else:
            res += '{},{}'.format(self.__limits[0], self.__limits[1])
        type_str = {
            RangeType.LINEAR: 'LIN',
            RangeType.EXPONENTIAL: 'EXP'
        }
        res += ',{}]'.format(type_str[self.__type])
        return res
 
 
class Unbounded(object):
    """
    Unbounded, but normalize, value of a model fitting parameter
 
    Parameters
    ----------
    normalization_factor: float, or a callable that receives the initial value parameter value and a source,
        and returns a float
        The world value which will be normalized to 1 in engine coordinates
    """
 
    def __init__(self, normalization_factor=1):
        """
        Constructor.
        """
        self.__normalization_factor = normalization_factor
        if hasattr(normalization_factor, '__call__'):
            self.__callable = normalization_factor
        else:
            self.__callable = lambda v, o: normalization_factor
 
    def get_normalization_factor(self, v, o):
        """
        Parameters
        ----------
        v : initial value
        o : object being fitted
 
        Returns
        -------
        float
            The world value which will be normalized to 1 in engine coordinates
        """
        return self.__callable(v, o)
 
    def __str__(self):
        """
        Returns
        -------
        str
            Human readable representation for the object
        """
        res = '['
        if hasattr(self.__normalization_factor, '__call__'):
            res += 'func'
        else:
            res += '{}'.format(self.__normalization_factor)
        res += ']'
        return res
 
 
class ParameterBase(cpp.Id):
    """
    Base class for all model fitting parameter types.
    Can not be used directly.
    """
 
    def __str__(self):
        """
        Returns
        -------
        str
            Human readable representation for the object
        """
        return '(ID:{})'.format(self.id)
 
 
class ConstantParameter(ParameterBase):
    """
    A parameter with a single value that remains constant. It will not be fitted.
 
    Parameters
    ----------
    value : float, or callable that receives a source and returns a float
        Value for this parameter
    """
 
    def __init__(self, value):
        """
        Constructor.
        """
        ParameterBase.__init__(self)
        self.__value = value
        self.__callable = value if hasattr(value, '__call__') else lambda o: value
 
    def get_value(self, o):
        """
        Parameters
        ----------
        o : object being fitted
 
        Returns
        -------
        float
            Value of the constant parameter
        """
        return self.__callable(o)
 
    def __str__(self):
        """
        Returns
        -------
        str
            Human readable representation for the object
        """
        res = ParameterBase.__str__(self)[:-1] + ', value:'
        if hasattr(self.__value, '__call__'):
            res += 'func'
        else:
            res += str(self.__value)
        return res + ')'
 
 
class FreeParameter(ParameterBase):
    """
    A parameter that will be fitted by the model fitting engine.
 
    Parameters
    ----------
    init_value : float or callable that receives a source, and returns a float
        Initial value for the parameter.
    range : instance of Range or Unbounded
        Defines if this parameter is unbounded or bounded, and how.
 
    See Also
    --------
    Unbounded
    Range
 
    Examples
    --------
    >>> sersic = FreeParameter(2.0, Range((1.0, 7.0), RangeType.LINEAR))
    """
 
    def __init__(self, init_value, range=Unbounded()):
        """
        Constructor.
        """
        ParameterBase.__init__(self)
        self.__init_value = init_value
        self.__init_call = init_value if hasattr(init_value, '__call__') else lambda o: init_value
        self.__range = range
 
    def get_init_value(self, o):
        """
        Parameters
        ----------
        o : object being fitted
 
        Returns
        -------
        float
            Initial value for the free parameter
        """
        return self.__init_call(o)
 
    def get_range(self):
        """
        Returns
        -------
        Unbounded or Range
        """
        return self.__range
 
    def __str__(self):
        """
        Returns
        -------
        str
            Human readable representation for the object
        """
        res = ParameterBase.__str__(self)[:-1] + ', init:'
        if hasattr(self.__init_value, '__call__'):
            res += 'func'
        else:
            res += str(self.__init_value)
        if self.__range:
            res += ', range:' + str(self.__range)
        return res + ')'
 
 
class DependentParameter(ParameterBase):
    """
    A DependentParameter is not fitted by itself, but its value is derived from another Parameters, whatever their type:
    FreeParameter, ConstantParameter, or other DependentParameter
 
    Parameters
    ----------
    func : callable
        A callable that will be called with all the parameters specified in this constructor each time a new
        evaluation is needed.
    params : list of ParameterBase
        List of parameters on which this DependentParameter depends.
 
    Examples
    --------
    >>> flux = get_flux_parameter()
    >>> mag = DependentParameter(lambda f: -2.5 * np.log10(f) + args.mag_zeropoint, flux)
    >>> add_output_column('mf_mag_' + band, mag)
    """
 
    def __init__(self, func, *params):
        """
        Constructor.
        """
        ParameterBase.__init__(self)
        self.func = func
        self.params = list(params)
 
 
def get_pos_parameters():
    """
    Convenience function for the position parameter X and Y.
 
    Returns
    -------
    x : FreeParameter
        X coordinate, starting at the X coordinate of the centroid and linearly limited to X +/- the object radius.
    y : FreeParameter
        Y coordinate, starting at the Y coordinate of the centroid and linearly limited to Y +/- the object radius.
    Notes
    -----
    X and Y are fitted on the detection image X and Y coordinates. Internally, these are translated to measurement
    images using the WCS headers.
    """
    param_range = Range(lambda v, o: (v - o.radius, v + o.radius), RangeType.LINEAR)
    return (
        FreeParameter(lambda o: o.centroid_x, param_range),
        FreeParameter(lambda o: o.centroid_y, param_range)
    )
 
 
class FluxParameterType():
    """
    Possible flux types to use as initial value for the flux parameter.
    Right now, only isophotal is supported.
    """
    ISO = 1
 
 
def get_flux_parameter(type=FluxParameterType.ISO, scale=1):
    """
    Convenience function for the flux parameter.
 
    Parameters
    ----------
    type : int
        One of the values defined in FluxParameterType
    scale : float
        Scaling of the initial flux. Defaults to 1.
 
    Returns
    -------
    flux : FreeParameter
        Flux parameter, starting at the flux defined by `type`, and limited to +/- 1e3 times the initial value.
    """
    attr_map = {
        FluxParameterType.ISO: 'isophotal_flux'
    }
    return FreeParameter(lambda o: getattr(o, attr_map[type]) * scale,
                         Range(lambda v, o: (v * 1E-3, v * 1E3), RangeType.EXPONENTIAL))
 
 
class Prior(cpp.Id):
    """
    Model a Gaussian prior on a given parameter.
 
    Parameters
    ----------
    param : ParameterBase
        Model fitting parameter
    value : float or callable that receives a source and returns a float
        Mean of the Gaussian
    sigma : float or callable that receives a source and returns a float
        Standard deviation of the Gaussian
    """
 
    def __init__(self, param, value, sigma):
        """
        Constructor.
        """
        cpp.Id.__init__(self)
        self.param = param.id
        self.value = value if hasattr(value, '__call__') else lambda o: value
        self.sigma = sigma if hasattr(sigma, '__call__') else lambda o: sigma
 
 
class ModelBase(cpp.Id):
    """
    Base class for all models.
    """
    
    def get_params(self):
        return []
 
 
class CoordinateModelBase(ModelBase):
    """
    Base class for positioned models with a flux. It can not be used directly.
 
    Parameters
    ----------
    x_coord : ParameterBase or float
        X coordinate (in the detection image)
    y_coord : ParameterBase or float
        Y coordinate (in the detection image)
    flux : ParameterBase or float
        Total flux
    """
 
    def __init__(self, x_coord, y_coord, flux):
        """
        Constructor.
        """
        ModelBase.__init__(self)
        self.x_coord = x_coord if isinstance(x_coord, ParameterBase) else ConstantParameter(x_coord)
        self.y_coord = y_coord if isinstance(y_coord, ParameterBase) else ConstantParameter(y_coord)
        self.flux = flux if isinstance(flux, ParameterBase) else ConstantParameter(flux)
        
    def get_params(self):
        return [self.x_coord, self.y_coord, self.flux]
 
 
 
class PointSourceModel(CoordinateModelBase):
    """
    Models a source as a point, spread by the PSF.
 
    Parameters
    ----------
    x_coord : ParameterBase or float
        X coordinate (in the detection image)
    y_coord : ParameterBase or float
        Y coordinate (in the detection image)
    flux : ParameterBase or float
        Total flux
    """
 
    def __init__(self, x_coord, y_coord, flux):
        """
        Constructor.
        """
        CoordinateModelBase.__init__(self, x_coord, y_coord, flux)
 
    def to_string(self, show_params=False):
        """
        Return a human readable representation of the model.
 
        Parameters
        ----------
        show_params: bool
            If True, include information about the parameters.
 
        Returns
        -------
        str
        """
        if show_params:
            return 'PointSource[x_coord={}, y_coord={}, flux={}]'.format(
                self.x_coord, self.y_coord, self.flux)
        else:
            return 'PointSource[x_coord={}, y_coord={}, flux={}]'.format(
                self.x_coord.id, self.y_coord.id, self.flux.id)
 
 
class ConstantModel(ModelBase):
    """
    A model that is constant through all the image.
 
    Parameters
    ----------
    value: ParameterBase or float
        Value to add to the value of all pixels from the model.
    """
 
    def __init__(self, value):
        """
        Constructor.
        """
        ModelBase.__init__(self)
        self.value = value if isinstance(value, ParameterBase) else ConstantParameter(value)
 
    def to_string(self, show_params=False):
        """
        Return a human readable representation of the model.
 
        Parameters
        ----------
        show_params: bool
            If True, include information about the parameters.
 
        Returns
        -------
        str
        """
        if show_params:
            return 'ConstantModel[value={}]'.format(self.value)
        else:
            return 'ConstantModel[value={}]'.format(self.value.id)
        
        def get_params(self):
            return [self.value]
 
 
class SersicModelBase(CoordinateModelBase):
    """
    Base class for the Sersic, Exponential and de Vaucouleurs models. It can not be used directly.
 
    Parameters
    ----------
    x_coord : ParameterBase or float
        X coordinate (in the detection image)
    y_coord : ParameterBase or float
        Y coordinate (in the detection image)
    flux : ParameterBase or float
        Total flux
    effective_radius : ParameterBase or float
        Ellipse semi-major axis, in pixels on the detection image.
    aspect_ratio : ParameterBase or float
        Ellipse ratio.
    angle : ParameterBase or float
        Ellipse rotation, in radians
    """
 
    def __init__(self, x_coord, y_coord, flux, effective_radius, aspect_ratio, angle):
        """
        Constructor.
        """
        CoordinateModelBase.__init__(self, x_coord, y_coord, flux)
        self.effective_radius = effective_radius if isinstance(effective_radius,
                                                               ParameterBase) else ConstantParameter(
            effective_radius)
        self.aspect_ratio = aspect_ratio if isinstance(aspect_ratio,
                                                       ParameterBase) else ConstantParameter(
            aspect_ratio)
        self.angle = angle if isinstance(angle, ParameterBase) else ConstantParameter(angle)
 
    def get_params(self):
        return CoordinateModelBase.get_params(self) + [self.effective_radius, self.aspect_ratio, self.angle]
 
class SersicModel(SersicModelBase):
    """
    Model a source with a Sersic profile.
 
    Parameters
    ----------
    x_coord : ParameterBase or float
        X coordinate (in the detection image)
    y_coord : ParameterBase or float
        Y coordinate (in the detection image)
    flux : ParameterBase or float
        Total flux
    effective_radius : ParameterBase or float
        Ellipse semi-major axis, in pixels on the detection image.
    aspect_ratio : ParameterBase or float
        Ellipse ratio.
    angle : ParameterBase or float
        Ellipse rotation, in radians
    n : ParameterBase or float
        Sersic index
    """
 
    def __init__(self, x_coord, y_coord, flux, effective_radius, aspect_ratio, angle, n):
        """
        Constructor.
        """
        SersicModelBase.__init__(self, x_coord, y_coord, flux, effective_radius, aspect_ratio,
                                 angle)
        self.n = n if isinstance(n, ParameterBase) else ConstantParameter(n)
 
    def to_string(self, show_params=False):
        """
        Return a human readable representation of the model.
 
        Parameters
        ----------
        show_params: bool
            If True, include information about the parameters.
 
        Returns
        -------
        str
        """
        if show_params:
            return 'Sersic[x_coord={}, y_coord={}, flux={}, effective_radius={}, aspect_ratio={}, angle={}, n={}]'.format(
                self.x_coord, self.y_coord, self.flux, self.effective_radius, self.aspect_ratio,
                self.angle, self.n)
        else:
            return 'Sersic[x_coord={}, y_coord={}, flux={}, effective_radius={}, aspect_ratio={}, angle={}, n={}]'.format(
                self.x_coord.id, self.y_coord.id, self.flux.id, self.effective_radius.id,
                self.aspect_ratio.id, self.angle.id, self.n.id)
 
    def get_params(self):
        return SersicModelBase.get_params(self) + [self.n]
 
 
class ExponentialModel(SersicModelBase):
    """
    Model a source with an exponential profile (Sersic model with an index of 1)
 
    Parameters
    ----------
    x_coord : ParameterBase or float
        X coordinate (in the detection image)
    y_coord : ParameterBase or float
        Y coordinate (in the detection image)
    flux : ParameterBase or float
        Total flux
    effective_radius : ParameterBase or float
        Ellipse semi-major axis, in pixels on the detection image.
    aspect_ratio : ParameterBase or float
        Ellipse ratio.
    angle : ParameterBase or float
        Ellipse rotation, in radians
    """
 
    def __init__(self, x_coord, y_coord, flux, effective_radius, aspect_ratio, angle):
        """
        Constructor.
        """
        SersicModelBase.__init__(self, x_coord, y_coord, flux, effective_radius, aspect_ratio,
                                 angle)
 
    def to_string(self, show_params=False):
        """
        Return a human readable representation of the model.
 
        Parameters
        ----------
        show_params: bool
            If True, include information about the parameters.
 
        Returns
        -------
        str
        """
        if show_params:
            return 'Exponential[x_coord={}, y_coord={}, flux={}, effective_radius={}, aspect_ratio={}, angle={}]'.format(
                self.x_coord, self.y_coord, self.flux, self.effective_radius, self.aspect_ratio,
                self.angle)
        else:
            return 'Exponential[x_coord={}, y_coord={}, flux={}, effective_radius={}, aspect_ratio={}, angle={}]'.format(
                self.x_coord.id, self.y_coord.id, self.flux.id, self.effective_radius.id,
                self.aspect_ratio.id, self.angle.id)
 
 
class DeVaucouleursModel(SersicModelBase):
    """
    Model a source with a De Vaucouleurs profile (Sersic model with an index of 4)
 
    Parameters
    ----------
    x_coord : ParameterBase or float
        X coordinate (in the detection image)
    y_coord : ParameterBase or float
        Y coordinate (in the detection image)
    flux : ParameterBase or float
        Total flux
    effective_radius : ParameterBase or float
        Ellipse semi-major axis, in pixels on the detection image.
    aspect_ratio : ParameterBase or float
        Ellipse ratio.
    angle : ParameterBase or float
        Ellipse rotation, in radians
    """
 
    def __init__(self, x_coord, y_coord, flux, effective_radius, aspect_ratio, angle):
        """
        Constructor.
        """
        SersicModelBase.__init__(self, x_coord, y_coord, flux, effective_radius, aspect_ratio,
                                 angle)
 
    def to_string(self, show_params=False):
        """
        Return a human readable representation of the model.
 
        Parameters
        ----------
        show_params: bool
            If True, include information about the parameters.
 
        Returns
        -------
        str
        """
        if show_params:
            return 'DeVaucouleurs[x_coord={}, y_coord={}, flux={}, effective_radius={}, aspect_ratio={}, angle={}]'.format(
                self.x_coord, self.y_coord, self.flux, self.effective_radius, self.aspect_ratio,
                self.angle)
        else:
            return 'DeVaucouleurs[x_coord={}, y_coord={}, flux={}, effective_radius={}, aspect_ratio={}, angle={}]'.format(
                self.x_coord.id, self.y_coord.id, self.flux.id, self.effective_radius.id,
                self.aspect_ratio.id, self.angle.id)
 
 
class ComputeGraphModel(CoordinateModelBase):
    """
    ComputeGraphModel model
 
    Parameters
    ----------
    models: string or list of strings, corresponding to path to Onnx format models,
     (specifying more than one allows using the most efficient model based on render size.)
    x_coord : ParameterBase or float
        X coordinate (in the detection image)
    y_coord : ParameterBase or float
        Y coordinate (in the detection image)
    flux : ParameterBase or float
        Total flux
    params : Dictionary of String and ParameterBase or float
        Dictionary of custom parameters for the ONNX model
    """
 
    def __init__(self, models, x_coord, y_coord, flux, params={}):
        """
        Constructor.
        """
        CoordinateModelBase.__init__(self, x_coord, y_coord, flux)
 
        ratio_name = "_aspect_ratio"
        angle_name = "_angle"
        scale_name = "_scale"
 
        for k in params.keys():
            if not isinstance(params[k], ParameterBase):
                params[k] = ConstantParameter(params[k])
 
        aspect_ratio = params[ratio_name] if ratio_name in params.keys() else 1.0
        angle = params[angle_name] if angle_name in params.keys() else 0.0
        scale = params[scale_name] if scale_name in params.keys() else 1.0
 
        self.aspect_ratio = aspect_ratio if isinstance(aspect_ratio,
                                                       ParameterBase) else ConstantParameter(
            aspect_ratio)
        self.angle = angle if isinstance(angle, ParameterBase) else ConstantParameter(angle)
        self.scale = scale if isinstance(scale, ParameterBase) else ConstantParameter(scale)
 
        params.pop(ratio_name, None)
        params.pop(angle_name, None)
        params.pop(scale_name, None)
 
        self.params = params
        self.models = models if isinstance(models, list) else [models]
 
    def to_string(self, show_params=False):
        """
        Return a human readable representation of the model.
 
        Parameters
        ----------
        show_params: bool
            If True, include information about the parameters.
 
        Returns
        -------
        str
        """
        if show_params:
            return 'ComputeGraph[x_coord={}, y_coord={}, flux={}, effective_radius={}, aspect_ratio={}, angle={}]'.format(
                self.x_coord, self.y_coord, self.flux, self.effective_radius, self.aspect_ratio,
                self.angle)
        else:
            return 'ComputeGraph[x_coord={}, y_coord={}, flux={}, effective_radius={}, aspect_ratio={}, angle={}]'.format(
                self.x_coord.id, self.y_coord.id, self.flux.id, self.effective_radius.id,
                self.aspect_ratio.id, self.angle.id)
 
    def get_params(self):
        return (CoordinateModelBase.get_params(self) + [self.scale, self.aspect_ratio, self.angle] +
            list(self.params.values()))
 
 
class WorldCoordinate:
    """
    Coordinates in right ascension and declination
 
    Parameters
    ----------
    ra : float
        Right ascension
    dec : float
        Declination
    """
 
    def __init__(self, ra, dec):
        """
        Constructor.
        """
        self.ra = ra
        self.dec = dec
 
 
def pixel_to_world_coordinate(x, y):
    """
    Transform an (X, Y) in pixel coordinates on the detection image to (RA, DEC) in world coordinates.
    Parameters
    ----------
    x : float
    y : float
 
    Returns
    -------
    WorldCoordinate
    """
    # -1 as we go FITS -> internal
    wc_alpha = pyston.image_to_world_alpha(x - 1, y - 1)
    wc_delta = pyston.image_to_world_delta(x - 1, y - 1)
    return WorldCoordinate(wc_alpha, wc_delta)
 
 
def get_sky_coord(x, y):
    """
    Transform an (X, Y) in pixel coordinates on the detection image to astropy SkyCoord.
 
    Parameters
    ----------
    x : float
    y : float
 
    Returns
    -------
    SkyCoord
    """
    coord = pixel_to_world_coordinate(x, y)
    sky_coord = SkyCoord(ra=coord.ra * u.degree, dec=coord.dec * u.degree)
    return sky_coord
 
 
def radius_to_wc_angle(x, y, rad):
    """
    Transform a radius in pixels on the detection image to a radius in sky coordinates.
 
    Parameters
    ----------
    x : float
    y : float
    rad : float
 
    Returns
    -------
    Radius in degrees
    """
    return (get_separation_angle(x, y, x + rad, y) + get_separation_angle(x, y, x, y + rad)) / 2.0
 
 
def get_separation_angle(x1, y1, x2, y2):
    """
    Get the separation angle in sky coordinates for two points defined in pixels on the detection image.
 
    Parameters
    ----------
    x1 : float
    y1 : float
    x2 : float
    y2 : float
 
    Returns
    -------
    Separation in degrees
    """
    c1 = get_sky_coord(x1, y1)
    c2 = get_sky_coord(x2, y2)
    return c1.separation(c2).degree
 
 
def get_position_angle(x1, y1, x2, y2):
    """
    Get the position angle in sky coordinates for two points defined in pixels on the detection image.
 
    Parameters
    ----------
    x1
    y1
    x2
    y2
 
    Returns
    -------
    Position angle in degrees, normalized to -/+ 90
    """
    c1 = get_sky_coord(x1, y1)
    c2 = get_sky_coord(x2, y2)
    angle = c1.position_angle(c2).degree
 
    # return angle normalized to range: -90 <= angle < 90
    return (angle + 90.0) % 180.0 - 90.0
 
 
def get_world_position_parameters(x, y):
    """
    Convenience function for generating two dependent parameter with world (alpha, delta) coordinates
    from image (X, Y) coordinates.
 
    Parameters
    ----------
    x : ParameterBase
    y : ParameterBase
 
    Returns
    -------
    ra : DependentParameter
    dec : DependentParameter
 
    See Also
    --------
    get_pos_parameters
 
    Examples
    --------
    >>> x, y = get_pos_parameters()
    >>> ra, dec = get_world_position_parameters(x, y)
    >>> add_output_column('mf_ra', ra)
    >>> add_output_column('mf_dec', dec)
    """
    ra = DependentParameter(lambda x, y: pixel_to_world_coordinate(x, y).ra, x, y)
    dec = DependentParameter(lambda x, y: pixel_to_world_coordinate(x, y).dec, x, y)
    return (ra, dec)
 
 
def get_world_parameters(x, y, radius, angle, ratio):
    """
    Convenience function for generating five dependent parameters, in world coordinates, for the position
    and shape of a model.
 
    Parameters
    ----------
    x : ParameterBase
    y : ParameterBase
    radius : ParameterBase
    angle : ParameterBase
    ratio : ParameterBase
 
    Returns
    -------
    ra : DependentParameter
        Right ascension
    dec : DependentParameter
        Declination
    rad : DependentParameter
        Radius as degrees
    angle : DependentParameter
        Angle in degrees
    ratio : DependentParameter
        Aspect ratio. It has to be recomputed as the axis of the ellipse may have different ratios
        in image coordinates than in world coordinates
 
    Examples
    --------
    >>> flux = get_flux_parameter()
    >>> x, y = get_pos_parameters()
    >>> radius = FreeParameter(lambda o: o.radius, Range(lambda v, o: (.01 * v, 100 * v), RangeType.EXPONENTIAL))
    >>> angle = FreeParameter(lambda o: o.angle, Range((-np.pi, np.pi), RangeType.LINEAR))
    >>> ratio = FreeParameter(1, Range((0, 10), RangeType.LINEAR))
    >>> add_model(group, ExponentialModel(x, y, flux, radius, ratio, angle))
    >>> ra, dec, wc_rad, wc_angle, wc_ratio = get_world_parameters(x, y, radius, angle, ratio)
    >>> add_output_column('mf_world_angle', wc_angle)
    """
    ra = DependentParameter(lambda x, y: pixel_to_world_coordinate(x, y).ra, x, y)
    dec = DependentParameter(lambda x, y: pixel_to_world_coordinate(x, y).dec, x, y)
 
    def get_major_axis(x, y, radius, angle, ratio):
        if ratio <= 1:
            x2 = x + math.cos(angle) * radius
            y2 = y + math.sin(angle) * radius
        else:
            x2 = x + math.sin(angle) * radius * ratio
            y2 = y + math.cos(angle) * radius * ratio
 
        return x2, y2
 
    def get_minor_axis(x, y, radius, angle, ratio):
        if ratio <= 1:
            x2 = x + math.sin(angle) * radius * ratio
            y2 = y + math.cos(angle) * radius * ratio
        else:
            x2 = x + math.cos(angle) * radius
            y2 = y + math.sin(angle) * radius
 
        return x2, y2
 
    def wc_rad_func(x, y, radius, angle, ratio):
        x2, y2 = get_major_axis(x, y, radius, angle, ratio)
        return get_separation_angle(x, y, x2, y2)
 
    def wc_angle_func(x, y, radius, angle, ratio):
        x2, y2 = get_major_axis(x, y, radius, angle, ratio)
        return get_position_angle(x, y, x2, y2)
 
    def wc_ratio_func(x, y, radius, angle, ratio):
        minor_x, minor_y = get_minor_axis(x, y, radius, angle, ratio)
        minor_angle = get_separation_angle(x, y, minor_x, minor_y)
 
        major_x, major_y = get_major_axis(x, y, radius, angle, ratio)
        major_angle = get_separation_angle(x, y, major_x, major_y)
 
        return minor_angle / major_angle
 
    wc_rad = DependentParameter(wc_rad_func, x, y, radius, angle, ratio)
    wc_angle = DependentParameter(wc_angle_func, x, y, radius, angle, ratio)
    wc_ratio = DependentParameter(wc_ratio_func, x, y, radius, angle, ratio)
 
    return ra, dec, wc_rad, wc_angle, wc_ratio
 
 
class ModelFitting:
    def __init__(self):
        self.constant_parameter_dict = {}
        self.free_parameter_dict = {}
        self.dependent_parameter_dict = {}
        self.frame_models_dict = {}
        self.prior_dict = {}
        self.constant_model_dict = {}
        self.point_source_model_dict = {}
        self.sersic_model_dict = {}
        self.exponential_model_dict = {}
        self.de_vaucouleurs_model_dict = {}
        self.onnx_model_dict = {}
        self.params_dict = {"max_iterations": 200, "modified_chi_squared_scale": 10, "engine": "",
                            "use_iterative_fitting": True, "meta_iterations": 5,
                            "deblend_factor": 0.95, "meta_iteration_stop": 0.0001}
 
    def _set_model_to_frames(self, group, model):
        for x in group:
            if isinstance(x, tuple):
                self._set_model_to_frames(x[1], model)
            else:
                if x.id not in self.frame_models_dict:
                    self.frame_models_dict[x.id] = []
                self.frame_models_dict[x.id].append(model.id)
 
    def _is_param_known(self, param):
        return param.id in self.constant_parameter_dict or \
               param.id in self.free_parameter_dict or \
               param.id in self.dependent_parameter_dict
 
    def _register_parameter(self, attr):
        if isinstance(attr, ConstantParameter):
            self.constant_parameter_dict[attr.id] = attr
        elif isinstance(attr, FreeParameter):
            self.free_parameter_dict[attr.id] = attr
        elif isinstance(attr, DependentParameter):
            self.dependent_parameter_dict[attr.id] = attr
            for param in attr.params:
                self._register_parameter(param)
 
    def _populate_parameters(self, model):
        for param in model.get_params():
            self._register_parameter(param)
 
    def _register_model(self, model):
        if isinstance(model, ConstantModel):
            self.constant_model_dict[model.id] = model
        elif isinstance(model, PointSourceModel):
            self.point_source_model_dict[model.id] = model
        elif isinstance(model, SersicModel):
            self.sersic_model_dict[model.id] = model
        elif isinstance(model, ExponentialModel):
            self.exponential_model_dict[model.id] = model
        elif isinstance(model, DeVaucouleursModel):
            self.de_vaucouleurs_model_dict[model.id] = model
        elif isinstance(model, ComputeGraphModel):
            self.onnx_model_dict[model.id] = model
        else:
            raise TypeError('Unknown model type {}'.format(type(model)))
 
    def add_model(self, group, model):
        """
        Add a model to be fitted to the given group.
 
        Parameters
        ----------
        group : MeasurementGroup
        model : ModelBase
        """
        if not isinstance(group, MeasurementGroup):
            raise TypeError(
                'Models can only be added on MeasurementGroup, got {}'.format(type(group)))
        if not hasattr(group, 'models'):
            group.models = []
        group.models.append(model)
        self._set_model_to_frames(group, model)
        self._populate_parameters(model)
        self._register_model(model)
 
    def add_prior(self, param, value, sigma):
        """
        Add a prior to the given parameter.
 
        Parameters
        ----------
        param : ParameterBase
        value : float or callable that receives a source and returns a float
            Mean of the Gaussian
        sigma : float or callable that receives a source and returns a float
            Standard deviation of the Gaussian
        """
        prior = Prior(param, value, sigma)
        self.prior_dict[prior.id] = prior
        if not self._is_param_known(param):
            self._register_parameter(param)
 
    def print_parameters(self, file=sys.stderr):
        """
        Print a human-readable representation of the configured model fitting parameters.
 
        Parameters
        ----------
        file : file object
            Where to print the representation. Defaults to sys.stderr
        """
        if self.constant_parameter_dict:
            print('Constant parameters:', file=file)
            for n, param in self.constant_parameter_dict.items():
                print('  {}: {}'.format(n, param), file=file)
        if self.free_parameter_dict:
            print('Free parameters:', file=file)
            for n, param in self.free_parameter_dict.items():
                print('  {}: {}'.format(n, param), file=file)
        if self.dependent_parameter_dict:
            print('Dependent parameters:', file=file)
            for n, param in self.dependent_parameter_dict.items():
                print('  {}: {}'.format(n, param), file=file)
 
    def set_max_iterations(self, iterations):
        """
        Parameters
        ----------
        iterations : int
            Max number of iterations for the model fitting.
        """
        self.params_dict["max_iterations"] = iterations
 
    def set_modified_chi_squared_scale(self, scale):
        """
        Parameters
        ----------
        scale : float
            Sets u0, as used by the modified chi squared residual comparator, a function that reduces the effect of large
            deviations.
            Refer to the SourceXtractor++ documentation for a better explanation of how residuals are computed and how
            this value affects the model fitting.
        """
        self.params_dict["modified_chi_squared_scale"] = scale
 
    def set_engine(self, engine):
        """
        Parameters
        ----------
        engine : str
            Minimization engine for the model fitting : levmar or gsl
        """
        self.params_dict["engine"] = engine
 
    def use_iterative_fitting(self, use_iterative_fitting):
        """
        Parameters
        ----------
        use_iterative_fitting : boolean
            use iterative model fitting or legacy
        """
        self.params_dict["use_iterative_fitting"] = use_iterative_fitting
 
    def set_meta_iterations(self, meta_iterations):
        """
        Parameters
        ----------
        meta_iterations : int
            number of meta iterations on the whole group (when using iterative model fitting)
        """
        self.params_dict["meta_iterations"] = meta_iterations
 
    def set_deblend_factor(self, deblend_factor):
        """
        Parameters
        ----------
 
        """
        self.params_dict["deblend_factor"] = deblend_factor
 
    def set_meta_iteration_stop(self, meta_iteration_stop):
        """
        Parameters
        ----------
 
        """
        self.params_dict["meta_iteration_stop"] = meta_iteration_stop
 
 
def print_model_fitting_info(group, show_params=False, prefix='', file=sys.stderr):
    """
    Print a human-readable representation of the configured models.
 
    Parameters
    ----------
    group : MeasurementGroup
        Print the models for this group.
    show_params : bool
        If True, print also the parameters that belong to the model
    prefix : str
        Prefix each line with this string. Used internally for indentation.
    file : file object
        Where to print the representation. Defaults to sys.stderr
    """
    if hasattr(group, 'models') and group.models:
        print('{}Models:'.format(prefix), file=file)
        for m in group.models:
            print('{}  {}'.format(prefix, m.to_string(show_params)), file=file)
    for x in group:
        if isinstance(x, tuple):
            print('{}{}:'.format(prefix, x[0]), file=file)
            print_model_fitting_info(x[1], show_params, prefix + '    ', file=file)