smooth_linear_amplitude.py 7 KB
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# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program 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 General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.
#
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# Copyright(C) 2013-2019 Max-Planck-Society
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#
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# NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik.
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import numpy as np
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from ..domain_tuple import DomainTuple
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from ..domains.power_space import PowerSpace
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from ..field import Field
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from ..operators.exp_transform import ExpTransform
from ..operators.offset_operator import OffsetOperator
from ..operators.qht_operator import QHTOperator
from ..operators.slope_operator import SlopeOperator
from ..operators.symmetrizing_operator import SymmetrizingOperator
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from ..sugar import makeOp
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def _ceps_kernel(dof_space, k, a, k0):
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    res = 1.
    for i in range(len(k0)):
        res += (k[i]/k0[i])**2
    return a**2/(res)**2
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def CepstrumOperator(target, a, k0):
    '''Turns a white Gaussian random field into a smooth field on a LogRGSpace.
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    Composed out of three operators:

        sym @ qht @ diag(sqrt_ceps),

    where sym is a :class:`SymmetrizingOperator`, qht is a :class:`QHTOperator`
    and ceps is the so-called cepstrum:

    .. math::
        \\mathrm{sqrt\_ceps}(k) = \\frac{a}{1+(k/k0)^2}

    These operators are combined in this fashion in order to generate:

    - A field which is smooth, i.e. second derivatives are punished (note
      that the sqrt-cepstrum is essentially proportional to 1/k**2).

    - A field which is symmetric around the pixel in the middle of the space.
      This is result of the :class:`SymmetrizingOperator` and needed in order to
      decouple the degrees of freedom at the beginning and the end of the
      amplitude whenever :class:`CepstrumOperator` is used as in
      :class:`SLAmplitude`.

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    The prior on the zero mode, or zero subspaces in the case of dim > 1,
    is the integral of the prior of all other modes along the corresponding
    axis.
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    Parameters
    ----------
    target : LogRGSpace
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        Target domain of the operator, needs to be non-harmonic.
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    a : float
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        Cutoff of smoothness prior (positive only). Controls the
        regularization of the inverse laplace operator to be finite at zero.
        Larger values for the cutoff results in a weaker constraining prior.
    k0 : float, list of float
        Strength of smothness prior in quefrency space (positive only) along
        each axis. If float then the strength is the same along each axis.
        Larger values result in a weaker constraining prior.
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    '''
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    a = float(a)
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    target = DomainTuple.make(target)
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    if a <= 0:
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        raise ValueError
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    if len(target) > 1 or target[0].harmonic:
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        raise TypeError
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    if isinstance(k0, float):
        k0 = (k0, )*len(target.shape)
    elif len(k0) != len(target.shape):
        raise ValueError
    if np.any(np.array(k0) <= 0):
        raise ValueError
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    qht = QHTOperator(target)
    dom = qht.domain[0]
    sym = SymmetrizingOperator(target)

    # Compute cepstrum field
    dim = len(dom.shape)
    shape = dom.shape
    q_array = dom.get_k_array()
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    # Fill all non-zero modes
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    no_zero_modes = (slice(1, None),)*dim
    ks = q_array[(slice(None),) + no_zero_modes]
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    cepstrum_field = np.zeros(shape)
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    cepstrum_field[no_zero_modes] = _ceps_kernel(dom, ks, a, k0)
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    # Fill zero-mode subspaces
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    for i in range(dim):
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        fst_dims = (slice(None),)*i
        sl = fst_dims + (slice(1, None),)
        sl2 = fst_dims + (0,)
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        cepstrum_field[sl2] = np.sum(cepstrum_field[sl], axis=i)
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    cepstrum = Field.from_global_data(dom, cepstrum_field)
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    return sym @ qht @ makeOp(cepstrum.sqrt())


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def SLAmplitude(target, n_pix, a, k0, sm, sv, im, iv, keys=['tau', 'phi']):
    '''Operator for parametrizing smooth amplitudes (square roots of power
    spectra).
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    The general guideline for setting up generative models in IFT is to
    transform the problem into the eigenbase of the prior and formulate the
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    generative model in this base. This is done here for the case of an
    amplitude which is smooth and has a linear component (both on
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    double-logarithmic scale).

    This function assembles an :class:`Operator` which maps two a-priori white
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    Gaussian random fields to a smooth amplitude which is composed out of
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    a linear and a smooth component.

    On double-logarithmic scale, i.e. both x and y-axis on logarithmic scale,
    the output of the generated operator is:

        AmplitudeOperator = 0.5*(smooth_component + linear_component)
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    This is then exponentiated and exponentially binned (in this order).
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    The prior on the linear component is parametrized by four real numbers,
    being expected value and prior variance on the slope and the y-intercept
    of the linear function.

    The prior on the smooth component is parametrized by two real numbers: the
    strength and the cutoff of the smoothness prior (see :class:`CepstrumOperator`).
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    Parameters
    ----------
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    n_pix : int
        Number of pixels of the space in which the .
    target : PowerSpace
        Target of the Operator.
    a : float
        Strength of smoothness prior (see :class:`CepstrumOperator`).
    k0 : float
        Cutoff of smothness prior in quefrency space (see :class:`CepstrumOperator`).
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    sm : float
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        Expected exponent of power law. FIXME
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    sv : float
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        Prior standard deviation of exponent of power law.
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    im : float
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        Expected y-intercept of power law. FIXME
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    iv : float
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        Prior standard deviation of y-intercept of power law.
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    Returns
    -------
    Operator
        Operator which is defined on the space of white excitations fields and
        which returns on its target a power spectrum which consists out of a
        smooth and a linear part.
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    '''
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    if not (isinstance(n_pix, int) and isinstance(target, PowerSpace)):
        raise TypeError

    a, k0 = float(a), float(k0)
    sm, sv, im, iv = float(sm), float(sv), float(im), float(iv)
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    if sv <= 0 or iv <= 0:
        raise ValueError
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    et = ExpTransform(target, n_pix)
    dom = et.domain[0]

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    # Smooth component
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    dct = {'a': a, 'k0': k0}
    smooth = CepstrumOperator(dom, **dct).ducktape(keys[0])
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    # Linear component
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    sl = SlopeOperator(dom)
    mean = np.array([sm, im + sm*dom.t_0[0]])
    sig = np.array([sv, iv])
    mean = Field.from_global_data(sl.domain, mean)
    sig = Field.from_global_data(sl.domain, sig)
    linear = (sl @ OffsetOperator(mean) @ makeOp(sig)).ducktape(keys[1])
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    # Combine linear and smooth component
    loglog_ampl = 0.5*(smooth + linear)
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    # Go from loglog-space to linear-linear-space
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    return et @ loglog_ampl.exp()