@@ -37,7 +37,7 @@ NIFTy comes with reimplemented MAP and VI estimators.
Free Theory & Implicit Operators
--------------------------------
A free IFT appears when the signal field :math:`{s}` and the noise :math:`{n}` of the data :math:`{d}` are independent, zero-centered Gaussian processes of kown covariances :math:`{S}` and :math:`{N}`, respectively,
A free IFT appears when the signal field :math:`{s}` and the noise :math:`{n}` of the data :math:`{d}` are independent, zero-centered Gaussian processes of known covariances :math:`{S}` and :math:`{N}`, respectively,
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@@ -49,7 +49,7 @@ and the measurement equation is linear in both signal and noise,
d= R\, s + n,
with :math:`{R}` being the measurement response, which maps the continous signal field into the discrete data space.
with :math:`{R}` being the measurement response, which maps the continuous signal field into the discrete data space.
This is called a free theory, as the information Hamiltonian
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@@ -96,8 +96,8 @@ These implicit operators can be combined into new operators, e.g. to :math:`{D^{
The invocation of an inverse operator applied to a vector might trigger the execution of a numerical linear algebra solver.
Thus, when NIFTy calculates :math:`{m = D\, j}`, it actually solves :math:`{D^{-1} m = j}` for :math:`{m}` behind the scenes.
The advantage of implicit operators to explicit matrices is the reduced memory requirements.
The reconstruction of only a Megapixel image would otherwithe require the storage and processing of matrices with sizes of several Terabytes.
The advantage of implicit operators compared to explicit matrices is the reduced memory consumption;
for the reconstruction of just a Megapixel image the latter would already require several Terabytes.
Larger images could not be dealt with due to the quadratic memory requirements of explicit operator representations.
The demo codes `demos/getting_started_1.py` and `demos/Wiener_Filter.ipynb` illustrate this.
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@@ -106,7 +106,7 @@ The demo codes `demos/getting_started_1.py` and `demos/Wiener_Filter.ipynb` illu
Generative Models
-----------------
For more sophisticated measurement situations, involving non-linear measuremnts, unknown covariances, calibration constants and the like, it is recommended to formulate those as generative models for which NIFTy provides powerful inference algorithms.
For more sophisticated measurement situations (involving non-linear measurements, unknown covariances, calibration constants and the like) it is recommended to formulate those as generative models for which NIFTy provides powerful inference algorithms.
In a generative model, all known or unknown quantities are described as the results of generative processes, which start with simple probability distributions, like the uniform, the i.i.d. Gaussian, or the delta distribution.
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@@ -129,7 +129,7 @@ NIFTy takes advantage of this formulation in several ways:
1) All prior degrees of freedom have unit covariance, which improves the condition number of operators that need to be inverted.
2) The amplitude operator can be regarded as part of the response, :math:`{R'=R\,A}`.
In general, more sophisticated responses can be constructed out of the composition of simpler operators.
In general, more sophisticated responses can be obtained by combining simpler operators.
3) The response can be non-linear, e.g. :math:`{R'(s)=R \exp(A\,\xi)}`, see `demos/getting_started_2.py`.
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@@ -168,7 +168,7 @@ Maximum a Posteriori
One popular field estimation method is Maximum a Posteriori (MAP).
It only requires to minimize the information Hamiltonian, e.g by a gradient descent method that stops when
It only requires minimizing the information Hamiltonian, e.g. by a gradient descent method that stops when
Here the domain of the integral :math:`\Omega = \dot{\bigcup_q} \; \Omega_i` is the disjoint union over smaller :math:`\Omega_i`, e.g. the pixels of the space, and :math:`s_i` is the discretized field value on the :math:`i`-th pixel.
Here the domain of the integral :math:`\Omega = \dot{\bigcup_q} \; \Omega_i` is the disjoint union over smaller :math:`\Omega_i`, e.g. the pixels of the space, and :math:`s_i` is the discretised field value on the :math:`i`-th pixel.
This introduces the weighting :math:`V_i=\int_{\Omega_i}\text{d}x\, 1`, also called the volume factor, a property of the space.
NIFTy aids you in constructing your own log-likelihood by providing methods like :func:`~field.Field.weight`, which weights all pixels of a field with their corresponding volume.
An integral over a :class:`~field.Field` :code:`s` can be performed by calling :code:`s.weight(1).sum()`, which is equivalent to :code:`s.integrate()`.
Volume factors are also applied automatically in the following places:
- :class:`~operators.harmonic_operators.FFTOperator` as well as all other harmonic operators. Here the zero mode of the transformed field is the integral over the original field, thus the whole field is weighted once.
- some response operators, such as the :class:`~library.los_response.LOSResponse`. In this operator a line integral is descritized, so a 1-dimensional volume factor is applied.
- In :class:`~library.correlated_fields.CorrelatedField` as well :class:`~library.correlated_fields.MfCorrelatedField`, the field is multiplied by the square root of the total volume in configuration space. This ensures that the same field reconstructed over a larger domain has the same variance in position space in the limit of infinite resolution. It also ensures that power spectra in NIFTy behave according to the definition of a power spectrum, namely the power of a k-mode is the expectation of the k-mode square, divided by the volume of the space.
- :class:`~operators.harmonic_operators.FFTOperator` as well as all other harmonic operators.
Here the zero mode of the transformed field is the integral over the original field, thus the whole field is weighted once.
- Some response operators, such as the :class:`~library.los_response.LOSResponse`.
In this operator a line integral is discretised, so a 1-dimensional volume factor is applied.
- In :class:`~library.correlated_fields.CorrelatedField` as well as :class:`~library.correlated_fields.MfCorrelatedField`.
Both describe fields with a smooth, a priori unknown correlation structure specified by a power spectrum.
The field is multiplied by the square root of the total volume of it domain's harmonic counterpart.
This ensures that the same power spectrum can be used regardless of the chosen resolution, provided the total volume of the space remains the same.
It also guarantees that the power spectra in NIFTy behave according to their definition, i.e. the power of a mode :math:`s_k` is the expectation value of that mode squared, divided by the volume of its space :math:`P(k) = \left\langle s_k^2 \right\rangle / V_k`.
Note that in contrast to some older versions of NIFTy, the dot product :code:`s.vdot(t)` of fields does **not** apply a volume factor, but instead just sums over the field components,
