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ift
NIFTy
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ba6dc97b
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ba6dc97b
authored
Aug 08, 2018
by
Martin Reinecke
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some documentation work
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012feede
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docs/source/code.rst
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ba6dc97b
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@@ -99,6 +99,7 @@ Combinations of domains
The fundamental classes described above are often sufficient to specify the
domain of a field. In some cases, however, it will be necessary to have the
field live on a product of elementary domains instead of a single one.
More sophisticated models also require a set of several such fields.
Some examples are:
- sky emission depending on location and energy. This could be represented by
...
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@@ -107,6 +108,9 @@ Some examples are:
- a polarised field, which could be modeled as a product of any structured
domain (representing location) with a four-element
:class:`UnstructuredDomain` holding Stokes I, Q, U and V components.
- a model for the sky emission, which holds both the current realization
(on a harmonic domain) and a few inferred model parameters (e.g. on an
unstructured grid).
Consequently, NIFTy defines a class called :class:`DomainTuple` holding
a sequence of :class:`Domain` objects, which is used to specify full field
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@@ -117,10 +121,15 @@ A :class:`DomainTuple` supports iteration and indexing, and also provides the
properties :attr:`~DomainTuple.shape`, :attr:`~DomainTuple.size` in analogy to
the elementary :class:`Domain`.
An aggregation of several :class:`DomainTuple`s, each member identified by a
name, is described by the :class:`MultiDomain` class.
Fields
======
Fields on a single DomainTuple
------------------------------
A :class:`Field` object consists of the following components:
- a domain in form of a :class:`DomainTuple` object
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@@ -148,14 +157,44 @@ that are not covered by the provided standard operations, its data content must
be extracted first, then changed, and a new field has to be created from the
result.
Fields living on a MultiDomain
------------------------------
The :class:`MultiField` class can be seen as a dictionary of individual
:class:`Field`s, each identified by a name, which lives on an associated
:class:`MultiDomain`.
Operators
=========
All transformations between different NIFTy fields are expressed (explicitly
or implicitly) in the form of :class:`Operator` objects. The interface of this
class is very minimalistic: it has a property called `domain` which returns
a `Domaintuple` or `MultiDomain` object specifying the structure of the
`Field`s or `MultiField`s it expects as input, another property `target`
describing its output, and finally an overloaded `apply` method, which can
take
- a `Field`/`MultiField`object, in which case it returns the transformed
`Field`/`MultiField`
- a `Linearization` object, in which case it returns the transformed
`Linearization`
This is the interface that all objects derived from `Operator` must implement.
In addition, `Operator` objects can be added/subtracted, multiplied, chained
(via the `__call__` method) and support pointwise application of functions like
`exp()`, `log()`, `sqrt()`, `conjugate()` etc.
Linear Operators
================
A linear operator (represented by NIFTy5's abstract :class:`LinearOperator`
class) can be interpreted as an (implicitly defined) matrix.
It can be applied to :class:`Field` instances, resulting in other :class:`Field`
instances that potentially live on other domains.
class) is derived from `Operator` and can be interpreted as an
(implicitly defined) matrix. Since its operation is linear, it can provide some
additional functionality which is not available for the more generic `Operator`
class.
Operator basics
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@@ -163,13 +202,13 @@ Operator basics
There are four basic ways of applying an operator :math:`A` to a field :math:`f`:
- direct
multi
plication: :math:`A\cdot f`
- adjoint
multi
plication: :math:`A^\dagger \cdot f`
- inverse
multi
plication: :math:`A^{-1}\cdot f`
- adjoint inverse
multi
plication: :math:`(A^\dagger)^{-1}\cdot f`
- direct
ap
plication: :math:`A\cdot f`
- adjoint
ap
plication: :math:`A^\dagger \cdot f`
- inverse
ap
plication: :math:`A^{-1}\cdot f`
- adjoint inverse
ap
plication: :math:`(A^\dagger)^{-1}\cdot f`
(
For linear operators
, inverse adjoint
multiplication
and adjoint inverse
multiplication
are equivalent.)
(
Because of the linearity
, inverse adjoint and adjoint inverse
application
are equivalent.)
These different actions of an operator ``Op`` on a field ``f`` can be invoked
in various ways:
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@@ -190,9 +229,6 @@ enhanced by this approach to support the complete set. This functionality is
provided by NIFTy's :class:`InversionEnabler` class, which is itself a linear
operator.
There are two :class:`DomainTuple` objects associated with a
:class:`LinearOperator`: a :attr:`~LinearOperator.domain` and a
:attr:`~LinearOperator.target`.
Direct multiplication and adjoint inverse multiplication transform a field
living on the operator's :attr:`~LinearOperator.domain` to one living on the operator's :attr:`~LinearOperator.target`, whereas adjoint multiplication
and inverse multiplication transform from :attr:`~LinearOperator.target` to :attr:`~LinearOperator.domain`.
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@@ -221,7 +257,7 @@ operators.
As an example, if ``A``, ``B`` and ``C`` are of type :class:`LinearOperator`
and ``f1`` and ``f2`` are of type :class:`Field`, writing::
X = A
*
B.inverse
*
A.adjoint + C
X = A
(
B.inverse
(
A.adjoint
))
+ C
f2 = X(f1)
will perform the operation suggested intuitively by the notation, checking
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