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AdjointOperator documentation #2278

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64015ba
add adjoint operator doc
epapoutsellis Jan 29, 2026
2cd7fbe
fix docs
epapoutsellis Jan 29, 2026
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47 changes: 28 additions & 19 deletions Wrappers/Python/cil/optimisation/operators/Operator.py
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Expand Up @@ -425,29 +425,38 @@ def dot_test(operator, domain_init=None, range_init=None, tolerance=1e-6, **kwar
return False

class AdjointOperator(LinearOperator):

r"""
Adjoint of a linear operator.

"""
The Adjoint operator :math:`A^{*}: Y^{*}\rightarrow X^{*}` of a linear operator :math:`A: X\rightarrow Y` defined as

.. math:: <x, A^* y> = <Ax, y>
Given a linear operator :math:`A: X \to Y` between inner-product spaces
:math:`(X,\langle\cdot,\cdot\rangle_X)` and :math:`(Y,\langle\cdot,\cdot\rangle_Y)`,
its adjoint :math:`A^{*}: Y \to X` is defined by
\[
\langle Ax,\, y\rangle_Y \;=\; \langle x,\, A^{*}y\rangle_X,
\qquad \forall x\in X,\; y\in Y.
\]

Parameters
----------
Parameters
----------
operator : LinearOperator
The operator :math:`A` whose adjoint is constructed.

operator : A linear operator
Examples
--------
Verify the adjointness relation for the gradient operator :math:`G` and its adjoint
(the negative divergence in many discretisations):
>>> ig = ImageGeometry(2, 3)
>>> G = GradientOperator(ig)
>>> div = AdjointOperator(G) # represents G*
>>> x = G.domain.allocate("random_int")
>>> y = G.range.allocate("random_int")
>>> lhs = G.direct(x).dot(y) # <Gx, y>_Y
>>> rhs = x.dot(div.direct(y)) # <x, G* y>_X
>>> lhs == rhs
True
"""

Examples
--------
This example demonstrates that :math:` LHS:=<Gx, y> =<x, G^* y>=:RHS`, where :math:`G` is the gradient operator.
>>> ig = ImageGeometry(2,3)
>>> G = GradientOperator(ig)
>>> div = AdjointOperator(G)
>>> x = G.domain.allocate("random_int")
>>> y = G.range.allocate("random_int")
>>> lhs = G.direct(x).dot(y)
>>> rhs = x.dot(div.direct(y))
>>> lhs == rhs # returns True
"""

def __init__(self, operator):
super(AdjointOperator, self).__init__(domain_geometry=operator.range_geometry(),
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3 changes: 3 additions & 0 deletions docs/source/optimisation.rst
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Expand Up @@ -331,6 +331,9 @@ A :code:`ScaledOperator` represents the multiplication of any operator with a sc
.. autoclass:: cil.optimisation.operators.SumOperator
:members:

.. autoclass:: cil.optimisation.operators.AdjointOperator
:members:


Trivial operators
-----------------
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