# AngularSpectrumPropagation
### *class* deepinv.physics.AngularSpectrumPropagation(img_size, H, pixel_size=1.0, dtype=torch.cfloat, device='cpu', \*\*kwargs)
Bases: [`LinearPhysics`](https://deepinv.org/api/stubs/deepinv.physics.LinearPhysics.html.md#deepinv.physics.LinearPhysics)
Linear angular-spectrum propagation with a user-defined transfer function.
This operator propagates a complex wave field $u$ according to
$$
P_H(u) = \mathcal{F}^{-1}\{H(f_x, f_y)\mathcal{F}\{u\}\}.
$$
The transfer function `H` can be supplied either as a tensor in the
unshifted ordering used by [`torch.fft.fft2()`](https://docs.pytorch.org/docs/stable/generated/torch.fft.fft2.html#torch.fft.fft2), or as a callable with
signature `H(fx, fy)`. In the latter case, the callable is evaluated once
at construction on frequency grids expressed in cycles per unit of
`pixel_size`.
This class models propagation of an already formed complex field. Object
transmission functions such as $T(\delta, \beta)=\exp(g(\delta,
\beta))$ should be represented by a separate nonlinear
[`deepinv.physics.Physics`](https://deepinv.org/api/stubs/deepinv.physics.Physics.html.md#deepinv.physics.Physics) operator and composed with this one.
#### NOTE
The FFT implementation imposes periodic boundary conditions. Pad the
field and define `H` on the padded grid when circular wrap-around is
not appropriate for the experiment.
* **Parameters:**
* **img_size** ([*tuple*](https://docs.python.org/3.12/library/stdtypes.html#tuple) *[*[*int*](https://docs.python.org/3.12/library/functions.html#int) *,* *...* *]*) – Shape of an unbatched input. The final two
entries are interpreted as `(height, width)`.
* **H** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) *,* *Callable*) – Transfer-function tensor or callable
`H(fx, fy)`. It must act pointwise without adding an output dimension. It may be (or return) a
scalar, have shape `(height, width)`, or contain leading dimensions that
broadcast to `img_size` (for example, `(channels, height, width)`).
* **pixel_size** ([*float*](https://docs.python.org/3.12/library/functions.html#float) *,* [*tuple*](https://docs.python.org/3.12/library/stdtypes.html#tuple) *[*[*float*](https://docs.python.org/3.12/library/functions.html#float) *,* [*float*](https://docs.python.org/3.12/library/functions.html#float) *]*) – Sampling pitch `dy, dx`. If `float`, then `dy, dy`, default 1.
* **dtype** ([*torch.dtype*](https://docs.pytorch.org/docs/stable/tensor_attributes.html#torch.dtype)) – Complex dtype used to evaluate and store `H`.
Default: `torch.cfloat`.
* **device** ([*torch.device*](https://docs.pytorch.org/docs/stable/tensor_attributes.html#torch.device) *,* [*str*](https://docs.python.org/3.12/library/stdtypes.html#str)) – Device on which the frequency grid and
transfer function are created. Default: `"cpu"`.
* **Examples:**
Define a quadratic phase transfer function directly:
```pycon
>>> import torch
>>> from deepinv.physics import AngularSpectrumPropagation
>>> propagation = AngularSpectrumPropagation(
... img_size=(1, 32, 48),
... pixel_size=(2e-6, 3e-6),
... H=lambda fx, fy: torch.exp(
... -1j * 1e-10 * (fx.square() + fy.square())
... ),
... )
>>> x = torch.randn(2, 1, 32, 48, dtype=torch.cfloat)
>>> propagation(x).shape
torch.Size([2, 1, 32, 48])
```
#### A(x, \*\*kwargs)
Propagate a complex field without noise or sensor nonlinearities.
* **Parameters:**
**x** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – complex input image, where the last two dimensions are `(height, width)`.
* **Returns:**
[`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) `y`, the measurements, of the same shape as `x`.
* **Return type:**
[*Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)
#### A_adjoint(y, \*\*kwargs)
Apply the Hermitian adjoint of the propagation operator.
* **Parameters:**
**y** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – complex input image, where the last two dimensions are `(height, width)`.
* **Returns:**
[`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) `x`, of the same shape as `y`.
* **Return type:**
[*Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)
#### A_dagger(y, rcond=None, \*\*kwargs)
Apply the Moore-Penrose pseudo-inverse.
The operator is diagonal in the Fourier domain, so the pseudo-inverse is closed-form, analogous to [`deepinv.physics.DecomposablePhysics`](https://deepinv.org/api/stubs/deepinv.physics.DecomposablePhysics.html.md#deepinv.physics.DecomposablePhysics).
* **Parameters:**
* **y** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – complex input image, where the last two dimensions are `(height, width)`.
* **rcond** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – frequencies whose magnitude `< rcond * max(abs(H))` are mapped to zero. If `None`, uses real component dtype machine precision.
* **Returns:**
[`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) `x`, of the same shape as `y`.
* **Return type:**
[*Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)
#### *static* frequency_grid(img_size, pixel_size=1.0, dtype=torch.float32, device='cpu')
Return unshifted spatial-frequency grids `(fx, fy)`.
Frequencies are in cycles per unit of `pixel_size` and follow the
ordering expected by [`torch.fft.fft2()`](https://docs.pytorch.org/docs/stable/generated/torch.fft.fft2.html#torch.fft.fft2).
* **Parameters:**
* **img_size** ([*tuple*](https://docs.python.org/3.12/library/stdtypes.html#tuple) *[*[*int*](https://docs.python.org/3.12/library/functions.html#int) *,* *...* *]*) – Shape whose final entries are
`(height, width)`.
* **pixel_size** ([*float*](https://docs.python.org/3.12/library/functions.html#float) *,* [*tuple*](https://docs.python.org/3.12/library/stdtypes.html#tuple) *[*[*float*](https://docs.python.org/3.12/library/functions.html#float) *,* [*float*](https://docs.python.org/3.12/library/functions.html#float) *]*) – Scalar pitch or
`(dy, dx)`.
* **dtype** ([*torch.dtype*](https://docs.pytorch.org/docs/stable/tensor_attributes.html#torch.dtype)) – Real floating-point dtype for the grids.
* **device** ([*torch.device*](https://docs.pytorch.org/docs/stable/tensor_attributes.html#torch.device) *,* [*str*](https://docs.python.org/3.12/library/stdtypes.html#str)) – Grid device.
* **Returns:**
The two grids `(fx, fy)`, each of shape `(height, width)`.
* **Return type:**
[tuple](https://docs.python.org/3.12/library/stdtypes.html#tuple)[[*Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor), [*Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)]
## Examples using `AngularSpectrumPropagation`: