.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/sampling/demo_diffusion_sde.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_sampling_demo_diffusion_sde.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_sampling_demo_diffusion_sde.py:


Posterior Sampling for Inverse Problems with Stochastic Differential Equations modeling.
====================================================================================================

This demo shows you how to use
:class:`deepinv.sampling.PosteriorDiffusion` to perform posterior sampling. It also can be used to perform unconditional image generation with arbitrary denoisers, if the data fidelity term is not specified.

This method requires:

* A well-trained denoiser with varying noise levels (ideally with large noise levels) (e.g., :class:`deepinv.models.NCSNpp`).

* A (noisy) data fidelity term (e.g., :class:`deepinv.sampling.DPSDataFidelity`).

* Define a drift term :math:`f(x, t)` and a diffusion term :math:`g(t)` for the forward-time SDE. They can be defined through the :class:`deepinv.sampling.DiffusionSDE` (e.g., :class:`deepinv.sampling.VarianceExplodingDiffusion`).

The :class:`deepinv.sampling.PosteriorDiffusion` class can be used to perform posterior sampling for inverse problems.
Consider the acquisition model:

.. math::
     y = \noise{\forw{x}}.

This class defines the reverse-time SDE for the posterior distribution :math:`p(x|y)` given the data :math:`y`:

.. math::
     d\, x_t = \left( f(x_t, t) - \frac{1 + \alpha}{2} g(t)^2 \nabla_{x_t} \log p_t(x_t | y) \right) d\,t + g(t) \sqrt{\alpha} d\, w_{t}

where :math:`f` is the drift term, :math:`g` is the diffusion coefficient and :math:`w` is the standard Brownian motion. The drift term and the diffusion coefficient are defined by the underlying (unconditional) forward-time SDE `sde`. The (conditional) score function :math:`\nabla_{x_t} \log p_t(x_t | y)` can be decomposed using the Bayes' rule:

.. math::
     \nabla_{x_t} \log p_t(x_t | y) = \nabla_{x_t} \log p_t(x_t) + \nabla_{x_t} \log p_t(y | x_t).

The first term is the score function of the unconditional SDE, which is typically approximated by a MMSE denoiser (`denoiser`) using the well-known Tweedie's formula, while the
second term is approximated by the (noisy) data-fidelity term (`data_fidelity`).
We implement various data-fidelity terms in `the documentations <https://deepinv.github.io/deepinv/user_guide/reconstruction/sampling.html#id2>`_.

.. GENERATED FROM PYTHON SOURCE LINES 38-46

---------------------------------------------------

Let us import the necessary modules, define the denoiser and the SDE.

In this example, we use the Variance-Exploding SDE, whose forward process is defined as:

.. math::
    d\, x_t = \sigma(t) d\, w_t \quad \mbox{where } \sigma(t) = \sigma_{\mathrm{min}}\left( \frac{\sigma_{\mathrm{max}}}{\sigma_{\mathrm{min}}}\right)^t

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.. code-block:: Python


    import torch
    import deepinv as dinv
    from deepinv.models import NCSNpp, EDMPrecond

    device = "cuda" if torch.cuda.is_available() else "cpu"
    dtype = torch.float64
    figsize = 2.5







.. GENERATED FROM PYTHON SOURCE LINES 55-88

.. code-block:: Python

    from deepinv.sampling import (
        PosteriorDiffusion,
        DPSDataFidelity,
        EulerSolver,
        VarianceExplodingDiffusion,
    )
    from deepinv.optim import ZeroFidelity

    # In this example, we use the pre-trained FFHQ-64 model from the
    # EDM framework: https://arxiv.org/pdf/2206.00364 .
    # The network architecture is from Song et al: https://arxiv.org/abs/2011.13456 .
    unet = NCSNpp(pretrained="download")
    denoiser = EDMPrecond(model=unet).to(device)

    # The solution is obtained by calling the SDE object with a desired solver (here, Euler).
    # The reproducibility of the SDE Solver class can be controlled by providing the pseudo-random number generator.
    num_steps = 150
    rng = torch.Generator(device).manual_seed(42)
    timesteps = torch.linspace(1, 0.001, num_steps)
    solver = EulerSolver(timesteps=timesteps, rng=rng)


    sigma_min = 0.02
    sigma_max = 20
    sde = VarianceExplodingDiffusion(
        sigma_max=sigma_max,
        sigma_min=sigma_min,
        alpha=0.5,
        device=device,
        dtype=dtype,
    )






.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Downloading: "https://huggingface.co/deepinv/edm/resolve/main/ncsnpp-ffhq64-uncond-ve.pt?download=true" to /home/runner/.cache/torch/hub/checkpoints/ncsnpp-ffhq64-uncond-ve.pt

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    100%|██████████| 240M/240M [00:00<00:00, 348MB/s]




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Reverse-time SDE as sampling process
--------------------------------------

When the data fidelity is not given, the posterior diffusion is equivalent to the unconditional diffusion.
Sampling is performed by solving the reverse-time SDE. To do so, we generate a reverse-time trajectory.

.. GENERATED FROM PYTHON SOURCE LINES 94-125

.. code-block:: Python


    model = PosteriorDiffusion(
        data_fidelity=ZeroFidelity(),
        sde=sde,
        denoiser=denoiser,
        solver=solver,
        dtype=dtype,
        device=device,
    )
    sample_seed_1, trajectory_seed_1 = model(
        y=None,
        physics=None,
        x_init=(1, 3, 64, 64),
        seed=1,
        get_trajectory=True,
    )
    dinv.utils.plot(
        sample_seed_1,
        titles="Unconditional generation",
        show=True,
        save_fn="sde_sample.png",
        figsize=(figsize, figsize),
    )
    dinv.utils.save_videos(
        trajectory_seed_1.cpu()[::10],
        time_dim=0,
        titles=["VE-SDE Trajectory"],
        save_fn="sde_trajectory.gif",
        figsize=(figsize, figsize),
    )








.. GENERATED FROM PYTHON SOURCE LINES 143-154

.. container:: image-row

   .. image-sg:: /auto_examples/images/sde_sample.png
      :alt: other example
      :srcset: /auto_examples/images/sde_sample.png
      :class: custom-gif

   .. image-sg:: /auto_examples/images/sde_trajectory.gif
      :alt: example learn_samples
      :srcset: /auto_examples/images/sde_trajectory.gif
      :class: custom-gif

.. GENERATED FROM PYTHON SOURCE LINES 156-158

When the data fidelity is given, together with the measurements and the physics, this class can be used to perform posterior sampling for inverse problems.
For example, consider the inpainting problem, where we have a noisy image and we want to recover the original image.

.. GENERATED FROM PYTHON SOURCE LINES 159-189

.. code-block:: Python


    x = sample_seed_1
    physics = dinv.physics.Inpainting(tensor_size=x.shape[1:], mask=0.4, device=device)
    y = physics(x)

    model = PosteriorDiffusion(
        data_fidelity=DPSDataFidelity(denoiser=denoiser),
        denoiser=denoiser,
        sde=sde,
        solver=solver,
        dtype=dtype,
        device=device,
    )

    # To perform posterior sampling, we need to provide the measurements, the physics and the solver.
    # Moreover, when the physics is given, the initial point can be inferred from the physics if not given explicitly.
    x_hat, trajectory = model(
        y,
        physics,
        seed=11,
        get_trajectory=True,
    )
    # Here, we plot the original image, the measurement and the posterior sample
    dinv.utils.plot(
        [x, y, x_hat],
        show=True,
        titles=["Original", "Measurement", "Posterior sample"],
        save_fn="posterior_sample.png",
        figsize=(figsize * 3, figsize),
    )



.. image-sg:: /auto_examples/sampling/images/sphx_glr_demo_diffusion_sde_001.png
   :alt: Original, Measurement, Posterior sample
   :srcset: /auto_examples/sampling/images/sphx_glr_demo_diffusion_sde_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 190-197

.. code-block:: Python

    dinv.utils.save_videos(
        trajectory[::10],
        time_dim=0,
        save_fn="posterior_trajectory.gif",
        figsize=(figsize, figsize),
    )








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We obtain the following posterior trajectory

.. container:: image-row

   .. image-sg:: /auto_examples/images/posterior_sample.png
      :alt: example learn_samples
      :srcset: /auto_examples/images/posterior_sample.png
      :class: custom-gif

   .. image-sg:: /auto_examples/images/posterior_trajectory.gif
      :alt: example learn_samples
      :srcset: /auto_examples/images/posterior_trajectory.gif
      :class: custom-gif

.. GENERATED FROM PYTHON SOURCE LINES 231-237

Varying samples
---------------

One can obtain varying samples by using a different seed.
To ensure the reproducibility, if the parameter `rng` is given, the same sample will
be generated when the same seed is used

.. GENERATED FROM PYTHON SOURCE LINES 237-256

.. code-block:: Python


    # By changing the seed, we can obtain different samples:
    x_hat_seed_111 = model(
        y,
        physics,
        seed=111,
    )
    dinv.utils.plot(
        [x_hat, x_hat_seed_111],
        titles=[
            "posterior sample: seed 11",
            "posterior sample: seed 111",
        ],
        show=True,
        save_fn="posterior_samples.png",
        figsize=(figsize * 2, figsize),
    )





.. image-sg:: /auto_examples/sampling/images/sphx_glr_demo_diffusion_sde_002.png
   :alt: posterior sample: seed 11, posterior sample: seed 111
   :srcset: /auto_examples/sampling/images/sphx_glr_demo_diffusion_sde_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 273-281

We obtain the following posterior trajectory

.. container:: image-row

   .. image-sg:: /auto_examples/images/posterior_samples.png
      :alt: example learn_samples
      :srcset: /auto_examples/images/posterior_samples.png
      :class: custom-gif

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Plug-and-play Posterior Sampling with arbitrary denoisers
---------------------------------------------------------

The :class:`deepinv.sampling.PosteriorDiffusion` class can be used together with any (well-trained) denoisers for posterior sampling.
For example, we can use the :class:`deepinv.models.DRUNet` for posterior sampling.
We can also change the underlying SDE, for example change the `sigma_max` value.

.. GENERATED FROM PYTHON SOURCE LINES 289-329

.. code-block:: Python


    sigma_min = 0.02
    sigma_max = 2.0
    rng = torch.Generator(device)
    timesteps = torch.linspace(1, 0.001, 200)
    solver = EulerSolver(timesteps=timesteps, rng=rng)

    denoiser = dinv.models.DRUNet(pretrained="download").to(device)

    sde = VarianceExplodingDiffusion(
        sigma_max=sigma_max, sigma_min=sigma_min, alpha=0.1, device=device, dtype=dtype
    )

    # As a plug-and-play sampling method, we can also change the data fidelity term.
    # But the sample quality depends on the quality of the denoiser and the data fidelity term.
    model = PosteriorDiffusion(
        data_fidelity=dinv.optim.L2(),
        denoiser=denoiser,
        sde=sde,
        solver=solver,
        dtype=dtype,
        device=device,
    )

    # To perform posterior sampling, we need to provide the measurements, the physics and the solver.
    x_hat, trajectory = model(
        y,
        physics,
        seed=11,
        get_trajectory=True,
    )
    # Here, we plot the original image, the measurement and the posterior sample
    dinv.utils.plot(
        [x, y, x_hat],
        show=True,
        titles=["Original", "Measurement", "Posterior sample DRUNet"],
        figsize=(figsize * 3, figsize),
        save_fn="posterior_sample_DRUNet.png",
    )




.. image-sg:: /auto_examples/sampling/images/sphx_glr_demo_diffusion_sde_003.png
   :alt: Original, Measurement, Posterior sample DRUNet
   :srcset: /auto_examples/sampling/images/sphx_glr_demo_diffusion_sde_003.png
   :class: sphx-glr-single-img





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We obtain the following posterior trajectory

.. container:: image-row

   .. image-sg:: /auto_examples/images/posterior_sample_DRUNet.png
      :alt: example learn_samples
      :srcset: /auto_examples/images/posterior_sample_DRUNet.png
      :class: custom-gif

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We can switch to a different denoiser, for example, the DiffUNet denoiser from the EDM framework.


.. GENERATED FROM PYTHON SOURCE LINES 361-397

.. code-block:: Python

    denoiser = dinv.models.DiffUNet(pretrained="download").to(device)

    sigma_min = 0.02
    sigma_max = 5.0
    rng = torch.Generator(device)
    timesteps = torch.linspace(1, 0.001, 200)
    solver = EulerSolver(timesteps=timesteps, rng=rng)
    sde = VarianceExplodingDiffusion(
        sigma_max=sigma_max, sigma_min=sigma_min, alpha=0.5, device=device, dtype=dtype
    )
    model = PosteriorDiffusion(
        data_fidelity=DPSDataFidelity(denoiser=denoiser),
        denoiser=denoiser,
        rescale=True,
        sde=sde,
        solver=solver,
        dtype=dtype,
        device=device,
    )

    # To perform posterior sampling, we need to provide the measurements, the physics and the solver.
    x_hat, trajectory = model(
        y,
        physics,
        seed=1,
        get_trajectory=True,
    )
    # Here, we plot the original image, the measurement and the posterior sample
    dinv.utils.plot(
        [x, y, x_hat],
        show=True,
        titles=["Original", "Measurement", "Posterior sample DiffUNet"],
        save_fn="posterior_sample_DiffUNet.png",
        figsize=(figsize * 3, figsize),
    )




.. image-sg:: /auto_examples/sampling/images/sphx_glr_demo_diffusion_sde_004.png
   :alt: Original, Measurement, Posterior sample DiffUNet
   :srcset: /auto_examples/sampling/images/sphx_glr_demo_diffusion_sde_004.png
   :class: sphx-glr-single-img





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We obtain the following posterior trajectory

.. container:: image-row

   .. image-sg:: /auto_examples/images/posterior_sample_DiffUNet.png
      :alt: example learn_samples
      :srcset: /auto_examples/images/posterior_sample_DiffUNet.png
      :class: custom-gif


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (12 minutes 20.718 seconds)


.. _sphx_glr_download_auto_examples_sampling_demo_diffusion_sde.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: demo_diffusion_sde.ipynb <demo_diffusion_sde.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: demo_diffusion_sde.py <demo_diffusion_sde.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: demo_diffusion_sde.zip <demo_diffusion_sde.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_