<a id="sphx-glr-auto-examples-optimization"></a>

# Optimization

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<!-- thumbnail-parent-div-open --><div class="sphx-glr-thumbcontainer" data-sgtags='["MRI", "Denoising"]' tooltip="This example shows how to use variational 3D denoisers for denoising a 3D image. We first apply a standard soft-thresholding wavelet denoiser to a 3D brain MRI volume, as well as a 3D TV denoiser. We then extend the wavelet denoiser objective to a redundant dictionary of wavelet bases, which does not admit a closed-form solution. We solve the denoising problem using the Dykstra-like algorithm.">![](auto_examples/optimization/images/thumb/sphx_glr_demo_3D_denoising_thumb.png)

[3D denoising of brain MRI with wavelet and TV priors](https://deepinv.org/auto_examples/optimization/demo_3D_denoising.html.md)

  <div class="sphx-glr-thumbnail-title">3D denoising of brain MRI with wavelet and TV priors</div>
</div><div class="sphx-glr-thumbcontainer" data-sgtags='["Denoising", "Inpainting"]' tooltip="In this example we use the expected patch log likelihood (EPLL) prior zoran2011learning. for denoising and inpainting of natural images. To this end, we consider the inverse problem y = Ax+\\epsilon, where A is either the identity (for denoising) or a masking operator (for inpainting) and \\epsilon\\sim\\mathcal{N}(0,\\sigma^2 I) is white Gaussian noise with standard deviation \\sigma.">![](auto_examples/optimization/images/thumb/sphx_glr_demo_epll_thumb.png)

[Expected Patch Log Likelihood (EPLL) for Denoising and Inpainting](https://deepinv.org/auto_examples/optimization/demo_epll.html.md)

  <div class="sphx-glr-thumbnail-title">Expected Patch Log Likelihood (EPLL) for Denoising and Inpainting</div>
</div><div class="sphx-glr-thumbcontainer" data-sgtags='["Deblurring"]' tooltip="This example shows how to use a standard TV prior for image deblurring. The problem writes as y = Ax + \\epsilon where A is a convolutional operator and \\epsilon is the realization of some Gaussian noise. The goal is to recover the original image x from the blurred and noisy image y. The TV prior is used to regularize the problem.">![](auto_examples/optimization/images/thumb/sphx_glr_demo_TV_minimisation_thumb.png)

[Image deblurring with Total-Variation (TV) prior](https://deepinv.org/auto_examples/optimization/demo_TV_minimisation.html.md)

  <div class="sphx-glr-thumbnail-title">Image deblurring with Total-Variation (TV) prior</div>
</div><div class="sphx-glr-thumbcontainer" data-sgtags='["Deblurring"]' tooltip="In this example, we show how to solve a deblurring inverse problem using an explicit prior.">![](auto_examples/optimization/images/thumb/sphx_glr_demo_custom_prior_thumb.png)

[Image deblurring with custom deep explicit prior.](https://deepinv.org/auto_examples/optimization/demo_custom_prior.html.md)

  <div class="sphx-glr-thumbnail-title">Image deblurring with custom deep explicit prior.</div>
</div><div class="sphx-glr-thumbcontainer" data-sgtags='["Inpainting"]' tooltip="This example shows how to use a standard wavelet prior for image inpainting. The problem writes as y = Ax + \\epsilon where A is a mask and \\epsilon is the realization of some Gaussian noise. The goal is to recover the original image x from the blurred and noisy image y. The wavelet prior is used to regularize the problem.">![](auto_examples/optimization/images/thumb/sphx_glr_demo_wavelet_prior_thumb.png)

[Image inpainting with wavelet prior](https://deepinv.org/auto_examples/optimization/demo_wavelet_prior.html.md)

  <div class="sphx-glr-thumbnail-title">Image inpainting with wavelet prior</div>
</div><div class="sphx-glr-thumbcontainer" data-sgtags='["Demosaicing"]' tooltip="This example reconstructs a full-resolution multispectral image from raw snapshot mosaiced measurements using various reconstruction algorithms.">![](auto_examples/optimization/images/thumb/sphx_glr_demo_multispectral_demosaicing_thumb.png)

[Multispectral demosaicing from raw sensor data](https://deepinv.org/auto_examples/optimization/demo_multispectral_demosaicing.html.md)

  <div class="sphx-glr-thumbnail-title">Multispectral demosaicing from raw sensor data</div>
</div><div class="sphx-glr-thumbcontainer" data-sgtags='["Tomography"]' tooltip="In this example we use patch priors for limited angle computed tomography. More precisely, we consider the inverse problem y = \\mathrm{noisy}(Ax), where A is the discretized Radon transform with 100 equispace angles between 20 and 160 degrees. For the reconstruction, we minimize the variational problem">![](auto_examples/optimization/images/thumb/sphx_glr_demo_patch_priors_CT_thumb.png)

[Patch priors for limited-angle computed tomography](https://deepinv.org/auto_examples/optimization/demo_patch_priors_CT.html.md)

  <div class="sphx-glr-thumbnail-title">Patch priors for limited-angle computed tomography</div>
</div><div class="sphx-glr-thumbcontainer" data-sgtags='["Tomography", "Deblurring", "PET/SPECT", "Low-photon imaging"]' tooltip="This example demonstrates how to solve Poisson inverse problems using the Maximum-Likelihood Expectation-Maximization (MLEM) algorithm sheppMaximumLikelihoodReconstruction1982, also known as the Richardson-Lucy algorithm in the deconvolution setting richardsonBayesianBasedIterativeMethod1972,lucyIterativeTechniqueRectification1974.">![](auto_examples/optimization/images/thumb/sphx_glr_demo_poisson_mlem_thumb.png)

[Poisson Inverse Problems with Maximum-Likelihood Expectation-Maximization (MLEM)](https://deepinv.org/auto_examples/optimization/demo_poisson_mlem.html.md)

  <div class="sphx-glr-thumbnail-title">Poisson Inverse Problems with Maximum-Likelihood Expectation-Maximization (MLEM)</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This code shows how to reconstruct a noisy and incomplete image using the deep image prior.">![](auto_examples/optimization/images/thumb/sphx_glr_demo_dip_thumb.png)

[Reconstructing an image using the deep image prior.](https://deepinv.org/auto_examples/optimization/demo_dip.html.md)

  <div class="sphx-glr-thumbnail-title">Reconstructing an image using the deep image prior.</div>
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