# Operators & Noise ## Operators Operators describe the forward model $z = A(x,\theta)$, where $x$ is the input image and $\theta$ are the parameters of the operator. The parameters $\theta$ can be sampled using random generators, which are available for some specific classes. Using automatic differentiation, we can compute derivatives w.r.t both the input $x$ or the parameters $\theta$. This is particular useful when dealing with blind inverse problems or parameter estimation. #### TIP The operator you are looking for is not on this list? **See** [Bring your own physics](https://deepinv.org/auto_examples/basics/demo_custom_physics.html.md#sphx-glr-auto-examples-basics-demo-custom-physics-py) for how to implement your own physics operator. #### Operators, Definitions, and Generators | **Family** | **Operators** | **Generators** | |----------------------------------|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| | Pixelwise | [`deepinv.physics.Denoising`](https://deepinv.org/api/stubs/deepinv.physics.Denoising.html.md#deepinv.physics.Denoising)


[`deepinv.physics.Inpainting`](https://deepinv.org/api/stubs/deepinv.physics.Inpainting.html.md#deepinv.physics.Inpainting)


[`deepinv.physics.Demosaicing`](https://deepinv.org/api/stubs/deepinv.physics.Demosaicing.html.md#deepinv.physics.Demosaicing)


[`deepinv.physics.Decolorize`](https://deepinv.org/api/stubs/deepinv.physics.Decolorize.html.md#deepinv.physics.Decolorize)


[`deepinv.physics.SpatialUnwrapping`](https://deepinv.org/api/stubs/deepinv.physics.SpatialUnwrapping.html.md#deepinv.physics.SpatialUnwrapping)

| [`BernoulliSplittingMaskGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.BernoulliSplittingMaskGenerator.html.md#deepinv.physics.generator.BernoulliSplittingMaskGenerator)


[`GaussianSplittingMaskGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.GaussianSplittingMaskGenerator.html.md#deepinv.physics.generator.GaussianSplittingMaskGenerator)


[`MultiplicativeSplittingMaskGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.MultiplicativeSplittingMaskGenerator.html.md#deepinv.physics.generator.MultiplicativeSplittingMaskGenerator)


[`Phase2PhaseSplittingMaskGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.Phase2PhaseSplittingMaskGenerator.html.md#deepinv.physics.generator.Phase2PhaseSplittingMaskGenerator)


[`Artifact2ArtifactSplittingMaskGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.Artifact2ArtifactSplittingMaskGenerator.html.md#deepinv.physics.generator.Artifact2ArtifactSplittingMaskGenerator)

| | Blur & Super-Resolution | [`deepinv.physics.Blur`](https://deepinv.org/api/stubs/deepinv.physics.Blur.html.md#deepinv.physics.Blur)


[`deepinv.physics.BlurFFT`](https://deepinv.org/api/stubs/deepinv.physics.BlurFFT.html.md#deepinv.physics.BlurFFT)


[`deepinv.physics.SpaceVaryingBlur`](https://deepinv.org/api/stubs/deepinv.physics.SpaceVaryingBlur.html.md#deepinv.physics.SpaceVaryingBlur)


[`deepinv.physics.TiledSpaceVaryingBlur`](https://deepinv.org/api/stubs/deepinv.physics.TiledSpaceVaryingBlur.html.md#deepinv.physics.TiledSpaceVaryingBlur)


[`deepinv.physics.Downsampling`](https://deepinv.org/api/stubs/deepinv.physics.Downsampling.html.md#deepinv.physics.Downsampling)


[`deepinv.physics.Upsampling`](https://deepinv.org/api/stubs/deepinv.physics.Upsampling.html.md#deepinv.physics.Upsampling)


[`deepinv.physics.DownsamplingMatlab`](https://deepinv.org/api/stubs/deepinv.physics.DownsamplingMatlab.html.md#deepinv.physics.DownsamplingMatlab)

| [`GaussianBlurGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.GaussianBlurGenerator.html.md#deepinv.physics.generator.GaussianBlurGenerator)


[`MotionBlurGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.MotionBlurGenerator.html.md#deepinv.physics.generator.MotionBlurGenerator)


[`DownsamplingGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.DownsamplingGenerator.html.md#deepinv.physics.generator.DownsamplingGenerator)


[`DiffractionBlurGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.DiffractionBlurGenerator.html.md#deepinv.physics.generator.DiffractionBlurGenerator)


[`ProductConvolutionBlurGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.ProductConvolutionBlurGenerator.html.md#deepinv.physics.generator.ProductConvolutionBlurGenerator)


[`TiledBlurGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.TiledBlurGenerator.html.md#deepinv.physics.generator.TiledBlurGenerator)


[`ConfocalBlurGenerator3D`](https://deepinv.org/api/stubs/deepinv.physics.generator.ConfocalBlurGenerator3D.html.md#deepinv.physics.generator.ConfocalBlurGenerator3D)


[`gaussian_blur`](https://deepinv.org/api/stubs/deepinv.physics.functional.gaussian_blur.html.md#deepinv.physics.functional.gaussian_blur), [`sinc_filter`](https://deepinv.org/api/stubs/deepinv.physics.functional.sinc_filter.html.md#deepinv.physics.functional.sinc_filter)


[`bilinear_filter`](https://deepinv.org/api/stubs/deepinv.physics.functional.bilinear_filter.html.md#deepinv.physics.functional.bilinear_filter), [`bicubic_filter`](https://deepinv.org/api/stubs/deepinv.physics.functional.bicubic_filter.html.md#deepinv.physics.functional.bicubic_filter)


[`Zernike`](https://deepinv.org/api/stubs/deepinv.physics.generator.Zernike.html.md#deepinv.physics.generator.Zernike)

| | Magnetic Resonance Imaging (MRI) | [`deepinv.physics.MRI`](https://deepinv.org/api/stubs/deepinv.physics.MRI.html.md#deepinv.physics.MRI)


[`deepinv.physics.MultiCoilMRI`](https://deepinv.org/api/stubs/deepinv.physics.MultiCoilMRI.html.md#deepinv.physics.MultiCoilMRI)


[`deepinv.physics.DynamicMRI`](https://deepinv.org/api/stubs/deepinv.physics.DynamicMRI.html.md#deepinv.physics.DynamicMRI)


[`deepinv.physics.SequentialMRI`](https://deepinv.org/api/stubs/deepinv.physics.SequentialMRI.html.md#deepinv.physics.SequentialMRI)


The above all also natively support 3D MRI.


[`deepinv.physics.NonCartesianMRI`](https://deepinv.org/api/stubs/deepinv.physics.NonCartesianMRI.html.md#deepinv.physics.NonCartesianMRI)

| [`GaussianMaskGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.GaussianMaskGenerator.html.md#deepinv.physics.generator.GaussianMaskGenerator)


[`RandomMaskGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.RandomMaskGenerator.html.md#deepinv.physics.generator.RandomMaskGenerator)


[`EquispacedMaskGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.EquispacedMaskGenerator.html.md#deepinv.physics.generator.EquispacedMaskGenerator)


[`PolyOrderMaskGenerator`](https://deepinv.org/api/stubs/deepinv.physics.generator.PolyOrderMaskGenerator.html.md#deepinv.physics.generator.PolyOrderMaskGenerator)


The above all also support k+t dynamic sampling.

| | Tomography | [`deepinv.physics.Tomography`](https://deepinv.org/api/stubs/deepinv.physics.Tomography.html.md#deepinv.physics.Tomography)


[`deepinv.physics.TomographyWithAstra`](https://deepinv.org/api/stubs/deepinv.physics.TomographyWithAstra.html.md#deepinv.physics.TomographyWithAstra)


[`deepinv.physics.PET`](https://deepinv.org/api/stubs/deepinv.physics.PET.html.md#deepinv.physics.PET)

| | | Ultrasound | [`deepinv.physics.UltrasoundPlaneWave`](https://deepinv.org/api/stubs/deepinv.physics.UltrasoundPlaneWave.html.md#deepinv.physics.UltrasoundPlaneWave)

| | | Remote Sensing & Multispectral | [`deepinv.physics.Pansharpen`](https://deepinv.org/api/stubs/deepinv.physics.Pansharpen.html.md#deepinv.physics.Pansharpen)


[`deepinv.physics.HyperSpectralUnmixing`](https://deepinv.org/api/stubs/deepinv.physics.HyperSpectralUnmixing.html.md#deepinv.physics.HyperSpectralUnmixing)


[`deepinv.physics.CompressiveSpectralImaging`](https://deepinv.org/api/stubs/deepinv.physics.CompressiveSpectralImaging.html.md#deepinv.physics.CompressiveSpectralImaging)

| | | Compressive | [`deepinv.physics.CompressedSensing`](https://deepinv.org/api/stubs/deepinv.physics.CompressedSensing.html.md#deepinv.physics.CompressedSensing)


[`deepinv.physics.StructuredRandom`](https://deepinv.org/api/stubs/deepinv.physics.StructuredRandom.html.md#deepinv.physics.StructuredRandom)


[`deepinv.physics.SinglePixelCamera`](https://deepinv.org/api/stubs/deepinv.physics.SinglePixelCamera.html.md#deepinv.physics.SinglePixelCamera)

| | | Radio Interferometric Imaging | [`deepinv.physics.RadioInterferometry`](https://deepinv.org/api/stubs/deepinv.physics.RadioInterferometry.html.md#deepinv.physics.RadioInterferometry)

| | | Single-Photon Lidar | [`deepinv.physics.SinglePhotonLidar`](https://deepinv.org/api/stubs/deepinv.physics.SinglePhotonLidar.html.md#deepinv.physics.SinglePhotonLidar)

| | | Dehazing | [`deepinv.physics.Haze`](https://deepinv.org/api/stubs/deepinv.physics.Haze.html.md#deepinv.physics.Haze)

| | | Phase Retrieval | [`deepinv.physics.PhaseRetrieval`](https://deepinv.org/api/stubs/deepinv.physics.PhaseRetrieval.html.md#deepinv.physics.PhaseRetrieval)


[`RandomPhaseRetrieval`](https://deepinv.org/api/stubs/deepinv.physics.RandomPhaseRetrieval.html.md#deepinv.physics.RandomPhaseRetrieval)


[`StructuredRandomPhaseRetrieval`](https://deepinv.org/api/stubs/deepinv.physics.StructuredRandomPhaseRetrieval.html.md#deepinv.physics.StructuredRandomPhaseRetrieval)


[`Ptychography`](https://deepinv.org/api/stubs/deepinv.physics.Ptychography.html.md#deepinv.physics.Ptychography)


[`PtychographyLinearOperator`](https://deepinv.org/api/stubs/deepinv.physics.PtychographyLinearOperator.html.md#deepinv.physics.PtychographyLinearOperator)

| [`build_probe`](https://deepinv.org/api/stubs/deepinv.physics.phase_retrieval.build_probe.html.md#deepinv.physics.phase_retrieval.build_probe)


[`generate_shifts`](https://deepinv.org/api/stubs/deepinv.physics.phase_retrieval.generate_shifts.html.md#deepinv.physics.phase_retrieval.generate_shifts)

| | Inverse Scattering | [`deepinv.physics.Scattering`](https://deepinv.org/api/stubs/deepinv.physics.Scattering.html.md#deepinv.physics.Scattering)

| | ## Wrappers Wrappers are operators that can be used to adapt existing operators to a new problem. #### Wrappers | **Family** | **Operators** | |------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| | Multiscale | [`deepinv.physics.PhysicsMultiScaler`](https://deepinv.org/api/stubs/deepinv.physics.PhysicsMultiScaler.html.md#deepinv.physics.PhysicsMultiScaler)


[`deepinv.physics.LinearPhysicsMultiScaler`](https://deepinv.org/api/stubs/deepinv.physics.LinearPhysicsMultiScaler.html.md#deepinv.physics.LinearPhysicsMultiScaler)

| | Padding/Cropping | [`deepinv.physics.PhysicsCropper`](https://deepinv.org/api/stubs/deepinv.physics.PhysicsCropper.html.md#deepinv.physics.PhysicsCropper)

| ## Noise distributions Noise distributions describe the noise model $N$, where $y = N(z)$ with $z=A(x)$. The noise models can be assigned to **any** operator in the list above, by setting the [`set_noise_model`](https://deepinv.org/api/stubs/deepinv.physics.Physics.html.md#deepinv.physics.Physics.set_noise_model) attribute at initialization. By default, the noise model is set to [`ZeroNoise`](https://deepinv.org/api/stubs/deepinv.physics.ZeroNoise.html.md#deepinv.physics.ZeroNoise). #### Noise Distributions and Their Probability Distributions | **Noise** | $y|z$ | |----------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------| | [`deepinv.physics.ZeroNoise`](https://deepinv.org/api/stubs/deepinv.physics.ZeroNoise.html.md#deepinv.physics.ZeroNoise) | $y=z$ | | [`deepinv.physics.GaussianNoise`](https://deepinv.org/api/stubs/deepinv.physics.GaussianNoise.html.md#deepinv.physics.GaussianNoise) | $y\sim \mathcal{N}(z, I\sigma^2)$ | | [`deepinv.physics.PoissonNoise`](https://deepinv.org/api/stubs/deepinv.physics.PoissonNoise.html.md#deepinv.physics.PoissonNoise) | $y \sim \mathcal{P}(z/\gamma)$ | | [`deepinv.physics.PoissonGaussianNoise`](https://deepinv.org/api/stubs/deepinv.physics.PoissonGaussianNoise.html.md#deepinv.physics.PoissonGaussianNoise) | $y = \gamma z + \epsilon$, $z\sim\mathcal{P}(\frac{z}{\gamma})$, $\epsilon\sim\mathcal{N}(0, I \sigma^2)$ | | [`deepinv.physics.LaplaceNoise`](https://deepinv.org/api/stubs/deepinv.physics.LaplaceNoise.html.md#deepinv.physics.LaplaceNoise) | $y\sim\text{Laplace}(z, b)$ | | [`deepinv.physics.LogPoissonNoise`](https://deepinv.org/api/stubs/deepinv.physics.LogPoissonNoise.html.md#deepinv.physics.LogPoissonNoise) | $y = -\frac{1}{\mu} \log(\frac{\mathcal{P}(\exp(-\mu z) N_0)}{N_0})$ | | [`deepinv.physics.UniformNoise`](https://deepinv.org/api/stubs/deepinv.physics.UniformNoise.html.md#deepinv.physics.UniformNoise) | $y\sim \mathcal{U}(z-a, z+b)$ | | [`deepinv.physics.SaltPepperNoise`](https://deepinv.org/api/stubs/deepinv.physics.SaltPepperNoise.html.md#deepinv.physics.SaltPepperNoise) | $y = \begin{cases} 0 & \text{if } z < p\\ x & \text{if } z \in [p, 1-s]\\ 1 & \text{if } z > 1 - s\end{cases}$ with $z\sim\mathcal{U}(0,1)$ | | [`deepinv.physics.GammaNoise`](https://deepinv.org/api/stubs/deepinv.physics.GammaNoise.html.md#deepinv.physics.GammaNoise) | $y = \mathcal{G}(\ell, x/\ell)$ | | [`deepinv.physics.FisherTippettNoise`](https://deepinv.org/api/stubs/deepinv.physics.FisherTippettNoise.html.md#deepinv.physics.FisherTippettNoise) | $p(y\vert x) = \frac{\ell^{\ell}}{\Gamma(\ell)}\mathrm{e}^{\ell(y-x)}\mathrm{e}^{-\ell\mathrm{e}^{(y-x)}}$ | | [`deepinv.physics.RicianNoise`](https://deepinv.org/api/stubs/deepinv.physics.RicianNoise.html.md#deepinv.physics.RicianNoise) | $y = \sqrt{(x + \sigma \epsilon_1)^2 + (\sigma \epsilon_2)^2}$, $\epsilon_1\sim\mathcal{N}(0,I)$, $\epsilon_2\sim\mathcal{N}(0,I)$ |