Note
Go to the end to download the full example code.
Patch priors for limited-angle computed tomography
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
\[\begin{equation*}
\label{eq:min_prob}
\underset{x}{\arg\min} \quad \datafid{x}{y} + \lambda g(x).
\end{equation*}\]
Here, the regularizier \(g\) is explicitly defined as
\[\begin{equation*}
g(x)=\sum_{i\in\mathcal{I}} h(P_i x),
\end{equation*}\]
where \(P_i\) is the linear operator which extracts the \(i\)-th patch from the image \(x\) and \(h\) is a regularizer on the space of patches. We consider the following two choices of \(h\):
The expected patch log-likelihood (EPLL) prior was proposed in “From learning models of natural image patches to whole image restoration”. It sets \(h(x)=-\log(p_\theta(x))\), where \(p_\theta\) is the probability density function of a Gaussian mixture model. The parameters \(\theta\) are estimated a-priori on a (possibly small) data set of training patches using an expectation maximization algorithm. In contrast to the original paper by Zoran and Weiss, we minimize the arising variational problem by simply applying the Adam optimizers. For an example for using the (approximated) half-quadratic splitting algorithm proposed by Zoran and Weiss, we refer to the denoising example…
The patch normalizing flow regularizer (PatchNR) was proposed in “PatchNR: learning from very few images by patch normalizing flow regularization”. It models \(h(x)=-\log(p_{\theta}(x))\) as negative log-likelihood function of a probaility density function \(p_\theta={\mathcal{T}_\theta}_\#\mathcal{N}(0,I)\) which is given as the push-forward measure of a standard normal distribution under a normalizing flow (invertible neural network) \(\mathcal{T}_\theta\).
import torch
from torch.utils.data import DataLoader
from deepinv.datasets import PatchDataset
from deepinv import Trainer
from deepinv.physics import LogPoissonNoise, Tomography, Denoising, UniformNoise
from deepinv.optim import LogPoissonLikelihood, PatchPrior, PatchNR, EPLL
from deepinv.loss.metric import PSNR
from deepinv.utils import plot
from deepinv.utils.demo import load_torch_url
from tqdm import tqdm
device = "cuda" if torch.cuda.is_available() else "cpu"
dtype = torch.float32
Load training and test images
Here, we use downsampled images from the “LoDoPaB-CT dataset”. Moreover, we define the size of the used patches and generate the dataset of patches in the training images.
url = "https://huggingface.co/datasets/deepinv/LoDoPaB-CT_toy/resolve/main/LoDoPaB-CT_small.pt"
dataset = load_torch_url(url)
train_imgs = dataset["train_imgs"].to(device)
test_imgs = dataset["test_imgs"].to(device)
img_size = train_imgs.shape[-1]
patch_size = 3
verbose = True
train_dataset = PatchDataset(train_imgs, patch_size=patch_size, transforms=None)
/home/runner/work/deepinv/deepinv/deepinv/utils/demo.py:225: FutureWarning: You are using `torch.load` with `weights_only=False` (the current default value), which uses the default pickle module implicitly. It is possible to construct malicious pickle data which will execute arbitrary code during unpickling (See https://github.com/pytorch/pytorch/blob/main/SECURITY.md#untrusted-models for more details). In a future release, the default value for `weights_only` will be flipped to `True`. This limits the functions that could be executed during unpickling. Arbitrary objects will no longer be allowed to be loaded via this mode unless they are explicitly allowlisted by the user via `torch.serialization.add_safe_globals`. We recommend you start setting `weights_only=True` for any use case where you don't have full control of the loaded file. Please open an issue on GitHub for any issues related to this experimental feature.
out = torch.load(BytesIO(response.content))
Set parameters for EPLL and PatchNR
For PatchNR, we choose the number of hidden neurons in the subnetworks and for the training batch size and number of epochs. For EPLL, we set the number of mixture components and the maximum number of steps and batch size for fitting the EM algorithm.
patchnr_subnetsize = 128
patchnr_epochs = 5
patchnr_batch_size = 32
patchnr_learning_rate = 1e-4
epll_num_components = 20
epll_max_iter = 20
epll_batch_size = 10000
Training / EM algorithm
If the parameter retrain is False, we just load pretrained weights. Set the parameter to True for retraining. On the cpu, this takes up to a couple of minutes. After training, we define the corresponding patch priors
Note
The normalizing flow training minimizes the forward Kullback-Leibler (maximum likelihood) loss function given by
\[\mathcal{L}(\theta)=\mathrm{KL}(P_X,{\mathcal{T}_\theta}_\#P_Z)= \mathbb{E}_{x\sim P_X}[p_{{\mathcal{T}_\theta}_\#P_Z}(x)]+\mathrm{const},\]where \(\mathcal{T}_\theta\) is the normalizing flow with parameters \(\theta\), latent distribution \(P_Z\), data distribution \(P_X\) and push-forward measure \({\mathcal{T}_\theta}_\#P_Z\).
retrain = False
if retrain:
model_patchnr = PatchNR(
pretrained=None,
sub_net_size=patchnr_subnetsize,
device=device,
patch_size=patch_size,
)
patchnr_dataloader = DataLoader(
train_dataset,
batch_size=patchnr_batch_size,
shuffle=True,
drop_last=True,
)
class NFTrainer(Trainer):
def compute_loss(self, physics, x, y, train=True):
logs = {}
self.optimizer.zero_grad() # Zero the gradients
# Evaluate reconstruction network
invs, jac_inv = self.model(y)
# Compute the Kullback Leibler loss
loss_total = torch.mean(
0.5 * torch.sum(invs.view(invs.shape[0], -1) ** 2, -1)
- jac_inv.view(invs.shape[0])
)
current_log = (
self.logs_total_loss_train if train else self.logs_total_loss_eval
)
current_log.update(loss_total.item())
logs[f"TotalLoss"] = current_log.avg
if train:
loss_total.backward() # Backward the total loss
self.optimizer.step() # Optimizer step
return invs, logs
optimizer = torch.optim.Adam(
model_patchnr.normalizing_flow.parameters(), lr=patchnr_learning_rate
)
trainer = NFTrainer(
model=model_patchnr.normalizing_flow,
physics=Denoising(UniformNoise(1.0 / 255.0)),
optimizer=optimizer,
train_dataloader=patchnr_dataloader,
device=device,
losses=[],
epochs=patchnr_epochs,
online_measurements=True,
verbose=verbose,
)
trainer.train()
model_epll = EPLL(
pretrained=None,
n_components=epll_num_components,
patch_size=patch_size,
device=device,
)
epll_dataloader = DataLoader(
train_dataset,
batch_size=epll_batch_size,
shuffle=True,
drop_last=False,
)
model_epll.GMM.fit(epll_dataloader, verbose=verbose, max_iters=epll_max_iter)
else:
model_patchnr = PatchNR(
pretrained="PatchNR_lodopab_small2",
sub_net_size=patchnr_subnetsize,
device=device,
patch_size=patch_size,
)
model_epll = EPLL(
pretrained="GMM_lodopab_small2",
n_components=epll_num_components,
patch_size=patch_size,
device=device,
)
patchnr_prior = PatchPrior(model_patchnr, patch_size=patch_size)
epll_prior = PatchPrior(model_epll.negative_log_likelihood, patch_size=patch_size)
Downloading: "https://drive.google.com/uc?export=download&id=1Z2us9ZHjDGOlU6r1Jee0s2BBej2XV5-i" to /home/runner/.cache/torch/hub/checkpoints/PatchNR_lodopab_small.pt
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Definition of forward operator and noise model
The training depends only on the image domain or prior distribution. For the reconstruction, we now define forward operator and noise model. For the noise model, we use log-Poisson noise as suggested for the LoDoPaB dataset. Then, we generate an observation by applying the physics and compute the filtered backprojection.
mu = 1 / 50.0 * (362.0 / img_size)
N0 = 1024.0
num_angles = 100
noise_model = LogPoissonNoise(mu=mu, N0=N0)
data_fidelity = LogPoissonLikelihood(mu=mu, N0=N0)
angles = torch.linspace(20, 160, steps=num_angles)
physics = Tomography(
img_width=img_size, angles=angles, device=device, noise_model=noise_model
)
observation = physics(test_imgs)
fbp = physics.A_dagger(observation)
Reconstruction loop
We define a reconstruction loop for minimizing the variational problem using the Adam optimizer. As initialization, we choose the filtered backprojection.
optim_steps = 200
lr_variational_problem = 0.02
def minimize_variational_problem(prior, lam):
imgs = fbp.detach().clone()
imgs.requires_grad_(True)
optimizer = torch.optim.Adam([imgs], lr=lr_variational_problem)
for i in (progress_bar := tqdm(range(optim_steps))):
optimizer.zero_grad()
loss = data_fidelity(imgs, observation, physics).mean() + lam * prior(imgs)
loss.backward()
optimizer.step()
progress_bar.set_description("Step {}".format(i + 1))
return imgs.detach()
Run and plot
Finally, we run the reconstruction loop for both priors and plot the results. The regularization parameter is roughly choosen by a grid search but not fine-tuned
lam_patchnr = 120.0
lam_epll = 120.0
recon_patchnr = minimize_variational_problem(patchnr_prior, lam_patchnr)
recon_epll = minimize_variational_problem(epll_prior, lam_epll)
psnr_fbp = PSNR()(fbp, test_imgs).item()
psnr_patchnr = PSNR()(recon_patchnr, test_imgs).item()
psnr_epll = PSNR()(recon_epll, test_imgs).item()
print("PSNRs:")
print("Filtered Backprojection: {0:.2f}".format(psnr_fbp))
print("EPLL: {0:.2f}".format(psnr_epll))
print("PatchNR: {0:.2f}".format(psnr_patchnr))
plot(
[
test_imgs,
fbp.clip(0, 1),
recon_epll.clip(0, 1),
recon_patchnr.clip(0, 1),
],
["Ground truth", "Filtered Backprojection", "EPLL", "PatchNR"],
)
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PSNRs:
Filtered Backprojection: 24.16
EPLL: 32.97
PatchNR: 33.66
Total running time of the script: (0 minutes 14.661 seconds)