Note
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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
\[\underset{x}{\arg\min} \quad \datafid{x}{y} + \lambda g(x).\]
Here, the regularizer \(g\) is explicitly defined as
\[g(x)=\sum_{i\in\mathcal{I}} h(P_i x),\]
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 by Zoran and Weiss[1]. 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 optimizer. For an example using the (approximated) half-quadratic splitting algorithm proposed by Zoran and Weiss[1], we refer to the example Expected Patch Log Likelihood (EPLL) for Denoising and Inpainting.
The patch normalizing flow regularizer (PatchNR) was proposed by Altekrüger et al.[2]. It models \(h(x)=-\log(p_{\theta}(x))\) as negative log-likelihood function of a probability 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
import deepinv as dinv
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 import load_torch_url
from tqdm import tqdm
device = dinv.utils.get_device()
dtype = torch.float32
Selected GPU 0 with 28319.25 MiB free memory
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, use_dict_output=True)
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. This takes up to a couple of minutes on a GPU. 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, epoch=None, **kwargs):
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 loss_total, 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)
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, device=device)
physics = Tomography(
img_width=img_size,
angles=angles,
device=device,
noise_model=noise_model,
normalize=False,
adjoint_via_backprop=False,
)
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 chosen 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: 19.92
EPLL: 31.84
PatchNR: 32.52
- References:
Total running time of the script: (0 minutes 2.821 seconds)
