Hi-GRPO: Optimized 3D Volumetric Imaging

• Hi-GRPO is a family of unsupervised inverse optimization methods that reconstruct high-fidelity 3D volumetric images while effectively reducing ring artifacts.
• The approach integrates a differentiable forward model with implicit neural fields and multiresolution hash encoding for robust, memory-efficient artifact correction.
• Empirical results demonstrate that Hi-GRPO outperforms traditional and supervised techniques, achieving higher PSNR and SSIM in both simulated and real CT imaging scenarios.

Hi-GRPO refers to the class of highly general, multi-parameter inverse problem optimization methods for reconstructing high-fidelity volumetric images from raw physical measurements, with specific application in 3D imaging modalities such as cone-beam computed tomography (CBCT) with strong non-ideal physical artifacts. This family of techniques is characterized by unsupervised, memory-efficient joint estimation frameworks in which imaging artifacts (e.g., detector gain non-uniformity, dead pixels) are modeled as learnable physical parameters, and the artifact-free image is represented via implicit neural fields. These methods are exemplified by recent proposals such as AR3D-R1 (also called Riner), which combines a physically accurate forward model, differentiable programming, and neural spectral bias for robust artifact reduction in large-scale 3D imaging problems (Wu et al., 8 Dec 2024).

1. Physical Model of Ring-Artifact Formation in 3D CBCT

In 3D CBCT, a volumetric object $\mu(x)$ is imaged by acquiring a series of 2D X-ray projections using a planar detector array $\mathbb{S}$ at a set of angles $\theta \in \Theta$. The idealized system assumes all detector channels have unit gain ($\alpha_s = 1$) and no defects; in practice, each detector may have a unique unknown gain ($\alpha_s \ne 1$) or may be non-responsive ($\alpha_s = 0$), leading to prominent ring artifacts and invalid signal channels.

The raw photon count at detector $s$ and angle $\theta$ follows from Lambert–Beer's law:

$$I(\theta, s) = \alpha_s I_0 \exp\left(-\int_{L(\theta, s)} \mu(x) dx\right)$$

where $L(\theta, s)$ is the path traced by a ray through the volume. The negative log-ratio, after masking dead channels, defines the physically-consistent forward model:

$$\hat \rho(\theta, s) = \left[ -\ln\left(\max(\alpha_s, 0) + \epsilon\right) + \sum_{x \in L(\theta, s)} \mu(x) \Delta x \right] m(s)$$

with $\epsilon = 10^{-8}$, $m(s) = 0$ for dead ($\alpha_s = 0$) and 1 otherwise. In operator form:

$\hat{y} = \mathcal{P}(x, \theta; \alpha)$

where $x \equiv \mu(\cdot)$ and $\alpha = \{\alpha_s\}_{s \in \mathbb{S}}$. Measurement noise is modeled as $$y(\theta,s) = \rho(\theta,s) + \epsilon_{\mathrm{meas}}$$.

2. Multi-Parameter Inverse Problem and Implicit Neural Regularization

The Hi-GRPO methodology, as instantiated by AR3D-R1, is framed as an unsupervised joint inverse problem:

• Recover the artifact-free attenuation volume $\mu(\cdot)$,
• Estimate channel responses $\{\alpha_s\}$, directly from raw noisy data $y$, without paired ground truth.

The problem is ill-posed due to the multiplicity of physical unknowns. Rather than explicit $L_2$ or total variation penalties, regularization is achieved by parameterizing $\mu$ as a neural field (MLP with multiresolution hash encoding), leveraging its spectral bias for implicit smoothness. Formally:

$$\min_{\Phi, \alpha}~ \mathcal{L}(\Phi, \alpha) = \sum_{\theta \in \Theta_{\text{sub}}} \sum_{s \in \mathbb{S}_{\text{sub}}} \left\| \rho(\theta, s) - \mathcal{P}(f_{\Phi}, \theta, s; \alpha) \right\|_{1}$$

where $f_{\Phi}$ is an MLP mapping $x \rightarrow \mu(x)$. Mini-batches $(\Theta_{\text{sub}}, \mathbb{S}_{\text{sub}})$ are sampled per iteration for stochastic optimization.

3. Differentiable Forward Model and Memory Efficiency

All operations in the forward model—ray sampling, MLP evaluation, log gain offset, dead-channel masking—are implemented via differentiable operators (e.g., ReLU, hash/Fourier encoding, log-sum) in frameworks such as PyTorch. Gradient backpropagation is enabled for both neural field weights $\Phi$ and physical gains $\alpha$.

Memory efficiency is a central design constraint. Hi-GRPO employs "ray marching" mini-batch optimization: at each step, only a small set of rays (e.g., $N_{\text{rays}}=80$), covering a fraction of angles and detector pixels, are processed. Volume-wide predictions are avoided; memory scales as $O(N_{\text{rays}} \cdot N_{\text{points}})$ instead of $O(V^3)$. This enables fitting to volumes as large as $512 \times 512 \times 80$ on commodity GPUs.

Training proceeds by stochastic ray sampling, forward projection, $L_1$ loss calculation against measurements, and simultaneous Adam-based updates of both field parameters and detector gains.

4. Neural Representation: Hash-Encoding Neural Fields

The artifact-free attenuation field is parameterized as an implicit neural field $x \mapsto \mu(x)$ with a multiresolution hash encoding architecture (following Müller et al. 2022):

• $L=16$ resolution levels,
• hash table size $T=2^{28}$,
• feature dimension $F=8$,
• two fully connected layers with 64 ReLU activations.

The hash encoding induces bias toward coarse geometric structures initially, enabling effective quick optimization and natural recovery of high-frequency details as training proceeds. No explicit denoising or additional regularization is applied; artifact removal arises from jointly optimizing the forward physical model and the neural field.

5. Empirical Performance and Benchmarks

Benchmarking is conducted on both simulated and real datasets:

• Simulated 2D/3D: DeepLesion, LIDC (2D fan-beam CBCT), AAPM (3D, $256\times256\times100$),
• Real: Bruker SKYSCAN 1276 micro-CT (Walnut: 2D FBCT, Chicken foot: 3D CBCT, $512\times512\times80$).

Ring artifacts are synthesized by randomly assigning $75\%$ of detectors random gains $\alpha_s \sim \text{Uniform}[0.75, 1.25]$ and inserting dead pixels ($\alpha_s = 0$), plus Poisson and Gaussian noise.

Performance is assessed using PSNR and SSIM. Key 3D AAPM results:

Method	PSNR (dB)	SSIM
FDK (linear)	$\approx17.96$	$0.799$
Restormer (DL-SOTA)	$\approx36.52$	$0.952$
AR3D-R1 (Hi-GRPO)	$\approx37.27$	$0.977$

Hi-GRPO consistently outperforms both supervised deep learning and traditional model-based RAR baselines by $1$–$3$ dB in PSNR (2D and 3D). Qualitatively, reconstructions exhibit clearer structural detail and effective suppression of ring artifacts.

6. Strengths, Limitations, and Extensions

Hi-GRPO avoids the domain-shift limitations of supervised RAR, as unsupervised direct fitting to measurement sinograms is agnostic to the synthetic/real divergence found in medical imaging data. The same code accommodates various geometries (2D fan-beam, 3D cone-beam, helical) via appropriate ray parameterization.

Limitations include per-scan optimization time (15–20 minutes on modern GPUs), as there is no feed-forward prediction. Potential accelerations could involve more compact encodings (e.g., K-planes, Gaussian Splatting) or transfer-based warm starting. Current models assume static objects and stationary detector gains; dynamic or time-varying gain models ($\alpha_s \rightarrow \alpha_s(\theta)$) would require further methodological extension.

7. Context and Research Impact

The Hi-GRPO paradigm represents a shift from paired, supervised, data- and memory-intensive approaches to unsupervised, physically-motivated, and scalable frameworks in 3D imaging artifact correction. By integrating differentiable physics, neural implicit representation, and memory-efficient inverse optimization, these methods deliver improved artifact suppression, better generalization, and the practical capacity to operate on real-world, large-scale volumetric scans directly from raw measurement data (Wu et al., 8 Dec 2024).