Diff-FBP: Shift-Variant CT Reconstruction

• Diff-FBP is a differentiable, shift-variant filtered backprojection model that integrates a physics-based algorithm into a neural network for cone-beam CT reconstruction.
• The method employs a PCA-based compressibility analysis to reduce the learned redundancy weight parameters by 97.25% without compromising reconstruction accuracy.
• Experimental results demonstrate faster convergence and reduced memory usage while maintaining high image quality for non-circular CBCT trajectories.

Searching arXiv for the primary Diff-FBP paper and closely related reconstruction work. Diff-FBP denotes a differentiable, shift-variant filtered backprojection model for cone-beam computed tomography that embeds a physics-based FBP algorithm into a neural network and learns the redundancy weights required for arbitrary, non-circular trajectories (Ye et al., 20 Jan 2025). In the formulation emphasized by "Compressibility Analysis for the differentiable shift-variant Filtered Backprojection Model" (Ye et al., 20 Jan 2025), Diff-FBP is both a reconstruction framework and a parameterized implementation of the Defrise and Clack shift-variant FBP algorithm. The 2025 contribution is not the original introduction of the framework but a compressibility analysis of its learned redundancy-weight layer, followed by a PCA-based reparameterization that reduces trainable parameters by 97.25% without compromising reconstruction accuracy (Ye et al., 20 Jan 2025).

1. Mathematical foundation and shift-variant reconstruction

Classical filtered backprojection in 2D parallel-beam CT reconstructs an object $f(\mathbf{x})$ from its Radon transform $p(\theta,s)$ by filtering each projection with a shift-invariant one-dimensional filter $h$, then backprojecting: $$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$ In this setting, the filter is shift-invariant along detector coordinate $s$ (Ye et al., 20 Jan 2025).

For non-circular CBCT trajectories, the reconstruction problem is governed instead by a shift-variant FBP formula derived by Defrise and Clack for arbitrary cone-beam trajectories: $$\begin{aligned} f(x) &= \int_{\Lambda} d\lambda \int_{S^2/2} d\theta \; \left[ -\frac{1}{4\pi^2} \, |a'(\lambda)\cdot\theta| \, \frac{1}{n(\theta,\lambda)} \right] \ &\quad \times \delta'\big( (x - a(\lambda))\cdot\theta \big)\; S(\theta,\lambda). \end{aligned}$$ Here $a(\lambda)$ is the 3D source trajectory, $a'(\lambda)$ its derivative, $\theta$ a half-sphere direction, $S(\theta,\lambda)$ a suitably preprocessed projection term, and $p(\theta,s)$0 the number of intersections of the trajectory with the plane orthogonal to $p(\theta,s)$1 and passing through $p(\theta,s)$2 (Ye et al., 20 Jan 2025).

The factor

$p(\theta,s)$3

is the redundancy weight. It compensates for plane-intersection multiplicity and source motion relative to the plane. This is the central source of shift-variance: the effective weighting depends on local geometry $p(\theta,s)$4, not merely on frequency or detector shift. In conventional circular short-scan CBCT, analogous roles are played by Parker weights and generalized redundancy weights; for arbitrary non-circular orbits, however, the weights become geometry-dependent and analytically complicated (Ye et al., 20 Jan 2025).

2. Differentiable formulation and operator structure

Diff-FBP recasts the Defrise-and-Clack reconstruction as a differentiable network composed primarily of known operators, with learnable weighting layers inserted where closed-form geometry compensation is difficult. In the recalled DRaCo/Diff-FBP framework, reconstruction is written as

$p(\theta,s)$5

Here $p(\theta,s)$6 is the cone-beam projection data; $p(\theta,s)$7 is a cosine-based weighting; $p(\theta,s)$8 and $p(\theta,s)$9 are 2D Radon and backprojection operators on an intermediate geometry; $h$0 is a differentiation operator; $h$1 is a sinogram-domain weighting; $h$2 is the learned redundancy-weight field; $h$3 is a detector-domain weighting; and $h$4 is the final 3D cone-beam backprojection (Ye et al., 20 Jan 2025).

A defining property of the method is that $h$5, $h$6, $h$7, and $h$8 are differentiable layers, for example via PyroNN, so the entire mapping $h$9 supports end-to-end backpropagation (Ye et al., 20 Jan 2025). This yields a physics-driven FBP-like reconstruction operator rather than a generic image-to-image network. The trainable component is correspondingly narrow in scope: Diff-FBP does not rely on a separate CNN that predicts redundancy weights from data. Rather, the redundancy weights themselves are free parameters of the network, optimized directly by gradient descent for a fixed known trajectory and training dataset (Ye et al., 20 Jan 2025).

This distinction is technically important. A common misconception is that Diff-FBP is a post-processing network attached to FBP or a generic learned reconstruction model. In the usage anchored by (Ye et al., 20 Jan 2025), it is instead a geometry-aware differentiable implementation of a shift-variant analytic inversion formula, with learnability concentrated in the weighting terms required by arbitrary trajectories.

3. Redundancy weights as the dominant parameterization

The principal burden in the original Diff-FBP model is the redundancy-weight layer $$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$0. The paper reports that the uncompressed layer contains

$$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$1

trainable parameters (Ye et al., 20 Jan 2025). The data block characterizes this as a giant tensor over projections and detector coordinates, with a memory footprint of approximately 400–450 MB in FP32 and a correspondingly high gradient-computation cost during training (Ye et al., 20 Jan 2025).

This scale motivated a compressibility analysis. The authors hypothesized that the learned redundancy-weight tensor lies on a low-dimensional manifold induced by the smoothness of the sinusoidal trajectory and the structure of geometric redundancy. They therefore trained the full Diff-FBP model, collected the learned redundancy weights, and applied principal component analysis to the resulting high-dimensional representation (Ye et al., 20 Jan 2025).

The PCA procedure was conventional: centering the weight samples, forming the covariance matrix, performing eigenvalue decomposition, selecting the top $$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$2 components, projecting into the reduced space, and reconstructing approximate weights from the retained basis (Ye et al., 20 Jan 2025). The empirical conclusion was that the eigenvalues decay rapidly and that the first few dozen principal components already capture most of the variance. In particular, reconstructions using only 10–30 principal components produce almost negligible degradation in image quality relative to the full weight field, as measured by MSE, PSNR, and SSIM on reconstructed images (Ye et al., 20 Jan 2025).

This suggests that, for the sinusoidal trajectory studied, the learned redundancy structure is strongly low-rank. Although the explicit parameter tensor is extremely large, its effective dimensionality is much smaller.

4. PCA-integrated compressed Diff-FBP

The compressibility result was converted into a new network parameterization by replacing direct optimization of $$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$3 with a PCA-layer reconstruction: $$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$4 where $$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$5 is a low-dimensional trainable coordinate vector, $$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$6 contains the retained principal components, and $$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$7 is the mean redundancy-weight vector (Ye et al., 20 Jan 2025). Incorporated into the full reconstruction pipeline, this becomes

$$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$8

Because the PCA layer is linear, the backward pass is correspondingly simple: gradients with respect to the low-dimensional coordinates propagate through multiplication by the fixed PCA basis. This preserves full differentiability while eliminating the need to optimize the original 113,184,000-parameter redundancy tensor directly (Ye et al., 20 Jan 2025).

The quantitative effect is summarized by the reported parameter counts and reconstruction metrics:

Model	Parameters	MSE / PSNR / SSIM
No PCA	113,184,000	$$p(\theta, s) = \int_{\mathbb{R}^2} f(\mathbf{x})\, \delta(\mathbf{x}\cdot\boldsymbol{\theta} - s)\, d\mathbf{x}, \qquad f(\mathbf{x}) = \int_{0}^{\pi} (h * p)(\theta, \mathbf{x}\cdot\boldsymbol{\theta})\, d\theta .$$9 / $s$0 / $s$1
PCA (50 comps)	14,450,960	$s$2 / $s$3 / $s$4
PCA (30 comps)	8,783,760	$s$5 / $s$6 / $s$7
PCA (10 comps)	3,116,560	$s$8 / $s$9 / $$\begin{aligned} f(x) &= \int_{\Lambda} d\lambda \int_{S^2/2} d\theta \; \left[ -\frac{1}{4\pi^2} \, |a'(\lambda)\cdot\theta| \, \frac{1}{n(\theta,\lambda)} \right] \ &\quad \times \delta'\big( (x - a(\lambda))\cdot\theta \big)\; S(\theta,\lambda). \end{aligned}$$0

The most compressed setting, with 10 principal components, reduces parameters from 113,184,000 to 3,116,560, corresponding to a 97.25% reduction (Ye et al., 20 Jan 2025). The training behavior changes accordingly: the uncompressed model converged after approximately 430 epochs, whereas the PCA-compressed model with 10 components converged after approximately 140 epochs (Ye et al., 20 Jan 2025). The paper attributes the speedup to fewer parameters, smaller gradient tensors, lower memory traffic, and the cheap linear cost of the PCA layer.

5. Experimental setting and reconstruction behavior

The reported experiments use a clinical C-arm system, Artis zeego (Siemens), with a sinusoidal trajectory, and cone-beam forward projection implemented in PyroNN (Ye et al., 20 Jan 2025). The training corpus consists of 30 simulated CBCT datasets with this geometry, split into 24 for training and 6 for validation. An additional test is performed on the Pancreatic-CT-CBCT-SEG dataset (TCIA) to evaluate behavior on real medical CT/CBCT data (Ye et al., 20 Jan 2025).

Training minimizes

$$\begin{aligned} f(x) &= \int_{\Lambda} d\lambda \int_{S^2/2} d\theta \; \left[ -\frac{1}{4\pi^2} \, |a'(\lambda)\cdot\theta| \, \frac{1}{n(\theta,\lambda)} \right] \ &\quad \times \delta'\big( (x - a(\lambda))\cdot\theta \big)\; S(\theta,\lambda). \end{aligned}$$1

with a one-cycle learning-rate policy (Ye et al., 20 Jan 2025). Within this regime, the compressed models show no substantial deterioration in reconstruction fidelity despite the large parameter reduction. Comparing the full model with PCA(10), the reported values change from MSE $$\begin{aligned} f(x) &= \int_{\Lambda} d\lambda \int_{S^2/2} d\theta \; \left[ -\frac{1}{4\pi^2} \, |a'(\lambda)\cdot\theta| \, \frac{1}{n(\theta,\lambda)} \right] \ &\quad \times \delta'\big( (x - a(\lambda))\cdot\theta \big)\; S(\theta,\lambda). \end{aligned}$$2, PSNR $$\begin{aligned} f(x) &= \int_{\Lambda} d\lambda \int_{S^2/2} d\theta \; \left[ -\frac{1}{4\pi^2} \, |a'(\lambda)\cdot\theta| \, \frac{1}{n(\theta,\lambda)} \right] \ &\quad \times \delta'\big( (x - a(\lambda))\cdot\theta \big)\; S(\theta,\lambda). \end{aligned}$$3 dB, and SSIM $$\begin{aligned} f(x) &= \int_{\Lambda} d\lambda \int_{S^2/2} d\theta \; \left[ -\frac{1}{4\pi^2} \, |a'(\lambda)\cdot\theta| \, \frac{1}{n(\theta,\lambda)} \right] \ &\quad \times \delta'\big( (x - a(\lambda))\cdot\theta \big)\; S(\theta,\lambda). \end{aligned}$$4 (Ye et al., 20 Jan 2025). The paper characterizes these differences as small and emphasizes that compression does not materially degrade reconstruction quality.

The qualitative findings align with the tabulated metrics. Reconstructions without PCA and with PCA using 10 components are described as visually very similar, with no significant worsening of artifacts or noise patterns. The corresponding redundancy-weight maps learned with and without compression are likewise described as remarkably similar (Ye et al., 20 Jan 2025). Taken together, the numerical and visual evidence supports the conclusion that the redundancy-weight layer is highly compressible for the sinusoidal trajectory considered.

6. Scope, implications, and relation to adjacent FBP literature

The practical significance of compressed Diff-FBP is that it makes a trajectory-aware, physics-consistent reconstruction model more tractable for non-standard CBCT orbits. The reported reduction in parameter count eases GPU memory constraints, the reduced convergence time makes retraining or fine-tuning more feasible, and the remaining analytic structure preserves a strong link to the underlying reconstruction geometry (Ye et al., 20 Jan 2025). The paper explicitly states that these changes greatly improve training speed and substantially enhance practical viability. It also states that further investigation is needed for more complex trajectories, which suggests that compression performance may depend on orbit class and that trajectory-specific PCA bases may be necessary (Ye et al., 20 Jan 2025).

Within the broader literature, Diff-FBP occupies a specific niche. It differs from fully learned reconstructions that map sinograms or preliminary images through large CNNs, and it also differs from compact differentiable FBP models that learn only a small filter parameterization. A closely related but distinct example is "Data-Driven Filter Design in FBP: Transforming CT Reconstruction with Trainable Fourier Series" (Sun et al., 2024), which implements a differentiable FBP block whose only learned part is a Fourier-series filter with 101 trainable scalar parameters. Another adjacent line is "Accelerated FBP for computed tomography image reconstruction" (Dolmatova et al., 2020), where IIR filtering and the Fast Hough Transform yield an analytic FBP backbone with $$\begin{aligned} f(x) &= \int_{\Lambda} d\lambda \int_{S^2/2} d\theta \; \left[ -\frac{1}{4\pi^2} \, |a'(\lambda)\cdot\theta| \, \frac{1}{n(\theta,\lambda)} \right] \ &\quad \times \delta'\big( (x - a(\lambda))\cdot\theta \big)\; S(\theta,\lambda). \end{aligned}$$5 addition operations, a structure that the source text explicitly identifies as natural for differentiable implementation. By contrast, "Generalized Backpropagation Algorithms for Diffraction Tomography" (Paladhi et al., 2016) concerns filtered backpropagation in a diffraction-tomography setting; it is mathematically related by the filter-plus-backprojection paradigm but belongs to a different physical model and uses the term in the lineage of diffraction-based FBPP rather than CBCT shift-variant FBP.

Accordingly, Diff-FBP is best understood not as a generic label for any differentiable reconstruction layer, but as a trajectory-specific differentiable realization of the Defrise-and-Clack shift-variant filtered backprojection framework in which the critical unknown is the redundancy-weight field (Ye et al., 20 Jan 2025). The 2025 compressibility analysis establishes that this field, although naively enormous, is effectively low-dimensional for the studied sinusoidal orbit. A plausible implication is that future trajectory-aware analytic–learned CBCT models may be designed directly around low-rank or similarly compressed parameterizations, rather than first learning a full high-dimensional redundancy tensor and compressing it afterward.