iLAMA-Net: Initialization-Augmented SVCT Reconstruction

• iLAMA-Net is an initialization-augmented deep learning framework for sparse-view CT that leverages a dual-domain variational approach.
• It employs a geometry-aware Init-Net to generate informed (x0, z0) initials, accelerating convergence and enhancing reconstruction stability.
• Experimental results demonstrate superior PSNR, SSIM, and RMSE metrics over baselines while using a compact, efficient network design.

iLAMA-Net is an initialization-augmented version of LAMA-Net for sparse-view CT (SVCT), introduced within the dual-domain variational framework of “LAMA-Net: A Convergent Network Architecture for Dual-Domain Reconstruction” (Ding et al., 30 Jul 2025). It combines a geometry-aware initialization network, termed Init-Net, with the learned alternating minimization algorithm (LAMA) unrolled as LAMA-Net. The underlying model jointly reconstructs an image variable $x \in \mathbb{R}^n$ and a measurement-domain variable $z \in \mathbb{R}^m$ from an observed sparse sinogram $s_0 \in \mathbb{R}^{n_0}$, using the forward operator $A \in \mathbb{R}^{m \times n}$ and a view-selection mask $P_0 \in \mathbb{R}^{n_0 \times m}$, with fidelity terms that are quadratic and therefore correspond to an additive Gaussian-noise assumption, or a least-squares relaxation, in both image-to-sinogram consistency and measured-view consistency (Ding et al., 30 Jul 2025).

1. Dual-domain formulation and reconstruction objective

The reconstruction problem is posed as a two-block, nonconvex, nonsmooth optimization problem:

$$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$

with

$$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$

Here $\Theta=(\theta_1,\theta_2)$ are learned parameters and $\lambda>0$ is a trade-off weight (Ding et al., 30 Jul 2025).

The learned regularizers are defined in the image and measurement domains as

$$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$

and

$z \in \mathbb{R}^m$0

The feature extractors $z \in \mathbb{R}^m$1 and $z \in \mathbb{R}^m$2 are lightweight CNNs of the form

$z \in \mathbb{R}^m$3

where $z \in \mathbb{R}^m$4 is a smoothed ReLU, “$z \in \mathbb{R}^m$5” denotes convolution, and $z \in \mathbb{R}^m$6 collects these CNN weights (Ding et al., 30 Jul 2025).

The rationale for the dual-domain model is explicit. Sparse-view CT suffers from strong streaking and aliasing that are structured in both the sinogram and image domains. Enforcing $z \in \mathbb{R}^m$7 and $z \in \mathbb{R}^m$8 uses physics on both domains, while learning complementary priors $z \in \mathbb{R}^m$9 and $s_0 \in \mathbb{R}^{n_0}$0 allows the model to denoise and de-alias in the sinogram, enforce image priors attuned to anatomy, and couple both through the data-consistency term $s_0 \in \mathbb{R}^{n_0}$1. The paper states that this dual coupling is empirically superior to single-domain approaches for aggressive undersampling (Ding et al., 30 Jul 2025).

2. LAMA and the unrolled LAMA-Net architecture

To handle the nonsmooth $s_0 \in \mathbb{R}^{n_0}$2 regularizers, LAMA adopts a smoothed surrogate

$s_0 \in \mathbb{R}^{n_0}$3

where

$s_0 \in \mathbb{R}^{n_0}$4

for $s_0 \in \mathbb{R}^{n_0}$5. Its gradient is

$s_0 \in \mathbb{R}^{n_0}$6

with $s_0 \in \mathbb{R}^{n_0}$7 and $s_0 \in \mathbb{R}^{n_0}$8 its complement (Ding et al., 30 Jul 2025).

For fixed $s_0 \in \mathbb{R}^{n_0}$9, LAMA uses a PALM-like alternating scheme with linearized proximal steps. In the measurement block,

$A \in \mathbb{R}^{m \times n}$0

followed by the linearized update

$A \in \mathbb{R}^{m \times n}$1

In the image block,

$A \in \mathbb{R}^{m \times n}$2

followed by

$A \in \mathbb{R}^{m \times n}$3

The gradients of the fidelity term are

$A \in \mathbb{R}^{m \times n}$4

These expressions define the data-consistency layers in the unrolled architecture (Ding et al., 30 Jul 2025).

LAMA includes sufficient descent checks. If the pair $A \in \mathbb{R}^{m \times n}$5 satisfies the stated sufficient descent conditions for some $A \in \mathbb{R}^{m \times n}$6, it is accepted. Otherwise, the algorithm falls back to a gradient BCD step with line search,

$A \in \mathbb{R}^{m \times n}$7

$A \in \mathbb{R}^{m \times n}$8

with repeated reduction $A \in \mathbb{R}^{m \times n}$9, $P_0 \in \mathbb{R}^{n_0 \times m}$0, until sufficient decrease is obtained (Ding et al., 30 Jul 2025).

LAMA-Net is obtained by unrolling this algorithm: each algorithm iteration is one network “phase,” and $P_0 \in \mathbb{R}^{n_0 \times m}$1 phases, for example $P_0 \in \mathbb{R}^{n_0 \times m}$2, form the network. The learned proximal terms are realized as residual corrections in both domains, while the fallback BCD descent with line search is implemented as a safeguard module. The paper distinguishes a U-module, corresponding to the linearized proximal residual steps, and a V-module, corresponding to the safeguarded fallback. Parameter sharing across all $P_0 \in \mathbb{R}^{n_0 \times m}$3 phases is used for memory efficiency and better generalization. Architectural details are explicit: $P_0 \in \mathbb{R}^{n_0 \times m}$4 has 4 convolutional layers, 32 channels, kernel $P_0 \in \mathbb{R}^{n_0 \times m}$5, stride 1, activation $P_0 \in \mathbb{R}^{n_0 \times m}$6, and padding to preserve size; $P_0 \in \mathbb{R}^{n_0 \times m}$7 has 4 convolutional layers, 32 channels, rectangular kernel $P_0 \in \mathbb{R}^{n_0 \times m}$8 with padding $P_0 \in \mathbb{R}^{n_0 \times m}$9, stride 1, and activation $$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$0. The use of rectangular kernels in the sinogram domain is stated to help capture along-view correlations aligned with CT geometry (Ding et al., 30 Jul 2025).

3. iLAMA-Net: initialization network and recurrent sinogram completion

iLAMA-Net augments LAMA-Net with an initialization network that produces informative $$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$1 from the sparse sinogram $$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$2. The motivation is that the objective is nonconvex and nonsmooth, and the paper states that better initialization substantially improves convergence speed, stability, and final accuracy (Ding et al., 30 Jul 2025).

The initialization mechanism is geometry-aware. It learns a recurrent mapping $$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$3 that advances sparse-view sinogram content by one angular step $$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$4:

$$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$5

Here $$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$6 is the downsampling factor, for example $$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$7 for $$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$8 views, and $$\min_{x,z} \ \Phi(x,z; s_0, \Theta) := f(x,z; s_0) + R(x; \theta_1) + Q(z; \theta_2),$$9 are concatenated to approximate a pseudo full-view sinogram (Ding et al., 30 Jul 2025).

Training of $$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$0 is performed by minimizing

$$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$1

After training, initialization is formed as

$$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$2

and

$$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$3

The LAMA iteration itself is unchanged; it simply starts from these learned initials:

$$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$4

This means that iLAMA-Net is not a new iterative rule but LAMA-Net supplied with a learned initialization pathway (Ding et al., 30 Jul 2025).

Init-Net is a CNN composed of 3 sequential blocks with skip connections; each block has 4 convolutional layers with $$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$5 kernels, stride 1, padding $$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$6, and ReLU activations. The paper states that these rectangular kernels align with sinogram geometry and capture along-view contextual correlations. Training uses ADAM for 100 epochs with learning rate $$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$7 on full sinograms split into $$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$8 partitions. Init-Net and LAMA-Net are trained independently, which the paper describes as avoiding leakage (Ding et al., 30 Jul 2025).

The paper also gives an interpretation of why the initialization matters. Theoretically, a better $$f(x,z; s_0) = \frac{1}{2}\|Ax-z\|_2^2 + \frac{\lambda}{2}\|P_0 z - s_0\|_2^2.$$9 is described as closer to feasible and high-quality regions, reducing the number of fallback steps, accelerating descent of $\Theta=(\theta_1,\theta_2)$0, and promoting rapid satisfaction of the gradient reduction criterion that triggers $\Theta=(\theta_1,\theta_2)$1-decrease. Empirically, the recurrent sinogram completion is stated to capture angular correlations missed by one-shot inpainting, producing cleaner $\Theta=(\theta_1,\theta_2)$2 through FBP of $\Theta=(\theta_1,\theta_2)$3, improving final PSNR and SSIM, and further reducing sinogram RMSE (Ding et al., 30 Jul 2025).

4. Convergence, Clarke stationarity, and stability claims

The paper provides a complete and rigorous convergence proof for LAMA. The relevant nonsmooth notion is the Clarke subdifferential $\Theta=(\theta_1,\theta_2)$4 of a locally Lipschitz function, defined via the Clarke directional derivative

$\Theta=(\theta_1,\theta_2)$5

with

$\Theta=(\theta_1,\theta_2)$6

A Clarke stationary point satisfies $\Theta=(\theta_1,\theta_2)$7 (Ding et al., 30 Jul 2025).

The main convergence theorem is formulated for the subsequence of iterations at which the reduction criterion is satisfied and $\Theta=(\theta_1,\theta_2)$8 is reduced. Let $\Theta=(\theta_1,\theta_2)$9 be generated by LAMA with arbitrary $\lambda>0$0, $\lambda>0$1, and $\lambda>0$2. Let $\lambda>0$3 be the subsequence at indices $\lambda>0$4 where the gradient threshold is satisfied and $\lambda>0$5 is reduced. Then the subsequence has at least one accumulation point, and every accumulation point $\lambda>0$6 is a Clarke stationary point of the original unsmoothed objective $\lambda>0$7 (Ding et al., 30 Jul 2025).

The proof sketch reported in the paper rests on several ingredients: with $\lambda>0$8 fixed, the inner loop yields sufficient decrease and bounded gradient norm; Lipschitz continuity of $\lambda>0$9 holds with $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$0; the line-search fallback guarantees finite termination at each iteration; the sequence is bounded by coercivity of $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$1; along the reduction subsequence, $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$2 as $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$3; and closedness of $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$4, together with the explicit forms of $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$5 and $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$6, yields Clarke stationarity of accumulation points (Ding et al., 30 Jul 2025).

A common misunderstanding is that this theorem implies global optimality. The paper explicitly frames the result as convergence of a reduction subsequence to a Clarke stationary point, which is weaker than global optimality. It also depends on the sufficient descent checks and the $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$7-schedule. This suggests that the theoretical guarantee is best understood as a stationarity and descent guarantee for a nonconvex learned variational method, rather than a claim of uniqueness or global minimization (Ding et al., 30 Jul 2025).

The paper further links convergence behavior to stability and robustness. Because each accepted step reduces $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$8 or is safeguarded by a line search, the reconstruction is described as avoiding uncontrolled amplifications often seen in unconstrained unrolled networks. This is used to explain the observed robustness to structured perturbations and noise (Ding et al., 30 Jul 2025).

5. Experimental protocol and quantitative performance in sparse-view CT

The reported experiments use the AAPM-Mayo Clinic Low-Dose CT Grand Challenge dataset and NBIA. The training and test split is formed by randomly selecting 500 image-sinogram pairs from AAPM-Mayo and 200 from NBIA, then splitting each set 80%/20%. The acquisition geometry uses 512 detector elements and 1024 full views, with sparse-view settings of 64 views (6.25%) and 128 views (12.5%). Images are of size $$R(x; \theta_1) = \|g_R(x; \theta_1)\|_{2,1} = \sum_i \|g_{R,i}(x; \theta_1)\|_2,$$9. The forward and backprojection operators are fan-beam distance-driven projector/backprojector from CTLIB. Sparse-view generation applies $z \in \mathbb{R}^m$00 to the full sinogram $z \in \mathbb{R}^m$01 to obtain zero-filled sparse $z \in \mathbb{R}^m$02, and $z \in \mathbb{R}^m$03 is used as a baseline and sometimes as $z \in \mathbb{R}^m$04 (Ding et al., 30 Jul 2025).

LAMA-Net is trained with

$z \in \mathbb{R}^m$05

with $z \in \mathbb{R}^m$06 and $z \in \mathbb{R}^m$07 phases. Optimization uses ADAM with learning rates $z \in \mathbb{R}^m$08 for the image network and $z \in \mathbb{R}^m$09 for the sinogram network. Training is recursive: 3 phases for 300 epochs, then adding 2 phases and fine-tuning for 200 epochs, up to 15 phases. For iLAMA-Net, $z \in \mathbb{R}^m$10 is trained with the recurrence loss for 100 epochs using ADAM at learning rate $z \in \mathbb{R}^m$11, then $z \in \mathbb{R}^m$12 are generated and LAMA-Net is trained on those initials (Ding et al., 30 Jul 2025).

Dataset / views	LAMA-Net	iLAMA-Net
AAPM-Mayo, 64 views	44.58 dB, SSIM 0.986, RMSE 0.68 $z \in \mathbb{R}^m$13	46.37 dB, SSIM 0.990, RMSE 0.52 $z \in \mathbb{R}^m$14
AAPM-Mayo, 128 views	50.01 dB, SSIM 0.995, RMSE 0.32 $z \in \mathbb{R}^m$15	51.02 dB, SSIM 0.996, RMSE 0.27 $z \in \mathbb{R}^m$16
NBIA, 64 views	41.40 dB, SSIM 0.976, RMSE 0.99 $z \in \mathbb{R}^m$17	42.11 dB, SSIM 0.979, RMSE 0.92 $z \in \mathbb{R}^m$18
NBIA, 128 views	45.20 dB, SSIM 0.988, RMSE 0.55 $z \in \mathbb{R}^m$19	47.28 dB, SSIM 0.992, RMSE 0.44 $z \in \mathbb{R}^m$20

Against the named baselines, the paper reports the following values. On AAPM-Mayo with 64 views: FBP 27.17 dB / 0.596, DDNet 35.70 dB / 0.923, LDA 37.16 dB / 0.932, DuDoTrans 37.90 dB / 0.952, LEARN++ 43.02 dB / 0.980, LAMA-Net 44.58 dB / 0.986, and iLAMA-Net 46.37 dB / 0.990. On AAPM-Mayo with 128 views: FBP 33.28 dB / 0.759, DDNet 42.73 dB / 0.974, LDA 43.00 dB / 0.976, DuDoTrans 43.48 dB / 0.985, LEARN++ 49.77 dB / 0.995, LAMA-Net 50.01 dB / 0.995, and iLAMA-Net 51.02 dB / 0.996. On NBIA with 64 views: FBP 25.72 dB / 0.592, DDNet 35.59 dB / 0.920, LDA 34.31 dB / 0.896, DuDoTrans 35.53 dB / 0.938, LEARN++ 38.53 dB / 0.956, LAMA-Net 41.40 dB / 0.976, and iLAMA-Net 42.11 dB / 0.979. On NBIA with 128 views: FBP 31.86 dB / 0.743, DDNet 40.23 dB / 0.961, LDA 40.26 dB / 0.963, DuDoTrans 40.67 dB / 0.976, LEARN++ 43.35 dB / 0.983, LAMA-Net 45.20 dB / 0.988, and iLAMA-Net 47.28 dB / 0.992. In all four cases, the paper also reports lower sinogram RMSE for iLAMA-Net than for LAMA-Net (Ding et al., 30 Jul 2025).

The initialization ablation on AAPM-Mayo isolates the contribution of Init-Net. Without initialization, using FBP as $z \in \mathbb{R}^m$21 and $z \in \mathbb{R}^m$22 as $z \in \mathbb{R}^m$23, LAMA-Net attains 44.58/0.986 at 64 views and 50.01/0.995 at 128 views. With a CNN inpainting initialization in the image domain only, the corresponding values are approximately 45.1/0.987 and approximately 50.7/0.995. With the geometry-aware sinogram recurrence, iLAMA-Net attains the best values, 46.37/0.990 and 51.02/0.996 (Ding et al., 30 Jul 2025).

6. Model size, robustness, limitations, and practical implications

The reported parameter counts are: DDNet $z \in \mathbb{R}^m$24, LDA $z \in \mathbb{R}^m$25, DuDoTrans $z \in \mathbb{R}^m$26, LEARN++ $z \in \mathbb{R}^m$27, LAMA-Net $z \in \mathbb{R}^m$28, and iLAMA-Net $z \in \mathbb{R}^m$29. The paper states that LAMA-Net and iLAMA-Net achieve SOTA accuracy with orders-of-magnitude fewer parameters than several baselines. A plausible implication is that the compactness follows from the use of small CNN regularizers and parameter sharing across phases, rather than transformer-scale representational capacity (Ding et al., 30 Jul 2025).

Qualitatively, LAMA-Net and iLAMA-Net are reported to suppress streaking while preserving fine edges and textures, and the sinogram RMSE improvements are taken as confirmation of better measurement-domain consistency. The paper also presents stability experiments. Under a structured perturbation in which “can u see it” text is injected into the ground truth, LAMA-Net reconstructs the perturbation faithfully without hallucinations elsewhere, and the difference heatmaps largely show the perturbation only. Under Gaussian perturbations, LAMA-Net consistently attains higher PSNRs against perturbed ground truths across several noise levels, and its difference images exhibit mostly isotropic, structureless noise rather than spurious edges (Ding et al., 30 Jul 2025).

The dual-domain coupling is further described as making the method less brittle as the number of views decreases, because $z \in \mathbb{R}^m$30 and $z \in \mathbb{R}^m$31 constrain both domains while $z \in \mathbb{R}^m$32 and $z \in \mathbb{R}^m$33 capture cross-domain priors. The ablation study indicates that iLAMA-Net further improves robustness by preventing bad local minima and reducing artifact propagation from extreme undersampling. This suggests that initialization is not merely a performance enhancement but also part of the method’s robustness profile (Ding et al., 30 Jul 2025).

Several limitations are explicit. The method requires accurate $z \in \mathbb{R}^m$34 and $z \in \mathbb{R}^m$35, so mismatch across scanners or geometries can degrade performance. The convergence theorem concerns a reduction subsequence and Clarke stationary points, not global optima. Nonconvexity remains fundamental because the learned priors make $z \in \mathbb{R}^m$36 nonconvex and allow multiple stationary points. The paper also notes that extreme out-of-distribution cases still pose challenges. Proposed future directions include multi-domain priors such as Fourier or wavefront domains, physics-informed constraints such as dose models, uncertainty quantification via Bayesian or ensemble approaches atop the convergent backbone, and joint learning of $z \in \mathbb{R}^m$37-calibration and priors for cross-scanner portability (Ding et al., 30 Jul 2025).

From an implementation standpoint, the paper gives a concrete inference procedure: initialize $z \in \mathbb{R}^m$38 with recurrent sinogram completion, set $z \in \mathbb{R}^m$39, perform $z \in \mathbb{R}^m$40 LAMA phases with the same accept-or-reject rules and optional $z \in \mathbb{R}^m$41 update, and return $z \in \mathbb{R}^m$42. It states that $z \in \mathbb{R}^m$43 phases balances accuracy and speed; $z \in \mathbb{R}^m$44, $z \in \mathbb{R}^m$45, $z \in \mathbb{R}^m$46, and $z \in \mathbb{R}^m$47 can be learned as trainable scalars per phase; safeguard parameters typically use $z \in \mathbb{R}^m$48, $z \in \mathbb{R}^m$49, $z \in \mathbb{R}^m$50, and $z \in \mathbb{R}^m$51; and inference time depends on the projector implementation and GPU. The experiments were run on an NVIDIA A100 80GB (Ding et al., 30 Jul 2025).