LAMA: CT Reconstruction via Alternating Minimization

• LAMA is a dual-domain CT reconstruction method that simultaneously recovers full-view sinogram and image data using learned variational regularizers.
• It employs an alternating minimization scheme that integrates Nesterov smoothing and residual updates to ensure energy descent and convergence to Clarke stationary points.
• The algorithm unrolls into a network with parameter sharing and an initialization step, improving stability, memory efficiency, and reconstruction accuracy.

LAMA, the Learned Alternating Minimization Algorithm, is a dual-domain method for sparse-view CT reconstruction in which the unknown image $x$ and an auxiliary full-view sinogram $z$ are reconstructed jointly from a measured sparse-view sinogram $s$. In its original formulation, LAMA is induced by a variational model with learnable nonsmooth nonconvex regularizers in both image and sinogram domains, parameterized as composite functions of deep networks, and minimized by combining Nesterov smoothing, residual learning architecture, and a safeguarded alternating scheme with convergence to Clarke stationary points (Ding et al., 2023).

1. Historical placement and naming

The name LAMA in this literature primarily denotes the CT reconstruction method introduced in "Learned Alternating Minimization Algorithm for Dual-domain Sparse-View CT Reconstruction" (Ding et al., 2023). Subsequent work restated the same core design as "Stable Dual-Domain Deep Reconstruction For Sparse-View CT" (Ding et al., 2024), unrolled it as LAMA-Net and iLAMA-Net in "LAMA-Net: A Convergent Network Architecture for Dual-Domain Reconstruction" (Ding et al., 30 Jul 2025), and generalized the underlying two-block scheme as LPAM in "A Learned Proximal Alternating Minimization Algorithm and Its Induced Network for a Class of Two-block Nonconvex and Nonsmooth Optimization" (Chen et al., 2024).

A recurrent source of ambiguity is that the acronym also appears in the supplied exposition of "Meta-learning based Alternating Minimization Algorithm for Non-convex Optimization" (Xia et al., 2020). That method is organized around meta-learned update rules implemented by recurrent networks and is presented as a proof of concept; the supplied summary states that it does not furnish a fully rigorous convergence theorem. By contrast, the CT-oriented LAMA literature is centered on dual-domain variational reconstruction, smoothing of $\ell_{2,1}$-type learned regularizers, residual PALM-style updates, and Clarke-stationarity guarantees.

This suggests that, within the recent inverse-problems literature, LAMA is best understood not as a generic synonym for learned alternating minimization, but as a specific family of convergent dual-domain reconstruction methods derived from a learnable two-block nonsmooth and nonconvex optimization model.

2. Variational model and dual-domain regularization

In the original CT formulation, the unknowns are the reconstructed image $x\in\mathbb{R}^n$, the reconstructed full-view sinogram $z\in\mathbb{R}^m$, and the measured sparse-view sinogram $s\in\mathbb{R}^{m_0}$. The forward operators are the full-view Radon transform $A:\mathbb{R}^n\to\mathbb{R}^m$ and a projection $P_0:\mathbb{R}^m\to\mathbb{R}^{m_0}$ that selects the measured views. The variational objective is

$$\min_{x,z}\; \underbrace{\tfrac12\|Ax-z\|_2^2+\tfrac{\lambda}{2}\|P_0z-s\|_2^2}_{f(x,z)\ \text{(data fidelity)}} +R(x;\theta_1)+Q(z;\theta_2).$$

The first term enforces image-data consistency through $z$0 and measurement consistency through $z$1 (Ding et al., 2023).

The regularizers are learnable and are defined as $z$2-norms of deep-feature maps,

$z$3

Each $z$4 is a small CNN that extracts $z$5-dimensional features at each of $z$6 spatial locations. In the CT implementation, the feature extractors are small CNNs with $z$7 convolutional layers, each followed by a smoothed ReLU activation; for the image net the kernels are $z$8 with $z$9 channels, and for the sinogram net the kernels are $s$0 with $s$1 channels. The stated role of these learned feature extractors is to replace hand-crafted finite differences by spatially coupled filters.

The dual-domain construction is central. Rather than regularizing only the reconstructed image or only the measured data, LAMA introduces a learned regularizer in each domain. In the later formulation, this is described as exploiting complementary redundancies and leveraging complementary information from both image and measurement domains. A plausible implication is that the auxiliary sinogram variable is not merely a bookkeeping device: it is the mechanism through which sinogram-domain priors and image-domain priors are coupled inside a single objective.

3. Smoothing, residual updates, and safeguarded alternating minimization

Because the $s$2 norm is nonsmooth at feature vectors of zero length, LAMA smooths each term $s$3 by a Huber-type surrogate

$s$4

which yields the smoothed objective

$s$5

For fixed $s$6, the gradient of the smoothed regularizer admits a closed-form expression, with the features below the threshold scaled by $s$7 and the others normalized by their Euclidean norms (Ding et al., 2023).

Algorithmically, LAMA is presented as an inexact proximal alternating linearized minimization scheme with primary residual updates, a safeguarding BCD step with backtracking, and periodic reduction of $s$8. With iteration index $s$9, the residual block first updates $\ell_{2,1}$0 through

$\ell_{2,1}$1

and then updates $\ell_{2,1}$2 through

$\ell_{2,1}$3

The step-size combinations satisfy

$\ell_{2,1}$4

and are described in the network realizations as learned step-sizes (Ding et al., 2024).

Acceptance of the residual block is controlled by an energy descent check together with a gradient-norm condition. If

$\ell_{2,1}$5

and the corresponding gradient condition holds, the residual iterate is accepted. Otherwise the method switches to a safeguarded BCD step with backtracking,

$\ell_{2,1}$6

$\ell_{2,1}$7

with $\ell_{2,1}$8 reduced by factor $\ell_{2,1}$9 until sufficient decrease is achieved. The smoothing parameter is reduced according to

$x\in\mathbb{R}^n$0

and otherwise remains unchanged.

The later LPAM formulation makes the same structure explicit at the level of a general two-block objective: gradient-descent previews, residual proximal linearization, a descent-and-gradient-control check, a simple BCD step with line search as safeguard, and automatic diminishing smoothing. In that generalization, the residual correction is stated to have the exact architecture of a two-layer ResNet (Chen et al., 2024).

4. Network realization, parameter sharing, and initialization

The algorithm is unrolled into a multi-phase network in which each phase mimics one outer iteration of LAMA (Ding et al., 2023). The “minus step-size times gradient” structure is implemented by residual connections, so the learned regularization update becomes a residual block rather than an opaque denoising module. In the later network formulations, each phase contains data consistency modules for both domains and learned residual modules for $x\in\mathbb{R}^n$1 and $x\in\mathbb{R}^n$2; each learned residual module is implemented by a CNN feature extractor, a channel-wise $x\in\mathbb{R}^n$3 norm, a back-propagation of that norm, and a residual update (Ding et al., 30 Jul 2025).

Parameter sharing is a defining design choice. In each LAMA phase the same $x\in\mathbb{R}^n$4 are reused for memory efficiency. This is explicitly identified as one reason the method reduces network complexity and improves memory efficiency. In the broader LPAM-net exposition, the same design is described as making the network interpretable because each module has a clear variational meaning: data step, prior step, and step-size (Chen et al., 2024).

The original CT implementation also includes an initialization network. It consists of five residual blocks, each with four $x\in\mathbb{R}^n$5 convolutional layers with 48 channels and ReLU, trained to map the sparse sinogram $x\in\mathbb{R}^n$6 to an initial full sinogram $x\in\mathbb{R}^n$7, after which FBP produces $x\in\mathbb{R}^n$8 (Ding et al., 2023). The 2025 extension iLAMA-Net makes this initialization stage explicit as an Init-Net that learns a pseudo full sinogram and its FBP image; the resulting pair $x\in\mathbb{R}^n$9 is then fed into LAMA-Net (Ding et al., 30 Jul 2025).

This evolution clarifies the architectural status of LAMA. It is simultaneously an optimization algorithm, an unrolled reconstruction network, and a framework for coupling learned initialization with convergent dual-domain refinement.

5. Convergence theory and interpretability

The convergence claims of LAMA are stated in terms of Clarke stationary points for the original nonsmooth and nonconvex objective. In the 2023 formulation, under standard Lipschitz and boundedness assumptions on $z\in\mathbb{R}^m$0, $z\in\mathbb{R}^m$1, and $z\in\mathbb{R}^m$2, together with the decreasing smoothing schedule $z\in\mathbb{R}^m$3, the iterates admit a subsequence converging to a Clarke stationary point of the original nonsmooth problem; the proof sketch proceeds by sufficient energy decrease, a uniform lower bound on backtracked step-sizes, summability of energy decrements, and passage to Clarke first-order conditions as $z\in\mathbb{R}^m$4 (Ding et al., 2023).

The later LAMA-Net paper states a stronger and more explicit theorem. Let $z\in\mathbb{R}^m$5 be the iterates of Algorithm 1, and form the subsequence at which $z\in\mathbb{R}^m$6 is decreased. That subsequence is bounded and has at least one accumulation point, and every such accumulation point is a Clarke stationary point of $z\in\mathbb{R}^m$7 (Ding et al., 30 Jul 2025). The paper describes this as a complete and rigorous convergence proof.

The broader LPAM theory organizes the analysis through smoothing conditions (C1)–(C4), a two-part descent/gradient lemma, and an automatic diminishing smoothing controller. For fixed $z\in\mathbb{R}^m$8, every inner iterate satisfies

$z\in\mathbb{R}^m$9

$s\in\mathbb{R}^{m_0}$0

with constants independent of $s\in\mathbb{R}^{m_0}$1, implying $s\in\mathbb{R}^{m_0}$2 at rate $s\in\mathbb{R}^{m_0}$3 for the smoothed problem (Chen et al., 2024).

These results are closely tied to interpretability claims. The network architecture follows the algorithm exactly, so the induced network is presented not as a free-form deep model but as an unrolled alternating-minimization procedure whose components correspond to data fidelity, learned regularization, step-size control, and safeguarding. The interpretability claim is therefore algorithmic rather than merely visual.

6. Empirical performance, stability, and broader significance

The reported CT experiments use the “2016 NIH-AAPM-Mayo” and NBIA CT collections, with sparse-view settings of 64 and 128 views, and evaluate PSNR, SSIM, number of parameters, and, in the later LAMA-Net paper, sinogram RMSE. Baselines include FBP, DDNet, LDA, DuDoTrans, and LEARN++ (Ding et al., 2023).

For 64-view Mayo data, the reported results are as follows (Ding et al., 30 Jul 2025):

Method	PSNR	SSIM / RMSE
DDNet	35.70	0.923 / 7.53
LDA	37.16	0.932 / 3.31
DuDoTrans	37.90	0.952 / 2.42
LEARN++	43.02	0.980 / 0.81
LAMA-Net	44.58	0.986 / 0.68
iLAMA-Net	46.37	0.990 / 0.52

The earlier paper reports the same 64-view Mayo trend in PSNR and SSIM, with FBP at approximately $s\in\mathbb{R}^{m_0}$4 dB and $s\in\mathbb{R}^{m_0}$5, DDNet at $s\in\mathbb{R}^{m_0}$6 and $s\in\mathbb{R}^{m_0}$7, LDA at $s\in\mathbb{R}^{m_0}$8 and $s\in\mathbb{R}^{m_0}$9, DuDoTrans at $A:\mathbb{R}^n\to\mathbb{R}^m$0 and $A:\mathbb{R}^n\to\mathbb{R}^m$1, LEARN++ at $A:\mathbb{R}^n\to\mathbb{R}^m$2 and $A:\mathbb{R}^n\to\mathbb{R}^m$3, and LAMA at $A:\mathbb{R}^n\to\mathbb{R}^m$4 and $A:\mathbb{R}^n\to\mathbb{R}^m$5; it also states that LAMA uses approximately $A:\mathbb{R}^n\to\mathbb{R}^m$6 M parameters, roughly $A:\mathbb{R}^n\to\mathbb{R}^m$7–$A:\mathbb{R}^n\to\mathbb{R}^m$8 fewer than DuDoTrans or LEARN++ while delivering $A:\mathbb{R}^n\to\mathbb{R}^m$9–$P_0:\mathbb{R}^m\to\mathbb{R}^{m_0}$0 dB higher PSNR (Ding et al., 2023). The 2025 report gives parameter counts of $P_0:\mathbb{R}^m\to\mathbb{R}^{m_0}$1 for LAMA-Net and $P_0:\mathbb{R}^m\to\mathbb{R}^{m_0}$2 for iLAMA-Net. This suggests that the architecture and reporting conventions changed across versions.

Qualitatively, LAMA is reported to better suppress streak artifacts and preserve fine vessels, and the 15-phase configuration is described as a good trade-off between accuracy and compute. The residual, or “inexact,” PALM steps outperform pure BCD in the ablation study (Ding et al., 2023).

A further empirical theme is stability. The later sparse-view CT paper reports robustness under structured “CAN U SEE IT”-mask perturbations and additive Gaussian noise with $P_0:\mathbb{R}^m\to\mathbb{R}^{m_0}$3 and $P_0:\mathbb{R}^m\to\mathbb{R}^{m_0}$4: LAMA-Net faithfully reconstructs the exact perturbation, yields the smallest reconstruction-to-true difference heat-maps, and remains top-ranked in PSNR versus noise level (Ding et al., 30 Jul 2025). The 2024 CT exposition explicitly ties this behavior to the convergence property, dual-domain learning, modest-sized CNNs rather than huge U-Nets or Transformers, and parameter sharing across phases (Ding et al., 2024).

In the broader optimization lineage, LPAM extends the same learned proximal alternating minimization template to a class of two-block nonsmooth and nonconvex problems and is stated to be easy to extend to multi-block problems (Chen et al., 2024). Within the supplied literature, this positions LAMA as both a specific sparse-view CT method and an instance of a more general program: unrolling convergent alternating schemes with learned nonsmooth regularizers while preserving explicit variational semantics.