Newton Deep Unfolding for Compressed Sensing

Changhua He; Xianchao Xiu

cs.CVSep 13, 2026

Newton Deep Unfolding for Compressed Sensing

Authors: Changhua He, Xianchao Xiu

Organizations: School of Mechatronic Engineering and Automation, Shanghai University

Abstract

Compressed sensing (CS) reconstructs images from highly limited measurements, but existing deep unfolding methods are typically driven by first-order optimization and weakly exploit the optimization states generated during reconstruction. To address these limitations, we propose a Newton deep unfolding network (NDU-Net), which, to the best of our knowledge, is the first deep unfolding framework that leverages second-order optimization for CS reconstruction. Specifically, NDU-Net introduces a Newton update (NU) module to estimate Newton-type update directions and generate optimization states that characterize the current reconstruction process. Furthermore, a Newton-guided multi-scale prior (MP) module is designed to incorporate these optimization states into multi-scale feature restoration, thereby enabling the learned prior to adapt to the current reconstruction stage. Experimental results under different CS ratios confirm that our proposed NDU-Net achieves promising reconstruction performance and exhibits enhanced robustness. Our code is available at https://github.com/xianchaoxiu/DNU-Net.

Figures & tables

Figure 1: PSNR versus reconstruction time on Set11 at CS ratio

=0.25

Figure 2: Comparison between typical first-order deep unfolding networks and our second-order NDU-Net.

Figure 3: Overall architecture of NDU-Net. (a) The $K$ -stage framework uses NU-derived optimization states $(\mathbf{g}^{k},\mathbf{r}_{N}^{k},\Delta\mathbf{x}^{k})$ to guide the MP module at each stage. (b) The NU module performs Newton-type data-consistency correction. (c) The MP module fuses Newton guidance with multi-scale features.

Figure 4: Multi-scale feature extractor in the MP module. Features from local, middle-scale, and large-scale branches are concatenated and fused.

Datasets	Methods	CS ratios				Average
		0.10	0.25	0.40	0.50
LIVE29	ISTA-Net+ (CVPR 2018)	25.05/0.7104	29.42/0.8648	32.55/0.9275	34.55/0.9514	30.39/0.8635
	TransCS (TIP 2022)	27.66/0.8196	32.05/0.9251	35.51/0.9633	37.74/0.9767	33.24/0.9212
	DPC-DUN (TIP 2023)	26.83/0.7812	31.17/0.9008	34.43/0.9477	36.43/0.9649	32.22/0.8987
	OCTUF (CVPR 2023)	28.23/0.8345	32.69/0.9327	36.19/0.9671	38.52 / 0.9795	33.91/0.9285
	CPP-Net (CVPR 2024)	28.46/0.8414	32.82/0.9343	36.20/0.9672	38.37/0.9790	33.96/0.9305

Table 1: Average PSNR/SSIM comparisons on four benchmark datasets under different CS ratios. Red bold and blue values denote the best and second-best results, respectively.

Figure 5: Visual comparisons of reconstructed images, including local detail enlargements and error heatmaps. From top to bottom, the CS ratios are $0.10$ , $0.25$ , and $0.40$ , respectively.

Figure 6: Robustness comparisons on LIVE29 at CS ratio

=0.10

, where the left corresponds to Gaussian noise and the right corresponds to salt-and-pepper noise.

Variants	LIVE29	Set11	BSD68
w/o NU	33.91/0.9307	36.92/0.9625	36.80/0.9505
w/o MP	33.16/0.9257	35.70/0.9545	36.21/0.9483
Full model	34.43/0.9409	37.34/0.9657	37.61/0.9599

Table 2: Ablation studies of the NU and MP modules at CS ratio

=0.25

Layers	PSNR	SSIM	Params
1	33.44	0.9351	1.001
2	35.73	0.9550	1.999
3	36.67	0.9612	2.996
4	37.02	0.9635	3.994
5	37.29	0.9653	4.991
6	37.34	0.9657	5.989

Table 3: Ablation studies of the number of unfolding stages on Set11 at CS ratio

=0.25

Compressed Sensing

Compressed sensing recovers signals from fewer measurements by exploiting sparsity, structure, and learned prior information.

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Dual-Path Hyperprior Informed Deep Unfolding Network for Image Compressive Sensing

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Harbin Institute of Technology

Image Reconstruction Deep Unfolding Image Inverse Problems

Open paper AskPDF arXiv

Recent Deep Unfolding Networks (DUNs) have significantly advanced Compressive Sensing (CS) by integrating iterative optimization with deep networks. However, existing DUNs still suffer from two challenges: 1) Reliance on a single measurement stream, which limits effective information interaction across distinct measurement subsets. 2) Uniform processing of all image regions, which overlooks varying reconstruction difficulties induced by diverse textures. To address these limitations, a novel Dual-Path Hyperprior Informed Deep Unfolding Network (DPH-DUN) is proposed, which partitions measurements into double subsets to enable hyperprior-guided reconstruction via a dual-path architecture. In the Deep Hyperprior Learning branch, a series of lightweight neural modules are designed to efficiently generate hyperprior knowledge of different domains, enabling collaborative guidance for the CS reconstruction. In the Hyperprior Informed Reconstruction branch, a deep unfolding framework with hyperprior guidance is constructed to iteratively refine reconstruction. Specifically, i) in the gradient descent step, a Hyperprior Informed Step Size Generation network is designed to dynamically generate spatially varying step maps, enabling adaptive fine-grained gradient updates. ii) In the proximal mapping step, two well-designed hyperprior informed attention mechanisms are introduced to dynamically focus on challenging regions via gradient-based hard and soft attentions, facilitating CS reconstruction accuracy. Extensive experiments demonstrate that the proposed DPH-DUN outperforms existing CS methods.

Abstract

Compressed Sensing

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