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Ke Liang

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6 publications

preprint2026arXiv

Combined Dictionary Unfolding Network with Gradient-Adaptive Fidelity for Transferable Multi-Source Fusion

Deep Unfolding Network-based methods have emerged as effective solutions for multi-source image fusion by combining model-driven iterative optimization with data-driven deep learning. However, most existing deep unfolding image fusion methods are derived from alternating minimization, which updates the features of different modalities separately. This design introduces considerable computational and memory overhead, limiting deployment on resource-constrained edge devices. To address this issue, we propose CDNet, a lightweight Combined Dictionary Unfolding Network for multi-source image fusion. Rather than introducing a new sparse coding prior or empirically compressing an existing fusion network, CDNet translates the unique-common decomposition prior of coupled dictionary learning into a structurally constrained joint unfolding architecture. The resulting CDBlock follows a block-sparse interaction topology and performs a model-derived joint update of common and modality-specific representations, thereby streamlining feature learning and improving efficiency.In addition, we design a compact High- and Low-frequency Image Fidelity loss for unsupervised training without ground-truth images. We evaluate CDNet on four tasks, including multi-exposure image fusion, infrared and visible image fusion, medical image fusion, and infrared and visible image fusion for semantic segmentation. Experimental results show that CDNet achieves competitive or superior fusion performance with high efficiency. For infrared and visible image fusion, CDNet outperforms competing methods on four of six metrics on the TNO dataset and five of six metrics on the RoadScene dataset. In particular, it surpasses the second-best method by 1.23 dB and 1.59 dB in PSNR on TNO and RoadScene, respectively.

preprint2015arXiv

Non-trivial mm-quasi-Einstein metrics on quadratic Lie groups

We call a metric mm-quasi-Einstein if RicXmRic_X^m (a modification of the mm-Bakry-Emery Ricci tensor in terms of a suitable vector field XX) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and left-invariant Riemannian metrics on quadratic Lie groups. First we prove that any left-invariant vector field XX such that the left-invariant Riemannian metric on a quadratic Lie group is mm-quasi-Einstein is a Killing field. Then we construct infinitely many non-trivial mm-quasi-Einstein metrics on solvable quadratic Lie groups G(n)G(n) for mm finite.

preprint2014arXiv

Non-trivial mm-quasi-Einstein metrics on simple Lie groups

We call a metric mm-quasi-Einstein if RicXmRic_X^m, which replaces a gradient of a smooth function ff by a vector field XX in mm-Bakry-Emery Ricci tensor, is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant metrics on simple Lie groups. First, we prove that XX is a left-invariant Killing vector field if the metric on a compact simple Lie group is mm-quasi-Einstein. Then we show that every compact simple Lie group admits non-trivial mm-quasi-Einstein metrics except SU(3)SU(3), E8E_8 and G2G_2, and most of them admit infinitely many metrics. Naturally, the study on mm-quasi-Einstein metrics can be extended to pseudo-Riemannian case. And we prove that every compact simple Lie group admits non-trivial mm-quasi-Einstein Lorentzian metrics and most of them admit infinitely many metrics. Finally, we prove that some non-compact simple Lie groups admit infinitely many non-trivial mm-quasi-Einstein Lorentzian metrics.

preprint2012arXiv

The decomposition of a Lie group with a left invariant pseudo-Riemannian metric and the uniqueness

In this paper, we discuss the decomposition of a Lie group with a left invariant pseudo-Riemannian metric and the uniqueness. In fact, it is a decomposition of a Lie group into totally geodesic sub-manifolds which is different from the De Rham decomposition on a Lie group. As an application, we give a decomposition of a Lie group with a left invariant pseudo-Riemannian Einstein metric, and prove that the decomposition is unique up to the order of the parts in the decomposition.

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