Generalized Alternating Projection Based Total Variation Minimization for Compressive Sensing

Abstract: We consider the total variation (TV) minimization problem used for compressive sensing and solve it using the generalized alternating projection (GAP) algorithm. Extensive results demonstrate the high performance of proposed algorithm on compressive sensing, including two dimensional images, hyperspectral images and videos. We further derive the Alternating Direction Method of Multipliers (ADMM) framework with TV minimization for video and hyperspectral image compressive sensing under the CACTI and CASSI framework, respectively. Connections between GAP and ADMM are also provided.

• The paper presents a novel GAP-based approach to directly solve TV minimization for compressive sensing.
• Empirical results indicate improved PSNR and reconstruction quality compared to ADMM-TV and other techniques.
• The method offers computational efficiency and practical benefits for real-time processing in imaging and surveillance applications.

Generalized Alternating Projection Based Total Variation Minimization for Compressive Sensing

The paper, authored by Xin Yuan, introduces a novel approach for solving the total variation (TV) minimization problem central to compressive sensing applications through the generalized alternating projection (GAP) algorithm. This work extends the utility of the GAP algorithm to TV minimization tasks and contrasts it with the Alternating Direction Method of Multipliers (ADMM), particularly in applications involving two-dimensional images, hyperspectral images, and video data.

Overview and Mathematical Formulation

The study builds upon the established competencies of the GAP algorithm in compressive sensing domains using transformation techniques such as wavelet or DCT domains. Despite the cited success, challenges regarding the selection of appropriate transformations and their computational load led to investigating the feasibility of deploying GAP directly for TV minimization. The proposed work formulates the compressive sensing problem as follows: $$\min_{x} \|(x)\|, \quad \text{subject to} \quad x = y$$ where $\|(x)\|$ denotes the TV norm. The formulation is refined into: $$\min_{x, C} \quad \text{subject to} \quad \|(x)\| \le C \quad \text{and} \quad x = y$$ where $C$ signifies the radius of the $\ell_1$-ball concerning the TV of the signal.

GAP-TV Algorithm and Implementation

The paper details the implementation of the GAP-TV algorithm, which iteratively updates variables to minimize the TV while maintaining proximity to measured data. The process iteratively refines $x$, the desired signal, by alternating projections between the constraint enforcing $x = y$ and the penalization of TV. Further, the study discusses an accelerated update process for $x$ and juxtaposes it with the ADMM approach that employs an auxiliary variable and regularization, illustrating the nuanced differences in updating strategies and parameter dependencies across these methodologies.

Empirical Evaluations and Results

The research provides empirical evidence of the algorithm's efficacy across 2D images, videos, and hyperspectral imaging. Quantitatively, the GAP-TV and its accelerated variant showcased superior or comparable performance metrics against existing methods like ADMM-TV, TVAL3, and TwIST in reconstructing images at varied compressive sensing ratios, with notably improved Peak Signal to Noise Ratio (PSNR) levels in reconstructive quality. Similarly, simulations on video and hyperspectral image compressive sensing highlight the algorithm's robust adaptability in different coding regimes such as the CACTI framework for videos and the CASSI framework for hyperspectral images.

Theoretical and Practical Implications

Theoretically, the integration of GAP with TV minimization offers a promising, parameter-free framework, optimizing computational efficiency and simplifying model implementation complexities in compressive sensing tasks. Practically, the approach holds significance for applications in lensless imaging systems and scenarios requiring real-time processing of high-dimensional data streams, where maintaining minimal visual artifacts and high fidelity is pivotal.

Future Prospects

The results and methodology proposed provide a foundation for further explorations into optimizing GAP-TV variants and exploring their potential applications, particularly in areas like medical imaging, remote sensing, and video surveillance, where compressive sensing plays a transformative role. Future research could harness the adaptability of GAP-TV in real-time scenarios or its integration with deep learning augmentations to further enhance performance metrics and extend applicability to broader datasets and more complex sensing configurations.