Compressive sensing by learning a Gaussian mixture model from measurements.

Area of Science:

• Signal Processing
• Machine Learning
• Statistical Modeling

Background:

• Compressive sensing (CS) typically requires accurate prior signal models.
• Obtaining precise Gaussian mixture model (GMM) signal statistics for CS is often challenging.
• Existing methods struggle with accurate GMM-based CS reconstruction without extensive training data.

Purpose of the Study:

• To develop a method for compressive sensing that learns the signal model in situ from measurements.
• To enable accurate GMM-based CS reconstruction without relying on external training datasets.
• To improve the performance of compressive sensing techniques in various applications.

Main Methods:

• Developed a maximum marginal likelihood estimator (MMLE) for GMM-based CS.
• Treated signals as random variables, integrating them out in the likelihood function.
• Extended the MMLE for GMMs with low-rank covariance matrices for computational efficiency.

Main Results:

• Achieved closed-form minimum mean squared error reconstruction.
• Demonstrated superior performance over state-of-the-art methods in image inpainting, high-speed video CS, and hyperspectral imaging.
• Validated the approach using real-world data from compressive cameras.

Conclusions:

• The proposed in situ GMM learning method significantly enhances compressive sensing performance.
• The MMLE provides accurate signal reconstruction even with incomplete linear measurements.
• The approach is robust and applicable to diverse imaging and sensing modalities.