Combining sparse bayesian learning (SBL) with compressed sensing (CS)

a new method of reconstruction for compressed images with contaminated measurements is presented. This method regards the process of image reconstruction as a linear regression model and the image to be reconstructed as the unknown weights of the regression model. By sparse bayesian learning

the weights are endowed with certain prior condition probability density function

which limits the complexity of the model and simultaneously introduces the hyper-parameters. With maximizing the marginal likelihood function of hyper-parameters

the optimal weights are acquired

i.e. the reconstructed image. Simultaneously

this method provides the posterior probability density and the error bars of estimated weights

which deduces the uncertainty of reconstruction. Experimental results show that the new method can acquire exact reconstruction and under the same relative error of reconstruction

it is superior to basis pursuit on reconstruction time and orthogonal matching pursuit on the number of measurements.