The Blade Tip Timing (BTT) method enables online real-time monitoring of rotating blade vibrations through non-contact vibration measurement. However, the blade vibration signals measured by BTT systems are severely undersampled. While sparse reconstruction theory can address the reconstruction of undersampled signals, traditional methods based on regularization often underestimate the amplitude of reconstructed signals, compromising effective monitoring of blade vibration states. To overcome this limitation, this paper introduces a non-convex regularization model incorporating the non-convex Cauchy penalty term into the cost function for reconstructing undersampled BTT signals of aeroengine compressor blades. A convex proximal splitting method is employed to optimize the objective function. By carefully selecting model parameters in the forward-backward (FB) algorithm, the convergence of the overall objective function is ensured. Simulations verify the accuracy of the proposed method in identifying multi-modal vibration parameters of the rotor. Comparative results demonstrate that this method significantly outperforms regularization methods in identifying multi-modal vibration amplitudes of blades.