Fourier transform infrared (FTIR) imaging spectrometers are often corrupted by the problems of band overlap and random noise during the infrared spectrum acquisition process. Such noise would degrade the quality of the acquired infrared spectrum, limiting the precision of the subsequent processing. In this paper, we present a novel blind reconstruction method with wavelet transform regularizations for infrared spectrum obtained from the aging instrument. Inspired by the finding that the wavelet coefficient distribution of the clean spectrum is sparser than that of the degraded spectrum, a blind reconstruction model for infrared spectrum is proposed in this paper to regularize the distribution of the degraded spectrum by total variation regularization. This method outperforms when suppressing random noise and preserving the spectral structure details. In addition, an effective optimization scheme is introduced in overcoming the issue of formulated optimization. The instrument response function and latent spectrum can be simultaneously estimated through the proposed method that can efficiently mitigate the effects caused by instrument degradation. Finally, extensive experiments on simulated and real noisy infrared spectra are carried out to demonstrate the superiority of the proposed method over the existing state-of-the-art ones. Thus, the reconstructed spectrum will better serve the feature extraction and educational robot infrared vision sensing in industrial applications.