Fourier Transform Infrared (FT-IR) spectroscopic imaging is a potentially valuable tool for diagnosing breast and prostate cancer, but its clinical deployment is limited due to long data acquisition times and vast storage requirements. To counter this limitation, we develop a sparse representation for FT-IR absorbance spectra using a learned dictionary. This sparse representation is used as prior knowledge in regularizing the compressed sensing inverse problem. The data size and acquisition time are directly proportional to the length of the measured signal, namely the interferogram. Hence, we model our measurement process as interferogram truncation, which we implement by low pass filtering and downsampling in the spectral domain. With a downsample factor of four, our reconstruction is adequate for tissue classification and provides a Peak Signal-to-noise Ratio (PSNR) of 41.92 dB, while standard interpolation of the same low resolution measurements can only provide a PSNR of 36.93 dB.