The identification and characterization of a mineral from its reflectance spectrum is based on the analysis of the continuum (i.e., the overall shape) and its diagnostic absorptions. The development of automatic methods to retrieve such useful information is a key challenge as new airborne and satelliteborne sensors are developed. We present a flexible and automatic deconvolution procedure able to deal with various minerals and sensors. The approach is based on a physical model and offers the possibility to include noise statistics. It is composed of three successive steps: (i) the continuum is pre-estimated and removed using a non-linear least-squares algorithm with adapted constraints; (ii) the absorptions are pre-estimated using a greedy algorithm; (iii) the continuum and absorption estimates are jointly refined using a non-linear least-squares procedure. Each step is validated first on synthetic spectra, including the sensitivity of the full procedure to instrumental noise. Then, the procedure is applied to diverse laboratory spectra. In most cases, absorption positions are recovered with an accuracy lower than 5 nm, enabling mineral identification. The potential of the proposed deconvolution procedure is finally assessed on hyperspectral images acquired over quarries during a dedicated airborne campaign. Here, minerals such as calcite and gypsum are clearly identified based on their diagnostic absorption features, including when they are in a mixture. Small changes in the shape of the kaolinite doublet are also clearly detected and could be related to crystallinity or mixture with other minerals such as gibbsite.