In recent years, there has been an increasing interest in miniature spectrometers for research and development. Especially, filter-array-based spectrometers have advantages of low cost and portability, and can be applied in various fields such as biology, chemistry and food industry. Miniaturization in optical filters causes degradation of spectral resolution due to limitations on spectral responses and the number of filters. Nowadays, many studies have been reported that the filter-array-based spectrometers have achieved resolution improvements by using digital signal processing (DSP) techniques. The performance of the DSP-based spectral recovery highly depends on the prior information of transmission functions (TFs) of the filters. The TFs vary with respect to an incident angle of light onto the filter-array. Conventionally, it is assumed that the incident angle of light on the filters is fixed and the TFs are known to the DSP. However, the incident angle is inconstant according to various environments and applications, and thus TFs also vary, which leads to performance degradation of spectral recovery. In this paper, we propose a method of incident angle estimation (IAE) for high resolution spectral recovery in the filter-array-based spectrometers. By exploiting sparse signal reconstruction of the L1- norm minimization, IAE estimates an incident angle among all possible incident angles which minimizes the error of the reconstructed signal. Based on IAE, DSP effectively provides a high resolution spectral recovery in the filter-array-based spectrometers.