We introduce a high-performance hyperspectral camera based on the Fourier-transform approach, where the two delayed images are generated by the Translating-Wedge-Based Identical Pulses eNcoding System (TWINS) [Opt. Lett. 37, 3027 (2012)], a common-path birefringent interferometer that combines compactness, intrinsic interferometric delay precision, long-term stability and insensitivity to vibrations. In our imaging system, TWINS is employed as a time-scanning interferometer and generates high-contrast interferograms at the single-pixel level. The camera exhibits high throughput and provides hyperspectral images with spectral background level of -30dB and resolution of 3 THz in the visible spectral range. We show high-quality spectral measurements of absolute reflectance, fluorescence and transmission of artistic objects with various lateral sizes.