Airborne Hyperspectral images can be efficiently obtained with imaging static Fourier transform spectrometers. However, to be effective on any location, this technology requires to know the relief of the scene. This is not a straightforward process, as the horizontal interference fringes on the images, which are necessary for spectrum construction, prevent efficient stereoscopic processing. We present a novel variational model for multiplicative image decomposition to separate the fringes from the panchromatic image of the scene. This multiplicative model is much more physically accurate than previous additive decomposition models inspired by cartoon-texture decomposition. It combines fully smoothed total variation operators and 1D Fourier transform. Smoothed total variation is adopted to avoid staircasing artifacts caused by traditional total variation regularisation. The use of a 1D Fourier transform is suggested by the geometry of the fringes, in order to circumvent the lack of horizontal periodicity in the interferometric pattern. We also present an optimization algorithm. Finally, a second algorithm is introduced, whose convergence is not mathematically guaranteed. However it systematically approaches the solution of the first one in much less computation time. Our experimental evaluation on real and simulated images shows that the proposed model separates fringes from the panchromatic image very accurately and that this accuracy significantly improves subpixel stereo matching results.