The knowledge of the spectral sensitivity of a colour camera is an essential requirement to calibrate cameras for a variety of illuminants and to parameterise colour constancy algorithms. The presented method estimates the spectral sensitivity by measuring an optimal set of known spectral stimuli. These are spectrally broadband colour samples generated by illumination of Roscolux transmissive filters of which there is a large number available. However, as a smaller set reduces measurement time in a practical setup, we show an approach to select a reduced set of optimal filters by evaluating Lagrange multipliers. We demonstrate practical results using this optimal set in an experimental setup to determine the spectral system sensitivity of a digital colour camera. The advantage of this new setup compared to existing approaches using test charts is its optical geometry using transmissive filters that allow precise measurements. A result of superior quality is achieved by additional optimisations of the measurement conditions. The quality of the measurements is judged by the prediction error using the estimated sensitivity curves for an independent set of colour samples.