Thesis: Gamma-ray spectrometry is one of the main techniques used for the measurement of radioactivity, which allows identifying and quantifying radionuclides. The objective of this thesis is to develop new spectrum analysis methods to improve the detection limits. In this context, the first contribution is investigating the activity estimation in gamma-ray spectrometry with spectral unmixing, which decomposes a measured spectrum into individual radionuclides' spectra. Contrary to standard methods, this approach allows accounting for the full spectrum analysis of a gamma-ray spectrum and the Poisson statistics underlying the detection process. By formulating the activity estimation as an inverse problem under non-negativity constraint, the sparse spectral unmixing is investigated to estimate the subset of active radionuclides and their activities jointly. The second contribution is the metrological use of the investigated spectral unmixing method, which further necessitates the evaluation of characteristic limits for decision making purposes and the instruments' calibration for quantitative analysis.