In this paper, we study and compare wavelet and PCA compression methods for spectral images. By spectral images we mean any kind of multi-, hyper-, or ultraspectral images. In our study, we handle spectral images by channels or slices. We clarify advantages and disadvantages of wavelet based compression methods. If spectra of spectral images are smooth then wavelets can compress the spectral images quite well. We use two dimensional wavelet transforms for both spatial directions and spectrum directions. We calculate the transform using different wavelets and try several threshold values and techniques. The tests were done using 14 different natural spectral images from our spectral image database and the results were compared to each other. In judging the quality of a compression method we used three different criteria: compression time, error (measured in various ways) and compression ratio.