Gaussian processes-prior systems generally consist of noisy measurements of samples of the putatively Gaussian process of interest, where the samples serve to constrain the posterior estimate. Here we consider the case where the measurements are instead noisy weighted sums of samples. This framework incorporates measurements of derivative information and of filtered versions of the process, thereby allowing GPs to perform sensor fusion and tomography, it allows certain group invariances (ie symmetries) to be weakly enforced, can be used to model heteroskedasticity in output variance, and under certain conditions it allows the dataset to be dramatically reduced in size. The method is applied to a sparsely sampled image, where each sample is taken using a broad and non-monotonic point spread function.