We present an introduction to spectral transformations on surfaces from a differential geometric perspective. We show that the Fourier series has an extension to smooth surfaces and that for triangulated surfaces, spectral transforms can be defined which approximate the Fourier coefficients of the underlying smooth manifold. This construction, which is based on a discrete approximation to the differential Laplacian operator, is compared to transforms

We improve Gaussian processes (GP) classification by reorganizing the (non-stationary and anisotropic) data to better fit to the isotropic GP kernel. First, the data is partitioned into two parts: along the feature with the highest frequency bandwidth. Secondly, for each part of the data, only the spectrally homogeneous features are chosen and used (the rest discarded) for GP classification. In this way, anisotropy of the data is lessened from the frequency point of view. Tests on synthetic data as well as real datasets show that our approach is effective and outperforms Automatic Relevance Determination (ARD). 1.

Many applications involve processing real data. It is ine cient to simply use a complex FFT on real data because arithmetic would be performed on the zero imaginary parts of the input, and, because of symmetries, output values would be calculated that are redundant. There are several approaches to developing special algorithms or to modifying complex algorithms for real data. There are two methods which use a complex FFT in a special way to increase e ciency [4], [14]. The rst method uses a length-N complex FFT to compute two length-N real FFTs by putting the two real data sequences into the real and the imaginary parts of the input to a complex FFT. Because transforms of real data have even real parts and odd imaginary parts, it is possible to separate the transforms of the two inputs with 2N-4 extra additions. This method requires, however, that two inputs be available at the same time. The second method [14] uses the fact that the last stage of a decimation-in-time radix-2 FFT combines two independent transforms of length N/2 to compute a length-N transform. If the data are real, the two half length transforms are calculated by the method described above and the last stage is carried out to calculate the total length-N FFT of the real data. It should be noted that the half-length FFT does not have to be calculated by a radix-2 FFT. In fact, it should be calculated by the most e cient complex-data algorithm possible, such as the SRFFT or the PFA. The separation of the two half-length transforms and