Mode Estimation in High-dimensional Spaces with Flat-top Kernels: Application to Image Denoising
Arnaud De Decker, John A. Lee, Damien François, Michel Verleysen · 2010
Abstract. Data denoising can be achieved by approximating the data distribution and replacing each data item with an estimate of its closest mode. This idea has already been successfully applied to image denoising. The data then consists of pixel intensities or image patches, that is, vectorized groups of pixel intensities. The latter case raises the issue of mode estimation in a high-dimensional space, since patches can contain about 10 to more than 100 pixels. This paper shows that the widely used Gaussian kernel is outperformed by flat-top kernels that are specifically tailored in order to fight the curse of dimensionality. 1 Mode Estimation Starting from a data sample, mode estimation can be achieved by first approximating the underlying data distribution. Next, a hill-climbing procedure can be run from any point in order to obtain an estimate of the closest mode. If X = [xi]1≤i≤N denotes the data sample, then the widely known Parzen’s window kernel probability density estimator (KPDE) [1] can be written as ˆp(xi) = C ∑ N j=1 Ψσ(‖xi − x j‖2 2 /2) , where C is a normalization factor that ensures that R ˆp(x)dx = 1 and kernel Ψσ is a positive and