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Hyperspectral image (HSI) classification requires spectral dimensionality reduction (DR) and spatial filtering. While common DR and denoising methods use linear algebra, we propose a tensorial method to jointly achieve denoising and DR. Tensorial processing models HSI data as a whole entity to treat jointly spatial and spectral modes. A multidimensional Wiener filter (MWF) was successfully applied to denoise multiway data such as color images. First, we adapt the quadtree decomposition to tensor data in order to take into account the local image characteristics in the MWF. We demonstrate that this novel version of the filter called adaptive multidimensional Wiener filtering (AMWF)-(K1, K2, K3) performs well as a denoising preprocessing to improve classification results. Then, we propose a novel method, referred to as AMWFdr-(K1, K2, D3) which performs both spatial filtering and spectral DR. Support vector machine is applied to the output of four-dimensionality and noise-reduction methods to compare their efficiency: the proposed AMWFdr-(K1, K2, D3), PCAdr, ICAdr, MNFdr, and DWTdr associated with Wiener filtering.