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Structure tensor images are obtained by a Gaussian smoothing of the dyadic product of gradient image. These images give at each pixel a <em>n</em>×<em>n</em> symmetric positive definite matrix SPD(<em>n</em>), representing the local orientation and the edge information. Processing such images requires appropriate algorithms working on the Riemannian manifold on the SPD(<em>n</em>) matrices. This contribution deals with structure tensor image filtering based on <em>L^p</em> geometric averaging. In particular, <em>L</em>¹ center-of-mass (Riemannian median or Fermat-Weber point) and <em>L</em>^∞ center-of-mass (Riemannian circumcenter) can be obtained for structure tensors using recently proposed algorithms. Our contribution in this paper is to study the interest of <em>L</em>¹ and <em>L</em>^∞ Riemannian estimators for structure tensor image processing. In particular, we compare both for two image analysis tasks: (i) structure tensor image denoising; (ii) anomaly detection in structure tensor images.