Quantile Filtering of Colour Images via Symmetric Matrices

Abstract Quantile filters, or rank-order filters, are local image filters which assign quantiles of intensities of the input image within neighbourhoods as output image values. Combining a multivariate quantile definition developed in matrix-valued morphology with a recently introduced mapping between the RGB colour space and the space of symmetric 2 × 2 matrices, we state a class of colour image quantile filters, along with a class of morphological gradient filters derived from these.We consider variants of these filters based on three matrix norms – the nuclear, Frobenius, and spectral norm – and study their differences. We investigate the properties of the quantile and gradient filters and their links to dilation and erosion operators. Using amoeba structuring elements,we devise image-adaptive versions of our quantile and gradient filters. Experiments are presented to demonstrate the favourable properties of the filters, and compare them to existing approaches in colour morphology.