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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. Using amoeba structuring elements, we devise image-adaptive versions of both filter classes. Experiments demonstrate the favourable properties of the filters.
Shape from Shading (SFS) is a fundamental task in computer vision. By given information about the reflectance of an object’s surface and the position of the light source, the SFS problem is to reconstruct the 3D depth of the object from a single grayscale 2D input image. A modern class of SFS models relies on the property that the camera performs a perspective projection. The corresponding perspective SFS methods have been the subject of many investigations within the last years. The goal of this chapter is to give an overview of these developments. In our discussion, we focus on important model aspects, and we investigate some prominent algorithms appearing in the literature in more detail than it was done in previous works.
In this paper we are concerned with robust structure-preserving denoising filters for color images. We build on a recently proposed transformation from the RGB color space to the space of symmetric 2×2 matrices that has already been used to transfer morphological dilation and erosion concepts from matrix-valued data to color images. We investigate the applicability of this framework to the construction of color-valued median filters. Additionally, we introduce spatial adaptivity into our approach by morphological amoebas that offer excellent capabilities for structure-preserving filtering. Furthermore, we define color-valued amoeba M-smoothers as a generalization of the median-based concepts. Our experiments confirm that all these methods work well with color images. They demonstrate the potential of our approach to define color processing tools based on matrix field techniques.
A Novel Approach for Computation of Morphological Operations Using the Number Theoretic Transform
(2022)
The fundamental operations of mathematical morphology are dilation and erosion. They are often implemented using a sliding window with the purpose to compute maximum respectively minimum of pixel values within the corresponding mask.
We reformulate the problem of morphological dilation respectively erosion of an image with a non-flat filter as a convolution of their umbras. To this end, we propose to make use of the number theoretic transform to compute the convolution in this setting. In contrast to other possible schemes, this transform represents a completely discrete computational approach. It allows exact convolution of sequences made up of integers. Therefore we propose by the combination of umbra framework and number theoretic transform a well-engineered combination.
There is no restriction on size or shape of the structuring element, and also flat and non-flat filters can be realised.