Complete lattice learning for multivariate mathematical morphology

• Extends mathematical morphology to multivariate vectors.
• Proposes an efficient strategy for complete lattice learning.
• Requires no prior assumption on background/foreground.
• Can integrate supervised information.
• Enables to perform patch-based morphological operations.

The generalization of mathematical morphology to multivariate vector spaces is addressed in this paper. The proposed approach is fully unsupervised and consists in learning a complete lattice from an image as a nonlinear bijective mapping, interpreted in the form of a learned rank transformation together with an ordering of vectors. This unsupervised ordering of vectors relies on three steps: dictionary learning, manifold learning and out of sample extension. In addition to providing an efficient way to construct a vectorial ordering, the proposed approach can become a supervised ordering by the integration of pairwise constraints. The performance of the approach is illustrated with color image processing examples.

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