This talk presents an overview of some
advances in two broad research directions in image analysis and computer vision
that share in common the properties of being nonlinear and geometric. The first
is based on morphological image operators and their lattice-theoretic generalizations
which have a rich algebraic structure. The second approach uses nonlinear PDEs some
of which are related to morphological operators and/or are derived from a variational
formulation. Both approaches and often
their combination are useful for multiscale edge-preserving smoothing, feature
detection, image simplification, structure+texture decomposition, segmentation,
and shape analysis. After a brief
synopsis of morphological operators on images and graphs, we shall continue with their PDE and
variational formulation. First we focus on a class of multiscale connected operators
with a combined local and global action where the PDE and the lattice approach
can harmoniously work together, the first to provide continuous-scale
isotropic-growth models with global constraints and the second to study discrete
algorithms for numerical implementations. Then, we describe their usage for
image simplification, structure+texture decomposition, and PDE-based image
segmentation implemented by levelset curve evolution and driven both by
watershed flooding and a texture oscillation energy. In an alternative scheme,
this energy approach and a modulation image model help us develop an efficient
unsupervised segmentation approach using region competition and weighted curve
evolution based on probabilistic cue integration. If time permits, we shall
also summarize some ongoing work in patch-based PDEs for tensor-based image
diffusions using a variational framework.
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