8 research outputs found
Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores
We provide the explicit solutions of linear, left-invariant,
(convection)-diffusion equations and the corresponding resolvent equations on
the 2D-Euclidean motion group SE(2). These diffusion equations are forward
Kolmogorov equations for stochastic processes for contour enhancement and
completion. The solutions are group-convolutions with the corresponding Green's
function, which we derive in explicit form. We mainly focus on the Kolmogorov
equations for contour enhancement processes which, in contrast to the
Kolmogorov equations for contour completion, do not include convection. The
Green's functions of these left-invariant partial differential equations
coincide with the heat-kernels on SE(2), which we explicitly derive. Then we
compute completion distributions on SE(2) which are the product of a forward
and a backward resolvent evolved from resp. source and sink distribution on
SE(2). On the one hand, the modes of Mumford's direction process for contour
completion coincide with elastica curves minimizing , related to zero-crossings of 2 left-invariant derivatives of the
completion distribution. On the other hand, the completion measure for the
contour enhancement concentrates on geodesics minimizing . This motivates a comparison between geodesics and elastica,
which are quite similar. However, we derive more practical analytic solutions
for the geodesics. The theory is motivated by medical image analysis
applications where enhancement of elongated structures in noisy images is
required. We use left-invariant (non)-linear evolution processes for automated
contour enhancement on invertible orientation scores, obtained from an image by
means of a special type of unitary wavelet transform
Analytic Solution of Stochastic Completion Fields
We use generalized particle trajectories to derive an analytic expression characterizing the probability distribution of boundary-completion shape. This is essential to the understanding of the perceptual phenomenon of illusory (subjective) contours. The particles' dynamics include Poisson-distributed ensembles of driving forces as well as particle decay. The resulting field, representing completed surface boundaries, is characterized by the fraction of particles at x with velocity x. The distributions are projectively covariant in the sense that fields calculated in any lower-dimensional projection correspond to the projections of fields calculated in any higher dimension. Being analytic, the relationship between velocity, diffusivity, and decay can be made readily apparent. 1 Introduction The phenomenon of illusory contours is much studied by visual psychologists and provides a compelling example of human visual competence not yet demonstrated by computer vision systems. Recently, ..
Analytic Solution of Stochastic Completion Fields
We use generalized particle trajectories to derive an analytic expression characterizing the probability distribution of boundary-completion shape. This is essential to the understanding of the perceptual phenomenon of illusory (subjective) contours. The particles' dynamics include Poisson-distributed ensembles of driving forces as well as particle decay. The resulting field, representing completed surface boundaries, is characterized by the fraction of particles at x with velocity x. The distributions are projectively covariant in the sense that fields calculated in any lower-dimensional projection correspond to the projections of fields calculated in any higher dimension. Being analytic, the relationship between velocity, diffusivity, and decay can be made readily apparent. 1 Introduction The phenomenon of illusory contours is much studied by visual psychologists and provides a compelling example of human visual competence not yet demonstrated by computer vision systems. Recently..