A differential geometry approach for biomedical image processing

Understanding Physiological and Degenerative Natural Vision Mechanisms to Define Contrast and Contour Operators

“…Such decomposition has been proven for a large class of Liénard systems [41]–[44]. The Thom-Sebastiani conjecture assumes that this result still holds by considering sufficiently regular systems.…”

“…In the neighbourhood of a stable singularity or of a limit cycle of the corresponding velocity vector field supposed to be continuous, let us suppose that we can decompose the system into two parts, a potential and a Hamiltonian one, such as: where the residue R(x,y) tends to 0 when (x,y) tends to the stable singularity or to the limit cycle. Such decomposition has been proven for a large class of Liénard systems [41] – [44] . The Thom-Sebastiani conjecture assumes that this result still holds by considering sufficiently regular systems.…”

“…Let us define the remarkable Gaussian line of a peak as the set of points where the mean Gaussian curvature of the peak vanishes ( Figure 10 ). Its equation writes [41] : …”

Multiscale modeling and imaging: the challenges of biocomplexity

“…The presence of positive loops is related to the observation of multiple attractors, which definitively gives to the positive loop another signification that to the negative loop, more related to the stability of the system (like in the classical Watt regulator, well known in cybernetics). In 1992, Kauffman [41] conjectured that the mean number of attractors for a Boolean genetic network with n genes and connectivity , was of order of [21]. This conjecture is now supported by real observations: we have about 30 000 genes in the human genome and about 200 different tissues, which can be considered as different attractors of the same dynamics.…”

“…This conjecture is now supported by real observations: we have about 30 000 genes in the human genome and about 200 different tissues, which can be considered as different attractors of the same dynamics. For Arabidopsis thaliana, there is different tissues (sepals, petals, stamens and carpels) [21] and for the lytic operon of the phage , , and there is (lytic and lysogenic) attractors [70]. If we consider the network ruling the gastrulation in Drosophila (cf.…”