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(e) (0 Fig.3.The zero-crossing points may be of the following types.(a) Ellip-tic.(b) Hyperbolic.(c) Parabolic.The zero-crossing lines may be of the
following types.(d) Elliptic umbilic.(e) Hyperbolic umbilic.(f) Para-bolic umbilic.
where the coefficients a,
{3,
j,and 0
are obtained by the Taylor expansion.It is easy to see that the set of points
The ZC lines may be a) an elliptic umbilic,if RA consists of three lines [see Fig.2(d)],b) a hyperbolic.umbilic,ifRA consists of a single real line [see Fig.2(e)] ,c) a parabolic umbilic,if RA consists of three lines,two of which are coincident [see Fig.2(f)],d) a symbolic umbilic,if RA consists of three coin-cident lines.5) If h(x,y) is a smooth function and in P
hex,y) depends on the fourth-order terms,the ZC lines have a complex shape that can be analyzed using results of [42].
Bifurcations of Zero Crossings:The isotopy theorem [50],[1] shows that transversal intersections are structur-ally stable,i.e.,that "transversal zero-crossings" are structurally stable:their topological properties do not change if the size (and thus the scale) of the filter is slightly changed.
If f(x,y) is a Morse function then Sf may meet So non-
elliptic Z.c.
(b) Fig.4.The two types of bifurcations that can occur for incre~sing
and
decreasing a in the case of Morse functions.(a) Left to right.(b) Right to left.
transverally,and these intersections are not structurally stable (observe that Morse functions are structurally sta-ble but not their intersections with So),If f is a Morse function,then Sf may meet So nontransversally at elliptic points and hyperbolic points.These intersections are not structurally stable and may change their topological prop-erties for small perturbations of f.More specifically,we may have two bifurcations:
a) Elliptic ZC:At elliptic ZC,a small perturbation of f may lead to the disappearance of the ZC or to the appearance of a contour of ZC constituted by a closed curve.
(e) (0 Fig.3.The zero-crossing points may be of the following types.(a) Ellip-tic.(b) Hyperbolic.(c) Parabolic.The zero-crossing lines may be of the
following types.(d) Elliptic umbilic.(e) Hyperbolic umbilic.(f) Para-bolic umbilic.
where the coefficients a,
{3,
j,and 0
are obtained by the Taylor expansion.It is easy to see that the set of points
The ZC lines may be a) an elliptic umbilic,if RA consists of three lines [see Fig.2(d)],b) a hyperbolic.umbilic,ifRA consists of a single real line [see Fig.2(e)] ,c) a parabolic umbilic,if RA consists of three lines,two of which are coincident [see Fig.2(f)],d) a symbolic umbilic,if RA consists of three coin-cident lines.5) If h(x,y) is a smooth function and in P
hex,y) depends on the fourth-order terms,the ZC lines have a complex shape that can be analyzed using results of [42].
Bifurcations of Zero Crossings:The isotopy theorem [50],[1] shows that transversal intersections are structur-ally stable,i.e.,that "transversal zero-crossings" are structurally stable:their topological properties do not change if the size (and thus the scale) of the filter is slightly changed.
If f(x,y) is a Morse function then Sf may meet So non-
elliptic Z.c.
(b) Fig.4.The two types of bifurcations that can occur for incre~sing
and
decreasing a in the case of Morse functions.(a) Left to right.(b) Right to left.
transverally,and these intersections are not structurally stable (observe that Morse functions are structurally sta-ble but not their intersections with So),If f is a Morse function,then Sf may meet So nontransversally at elliptic points and hyperbolic points.These intersections are not structurally stable and may change their topological prop-erties for small perturbations of f.More specifically,we may have two bifurcations:
a) Elliptic ZC:At elliptic ZC,a small perturbation of f may lead to the disappearance of the ZC or to the appearance of a contour of ZC constituted by a closed curve.
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