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Figure 1: A snake with traditional potential forces cannot move into the concave boundary region.




Figure 2: A snake with GVF external forces moves into the concave boundary region.




Figure 3: A GVF snake converges to the same result from either the inside or the outside.




Figure 4: A GVF snake can also be initialized across the object boundary.



Figure 5: A GVF snake correctly reconstructs the subjective contour.




Figure 6: A GVF snake can find the boundary represented by scattered points.




 \begin{landscape}% latex2html id marker 1991
..., and (h) 600 iterations.}

Figure 8: Stream lines of the particles shown in Fig. 7.


Figure 9: Stream lines of particles in (a) a potential force field and (b) a GVF field.



Figure: (a) A $64\times 64$ image of the U-shape object corrupted by additive white Gaussian noise (6 dB) (b) The edge map $\vert\nabla (G_\sigma *I)\vert$ with $\sigma = 1.5$ (normalized to the range [0,1]). (c) The computed GVF. (d) Initial and intermediate contours (gray curves) and the final contour (white curve) of a GVF snake.





Figure: (a) A ( $160\times 160$) magnetic resonance image of the left ventrical of a human heart (short-axis section). (b) The edge map $\vert\nabla (G_\sigma *I)\vert$ with $\sigma = 2.5$ (normalized to the range [0,1]). (c) The computed GVF (shown subsampled by a factor of two). (d) Initial and intermediate contours (gray curves) and the final contour (white curve) of the GVF snake.





 \begin{landscape}% latex2html id marker 2183
..., and (h) 100 iterations.}

Figure 13: The magnitude of the GVF field gives information about the medial axis of a shape.

\fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_disk.eps}}} \fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_disk1.eps}}}

\fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_square.eps}}} \fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_square1.eps}}}

\fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_rectangle.eps}}} \fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_rectangle1.eps}}}

\fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_core.eps}}} \fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_core1.eps}}}

\fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_saw.eps}}} \fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_saw1.eps}}}

\fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_screw.eps}}} \fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_screw1.eps}}}

\fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_blob.eps}}} \fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_blob1.eps}}}

\fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_dumbbell.eps}}} \fbox{\resizebox{3.6cm}{3.6cm}{\includegraphics{images/core_dumbbell1.eps}}}

Figure: The medialness map ${\cal M}^1(x,y)$(q=0.05) of a teardrop shape, shown as (a) a gray-level image and (b) a surface plot.



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Chenyang Xu