1. Extract Object Boundaries in Noisy Images
Quming Zhou

2. Sample Images

3. Level Set An implicit form of a closed curve:
By the chain rule,
The movement equation of a close curve:
with

4. Edge Map
Edge Map points toward the closet boundary pixel with its magnitude represent the total gradient energy in the half plane.
The local edge vector at pixel s along the orientation θ is given as
The edge map for pixel s is defined by

5. The Integration Range
The intensity difference between pixel and pixel is
To quantify the prediction of the boundary , an index is defined as
The maximize the integration of in the corresponding half plane as

6. Edge Map
The direction of the edge map points to its nearest boundary as its magnitude varies with the distance from the boundary.

7. Speed Function
A scaling function of the edge map is given as
where is the angle between the edge map and the outward normal vector of the curve.
The speed function in the outward normal direction of the curve is
where c and ε are constants, k is the curvature.

8. Narrow Band Level Set The total computational complexity
2D image domain is
Narrow band is , where m is the number of pixels in the narrow band.

9. Example I

10. Example II

11. Example III

12. Conclusion
The proposed approach uses both the edge direction and the gradient magnitude to overcome the problems resulting from weak edges. The results are significantly superior than results obtained using edge magnitude alone.