Extract Object Boundaries in Noisy ImagesSample ImagesLevel SetEdge MapThe Integration RangeSlide 6Speed FunctionNarrow Band Level SetExample IExample IIExample IIIConclusionExtract Object Boundaries in Noisy ImagesQuming ZhouSample ImagesLevel SetAn implicit form of a closed curve:By the chain rule,The movement equation of a close curve: with 0))),,(( tyxr0|| FNFrtttrt0|| Ft0)0,,( rtyx Edge MapEdge 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 asThe edge map for pixel s is defined by ))sin()()cos()((),( sgsgsEyx''),()( dsEsMThe Integration RangeThe intensity difference between pixel and pixel is To quantify the prediction of the boundary , an index is defined asThe maximize the integration of in the corresponding half plane as),(ssyxs )sin,cos('dydxsss|)()sin,cos(|),(ssssyxIdydxIsDiff ),(sP),(),(),(),(),(sDiffsDiffsDiffsDiffsP'),(sP'''),(maxarg'dsPEdge MapThe direction of the edge map points to its nearest boundary as its magnitude varies with the distance from the boundary.Speed FunctionA 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.)(MgotherwiseMifMifMMg22||)5~3(10cos5.0||10cos||1)()(/)( MgkcFNarrow Band Level SetThe total computational complexity2D image domain is Narrow band is , where m is the number of pixels in the narrow band. )(3N)(2mNCurve Position Outside Band Inside Band =0 <0 >0Example IExample IIExample IIIConclusionThe 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
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