About snake equation (5)

        Curvature ： The curve has curvature , The curve with negative curvature extends outward , The curvature is positive and the curve extends inward . The curvature force is the second derivative of the curve . Mathematical properties of curvature force driven evolution ： Any simple curvature , A circle is formed by the force of curvature , And then disappear .（ The curvature force of all points of the circle points to the center ）  

      Describe the geometric characteristic parameters of the curve ： Unit normal 、 Curvature .

Curve evolution process ： The curve is acting F Driven by , Towards the normal N At speed v Evolution . v For regular inward evolution , If it is negative, it will evolve outward .

When the curve reaches the target contour , Force balance ,v zero , The least energy , The curve does not change .

      key problem ：1. How to represent a curve        

                      2. How to construct force The active contour model is derived from the problem ,

  For the first question ：1. Represent the contour with parameters , Parametric active contour model .snake Model .2. Represent the outline in a set , Assemble active contour models .level set

level set （ Level set ） Algorithm  

                            

  The first formula , The evolution of the curve , Among them V How to choose , We don't care about ,V The way we choose determines whether we want to do image segmentation .

Second formula , Is the evolution of the surface , Evolution of surfaces , If you follow this formula , That is, the point on the surface ,z=f(x,y) The value of , As time changes, it satisfies the right side of this formula . So every time , Take this surface level 0, Those points must be the curve in the current state we want above .

.............. branch .. cut .. Line .....................

Classification of active contour models Parametric active contour model                    

                                      Geometric active contour model (Geometric Active Contour Model)           

Active contour model （Snake Model ） It is a deformable parameter curve and the corresponding energy function , The goal is to minimize the energy objective function , Control parameter curve deformation , The closed curve with minimum energy is the target contour .

Geometric active contour model Based on curve evolution theory and level set method (Level Set), It is to express the low dimensional evolution curve or surface through a high-dimensional function surface , The evolving curve or surface is expressed as an indirect expression of the zero level set of high-dimensional function surfaces , The evolution equation of the evolution curve or surface is transformed into the evolution partial differential equation of the high-dimensional level set function .