chan vess segmentation

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sign illustration

• f : denote the given grayscale image, f(x),x is point

• t : iteration t's border matrix, binary bool ?, describe as iteration th 's edge result, is closed curve mask

• ε : Heaviside regularization,custom params, = 1

• x : pixel/position in image,2D

• φ : level set function, φ(i,j),φ_i,j : φ value at point(i,j)

• C : an edge set curve

  

  image

  Boundary C represented as the zero level set of implicit function φ(x, y)¹

• H : Heaviside function, for caculate energy function

• H_ε(t,p) : regularised heavyside function, is matrix , size same to image,∈[0,1]

  for example: if border mark 1, unkown area mark 0, that trans to 1 and 0.5

• operator illustration: : sum for all H matrix value —— np.sum(H), is a scale value

• c_inner c_outer : the optimal values c₁ and c₂

• u : piecewise-smooth function

• The third and fourth terms penalize discrepancy between the piecewise constant model u and the input image f.

• divergence 散度计算:

  def div(fx,fy):
  # 图像的散度对某幅图像连续求2次梯度得到的zx_x和zy_y的和是等于散度的。换句话说div(z) = zx_x +zy_yd
      Fyy,fyx=np.grad(fy)
      Fxy,fxx=np.grad(fx)
      return fxx+fyy

• discretize the curvature 曲率计算 :

 
def curvature(f):
    fy,fx=grad(f)
    Norm=np.sqrt(fx**2+fy**2)
    Nx=fx/(Norm+1e-8)
    Ny=fy/(Norm+1e-8)
	return div(Nx,Ny)

img

• ∇_x⁺ denotes forward difference in the x dimension : f(x+1)-f(x)

  ∇_x⁻ denotes backward difference : f(x)-f(x-1)

  ∇_x⁰ =(∇_x⁺ + ∇_x⁻)/2 is central difference : (f(x+1)-f(x-1))/2

  process step

• φ update function:

  φ value at point(i,j) φ_i,j

  length penalty μ =0.2,

  area penalty ν = 0,

  time step dt = 0.5,

  convergence tolerance tol = 10e−3

• algorithm

  

  image-20220411145745250

Reference

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1. Deep Learning of Unified Region, Edge, and Contour Models for Automated Image Segmentation↩︎