To achieve the minimization of energy functions, the vector field
v(
x, y) = [
u(
x,y),
v(
x,y)] that satisfies
![hir-16-36-i002](/upload/SynapseData/ArticleImage/1088hir/hir-16-36-i002.jpg)
is used. Therefore, in the GVF, the modified method in formula (5) is used as the external energy, where f represents the edge map and differentiates image
I(
x, y) and ▽
fis the gradient of the edge map (7).
uis a weighted value that is involved in determining the degree of the diffusion of the vector field.
Based on the solution of Euler's equation using differential and integral calculus with permutation, the GVF is defined as formulas (6) and (7), and through the iteration of these, u, v will be obtained. Once the external energy has been defined, in areas with no edge information, the external force will be smoothly transformed to diffuse edge information, and in areas with many pieces of edge information, variable contours will be induced. Therefore, the target image necessary for clinical diagnoses can be more precisely segmented compared to traditional snakes.
1) Characteristics of GVF parameters
The GVF can adjust the degree of transformation by using four internal energy parameters. In formula (3), the parameters are α, β, γ, κ (α: elasticity, β: hardness, γ: viscosity, κ: weighted value of external force), and there is the parameter µ and numerical iteration in the formation of the vector field. It is difficult to find the optimum state through combinations of individual parameters, and even if the optimum state is found, the contour is drawn toward organs other than the target image if the default values are not accurate. The parameters obtained from formulas (3) and (5) are described below.
(1) Numerical problem of GVF energy: Next, formula (3) is expressed by formula (8) Euler equation, so it is expressed numerically.
When organized, it is represented by formula (9). When this has been organized into an Amatrix, the entire formula FINT + FEXT= 0 is represented by Ax + fx(x, y) = 0 and Ay + Fy(x, y) = 0. In this case, if weighted values α and β are adjusted, the characteristics of the matrix will be changed. The properties of elasticity and hardness can be adjusted. fx(x, y) is obtained from formula (6) and Fy(x, y), from formula (7). While converging in the scenario of energy minimiza tion through iteration, the coordinates of the contour being changed will be in the form of Ax = b, where xtrepresents the final contour coordinates to be obtained, Ais the internal energy matrix and b is the information obtained from the external energy. Therefore, the state is developed into formulas (10) and (11), and contour coordinates are obtained through iteration. xt-1represents the contour coordinates in the stage before iteration, xt, and the values may be adjusted with the weighted value γ. This shows a phenomenon physically similar to the characteristics of viscosity, and κ is designed to give certain weighted values to the calculated external force.
For these individual parameters, certain values may be used as necessary. Therefore, the characteristics of each parameter necessary for this study are explained below.
(2) Analysis of the characteristics of parameter µ: µ and numerical iteration are parameters that form vector fields and are involved in diffusion. In the following experiment, the shape of the human chest was virtually constructed, the weighted values of elasticity, hardness, viscosity and external force were kept constant, and µ values were changed while the experiment was conducted.
The default value was transformed into a round shape using formula (12), and the experiment was conducted while changing µ values (
Figure 1). In the artificial model, the black area represents the lung and the grey area, the heart. On the back of the heart, the aorta was modeled. The red area represents the process of convergence of the edge, and the yellow area represents the default value.
In
Figure 1A, a large part of the default value is included in the heart and small parts are included in the chest wall outside the heart. When µ = 0.02, the vector is well-diffused, and thus the edge is pulled toward the heart, while when µ = 0.2, many forces that are pulling the edge toward the lung appear confusingly as shown in
Figure 1D. This fact shows that the default value at the boundary between the heart and the lung is strongly pushed toward the lung.
Eventually, as µ becomes larger, the vector field is diffused and becomes larger. And, if the vector field is excessively diffused to become complicated, a balance of power is reached with an edge in another surrounding organ, and the contour for convergence will become unable to progress further between these two edges.
(3) Analysis of the characteristics of parameter alpha: Alpha is involved in the adjustment of elasticity characteristics. The green line is the result finally obtained through the process of transformation after the setup of the default value. When the alpha value decreases, convergence of elasticity will decrease, and if the alpha value increases, elasticity will increase and the vector field will flexibly converge even in large areas (
Figure 2). However, if the alpha is too large, the transformation cannot form the optimum image result, and the edge will excessively converge so that no edge can be found.
(4) Analysis of the characteristics of parameters beta, gamma and kappa: Like parameter alpha, parameter beta is a factor that adjusts the characteristics of transformation when the contour converges, and it adjusts the hardness of edge values. However, more constant values should be applied to these values compared to parameter alpha in order to obtain an optimum result. Gamma denotes viscosity and kappa is the weighted value of external force. More constant values should be applied to these variables compared to parameter alpha in order to obtain an optimum result. Even if new experimental images are tested, the range of adjusted values is not greater than that of alpha. Therefore, there is little necessity to adjust these after the default setting. The optimum result should be found out through parameters alpha and µ.