Curve and Surface Smoothing Using a Modified Cahn-Hilliard Equation

We present a new method using the modified Cahn-Hilliard (CH) equation for smoothing piecewise linear shapes of twoand three-dimensional objects. The CH equation has good smoothing dynamics and it is coupled with a fidelity term which keeps the original given data; that is, it does not produce significant shrinkage.Themodified CH equation is discretized using a linearly stable splitting scheme in time and the resulting scheme is solved by using a Fourier spectral method. We present computational results for both curve and surface smoothing problems. The computational results demonstrate that the proposed algorithm is fast and efficient.


Introduction
For many computational purposes such as visualization of scientific data and registration of multimodal medical data, smooth curves and surfaces must be approximated by polygonal curves and polyhedral surfaces, respectively.However, these approximation algorithms generate faceted curves and surfaces [1].In medical applications, Vollmer et al. [2] presented an improved Laplacian technique for smoothing polygonal surface meshes that avoids the wellknown problem of deformation and shrinkage caused by many smoothing methods as shown in Figure 1.Using Beziersmoothed boundary representation, Anflor and Marczak [3] studied topological optimization of anisotropic heat conducting devices.
To prevent the mesh from shrinking, Liu et al. [4] proposed a volume-constrained smoothing algorithm for triangular meshes, which preserves exactly the mesh volume during the smoothing process.Wei et al. [5] proposed an effective mesh smoothing algorithm which preserves morphology on polygonized isosurfaces of inhomogeneous binary volumes.Recently, various smoothing methods have been developed [6,7].
In this paper, we propose a new method using a modified Cahn-Hilliard (CH) equation for smoothing piecewise linear shapes of two-and three-dimensional objects.Let (x) be a given mosaic binary data, that is, either plus one or minus one.Here, x = (, ) or x = (, , ).We want to construct smooth contour line or isosurface from the mosaic binary data by using the modified CH equation: where  is the fidelity parameter and () = 0.25( 2 − 1) 2 .When  = 0, then (1) becomes the CH equation which was originally proposed by Cahn and Hilliard [8,9] as a phenomenological model of phase separation in a binary alloy.See [10] and the references therein for physical, mathematical, and numerical derivations of the CH equation.The various applications of the CH equation can be seen in [11].Equation (1) was used in image inpainting with a space-dependent (x) [12].
This paper is organized as follows.In Section 2, the discrete equations for the governing equations are presented.In Section 3, we present various computational examples.Finally, in Section 4, conclusions are drawn.Figure 1: (a)-(e) Origin, noisy mesh, three iterations of the algorithm in [2].Reprinted from Vollmer et al. [2] with permission from the Wiley Online Library.

Algorithm
We use an unconditionally stable Fourier spectral method for the modified CH equation ( 1) in two-dimensional space Ω = (, ) × (, ).Fourier spectral method has been applied in many studies [13,14].

Circular Mosaic Shape.
In this test problem, we will show the smoothing effect of our proposed method.Therefore, we choose a coarse mesh grid 20 × 20 on the computational domain Ω = (−1, 1) × (−1, 1).The initial configuration is which represents a circular mosaic shape.The numerical parameters ℎ = 0.1,  = ℎ, Δ = 0.1, and  = 1000 are used.Figure 2 shows the temporal evolution at  = 0, Δ, and 10Δ.Here, the shaded area is the filled contour with  ≥ 0.

Effect of 𝜆.
Next, we investigate the effect of parameter  on the smoothing dynamics.On the computational domain Ω = (−1, 1) × (−1, 1), the initial configuration is given as where  = tan −1 (/) if  > 0;  = +tan −1 (/) otherwise.We use the following parameters: ℎ = 2/30,  = ℎ, and Δ = 0.1.With the initial configuration as shown in Figure 3(a), we obtain the various numerical results (see Figures 3(b)-3(d)) at time  = 20Δ with respect to different .In this test, we use three different  = 10 6 , 10 4 , and 10 2 .As  is large, (, ) does not evolve and is close to the initial mosaic profile.However, if  is too small, then the CH dynamics dominates and the profile becomes a circular shape which has the minimum interfacial length.

Effect of 𝜀.
In our proposed model, parameter  determines the thickness of the interface of two phases, that is, interfacial region of transition profile.Therefore, it is one of  the important factors on the smoothing process.Now, we study the effect of  on the smoothing dynamics.
Figures 4(b)-4(d) represent the numerical configurations at  = 20Δ with  = 4ℎ,  = ℎ, and  = ℎ/4, respectively.As shown in Figure 4(b), large value of  causes wide transition layer.Therefore, it overly smooths the curve.On the other hand, if  is too small, then the phase-field has abrupt transition and the profile becomes mosaic.

2D Various Shapes.
To demonstrate the robustness of the proposed algorithm, we consider various complex shapes on two-dimensional space Ω = (−1, 1) × (−1, 1).For the numerical test, we use  = 30, ℎ = 2/,  = ℎ,  = 10000, and Δ = 0.1.The initial configurations are represented in Figure 5(a) and its corresponding smooth results in Figure 5(b).Through these results, we can see that our proposed algorithm is a good strategy to remove staircases while preserving the initial feature.

3D Various Shapes.
In this section, we consider various morphologies on the three-dimensional space.

Volume Preservation.
Laplacian smoothing is a wellknown algorithm to smooth a polygonal mesh.However, this method can cause volume shrinkage or shape distortion [5].However, our proposed method is a feature-preserving approach without severe volume changes.In this section, we demonstrate that our smoothing approach preserves the initial volume.We first define the volume of an object using triangles on the surface.For  = 1, 2, . . ., , let Tri  = (X  , X  , X  ) = ((  ,   ,   ), (  ,   ,   ), (  ,   ,   )) be a triangle consisting of three points on a triangular mesh.Here,  is the number of triangles.We have the following formula for the volume of the polyhedron as See [17] for more detailed explanation about the polyhedron volume.Also, we define the percent of relative error of volume (X) with the initial volume (X 0 ) as Now, we consider three-dimensional morphologies such as to holed tetrahedron, many spheres, and holed sphere on Ω = (−1, 1) × (−1, 1) × (−1, 1); see Figure 7(a).On a 40 × 40 × 40 mesh grid, we perform the numerical test with ℎ = 0.1,  = ℎ, Δ = 0.1, and  = 5000.We obtain smooth results after 10 iteration steps by our proposed method as shown in Figure 7(b).Also, to show the effectiveness of our method, we compute the temporal evolutions of the relative percent error; see Figure 7(c).
Next, to show the robustness of the proposed method, we perform surface smoothing problem with noise in Figure 8.The initial conditions are the same as in Figure 7 with 30 percent noise.The other parameters are the same as the previous test.Figure 8 also shows the smooth numerical results.Even with noise, our smoothing approach has robust and effective results.

Complex 3D Morphology.
As the final test, we take Armadillo, Stanford bunny, and happy Buddha models as the initial conditions.For all tests, the parameters  = 10 4 , Δ = 0.01, and  = ℎ are used.The others are stated in Table 1.The columns in Figure 9 represent the initial condition and numerical results after 1 and 3 iterations.As we expected, the numerical results are smooth and preserve the initial feature.
To show the effectiveness of the proposed smoothing approach, we compare results in [5] with those from our proposed method.Figure 10 illustrates CPU time in seconds against the total number of vertices of a model.As shown in Figure 10, under the similar number of vertex points, our proposed method gets smooth results and a reduction of over two orders of magnitude in CPU time.
Also, in Figure 11, we represent the percent of the relative error of volume ( err (X)) which is stated in (12).Through these results, we can see that the proposed method implements the smoothing procedure almost without total volume loss.

Conclusion
In this paper, we have introduced a novel method using the modified CH equation for smoothing piecewise linear shapes of two-and three-dimensional objects without producing significant shrinkage.The main advantages of the proposed method are very fast and easy to implement because we use a fast Fourier transform.Various computational experiments for both curve and surface smoothing problems were presented.The results demonstrate that the proposed algorithm is effective and efficient for smoothing curves and surfaces.
Figure2shows the temporal evolution at  = 0, Δ, and 10Δ.Here, the shaded area is the filled contour with  ≥ 0. Figures 2(d)-2(f) show the values of (, ) at the part of the red box in Figures 2(a)-2(c).As shown in Figure 2(c), we can see the smooth interface after 10 iterations.

Figure 7 :
Figure 7: (a) Three different initial conditions.(b) Smoothed results after 10 iterations by the proposed method.(c) Percentage of relative error of volume against iteration steps.

Figure 8 :
Figure 8: (a) Three different initial condition with 30% noise.(b) Smoothed results after 10 iterations by the proposed method.

Table 1 :
Domain size, mesh grid points, and spatial step size used in Armadillo, stanford bunny, and happy buddha models.