Mesh denoising algorithm (two-step framework)

The two-step-based mesh denoising algorithm, as its name implies, consists of two steps: First, filter the normal direction of the mesh surface, get the adjusted grid method to the information, and then update the vertex coordinate position according to the adjusted method, and then introduce three articles of this type.

[Sun et al. 2007] in this paper, the current normal filtering method and vertex coordinate updating method are introduced, and then the method of normal filtering and vertex coordinate updating is presented.

The method of normal filtering:

1. Mean filter (mean filter): ni ' = normalize (σj∈n (i) AJ NJ/σj∈n (i) Aj), the mean filter destroys the detail characteristics of the mesh.

2. Median filter (median filter): ni ' = arg mediannj {wj⊙∠ (NI, NJ): J∈n (i)}, median filtering preserves the detail characteristics of the mesh, but when the grid noise is large, the effect is not good.

3.α-Censored Filter (alpha-trimming filter): ni ' = normalize (Σj∈n (i) iα (j) · AJ NJ), ∠ (NI, NJ) The angle value before the α ratio and the posterior α ratio of iα (j) is 0, when α= 0 o'clock is equivalent to mean filter, when α= 0.5 is equivalent to median filter.

4. Fuzzy vector median filter (median filter): First use vector median filter nm = arg minnj {σk∈n (i) d (NJ, NK): J∈n (i)}, where D (NJ, NK) is the LP norm or ∠ of NJ and NK (NJ, N k) Angle value, then use Gaussian filter ni ' = normalize (Σj∈n (i) NJ exp (-D (NJ, NM) 2/2σ2)). The fuzzy vector median filter can get the normal direction with small error, but it will increase the computational amount.

5.[sun et al. 2007] Article of the normal filtering method: ni ' = normalize (σj∈n (i) HJ NJ), the weight is defined as, where N (i) is a triangular slice of the neighborhood triangle fi, there are two options, see, f (x) = X2,0≤t≤1,t value determines the neighborhood Triangular slices n (i) How many triangular pieces of the method will be involved in the calculation, when t = 1 o'clock, only with NI equal to the neighborhood triangular slice method will participate in the calculation, and when t = 0 o'clock, all the adjacent triangular slice method will participate in the calculation. When the mesh has a distinct edge, t can take a relatively small value, and when the mesh is smooth, t can take a relatively large value.

Ⅰ is based on the Triangle neighborhood of vertex (vertex based); Ⅱ represents a triangular slice neighborhood based on edge (Edge based)

Vertex coordinate Update method:

1. Solving a linear Equation group:

2. Gradient Descent method:

where NV (i) is the 1-ring neighborhood vertex set of vertex i.

3. [Sun et al. 2007] Article vertex Update method:

Wherein FV (i) is a 1-ring neighborhood triangular set of vertex i.

Effect:

[Zheng et al. 2011] In this paper, two methods of triangulation are proposed, and the vertex coordinate updating method is used in the same way as the [Sun et al. 2007] article.

1.Local and iterative Scheme:

The method uses a bilateral filter for the normal direction: nit+1 = Ki σj∈n (i) Wij NJT, where N (i) is the neighborhood triangular set of triangle fi, Ki is the normalized coefficient, wij =ξij WC Ws,ξij for triangular FJ area Si, Space domain nuclear WC (| | ci–cj| |) = exp (-| | ci–cj| | 2/2ΣC2), the domain core ws (| | ni–nj| |) = exp (-| | ni–nj| | 2/2ΣS2).

2. Global and non-iterative Scheme:

This method minimizes the objective function by solving the linear equations, and the method is a corresponding implicit updating vertex coordinate method, relative to the above explicit updating vertex coordinates, the objective function is as follows:

where Ai = Si/mean ({S}), the upper formula is equivalent to solving a linear equation group, but the equations are not rank, need to join the soft constraint: Ed =σi ai | | Ni ' –ni| | 2. The final optimization target is: argmin (1-λ) · Es +λ Ed, wherein λ∈[0, 1], is used to control the smoothness of the mesh.

Effect:

[Zhang et al. 2015] A method of triangulation is proposed, which is similar to the method of [Zheng et al. 2011] article, and the difference is that [Zheng et al. 2011] The original method is used as the domain information in the bilateral filter, and [Zhang et al. 2015] In this paper, the original normal direction filter (guided filter) is used to obtain the new method as the domain information. The vertex coordinate Update method also uses the same method as the [Sun et al. 2007] article.

Where NI is the triangular chip set of triangle FI, WI is normalized coefficient, AJ is the area of triangular slice FJ, space domain kernel KS and range kernel kr are Gaussian kernel functions.

Key how to calculate the triangular plate FI guidance Method (guidance) GI, the article proposed for the triangular piece of fi, in its vicinity to find a region containing a triangular piece of fi, so that the area of the triangular plate changes to the minimum, and then the region of all triangular pieces of the method to carry out a weighted average of the triangular film as the guidance method. Specifically, the definition of Pk for the triangular slice set {FK}, which has a common vertex with the triangle, then the candidate region of the Triangle Fi (candidate patch) is C (FI) = {PK∣FI∈PK}. For each region P∈c (FI), the normal consistency of the triangular plates in the area P is calculated (consistency): H (P) =ф (p) · R (P), where Ф (p) is the maximum deviation of the normal direction of the area P: Ф (P) = Maxfj,fk∈p | | nj–nk| |,r (p) is the relative degree of variation of the normal direction of the adjacent triangular plates in the region P:, in the formula φ (EJ) = | | nj1–nj2| |. Then, the p* of the minimum region of the consistency function H (P) is selected, and the triangular slice method is weighted to the area, that is, the Guide method of the triangular plate fi gi:gi = normalize (σfj∈p* AJ NJ) is obtained.

Effect:

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Reference documents:

[1] Xianfang Sun; Rosin, P.L.; Martin, R.R.; Langbein, F.C, "Fast and effective feature-preserving Mesh denoising," Visualization and computer Graphics, IEEE Transact Ions, vol.13, No.5, pp.925-938, Sept.-oct. 2007.

[2] Youyi Zheng; Hongbo Fu; Au, o.k.-c.; Chiew-lan Tai, "bilateral Normal Filtering for Mesh denoising," Visualization and computer Graphics, IEEE transactions, VO L.17, No.10, pp.1521-1530, Oct. 2011.

[3] Wangyu Zhang, Bailin Deng, Juyong Zhang, Sofien bouaziz, Ligang Liu, "Guided Mesh Normal Filtering," Computer Graphics Forum (Proc Pacific Graphics), 34 (7): 23–34, 2015.

Mesh denoising algorithm (two-step framework)