1 Introduction

In order to solve the image segmentation problem,the level set method was proposed by Osher and Sethian ^[1]. It is a kind of method for solving curve evolution,and adopts the method of implicit plane and partial differential equations (PDE) for expressing closed curves. During curve evolution,the level set method employs zero level set function (LSF) to generate a higher dimensional function,which avoids the track and parameterization. Level set method has been applied widely in various fields,such as image processing,fields of physics,fluid dynamics,materials science,etc.

Curve evolution models based on the level set method are in general divided into two types: the parameter model and the geometric characteristic model. The parametric active contours ^{[2, 3]} are explicitly constituted by Lagrangian equation. Because the Snake model ^[2] can easily fall into local extremum and is too sensitive to deal with topology structural changes in initial position,it is difficult to use the model for complex image segmentation. The geometric active contour ^{[4, 5, 6]} is implicitly constituted by Eulerian equation,which is effective in closed moving interface of geometrical and topological change. Li,et al. ^[7] presented an edge-based method, which used the local edge information to draw the active contour in the object boundaries. The energy function introduced a penalty term to maintain a signed distance function (SDF) during the evolution,however,it should preset the form of curve expansion or shrink. The edgebased image segmentation method is not robust enough for noise,and the edge leakage easily occurs on the weak edges. Chan and Vese ^[8] proposed the C-V model using a variational LSF. This C-V model is with larger convergence range and more flexible initialization. Vemuri and Chen ^[9] proposed the region-based piecewise smooth (PS) model which combined shape-prior information,and could be employed to achieve image fusion and segmentation. The region-based active contour models depend on intensity homogeneity in each segmentation region. However,the intensities are inhomogeneous from different modalities in real images. The results of edgebased and region-based methods considered the image data information are unsatisfactory,such as some covered target and similar grayscale images. To segment these kinds of intensity inhomogeneous images,Wang,et al. ^[10] presented the preprocessing step. Michailovich,et al. ^[11] proposed an active contour model used Bhattacharyya gradient flow to distinguish the intensity distributions between inside and outside. Zhou,et al. [12, 13] used the accurate image capturing control and achieved the fast bottle cap defect detection. Further, Zhou,et al. ^[14] presented the level set image segmentation with Bayesian analysis to solve the image tackle problem,and Zhou,et al. ^{[15, 16]} alslo employed gradient vector flow with mean shift for skin lesion image segmentation. However,the applications are constrained.

This paper proposes a new double regularization control (DRC) method,which is region-based active contour. This method makes use of a variational level set evolution function containing internal energy,external energy,length smooth regularization and potential penalty function regularization terms. DRC method needs to define energy function in terms of region contour. The regional control applies the level set evolution function,and the minimizing fitting energy function uses Gauss kernel function which recognizes the intensity information from initial small region to entire region. The length smooth regularization term is used for smooth curves and to avoid smaller region appearance. It is important that the potential penalty function regularization term maintains an approximate SDF during the evolution. In order to verify the DRC method,a number of tablet packaging images are employed in experiment studies. The results show that the DRC method is a simple and efficient numerical scheme. It can be used to implement the intensity inhomogeneous image segmentation,improve the computational efficiency and numerical accuracy,and reduce the number of iterations. 2 Region-based level set methods 2.1 Traditional C-V method

Based on the simplified Mumford-Shah model ^[17]and variational method,Vese and Chan ^[8] established a variational level set method,which was the two-phase piecewise constant image segmentation. Let be the boundary,and I(x,y) Ω: → R be a given gray image, where (x,y) represents an arbitrary point. The minimized fitting energy function of C-V is

The first two terms depict the region fitting energy in Inside (C) and Outside (C) ,and the last term is the regularization term. Here,Inside (C) and Outside (C) arethetargetand background regions on the contourC,respectively. Two constants c₁ and c₂ are the image intensities in Inside (C) and Outside (C),and |C| is the length of the contour C. µ,λ₁ and λ₂ are positive constants,and usually the coefficients satisfy λ₁ λ₂. For the energy minimized problem,the C-V method needs to solve a level set evolution equation

where φ(x,y) denotes the LSF replacing contourC,and δ_ε (φ) is Dirac function ^[8]. Since the function does not contain any local intensity information,if the intensities within Inside (C) or Outside (C) are inhomogeneous,the optimal constants c₁ and c₂ will be far away from the original image data.

C-V method cannot segment unevenly grayscale image, since it is sensitive to the restriction of an initial contour. Thus,Vese and Chan ^[18] proposed a piecewise smooth (PS) method,whose energy function is expressed as

where u⁺ and u^- are two smooth functions,which are defined on the inside and outside regions,respectively. H(φ) denotes the Heaviside function,which will be introduced in Section 3.2.1. The first two terms are data fitting terms,which represent the approximation sub-region of u⁺ and u^-. The PDE iterationso ⁺ and ^- are used for calculating the minimized level set energy function φ.Thus,u⁺ andu have to be updated in each iteration,from sub region to entire image. It is necessary to degrade level set evolution during the iteration. Unfortunately,the modified PS method increases the computation complexity and hinders its practical application. 2.2 Other level set methods

The region-scalable fitting (RSF) energy ^[19] and distance regularized level set evolution (DRLSE) ^[20] methods are introduced in this section.

The energy minimization function of the RSF smethod is

where H is replaced with H H_∈,M_&isin,1;=H_∈(φ) and M_∈,2=1-H_∈(φ). The gradient flow equation is shown as

The energy functional of the DRLSE method is approximated as

where μ,λ and a are the positive coefficient. The energy function can be minimized by the gradient flow

3 Double regularization control model

In traditional region-based active contour models,global image information is used to promote the approximation evolution of the real contour curve. However,it cannot get the desired results when segmenting the uneven grayscale images. Based on level set,a novel DRC method is proposed here using local image information. It contains region control energy and regularization term. 3.1 Region control energy

Uneven grayscale images are joined into smooth function field in the original images ^[21],the data model is established by

where U(x,y) is the gray even original image,V(x,y) the gray uneven field,and N(x,y) the noise,in general, N(x,y). Supposing θ(·) is a mapping from observed space to a high dimensional feature space ,where K represents kernel function and R denotes region. Let C(s): ^{[0, 1]} → Ω ⊂ R² ,the contour cdivides the original image into two regions R₁=R_In,C(s) and R₂=R_out,C(s),in which R₁=R_In,C(s) and R₁=R_out,C(s) denote the inside and outside on contour C,respectively. The image I(x,y) is divided into inside and outside regions,which has two approximated image intensity parameters,g₁(x₁,y₁) and g₂(x₂,y₂). According to the C-V method,the control minimized energy function of local region is expressed as

Let H(φ) denote Heaviside function and the zero level set function (φ(x,y)) represent boundary curve C. The local region control energy function can be rewritten as

The ||θ(I)-θ(g)||² fails to non-robustness,which is similar to the Euclidean distance vector of the data space. Therefore,the distance vector involves Gaussian kernel function as a flexible boundary,which is given as

Hence,the two-dimension data space is transformed into one-dimension,and the expression is

It is obvious that the evolution contour Cwill be naturally divided into the target region and background region. When K(I,g_i)=1,the kernel function approaches to 0. Let ,and the local region control energy function can be rewritten as

The Gaussian kernel function decreases drastically to 0. When |I-g_i > 3σ,the evolution curve is unbounded,and the contour is then far away from the regions. Therefore, the condition {I: |I-g_i| ≤ 3σ} fgis the localized energy around I(x,y) in the neighboring region.

By developing the local energy term,it enhances the gray contrast of the target and background. The method refers to the difference between the local region control kernel function and the original image convolution. 3.2 Regularization control term

The regularization term contains length smooth and potential penalty function terms which will be introduced in this subsection. 3.2.1 Length smooth regularization term

In order to make the evolution contour smooth enough and avoid the small region appearance as far as possible,the length smooth regularization term is proposed,which is given by

where H(φ) represents Heavide function,and δ(φ) expresses Dirac Delta function. According to Ref. ^[8],H_ε(φ) and δ_ε(φ) are shown as

3.2.2 Potential penalty function regularization term

During the level set evolution,the penalty energy term is designed to maintain an approximate SDF,

Thus the regularization energy function term can be represented as

where μ indicates the coefficient of length smooth term,and μ=σ × 255² (σ ∈^{[0, 1]}). In general,v is the coefficient of penalty regularization term,which is a positive constant. In order to increase the computational efficiency and decrease the number of iterations,a kind of potential function termg η(|∇φ|) is defined according to Ref. ^[19] and energy function (17). The penalty energy term is rewritten as dxdy,where gis the potential penalty function and . In the DRC method,a novel potential function is designed as

The calculated results show that when s=0 and s=1, η(s) has the minimum value. η'(s) is the first-order derivative of η(s) and the sign of potential penalty function satisfies l(s)=η'(s)/s,which are expressed as

3.3 Level set evolution equation

The energy fitting function of the DRC method is expressed as follows

The energy functional equation is determined by region parameters {g_i},i=1,2. According to Eq. (20),the regularization terms do not depend on region parameters when the level set function φ is fixed. Therefore,solving the inverse of local region parameters will be transformed into the local region parameter kernel function. This method is called fixed point iteration (FPI),the following equation is obtained as

Depending on RBF kernel function property,the necessary condition of energy function minimum satisfies g_i-ƒ_Ri(g_i)=0 where

Assuming the boundary contourCtakes the representation of zero level set function (φ(x;y)),the regional parameter gi fg;i¼1;2 can be expressed as follows

3.3.1 Update gradient flow of regularization term

The gradient flow is involved to update the regularization term. Specifically,the gradient flow of the length smooth regularization term is represented as

Similarly,in terms of Eq. (17),the gradient flow of penalty term is demonstrated as

wherer ∇²φ is Laplace operator. The curvature of contourC is expressed as div . Because the drastic increase of |∇ φ| will cause oscillation on level set function,the evolution curve will generate some ‘‘peaks’’ and ‘‘valleys’’. The improvement of the DRC method is to use η(|∇ φ|) replacing ,and the gradient flow of the potential penalty function is denoted as

Therefore,vP(φ) can be rewritten as vp(φ) and satisfies the requirement vl(φ) < v,which is the diffusion rate for the potential function η(|∇ φ|). From Eqs. (19) and (26), the diffusion rate is estimated according to the fundmental

In order to obtain the zero level set,|∇ φ| is satified the following three cases:

The key of the potential penalty function is the boundary corresponding to the diffusion rate which is restricted in the case |∇ φ| < 1/2. The contour evolutions are bilaterally extended,which are involved into the penalty term. Due to introducing potential penalty function η(|∇ φ|),the zero level set contour distortion can be avoided by both full domains. 3.3.2 Gradient flow for energy minimization

In order to ensure the level set evolution function φ,the DRC method should be able to maintain regional parameters {g_i},i=1,2 unchanged. Therefore,solving the inverse of the energy functional E_K;DRC region parameters needs to compute region parameters,which are generated from the local region term inside and outside,the length smooth and the potential penalty function regularization term. The level set curve evolution function is given as

Supposing h is the spacing of discrete grid mesh and Δt is the time step,the discrete level set function/passesntime periods,and the level set function can be then expressed by φ(ih; jh ; nΔt) corresponding to the image grid point (i,j). Figure 1 indicates an eight-neighborhood model of the discrete grid mesh.

Consequently,the forward differential discrete level set function can be updated using the following form

where is the numerical approximation of level set evolution function corresponding to the right side of evolution Eq. (29). If k represents the curvature of evolution curve,the numerical calculation is implemented using the second-order central difference form ^[8]. The curvature expression can be then represented as

where φ_x,φ_y,φ_xx,φ_yy and φ_xy are demonstrated by the following equations

The level set evolution Eq. (29) can be updated by

3.4 Algorithm procedure

As shown in Fig. 2,the algorithm procedure of the DRC method based on level set for image segmentation is summarized by the following four steps.

Step 1 Set initialization parameters. Equation (33)needsto define weight coefficients λ₁ and λ₂,which denote local region control term of inside and outside regions. Let Δt be the time step,hthe grid spacing,e the regularization parameter of Heavide function,K the Gaussian kernel function parameters,μ the length smooth term coefficient,and v the potential penalty function term coefficient.

Step 2 Set the initialized curve of level set evolution function φ. Assuming the parameter c₀ is the normal amount,R₁ and R₂ denote the inside and outside regions of the zero level set function φ(x,y,t),respectively. Signed distance function is defined as

Step 3 Update evolution parameters. According to Eq. (33),the evolution curve referring to Eq. (29) describes the discrete form during the evolution.

Step 4 Determine the termination conditions of level set evolution curve. If the termination condition is satisfied,the image segmentation result can be achieved. Otherwise,the next iteration of the initial level set function adopts the level set function φⁿ⁺¹=φⁿ,and goes to Step 3. 4 Experimental studies and discussions

The three groups of experimental images are derived from aluminum-plastic foamed mask package. There are a series of complex processes,such as stirring,dosing,sealing,and cutting,in preparation of tablets and capsules. These may incur some defects,i.e.,lacking of tablets,breakage,crack, and mixing contaminant. In order to overcome the disadvantage of traditional manual operation,machine vision technique is used to replace the human visual judgement in making the tablet packaging image segmentation. Here,the segmentation contains the following three steps. Firstly,a fixed camera is used for getting the tablet packaging image, and it is transmitted to computer. Secondly,the image processing controller is used to implement the image segmentation for the DRC. Finally,the segmentation result is matched with the previously stored template. The detection procedure will automatically remove the defective packages. Tablet packaging image requires segmentation for noise,and light intensity and movement time interval are more sensitive.

The experimental environment is given as: Matlab 2010 simulation software,Windows XP operating system,Intel 2.2 GHz CPU,and 3 GB capacity memory. In order to verify the effectiveness of DRC method,a number of different types of uneven grayscale images are selected.

The experiments compare DRC with C-V,RSF and DRLSE. The final level set function is the level set curve evolution function given by Eqs. (2),(5),(7) and (33). Initializing parameters are set to validate the three groups of experiments. The time step parameter is Δt = 0.2; the grid spacing parameter of original image layer is h = 1; regularization parameter of Heavide function is ε = 1:0, the standard deviation parameter of Gaussian kernel function is σ = 0.3. In addition,the initialized level set function is φ₀(x,y) with c₀ = 2 in the region Ω,and the length smooth coefficient is μ = 0.03 × 255 × 255. 4.1 The first experiment

Figure 3 shows the simulation of experiment one with a tablet for tablet packaging image,which size is 200 × 141 pixels.

The experimental results are illustrated as following. (i) The initial level set contour of the C-V method is a circle,and the searching result basically segments the tablet contour. According to the final level set function,the tablet contour is clearer than the other region contour.

(ii) RSF method initializes level set contour to be a rectangle,and the method is able to search the tablet packaging contour. RSF method can divide contour,e.g,the tablet,production date and other decorative parts. However,image segmentation results are not in priority. The final level set function shows that the segmentation effect is not obvious.

(iii) DRLSE method initializes the level set contour to be a rectangle as well. The level set segmentation contour shows that the tablet contour is not segmented,and the results do not meet the requirement of the image segmentation.

(iv) The initial contour is a rectangle with the proposed DRC method,and it is high efficiency and well effect. Figures3(h) and 3(l) show that the tablet contour,production date and other decorative parts can be well segmented. The final level set function demonstrates that the image contour is prominent.

The comparisons of the measure parameters using level set evolution function are listed in Table 1.

4.2 The second experiment

The effect of two tablets for tablet packaging image segmentation is shown in Fig. 4,and the image size is 1939128 pixels. The C-V method does not well segment two tablets contour,and the texture region participates image segmentation together. Based on the final level set function Fig. 4(i),it cannot distinguish the image segmentation effect. The RSF method shows that the tablets can be segmented approximately,which contain the contour of two tablets and production date. The texture contour is very serious interference. By the same way,the final level set function Fig. 4(j) shows clearly the effect of texture segmentation. The DRLSE method reduces image contour gradually,without partitioning the tablets packaging image. Although the light intensity of the image leads to grayscale uneven using the DRC method,the contour of two tablets and the production date can be clearly segmented. The final level set function Fig. 4(l) demonstrates the image segmentation contour clearly. Table 2 displays the measurement parameters involving level set function.

4.3 The third experiment

Ten tablets is employed to make tablet packaging image segmentation,and the image size is 184 × 81 pixels. The experimental result is shown in Fig. 5. The measurement parameters better verify the advantage of the DRC method (see Table 3).

According to the image segmentation results,it concludes that the RSF method is effective to tablets,but it does not deal with texture noise. The proposed DRC method has better computational efficiency and numerical accuracy than the CV and DRLSE methods. Particularly, the segmentation algorithm has better robustness to noise, and image segmentation contour is much clearer as well. 5 Conclusions

This paper proposes a novel image segmentation method DRC. Based on the level set evolution method,this method consists of three terms: the local region control term,the length smooth regularization term and potentia l penalty function regularization term. Introducing the local regional control term of image information is able to quickly segment uneven grayscale images. The length smooth regularization term is used for smoothing curves and avoiding smaller region appearance. The energy penalty term maintains an approximate SDF during the evolution. The potential penalty function is embedded into the penalty term,which is used for image segmentation in both full domains. The tablet packaging images are employed to make the verification. Comparing with other level set methods,the advantages of the DRC method are less time consuming,higher computational efficiency and more accurate in segmentation. In particular,the profile curve is not sensitive to the initial conditions,the texture segmentation and the de-noising effect is much better. The future work is to make mean shift based gradient vector flow algorithm for the tablet packaging image segmentation.

Acknowledgments The authors would like to thank the Science and Technology Commission of Shanghai Municipality (Grant No.14YF1408600),the Shanghai Municipal Commission of Economy and Informatization under Shanghai Industry University Research Collaboration (Grant No.CXY-2013-71),the Natural Science Foundation of Shandong Province (Grant No.ZR2012FM008),the Science and Technology Development Program of Shandong Province (Grant No.2013GNC11012),and the National Natural Science Foundation of China (Grant No.61100115).