a novel global energy and local energy-based legendre...

Research ArticleA Novel Global Energy and Local Energy-Based LegendrePolynomial Approximation for Image Segmentation

Feng Hu,1 Mengyun Zhang,2 and Bo Chen 2,3

1School of Statistics, Qufu Normal University, Qufu 273165, China2Shenzhen Key Laboratory of Advanced Machine Learning and Applications, College of Mathematics and Statistics,Shenzhen University, Shenzhen 518060, China3Guangdong Key Laboratory of Intelligent Information Processing, Shenzhen University, Shenzhen 518060, China

Correspondence should be addressed to Bo Chen; [email protected]

Received 26 April 2020; Accepted 19 September 2020; Published 5 October 2020

Academic Editor: Chuanjun Chen

Copyright © 2020 Feng Hu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Active contour model (ACM) is a powerful segmentation method based on differential equation. This paper proposes a noveladaptive ACM to segment those intensity inhomogeneity images. Firstly, a novel signed pressure force function is presentedwith Legendre polynomials to control curve contraction. Legendre polynomials can approximate regional intensitiescorresponding to evolving curve. Secondly, global term of our model characterizes difference of Legendre coefficients, and localenergy term characterizes fitting evolution curve of interested region. Final contour evolution will minimize the energy function.Thirdly, a correction term is employed to improve the performance of curve evolution according to the initial contour position,so wherever the initial contour being in the image, the object boundaries can be detected. Fourthly, our model combines theadvantages of two classical models such as good topological changes and computational simplicity. The new model can classifyregions with similar intensity values. Compared with traditional models, experimental results show effectiveness and efficientlyof the new model.

1. Introduction

Image segmentation [1], image denoising [2, 3], and imagereconstruction [4] are all basic tasks in image processingfield. The goal of image segmentation is to partition an imageset into several meaningful subsets according to different fea-ture information. It is difficult to segment images with noise,low contrast, and intensity inhomogeneity. Many segmenta-tion methods based on partial differential equation (PDE)model [1] have been proposed with the development ofPDE [5–10] and stochastic theory [11, 12]. One of the famoussegmentation methods is ACM, which can extract the desiredobject by an evolving curve based on a variational frame-work. The existing ACM methods classify with edge ACM[1] and region ACM [13].

The edge ACM utilizes image gradient to guide the con-tours toward the boundaries of desired objects, and segmen-tation result relies on the location of the initial contour.Region model applies intensity and texture information to

guide curve evolution. Chan-Vase (CV) [13] model is a clas-sical region-based model, which is derived from Mumford-Shah (MS) model [14].

Intensity inhomogeneity image segmentation is still achallenging problem. Li et al. propose a local binary fitting(LBF) [15] model with a kernel function to avoid reinitializa-tion and perform better than original CV model. Wang et al.propose a local Gaussian distribution fitting (LGDF) [16]model with local intensity mean and variance information.Zhou et al. introduce a global and local intensity information(LGIF) [17] model, which can achieve high segmentationaccuracy while with heavy computational complexity. Wanget al. propose a local CV (LCV) [18] model to deal with thecomputation problem. LCV model cannot perform well onintensity inhomogeneous images and sometimes leads toedge leakage because the average convolution operator isemployed in local region. Li et al. propose a LIC model [19]based on K-means clustering method. Zhang et al. developLIC model and propose a locally statistical ACM (LSACM)

HindawiJournal of Function SpacesVolume 2020, Article ID 2061841, 12 pageshttps://doi.org/10.1155/2020/2061841

https://orcid.org/0000-0003-2970-4813https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/2061841

[20]. LSACM collects pixels belonging to the same class andrealizes soft segmentation. LIC model and LSACM modelare all with high computational cost. Mukherjee and Acton[21] propose a L2S model to deal with intensity inhomogene-ity images. L2S model is also with heavy computational com-plexity due to Legendre basis functions. Shi and Pan [22]present a LGBF to deal with intensity inhomogeneous imagesegmentation problem.

This paper proposes a novel Legendre polynomialapproximation with adaptive global energy based on our pre-vious model [23]. For presenting a qualitative expansion onprevious work, this paper compares our model with repre-sentative models on noisy images, low-contrast images,real-world images, neuron images, and dendritic spineimage, and the experiments also give a clear description ofthe effect of correction term on the initial contour position.The detailed process reasoning of Legendre coefficient ofcontour is given in our work. The main contributions of thispaper are presented as follows. Firstly, a novel signed pres-sure force (SPF) function with Legendre polynomials is pro-posed. Regions are represented by a set of Legendre basisfunction. Secondly, our model has a global term and a localterm. The local term is motivated by SBGFRLS model andGAC model and then combines the advantages of them.The global term is designed by utilizing the Legendre basisfunction and difference between foreground and back-ground. Thirdly, a correction term is applied to control thedirection of curve evolution. Experimental results show thatour model is robust and efficient to segment intensity inho-mogeneity images.

This rest of this paper is organized as follows. In Section2, we review L2S and SBGFRLS models briefly. The newmodel and corresponding algorithm are presented in Section3. In Section 4, the proposed model is validated by someexperiments on synthetic and real images and compared withother ACM. Conclusion is drawn in Section 5.

2. The Related Works

2.1. The SBGFRLS Model. Zhang et al. propose the SBGFRLSmethod [24] in 2009, which incorporates the advantages ofGACmodel and CV model. New SPF function is constructedaccording to intensity information inside and outside thecontour curve. Let Ω be a bounded open subset of R2 and I: ½0, a� × ½0, b�→ R+ be a given image. The level set evolutionequation is as follows:

∂ϕ∂t

= SPF I xð Þð Þ ⋅ α ∇ϕj j, x ∈Ω, ð1Þ

where α is the balloon force, which controls the evolutionspeed of the level set. The SPF function is defined as follows:

SPF I xð Þð Þ = I xð Þ − c1 + c2/2max I xð Þ − c1 + c2/2j jð Þ, x ∈Ω, ð2Þ

where c1 and c2 are mean grey values and can be computed as

c1 ϕð Þ =ÐΩI xð Þ ⋅H ϕð ÞdxÐΩH ϕð Þdx ,

c2 ϕð Þ =ÐΩI xð Þ ⋅ 1‐H ϕð Þð ÞdxÐΩ1 −H ϕð Þð Þdx :

ð3Þ

The model utilizes region information to stop the curveevolution, which is more efficient and less sensitive to noise.SBGFRLS model and CV model cannot deal with imageswith intensity inhomogeneity because they all consider globalimage intensities merely.

2.2. The L2S Model. Mukherjee and Acton [21] propose aLegendre polynomial-based segmentation model. Theyreplace c1 and c2 in the original CV model as two smoothfunctions cm1ðxÞ and c2m. Then, the energy function of theL2S is as follows:

EL2S ϕ, A, Bð Þ =ðΩ

f xð Þ − ATP xð Þ�� 2H ϕ xð Þð Þdx + λ1 Ak k22+ðΩ

f xð Þ − BTP xð Þ�� 2 1 −H ϕ xð Þð Þð Þdx+ λ1 Bk k22 + ν

ðΩ

δε ϕð Þ∇ϕ∇ϕj j dx:

ð4Þ

whereλ1 ≥ 0, λ2 ≥ 0are fixed scalars. A = ðα0,⋯,αNÞT and B= ðβ0,⋯,βNÞT are the coefficient vectors for the inside regionand outside region, respectively. The last term in Eq. (7) is theregularization item, which introduces smoothness in the zerocurve. Let perform∂EL2S/∂A = 0, ∂EL2S/∂B = 0, soÂ and B̂(Appendix A gives a detailed calculation of coefficient vec-tors, and Appendix B gives the existence and computabilityof coefficient vectors) are, respectively, computed as follows:

Â = K + λ1Ι½ �−1P, B̂ = L + λ2Ι½ �−1Q,K½ �i,j =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH ϕ xð Þð Þ

pPi xð Þ,


pPj xð Þ

D E,

L½ �i,j =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −H ϕ xð Þð Þ

pPi xð Þ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −H ϕ xð Þð Þ

pPj xð Þ

D E:

ð5Þ

h,i denotes the inner product operator. The vectors P and Qare obtained as P =

ÐΩPðxÞf ðxÞHðϕðxÞÞdx, Q = Ð

ΩPðxÞf ðxÞ

ð1‐HðϕðxÞÞÞdx. By minimizing Eq. (7), the correspondingvariational level set formulation is as follows:

∂ϕ∂t

= − f xð Þ − ÂTP xð Þ�� 2 + f xð Þ − B̂TP xð Þ�� 2� �δε ϕð Þ

+ νδε ϕð Þ div∇ϕ∇ϕj j

� �:

ð6Þ

2 Journal of Function Spaces

The model approximates foreground and background by

computing ÂTPðxÞ and B̂TPðxÞ, respectively, which can per-

form well for images with inhomogeneous intensity.

3. A Novel Adaptive Segmentation Model

3.1. Model Construction. Motivated by the SBGFRLS model,we propose a novel ACM based on Legendre polynomial.The energy function of our model includes two parts, thelocal term and the global term:

E = EL + EG, ð7Þ

(a)

(b)

(c)

Figure 1: Segmental results for images with noise: (a) CV model segmentation results with noise image variance of 0.1, 0.2, and 0.3; (b)SBGFRLS model segmentation results with noise image variance of 0.1, 0.2, and 0.3; (c) Our model segmentation results with noise imagevariance of 0.1, 0.2, and 0.3.

0.94

0.95

0.96

0.97

0.98

CV SBGFRLS Our model

0.10.20.3

Figure 2: The corresponding Dice values of the segmental results inFigure 1.

3Journal of Function Spaces

(a)

(b)

(c)

Figure 3: Continued.


where EL and EG denote the local term and the global term,respectively.

3.1.1. The Local Energy. The local energy term EL is derivedfrom the SBGFRLS model, which utilizes the global imagemean intensity values inside and outside of the evolvingcurve C, respectively. In order to smooth the level set func-tion to maintain the interface regular, div ð∇ϕ/j∇ϕjÞ is curva-ture of evolving curve C is introduced. The level setformulation is as follows:

∂ϕ∂t

= SPF I xð Þð Þ div ∇ϕ∇ϕj j� �

+ α� �

∇ϕj j, x ∈Ω: ð8Þ

The balloon force α can be regarded as a correction term,which is responsible for controlling the contour shrinking orexpanding rate; then, the term div ð∇ϕ/j∇ϕjÞ + α can be neg-ative value or positive value, so we can improve the perfor-

mance of the SPF function to some extent. The SPFfunction will drive the contour to expand or shrink accordingto the location of the region of interest. However, the finalobtained curve C can hardly extract local image feature forimages with intensity inhomogeneity. The main reason isthat the SPF model is constructed based on the intensityinside and outside the objects that are homogeneous. Tosolve these problem, our model introduces the Legendrepolynomials to replace the scalars c1 and c2 in Eq. (5) bytwo smooth functions cm1 ðxÞ and cm2 ðxÞ, which can representby a linear combination of a few Legendre basis functions:

cm1 xð Þ =〠αkPk xð Þ, c2m xð Þ =〠βkPk xð Þ, ð9Þ

which can make these functions have the smoothness andflexibility, where Pkis a multidimensional Legendre

(d)

(e)

Figure 3: The comparisons of LBF model, LIC model, L2S model, LSACM model, and our proposed model on segmenting images withintensity inhomogeneity: (a) results of LBF model; (b) results of LIC model; (c) results of L2S model; (d) result of LSACM model; (e)result of our proposed model.


polynomial with degree k, and the 2-D polynomial isdefined as follows:

pk x, yð Þ = Pk xð ÞPk yð Þ, X = x, yð Þ ∈Ω ⊂ −1, 1½ �2,

Pk xð Þ =12k

〠k

i=0

k

i

!2x − 1ð Þk−i x + 1ð Þi:

ð10Þ

We can infer from the above equation that the highestdegree of the 1D basis is m, and the highest degree of 2Dbasis is ðm + 1Þ2. When m = 0, two smooth functions willreduce to the scalars c1 and c2 in Eq. (5). Legendre poly-nomials’ primary objective is to perform segmentation inthe presence of an inhomogeneity intensity field. So thenew SPF function is defined as follows:

SPFnew I xð Þð Þ = I xð Þ − ÂTP xð Þ + B̂TP xð Þ/2

max I xð Þ − ÂTP xð Þ + B̂TP xð Þ/2�� , x ∈Ω,

ð11Þ

where ÂTPðxÞ and B̂TPðxÞ are approximate to the gray

values inside and outside of the evolving curve. j∇ϕj canbe replaced by δðϕÞ in (19) to increase the speed of curveevolution, and then the local term is as follows

EL = δ ϕð Þ ⋅ SPFnew I xð Þ div ∇ϕ∇ϕj j� �

+ α� ��

x ∈Ω: ð12Þ

For each category of images, a correction term willdecide the curve to evolve from inside to outside or fromoutside to inside, so that the initial contour being any-where in the image can detect the object boundaries. Atthe same time, a new SPF can effectively drive curve tostop contours at weak edge, even images in the presenceof inhomogeneity intensity.

3.1.2. The Global Energy. The main role of global energyaffects the speed and accuracy of the evolution curve. There-fore, it is necessary to choose an appropriate global term.Most of parameters are selected manually, but we designan adaptive parameter strategy based on the differencebetween the foreground and background. The foregroundand the background can be modeled by a set of Legendrebasis functions in our model and can be represented in alower dimensional subspace. When the gray value of back-ground is less than the gray value of foreground, a novel

adaptive global term can be written by utilizing Legendrebasis functions as follows:

EG = mean Â − B̂

�

min Â − B̂

�

−max Â − B̂

� , ð13Þ

where mean(•), max(•), and mic(•) are the average function,the maximum function, and the minimum function, respec-tively, where Â and B̂ in Eq. (19) denote the coefficient vectorsfor the foreground and background. Each index satisfies thefollowing relationship: max ðÂ − B̂Þ >meanðÂ − B̂Þ >min ðÂ− B̂Þ, and the data shows that the global term has values inthe range ½−1, 1�. An adaptive term can better meet thedynamic changes of the evolution curve than the constantterm, which can improve the speed and accuracy ofmodel seg-mentation. When the background gray value is greater thanthe foreground gray value, the global term can be set as zero.

Then, the final proposed model is as follows:

E = δ ϕð Þ ⋅ SPFnew I xð Þð Þ div ∇ϕ∇ϕj j� �

+ α� �

+ mean Â − B̂

�

min Â − B̂

�

−max Â − B̂

� ð14Þ

3.2. Algorithm Procedure

(a) Initialization:α, m, and σ

(b) Initialize the level set function ϕ as

ϕ x, t = 0ð Þ =−ρ x ∈ outside Cð Þ0 x ∈ Cρ x ∈ inside Cð Þ

ρ > 0

8>><>>: ð15Þ

(c) Compute A, B, and SPFnewðIðxÞÞby Eq. (11) and Eq.(17), respectively

(d) Evolve the level set function according to Eq. (20)

(e) Let ϕ = 1 if ϕ > 0; otherwise, ϕ = −1(f) Regularize the level set function ϕwith a Gaussian fil-

ter, i.e., ϕ = ϕ ∗Gσ(g) Check whether the evolution of the level set has con-

verged. If not, return to stage d

The step (e) serves as an optional segmentation procedure.

4. Experimental Results

In this section, the experimental results of our proposedmodel will be presented on a series of synthetic and realimages. All experimental are implemented in MatlabR2014a on a 3.30-GHz PC. The initial contour can be cho-sen as rectangle, ellipse, and multiball manually according

Table 1: The corresponding DICE values of the segmental results inFigure 3.

LBF LIC L2S LSACM Ours

0.6198 0.9061 0.8532 0.7893 0.9348

0.9654 0.8994 0.8823 0.9033 0.9311

0.9759 0.8669 0.8238 0.9035 0.9366


to image. The correction term α is important for curve evo-lution and should be set according to each category ofimages. We choose m = 1 and find that images with inten-sity inhomogeneity can be adequately modeled. When m> 3, it will require inversion of a larger matrix and needcomputation more expensive. To measure the quality ofthe extracted objects in segmentation, the Dice coefficientsare used to evaluate the corresponding segmentation resultsof both our method and the other method. The Dice indexD ∈ ½0, 1� represents the difference between the segmentalresult R1 and the ground truth R2, and larger value impliesbetter segmentation results. The Dice is defined as

D R1, R2ð Þ =2Area R1 ∩ R2ð Þ

Area R1ð Þ + Area R2ð Þ: ð16Þ

4.1. Results for Noisy Images. Figure1 shows the effective-ness of our model for noisy image segmentation. The image(two objects, [25]) in the first, second, and third row showsthe corresponding segmentation results by CV model,SBGFRLS model, and our model. The first column, secondcolumn, and third column are images with Gaussian noiseof standard deviations 0.1, 0.2, and 0.3, respectively. Asshown in Figure2, the Dice value of our model is more sta-

ble with the variance increasing and our model can obtainbetter results than other models.

4.2. Robustness to Intensity Inhomogeneity. Figure 3 showsthe segmentation results for yeast fluorescence micrographimage and two real blood vessel images with intensity inho-mogeneity. The edges of these images are difficult to distin-guish clearly, which makes the segmentation of such imageschallenging. For quantitatively evaluation, we give a compar-ison of the Dice value in Table 1; it can be seen from Figure 3and Table 1 that our proposed model is proved to be moreefficient in segmenting images with intensity inhomogeneityand more accurate in terms of segmentation accuracy thanthe other four methods.

Table 2: Running time and the DICE value for images shown inFigure 4.

ImageCPU running time

LCV LSACM Ours

First 1.9375 37.2031 2.4375

Second 7.1719 35.8125 7.7031

Image The DICE value

First 0.4708 0.9745 0.9757

Second 0.9607 0.9966 0.9968

(a) (b) (c)

Figure 4: Detected contour of regions of interest by LCV model, LSACM model, and our model: (a) results of LCV model; (b) results ofLSACM model; (c) results of our model.


(a) (b) (c) (d)

Figure 5: The comparisons of LSACM model, LBF model, LIC model, and our model on segmenting images with the intensityinhomogeneity: (a) results of LSACM model; (b) results of LBF model; (c) results of LIC model; (d) results of our model.

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 6: The process of segmentation using our model: (a) initial contour; (b) final contour, 20 iterations; (c) final contour, 40 iterations; (d)final contour, 100 iterations; (e) initial contour; (f) final contour, 60 iterations; (g) final contour, 120 iterations; (h) final contour, 240 iterations.


(a)

(b)

(c)

(d)

Figure 7: Continued.


4.3. Results for Low-Contrast Images. Figure 4 and Table 2show the effectiveness of our model for low-contrast images[26]. The first column, second column, and third columnshow the final contours by LCV model, LSACM model, andour model. Figure 4 shows the two low-contrast images withsome weak boundaries, and our model obtains smoothercurve and detects well the object’s boundary, but the LCVmodel and LSACM model can obtain incomplete bound-aries. As shown in Table 2, our model can effectively segmentimages with weak boundary and low contrast.

4.4. Results for Real-World Image. Figure 5 shows the abilityof dealing with intensity inhomogeneous images (ultrasoundmedical image [24], skin image [25], brain MRI image [24],and Europe-night-light image [13]) in the real world, whichcaptured from the digital camera. In order to further showthe advantage of our model, we compare the segmentationresults with the three models (LSACM, LBF, and LIC). Ourmodel extracts the object contours accurately, whereas theother three methods produce oversegmentation and under-segmentation. For example, the result of the first image usingLSACM, LBF, and LIC shows that the edge of image is over-segmented; the result of the second image using LSACM,LBF, and LIC shows that the edge of the cancer is overseg-mented; the result of the third image using LSACM is over-segmented; and the result of the third image using LBF andLIC is undersegmented. Because the local term and globalterm are considered simultaneously, our model performsbetter than the other methods.

4.5. The Influence of Parameter α. Figure 6 shows the flexibil-ity of the initial contour position in our model. Different ini-tial contour position in image [24] is tested for the newmodel. The sign of parameter α is selected according to theimage’s characteristics, such as intensity and the position ofinitial contour. Since the gray value of the background ismuch larger than the object, the negative parameter α onlyneeds 100 iterations in our experiment of first row, and thepositive parameter α needs 480 iterations in our experimentof second row. The corresponding segmentation process of

curve evolution illustrates the appropriate parameters thatcan speed up the evolution of curve.

4.6. Application on Neuron Images and Dendritic SpineImage. Figure 7 presents the results for dendritic spineimage and neuron images which imaged by confocal micro-scope [21]. We validated the performance of our model ona set of neuron images by comparing with other methods.CV model and LBF model failed to obtain the object trueboundary. The fourth and fifth rows in Figure 7 show thatthe L2S model and our model can extract similar results,but some fine details have been erased through L2S model.That suggests that our model based on local term andglobal term can be more effective than L2S model basedon global term, which significantly improves the segmenta-tion accuracy.

5. Conclusion

A novel adaptive segmentation model for images in the pres-ence of low contrast, noise, weak edge, and intensity inhomo-geneity is proposed in this paper. Regions are represented bya set of Legendre basis function, so Legendre polynomials areintroduced to deal with intensity inhomogeneous image seg-mentation problem. The local and global information are allconsidered, and GAC model and SBGFRLS model are com-bined in our model. Our model has good topological changesand computational simplicity. The evolution direction can bechosen adaptively according to the parameter α, and theshape of initial contour can be selected on rectangular orelliptical manually. Experimental results show that ourmodel is available and effective.

Appendix

A. The Process of Solving Â and B̂

In this appendix, we deduce the corresponding coefficients Âand B̂ in Eq. (11). Keeping ϕ fixed and minimizing the energyEL2Sðϕ, A, BÞ with respect to the constant A andB. Weperform∂EL2S/∂A = 0, so the Eq. (7) can be expresses as

(e)

Figure 7: Comparison results of our model with CV, LBF, and L2S models: (a) initial contours; (b) final segmentation results using the CVmodel; (c) final segmentation results using the LBF model; (d) final segmentation results using the L2S model; (e) final segmentation resultsusing our model.


∂ EL2S

�∂A

= 2ðΩ

∂ −ATP xð Þ �∂A

∗

� f xð Þ − ATP xð Þ �H ϕ xð Þð Þdx + 2λ1A = 0ðA:1Þ

then

2ðΩ

−P xð Þf xð ÞH ϕ xð Þð Þ + P xð ÞP xð ÞTAH ϕ xð Þð Þ�

dx + 2λ1A = 0

ðA:2ÞðΩ

P xð ÞP xð ÞTAH ϕ xð Þð Þdx + λ1A =ðΩ

P xð Þf xð ÞH ϕ xð Þð Þð Þdx

ðA:3ÞðΩ

P xð ÞP xð ÞTH ϕ xð Þð Þdx + λ1Ι� �

A =ðΩ

P xð Þf xð ÞH ϕ xð Þð Þð Þdx

ðA:4Þ

Let

P =ðΩ

P xð Þf xð ÞH ϕ xð Þð Þdx,

K =ðΩ

P xð ÞP xð ÞTH ϕ xð Þð Þdx,

K + λ1Ι½ �Â = P,Â = K + λ1Ι½ �−1P:

ðA:5Þ

½K� is N ×N Gramian matrices, whose (i,j) th entry canbe represented as

K½ �i,j =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH ϕ xð Þð Þ

pPi xð Þ,


pPj xð Þ

D E: ðA:6Þ

Then, we perform ∂EL2S/∂B = 0, so the Eq. (7) can beexpresses as

∂ EL2S

�∂B

= 2ðΩ

∂ −BTP xð Þ �∂A

∗ f xð Þ − BTP xð Þ �� 1 −H ϕ xð Þð Þð Þdx + 2λ1B = 0

ðA:7Þ

then

2ðΩ

−P xð Þf xð ÞH ϕ xð Þð Þ + P xð ÞP xð ÞTB 1 −H ϕ xð Þð Þð Þ�

dx + 2λ1B = 0

ðA:8ÞðΩ

P xð ÞP xð ÞTB 1 −H ϕ xð Þð Þð Þdx + λ1B =

ðΩ

P xð Þf xð Þ 1 −H ϕ xð Þð Þð Þð Þdx

ðA:9ÞðΩ

P xð ÞP xð ÞT 1 −H ϕ xð Þð Þð Þ� �

dx + λ1ΙÞB =ðΩ

P xð Þf xð Þ 1 −H ϕ xð Þð Þð Þð Þdx

ðA:10Þ

Let

Q =ðΩ

P xð Þf xð Þ 1 −H ϕ xð Þð Þð Þdx,

L =ðΩ

P xð ÞP xð ÞT 1 −H ϕ xð Þð Þð Þdx,

L + λ1Ι½ �B̂ =Q,B̂ = L + λ1Ι½ �−1Q:

ðA:11Þ

B. Coefficient Vectors are Invertibleand Existing

However, computing the coefficient vectors needs a matrixinversion step. Here, we concretely show that the coefficientvectors Â and B̂ are invertible if the matrices ½K� and ½L� in(11) are invertible. ½L� is N ×NGramian matrices, whose(i,j)th entry are obtained as the Eq. (11). Since ½K� and ½L�are a Gramian matrix, it is full rank if the poly-nomials

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðϕðxÞÞp PðxÞ, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 −HðϕðxÞÞÞp PiðxÞði = 1,⋯,NÞ are

linearly independent. The regularized version of the Heavi-side function is as HðϕÞ = 1/2ð1 + ð2/πÞ arctan ðϕ/εÞÞ, so itis easy to find that the functions HðϕðxÞÞ and 1 −HðϕðxÞÞare bounded in (0, 1). Since the polynomials PiðxÞ are linearlyindependent themselves, it clearly shows that the polyno-mials

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðϕðxÞÞp PiðxÞ, ði = 1,⋯,NÞ are linearly independent.

So the coefficient vectors are existing.

Data Availability

All experimental images come from reference literatures, andwe also point out the source one by one in the manuscript.

Conflicts of Interest

The authors declare there is no conflict of interest.

Authors’ Contributions

All authors typed, read, and approved the final manuscript.

Acknowledgments

This paper is partially supported by the Natural ScienceFoundation of Shandong Province of China (ZR2017MA012),the Natural Science Foundation of Guangdong Province(2018A030313364), the Special Innovation Projects of Univer-sities in Guangdong Province (2018KTSCX197), the Scienceand Technology Planning Project of Shenzhen City(JCYJ20180305125609379), the Natural Science Foundationof Shenzhen (JCYJ20170818091621856), and the ChinaScholarship Council Project (201508440370).

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A Novel Global Energy and Local Energy-Based Legendre Polynomial Approximation for Image Segmentation1. Introduction2. The Related Works2.1. The SBGFRLS Model2.2. The L2S Model

3. A Novel Adaptive Segmentation Model3.1. Model Construction3.1.1. The Local Energy3.1.2. The Global Energy

3.2. Algorithm Procedure

4. Experimental Results4.1. Results for Noisy Images4.2. Robustness to Intensity Inhomogeneity4.3. Results for Low-Contrast Images4.4. Results for Real-World Image4.5. The Influence of Parameter α4.6. Application on Neuron Images and Dendritic Spine Image

5. ConclusionAppendixA. The Process of Solving A&Hat; and B&Hat;B. Coefficient Vectors are Invertible and ExistingData AvailabilityConflicts of InterestAuthors’ ContributionsAcknowledgments

a novel global energy and local energy-based legendre...

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