Multidimensional Bernstein polynomials and Bezier curves: Analysis of machine learning algorithm for facial expression recognition based on curvature

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Abstract

In this paper, by using partial derivative formulas of generating functions for the multidimensional unification of the Bernstein basis functions and their functional equations, we derive derivative formulas and identities for these basis functions and their generating functions. We also give a conjecture and some open questions related to not only subdivision property of these basis functions, but also solutions of a higher-order special differential equations. Moreover, we provide an implementation for a real world problem of human facial expression recognition with the help of curvature of Bezier curves whose machine learning supported by statistical evaluations on feature vectors using in the aforementioned machine learning algorithm.

Introduction

Bezier curves, which are constructed with the help of Bernstein basis functions, have a wide variety of applications in not only mathematics, engineering and medicine, but also almost all areas in recent years. Therefore, a variety of new applications and methods have been developed recently on these concepts (cf. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]).

In this paper, in order to give a new dimension to these concepts, we investigate some theoretical properties of multidimensional unification of the Bernstein basis functions. We first give higher-order partial differential equations (PDEs) for these basis functions and their generating functions. Furthermore, by using these PDEs, we derive some recurrence relations for the multidimensional unification of the Bernstein basis functions. We also provide an image processing implementation of Bezier type curves for human facial expression recognition. By using C++ programming language with an computer vision library and machine learning algorithm, we give a facial expression classifier to show how Bezier type curves are used. While selecting the feature vectors to be used in the machine learning algorithm, we give a new method related to curvature of Bezier curves. Thanks to this method, a new dimension has been gained to the studies related to the facial recognition algorithms presented up to now.

Before giving implementation and computation, in order to convey the potential and motivation of the implementation in our work to the reader, we introduce not only our new results and identities related to multidimensional unification of the Bernstein basis functions and their generating functions, but also some open questions related to these results and Bezier type curves.

Here, we investigate some theoretical properties of multidimensional unification of the Bernstein basis functions.

In [25], Simsek defined the multidimensional(or m-dimensional) unification of the Bernstein basis functions Gn(x;b,s,k) as follows:Gn(x;b,s,k)=(nb1s1k1,b2s2k2,,nj=1m1(bjsjkj))×2j=1mbj(1sj)v=1m(xvbvsvkv)(1j=1mxj)nj=1m(bjsjkj),where x=(x1,x2,,xm); xj ∈ [0, 1], b=(b1,b2,,bm), s=(s1,s2,,sm) and k=(k1,k2,,km) with bj,sj,kjN0, j=1,,m andj=1m(bjsjkj)nand also(nb1s1k1,b2s2k2,,bmsmkm)=n!v=1m(bvsvkv)!(nj=1m(bjsjkj))!.

Generating functions for the multidimensional unification of the Bernstein basis functions Gn(x;b,s,k) are given by (cf. [25]):FB(t,x;b,s,k)=n=0Gn(x;b,s,k)tnn!=2j=1m(bjkj)v=1mxvbvsvkv(t2)j=1mbjsjkjj=1m(bjsjkj)!et(1j=1mxj),Note that there is one generating function for each value of b,s and k.

Remark 1

If we set s=1=(1,1,,1) and k=0=(0,0,,0) in (1.1), we haveGn(x;b,1,0)=(nb1,b2,,bm)v=1mxvbv(1j=1mxj)nj=1mbj=Pb1,b2,,bmn(x1,x2,,xm)where(nb1,b2,,bm)=n!b1!b2!bm!(nj=1mbj)!(cf. [17, p. 51, Eq. (13)], [4]).

Remark 2

If we set m=1 and k1=0 in (1.2), we haveFB(t,x1;b1,s1,0)=2b1x1b1s1(t2)b1s1et(1x1)(b1s1)!=n=0G(b1,s1,x1)tnn!which G(b1,s1,x1) denotes the unification of the classical Bernstein basis function given byG(b1,s1,x1)=(nb1s1)x1b1s1(1x1)nb1s12b1(s11)where x1[0,1] and n,b1,s1N0 with n ≥ b1s1 (cf. [31], [30, Theorem 1.1]). Hence, one can infer thatGn(x1;b1,s1,0)=G(b1,s1,x1).Moreover, if we set m=1 into (1.1), we getGn(x1;b1,s1,k1)=(nb1s1k1)x1b1s1k1(1x1)nb1s1+k12b1(s11).so that,Gn(x1;b1,s1,k1)=G(b1k1,s1k1,x1)which is given by Simsek in [30, Eq. (22)].

Remark 3

If we set m=1, s1=1 and k1=0 in (1.2), we haveFB(t,x1;b1,1,0)=(x1t)b1b1!et(1x1)=n=0Bb1n(x1)tnn!which Bb1n(x1) denotes the classical Bernstein basis function given byBb1n(x1)=(nb1)x1k(1x1)nb1(cf. [1], [21], [26], [27], [29], [30]), so that, when n < b1Bb1n(x1)=0(cf. [17]; see also the references cited therein). Hence, one can infer thatGn(x1;b1,1,0)=Bb1n(x1).

In order to give an implementation of Bezier curves for human facial expression recognition in the next sections, we need to recall some properties of these curves which are related to Bernstein basis functions.

In [31], by using unification of Bernstein basis functions, Simsek unified nth degree of Bezier type curves with n+1 control points Pb, s as follows:Bn(b,s,x)=0b,sn;bsnPb,sG(b,s,x).

From the above curves, for s=1, we easily get nth degree of Bezier curves with n+1 control points Pb=Pb,1 byB(n,x):=Bn(b,1,x)=b=0nPbBbn(x).

Further properties of these curves are studied by many authors in many different areas (cf. [8], [9], [11], [14], [24], [27], [28], [29], [30], [31]).

Section snippets

PDEs for the multidimensional unification of the Bernstein basis functions and their generating functions

In this section, we give higher-order partial derivative formulas for the multidimensional unification of the Bernstein basis functions and their generating functions. By using these partial derivative equations, we derive recurrence formulas and identities for these basis functions.

Differentiating both side of (1.2), with respect to xr; 1 ≤ r ≤ m, we get the following theorem whose proof can be verified by direct derivative, with respect to xr; 1 ≤ r ≤ m, or induction on r.

Theorem 1

Let 1 ≤ r ≤ m. We

Bezier curves with application of facial expression recognition based on curvature

It is well known that people are using facial expressions as complementary during communicating between each other in addition to verbal communication. Therefore, usage of facial expressions is now very common in the human and computer interaction applications. There exist many studies on facial expression recognition using Bezier curves with Hausdorff distance. On the other hand, in order to present a new dimension to the studies associated with the facial recognition algorithms given up to

Machine learning supported by statistical evaluations

In this section, by using statistical methods on feature vectors used during machine learning, it is shown how meaningful they are.

Conjecture and open questions

In this section, we state some open questions and conjecture related to subdivision property of the multidimensional unification of Bernstein (type) basis functions.

If y is a scaler, by using (2.8), in Section 2.2, we give some properties of subdivision property for the multidimensional unification of the Bernstein basis functions. However, if we replace y by y, the following conjecture and open questions appeared for subdivision property of the multidimensional unification of the Bernstein

Acknowledgments

The third author was supported by the Scientific Research Project Administration of Akdeniz University.

We thank the referees for their comments and valuable suggestions.

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