Yan-Li Liu et al.:Pores-Preserving Face Cleaning 561 as(T2-1)/3. 4.1 Extract the Local Property Compared with interpolation-based EMD,in our adaptive local mean algorithm,the approximation to Since our characterization and cleaning operation the local mean of image data is much more accurate, are only performed on skin parts of face,for a given therefore the problem of overshoot or undershoot of image,we first need to know which regions on the im- interpolation-based EMD is reduced.On the other age are skin parts.This information can be obtained hand,by setting a maximum neighborhood in sifting either by requiring users to indicate the skin regions, process,the minimal frequency of IMF is kept un- or by using facial feature detection algorithms to au- der control and hence image overdecomposition is pro- tomatize the process.In the paper,we manually mark hibited.Integrating both adaptive local mean algo- out the contour of the face,eyes and lips that are es- rithm and neighborhood limitation schemes,more ac- sential to the character of individuals which we do not curate frequency and amplitude can be achieved by intend to adjust.In the following,the operations are our method,laying a foundation to make quantitative all performed on the skin parts.Assuming that we energy-space analysis of facial images. have decomposed I into K IMFs with the ALNEMD algorithm described in the above section,now we apply Table 1.Adaptive Local Mean Algorithm Riesz transform to each IMF. Defining x=(1,r2)T,r is the pixel set The l-th Decomposition in Sifting Process of I,the Riesz kernel in the spatial domain is given for every pixel p of hk.1-1 as R(z)=z/2m3.We can see from the kernel that set T=3,flag false,ur; the integration in the Reisz transform is extremely lo- let p.r be the T x T neighborhood of p cal,which is a crucial requirement for analyzing non- while T<S stationary signals.The combination of a signal and its search for all local extreme points in p.T; Riesz transformed is called the monogenic signal.Let if NumExtreme(p）>r and Symmetry(p）==tue fu(z)be the k-th IMF of the image,the corresponding Cmeam)）=TxT∑qen.rhk-1(g monogenic signal thus takes the following form: flag =true; fk.M()=fr()+(R*fk)()=fk()+fk.R() break; else (6(t)+t/2)fk(-t)dt. (3) T=T+2: end if From the monogenic signal,meaningful local ampli- end while tude and local phase can be extracted.The local am- if flag==false plitude of fi.M()is: 1 Cmean(p）=x5∑qe.rhkI-1(gh end if f,M(川=VfM() end for =Vf(x)+lfk.R()2 (4) 4 Algorithm of Face Cleaning As local amplitude offers image's energetic informa- We will present in this section our face cleaning al- tion,we use the square of local amplitude,namely local energy,to characterize the oscillatory degree of every gorithm.First,we give an overview of the algorithm pixel x at level k. Formally,for the facial image I to be processed,we first use ALNEMD to decompose it into a series of both fre- LEk()=lfk.M()2. (5) quency and amplitude modulated IMFs.Next,Riesz transform is applied to every IMF,generating a corre- According to the theory of EMD.with the increase sponding monogenic signal22]which is the extension of of k,LEk(r)decreases.In order to study the behavior the analytical signal from 1D to 2D.Built upon mono- of LEL(x),for a given pixel x,across different levels k genic signals,the meaningful local energy of every pixel in a unified fashion,we normalize LEk(r),x E at ev- is extracted.By analyzing normalized local energy,we ery level,getting the normalized local energy NLE(). propose a function to quantitatively measure the im- Being a relative value,NLEk(z)reflects the relative perfect degree of image.The cleaning result is finally weight of local energy of pixel z at level k.Figs.1(b), produced after the coefficients of IMFs are adaptively 1(c),1(d)show the first three normalized local energy adjusted under the guide of imperfect degree. maps of Fig.1(f),a selected part of Fig.1(a).Yan-Li Liu et al.: Pores-Preserving Face Cleaning 561 as (T 2 − 1)/3. Compared with interpolation-based EMD, in our adaptive local mean algorithm, the approximation to the local mean of image data is much more accurate, therefore the problem of overshoot or undershoot of interpolation-based EMD is reduced. On the other hand, by setting a maximum neighborhood in sifting process, the minimal frequency of IMF is kept un￾der control and hence image overdecomposition is pro￾hibited. Integrating both adaptive local mean algo￾rithm and neighborhood limitation schemes, more ac￾curate frequency and amplitude can be achieved by our method, laying a foundation to make quantitative energy-space analysis of facial images. Table 1. Adaptive Local Mean Algorithm The l-th Decomposition in Sifting Process for every pixel p of hk,l−1 set T = 3, flag = false, µT ; let Ωp,T be the T × T neighborhood of p; while T < Sl search for all local extreme points in Ωp,T ; if NumExtreme(p) > µT and Symmetry(p) == ture emean(p) = 1 T × T P q∈Ωp,T hk,l−1(q); flag = true; break; else T = T + 2; end if end while if flag == false emean(p) = 1 Sl × Sl P q∈Ωp,T hk,l−1(q); end if end for 4 Algorithm of Face Cleaning We will present in this section our face cleaning al￾gorithm. First, we give an overview of the algorithm. Formally, for the facial image I to be processed, we first use ALNEMD to decompose it into a series of both fre￾quency and amplitude modulated IMFs. Next, Riesz transform is applied to every IMF, generating a corre￾sponding monogenic signal[22] which is the extension of the analytical signal from 1D to 2D. Built upon mono￾genic signals, the meaningful local energy of every pixel is extracted. By analyzing normalized local energy, we propose a function to quantitatively measure the im￾perfect degree of image. The cleaning result is finally produced after the coefficients of IMFs are adaptively adjusted under the guide of imperfect degree. 4.1 Extract the Local Property Since our characterization and cleaning operation are only performed on skin parts of face, for a given image, we first need to know which regions on the im￾age are skin parts. This information can be obtained either by requiring users to indicate the skin regions, or by using facial feature detection algorithms to au￾tomatize the process. In the paper, we manually mark out the contour of the face, eyes and lips that are es￾sential to the character of individuals which we do not intend to adjust. In the following, the operations are all performed on the skin parts. Assuming that we have decomposed I into K IMFs with the ALNEMD algorithm described in the above section, now we apply Riesz transform to each IMF. Defining x = (x1, x2) T, x ∈ Ω, Ω is the pixel set of I, the Riesz kernel in the spatial domain is given as R(x) = x/2π|x| 3 . We can see from the kernel that the integration in the Reisz transform is extremely lo￾cal, which is a crucial requirement for analyzing non￾stationary signals. The combination of a signal and its Riesz transformed is called the monogenic signal. Let fk(x) be the k-th IMF of the image, the corresponding monogenic signal thus takes the following form: fk,M(x) = fk(x) + (R ∗ fk)(x) = fk(x) + fk,R(x) = Z Ω (δ(t) + t/2π|t| 3 )fk(x − t)dt. (3) From the monogenic signal, meaningful local ampli￾tude and local phase can be extracted. The local am￾plitude of fk,M(x) is: |fk,M(x)| = q f 2 k,M(x) = q f 2 k (x) + |fk,R(x)| 2. (4) As local amplitude offers image’s energetic informa￾tion, we use the square of local amplitude, namely local energy, to characterize the oscillatory degree of every pixel x at level k. LEk(x) = |fk,M(x)| 2 . (5) According to the theory of EMD, with the increase of k, LEk(x) decreases. In order to study the behavior of LEk(x), for a given pixel x, across different levels k in a unified fashion, we normalize LEk(x), x ∈ Ω at ev￾ery level, getting the normalized local energy NLEk(x). Being a relative value, NLEk(x) reflects the relative weight of local energy of pixel x at level k. Figs. 1(b), 1(c), 1(d) show the first three normalized local energy maps of Fig.1(f), a selected part of Fig.1(a)