Mabelle Sayah

Analyzing and Comparing Basel's III Sensitivity Based Approach for the interest rate risk in the trading book

Keywords: Capital charge, Sensitivity Based approach, Basel III, GARCH, PCA, ICA, Dynamic Nelson Siegel, bonds portfolio, trading book, interest rate risk

A bank s capital charge computation is a widely discussed topic with new approaches emerging continuously. Each bank is computing this figure using internal methodologies in order to reflect its capital adequacy; however, a more homogeneous model is recommended by the Basel committee to enable judging the situation of these financial institutions and comparing different banks among each other.

In this paper, we compare different numerical and econometric models to the sensitivity based approach (SBA) implemented by BCBS under Basel III in its February 2015 publication in order to compute the capital charge, we study the influence of having several currencies and maturities within the portfolio and try to define the time horizon and confidence level implied by Basel s III approach through an application on bonds portfolios.

By implementing several approaches, we are able to find equivalent VaRs to the one computed by the SBA on a pre-defined confidence level (97.5 %). However, the time horizon differs according to the chosen methodology and ranges from 1 month up to 1 year.

I. Introduction

Commercial banks are a key component of today's financial and economic system. Banks allocate funds from depositors to borrowers, convert maturities, and provide financial products. These services among others enhances the efficiency of the overall economy. Given this crucial role, adequate regulations should apply to monitor banks risks.

Since its first issuance in 1988, Basel has been the main banking regulation authority initializing with some main credit risk rules. In 1996, the market amendment was issued setting the basic standards regarding trading assets and the value at risk computation methodology. It also divided the risk into market and credit risk; market risk being divided between equity, interest rate, foreign exchange, commodity, and option risk with a standardized capital computation approach treating each asset class separately. In 2006, Basel II came along with more 'personalized' approaches such as internal rating based IRB for credit risk, internal models for OTC derivatives exposure along with the introduction of the operational risk charge.

However,these regulations did not prevent major crisis hitting the international market causing huge losses in different sectors (cf. [START_REF] Baptistab | A comparison of the original and revised Basel market risk frameworks for regulating bank capital, Shu Yanc[END_REF]). As a response to this shortage, Basel 2.5 was created beginning 2011 as a response adding more capital to the trading book (especially on the poorly modeled products). Basel 2.5 additions targets the stressed VaR concept, the incremental risk charge and few new standard rules regarding the banking book.

In 2013, after a thorough observation of the consequences of the crisis and the attempt for correction made by 2.5, Basel III was introduced. The main functions of this issuance are: increasing capital for counterpart exposures, tightening the definition of the bank's capital, adding buffers 1 for liquidity and introducing a new leverage ratio (cf. BCBS (2014)). In parallel, increasing the trading book capital requirement under Basel 2.5 was required following the crisis and was not well designed: the calculation remains non-risk sensitive and highly conservative, and differences in model approval persist between jurisdictions.

As a result, in May 2012 the Basel Committee published a Consultative Paper on a 'Fundamental Review of the Trading Book' (FRTB) to improve this framework then in February 2015, "Fundamental review of the trading book: outstanding issues", the BCBS exposed the weaknesses of Basel's previous approaches. It suggested some major changes to the trading book to be implemented by 2018: Modeling Issues (stressed expected shortfall, liquidity adjustments, diversification, default and migration risk, and non-modelable risk factors), scope and approval process (boundary of the trading book, desk level model approval, model testing, model independent assessment tool) and new standard rule for capital charges computation.

The standard rule suggested by the FRTB, is based on sensitivities hence, it is called the "Sensitivity based approach" (SBA): it was implemented as the 'homogeneous' method for capital charge computation across all banks. The SBA is based on percentages and correlations between different maturities and currencies (cf. BCBS (2015)). The existing standard rules poorly reflected hedging or diversification thus inflating the trading book capital level. The SBA is simple yet risk sensitive which is already a big improvement. SBA is a standardized method that reflects the risk resulting from: Interest rate, credit spread, equity, commodity, foreign exchange, default, and options risk.

Comparing regulatory approaches, an obvious contrast between Basel and Solvency is noted: Solvency II has a similar three pillar structure as Basel's Accords. The target capital requirements (called SCR) are described under the first pillar and refer to all types of risks an insurance undertaking is exposed to: market risk (interest rate risks, equity risk, property risk, spread risk, concentration risk, and currency risk) and counterpart default risk. Both frameworks take diversification effects into account and use square root formulas. However, these aggregation approaches are applied at different levels: a considerably stronger risk differentiation is shown under Basel III. For example, the SBA equity risk distinguishes 10 risk categories in order to assign the risk weights, in contrast to one single shock for all listed equities under Solvency II. Under the interest rate risk, SBA is Basel's III approach whereas in solvancy II a shocked scenarios based computation sets the capital charge(cf. LAAS and Siegel (2015)).

Seemingly, the SBA has several add-ons regarding sensitivity and diversifying considerations however it still has some issues (specifying the coefficients) and details concerning the aggregations that need to be tweaked; for instance the figure under the square root is sometimes negative(cf. ISDA (2015)). In this paper we aim to focus on the suggested standard rule by the FRTB in order to compare it with other econometric models to find an equivalent capital charge computation technique with few additional details such as time horizon and confidence level for a given capital. Among the different risk modulations, we chose to focus on the interpretation of the interest rate calculation.

According to Oxford Dictionary of Economics, interest rate is defined as "The charge made for the loan of financial capital expressed as a proportion of the loan". More formally, Basel Committee on Banking Supervision (cf. BCBS ( 2004)) indicated that interest rate risk (IRR) is the exposure of a bank's financial condition to adverse movements in interest rate.

The sound IRR management conducted by Basel Committee on Banking Supervision had been the source on which analysts rely to evaluate the activities of bank's risk management of interest rate (cf. BCBS (2004)). In the guideline, the committee offers four basic elements of IRR: appropriate board oversight, comprehensive internal controls, adequate policies and appropriate risk measure. The SBA falls under this forth element in modeling the interest rate risk in the trading book.

Our aim in this paper is to understand the computation of the interest rate risk in banks based on BASEL's III approach. However, the SBA remains relatively vague and the choices of its coefficient and correlation parameters are not robustly detailed and documented. Hence, studying the interest rate risk from an econometric point of view in order to compare and contrast the results is critical and highly important.

Based on different central banks approaches for term structure interest rate, we selected few econometric models. The main idea was to reduce the dimensions of the database, study the dynamics of these 'reduced' factors and then conclude on the wider data range dynamic. We introduce the methods used to derive capital requirement and compare them with SBA's in order to conclude on some equivalence between them.

Having this objective in mind, the structure of the work is as follows: We started by modeling each interest rate curve on a stand-alone maturity basis using a GARCH approach. It is worth noting that by doing so, we dropped a crucial information which is the strong correlation between these maturities, however we needed this phase as a starting point and a comparison threshold. Modeling the volatility of term structures using GARCH processes has become a current practice due to its numerous advantages relative to alternative models. GARCH-methods are a way of investigating how a function of past returns, in a specific financial series, should be constructed and mapped onto the second moment (cf. [START_REF] Hull | Options, Futures and Other Derivatives[END_REF]). Proposed by [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[END_REF] and then generalized by [START_REF] Bollerslev | Generalized Autoregressive Conditional Heteroskedasticity[END_REF], GARCH models explain high frequency financial data series through the autoregressive conditional heteroskedasticity and can model simultaneously conditional mean and conditional variance (cf. [START_REF] Edison | Foreign Exchange Intervention and the Australian Dollar: Has It Mattered?[END_REF]. Two parameters for orders could be used in order to optimize the results of the tests regarding GARCH coefficients convergence. However, in practice, low orders are more frequently used. The first-order (p=q=1) GARCH model (cf. [START_REF] Taylor | Modelling financial time series[END_REF]) has become the most popular GARCH model. Secondly, we introduced the component approaches starting with the Principal component analysis (PCA) which is one of the multivariate analysis techniques usually used for correlation studies, data reduction and efficiency assessment (cf. Levieuge et al. 1 (2010)). This method incorporates the interdependence between term structures maturities: it considers the correlated curves and generates new non-correlated variables. Each factor is related to a loading and a cumulative variance defining the variance explained by each one of the new variables. PCA creates the same number of term structures included in the model however, we need to choose the reduced number of factors that we want to handle. In this work, we chose to cover 98% of the variance, by considering two or three factors. Using a GARCH model, we only project the chosen factors and not the loadings; then we re-create the entire data from the projected factors and previously observed loadings.

Thirdly, was the implementation of the Independent component analysis (ICA): it provides a mechanism of decomposing a given signal into statistically independent components. PCA uses only second order statistical information however, ICA uses higher order (kurtosis) for separating the signals which permits more conclusive results in financial data (cf. [START_REF] Comon | Independent component analysis -a new concept[END_REF]). A drawback in the ICA is its inability to indicate the data variance coverage for each factor, therefore the modeler has to define the number of factors to be considered; in this paper we chose to include three ICA factors.

Regarding the last approach, a factor model is suggested: the Dynamic Nelson Siegel. No GARCH processes are used, instead a mix of Nelson Siegel estimation and ARIMA processes projection are put in practice. Factor models for the yield curve, such as Nelson- [START_REF] Siegel | Long-term behavior of yield curves[END_REF], its dynamic version (cf. [START_REF] Diebold | Forecasting the Term structure of government bond yields[END_REF]) and its arbitrage-free counterpart proposed by Christensen, Diebold and Rudebusch (2011), have been extensively applied to forecast bond yields. We used the Dynamic Nelson Siegel due to its flexibility in representation especially for the long term projection. By fitting the curves, projecting the factors using Diebold method and the loadings employing an ARIMA process, we are able to reconstruct the curves from which we concluded the capital requirement.

The capital charge using the previously mentioned methods, except SBA, can be computed on a certain confidence level basis and for a given time horizon; therefore comparing these methods to SBA would determine a common time horizon and confidence level, reaching the purpose of this paper. We start by explaining in details the procedure of the SBA, the correlation between the duration of the portfolios and the capital charge required by this procedure then compare the methods using different approaches. These latters will be based on bonds portfolios denoted in: euros, dollars and Turkish lira from the French, German, US and Turkish governments yields respectively, for maturities between 1 month and 30 years.

In this paper we proceed as follows: in Section 2 we provide a detailed description of the sensitivity based approach through hypothetical portfolios, we also show the link between this capital charge computation and the portfolios duration. In section 3, we proceed with an overview of four different approaches then we explain how we computed the capital charge using these processes. In section 4 we present the empirical analysis, describe the data, estimate the models, compute the VaRs and conclude with analyzing, comparing and back-testing the capital requirement calculations. In section 5 we offer some interpretation and conclusive remarks.

II. Sensitivity Based approach (SBA)

II.1. Introducing the approach

This new method would require banks to use prices and rate sensitivities in order to compute their capital charge. This revised (sensitivity-based) standardized approach would capture more granular or complex risk factors across different asset classes in the trading book (cf. BCBS ( 2015)). It builds on the standardized framework tested in the trading book QIS conducted in the second half of 2014. (cf. Basel ( 2014)) The proposed methodology covers the delta and optionality risk: general interest rate risk, credit spread risk of non-securitization and securitization exposures, equity, commodity and FX risk. Vega and curvature risk measurements are under development in order to measure the sensitivity of the value of an option with respect to a modification in volatility and the rate of change of delta.

II.2. Implementation reasons

• The approach must provide a method for calculating capital requirements for banks with a level of trading activity that does not require sophisticated measurement of market risk. • It provides a fall back in the event that a bank s internal model is deemed inadequate, including the potential use as add-on or floor to an internal model-based charge. • The approach should facilitate consistent and comparable reporting of market risk across banks and jurisdictions.

II.3. Computational steps

We first compute the net sensitivity of the bond (relative 1 bps change) and multiply it by its corresponding risk weight in order to get the weighted sensitivity. We note that for each maturity a different risk weight is allocated based on a matrix provided by the Basel committee. For each currency, the 'average' is computed as the square root of the sum of squared single weighted sensitivities and double products of these latter weighted by given, maturity based, correlation coefficients. Aggregation on a portfolio level is another sum of the squared capital charges computed for each currency plus the double products weighted by a factor of 0.5 fixed by Basel (between currencies).

1. Get the observed yield and price on the market. 2. Compute the net sensitivity of each instrument and recalculate the price.

3. Based on the matrix imposed by Basel, get the weighted sensitivities. 4. Form buckets by sorting each currency in a separate bucket. 5. For each bucket compute the following:

K bucket = ∑ N i=1 WS 2 i + ∑ N i=1 ∑ j =i ρ ij WS i WS j 6. Compute the capital charge: Capital Charge = ∑ N i=1 K 2 i +0.5 ∑ N i=1 ∑ j =i S i * S j initial portfolio value where S i = ∑ WS t

II.4. Hypothetical example

Let us consider a hypothetical portfolio composed of only one zero coupon bond. The portfolio is studied on a rolling basis and the bond does not include optionality. In this paper, we do not consider the effect of currency or default risk.

Price P = 100e -r T T Modified price P = 100e -r T T-0.0001T Sensitivity S = P -P 0.0001 Weighed sensitivity WS = RW * S Capital Charge WS P
The previous example proves that the capital charge under the SBA is only dependent of the zero bond maturity (therefore duration) and the associated risk weight of this particular maturity. We analyzed the three following cases: a portfolio consisting of one bond, two same currency bonds and different currencies bond in order to come up with one generalized approach.

Step by step computation and numerical examples can be found in Appendix 1.

Hereafter we consider a portfolio combining three bonds: two in a same currency and a third in a different one, we obtain the following:

The price of the three bonds being: P i = ∑ C it e -r it t and the durations D i = ∑ tC it e -r it t P i with C it being the cashflows and r it the interest rate. We compute the net sensitivity NS i and the weighted sensitivity WS i :

NS i = ∑ t C it e -r it t -∑ t C it e (-r it -0.01%)t 0.0001 and WS i = RW i 0.0001 ∑ t C it e-r it t(1 -e -0.0001t ) = RW i * D i * P i .
Having two different currencies, two buckets are created and a K α i is computed for each:

K α 1 = WS 2 1 + WS 2 2 + 2ρ 12 WS 1 WS 2 in the second bucket having only one bond, K α 2 equals WS 3 .
Bringing these together, SBA demands a 0.5 coefficient for the correlation and a sum of square to compute the squared capital charge:

CC = ∑ 3 i=1 RW 2 i P 2 i D 2 i ∑ 3 i=1 P i (1 + 2ρ 1,2 RW 1 D 1 P 1 RW 2 D 2 P 2 +RW 1 D 1 P 1 RW 3 D 3 P 3 +RW 2 D 2 P 2 RW 3 D 3 P 3 2(∑ 3 i=1 RW 2 i D 2 i P 2 i )
).

III. Equivalent interest rate risk assessment methods

Computing the capital charge of a given portfolio needs the computation of the value at risk using the traditional approaches. Therefore, we will compute the value at risk combining different known methods in order to compare them with the SBA results.

III.1. ARCH-GARCH

ARCH methods were introduced by Engle in 1982 (cf. [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[END_REF]) then generalized by Bollerslev in 1986 (cf. Bollerslev and Tim (2008)) as a conditional variance prediction model, especially useful when the volatility of the financial data is the main issue.

Let X t denote a real-time stochastic process; the GARCH(p,q) process is given by:

X t |σ t ∼ N(0, σ 2 t ) σ 2 t = α 0 + ∑ p i=1 α i X 2 t-i + ∑ q j=1 β j σ 2 t-j
The GARCH approach has been used in modeling financial time series, test financial theories and interpret key features of a given data in a time-dependent matter.

In our bonds portfolio we consider government yield curves with maturities ranging from 3 months to 30 years, compute the return and apply GARCH models for each maturity. The orders of a GARCH process play a major role in determining the results: q can be based on model selection tests such as the autocorrelation function of the squared residuals; however with large q, estimation error might increase. Finding both p and q parameters can be facilitated through time series testing. In our paper GARCH parameters are determined on the basis of coefficients significance and Jarque Berra test (cf. [START_REF] Bollerslev | Glossary to ARCH (GARCH)[END_REF]): skewness and kurtosis are used for constructing Jarque Berra's test statistic to find whether the coefficients of skewness and excess kurtosis are jointly nul(cf. [START_REF] Jarque | Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo evidence[END_REF]).

When estimating GARCH models, computer-based softwares have to be used. Different softwares have different functionalities, drawbacks and features. Several works present these GARCH estimating software packages and compare them (cf. Brooks et al. (2001)). In our work, the 'tseries' R-package is used to estimate these models. We start by estimating the GARCH process for each maturity, after having adequately selected the parameters. Since the projection requires initial values for each yield curve, we adopted the traditional way of having the last observed yield as the first data point of projection and the volatility as the historically observed volatility for each curve. We project for one year, fixing a year as 252 days, using Monte Carlo simulations then extract the value at risk for different confidence levels from 95% to 99.9 %. The capital charge percentage was computed on a mean relative figure basis, i.e. the mean of all simulations is extracted of the VaR and divided again by the mean (10,000 simulations were in order).

III.2. PCA-GARCH

Principal component analysis is mainly used to reduce the dimension of the handled data (cf. [START_REF] Jackson | A user's guide to principal components[END_REF]): This is useful in visualizing extraction. PCA can be derived from a number of starting points and optimization criteria. The most important of these are minimization of the mean-square error in data compression, finding mutually orthogonal directions in the data having maximal variances, and decorrelation of the data using orthogonal transformations.

The transformation is defined by a set of p-dimensional vectors of weights or loadings w (k) = (w 1 , . . . , w p ) (k) that map each row vector x (i) of X to a new vector of principal component scores

t (i) = (t 1 , . . . , t p ) (i) , given by t k(i) = x (i) • w (k)
Therefore instead of having 15 yield curves with different maturities, using the PCA process we reduce it to a number of principle factors that represent more than 98% of our data.

Having these factors (in most cases 2 or 3) we project them on a one year basis using GARCH models and choose their parameters based on the previously mentioned details. After rebuilding the 15 maturities using the projected factors and the previously computed loadings, we generate the VaR and capital charge of our portfolios using the same way selected in the previous methodology for the capital charge computation. This process adds the correlation between different maturities and shows the interdependency between all tenors even when the portfolio does not include the entirety of maturities.

III.3. ICA-GARCH

Independent components analysis, (cf. [START_REF] Comon | Independent component analysis -a new concept[END_REF]), is another method to reduce dimensions that has the same functionality as PCA except for the difference in the determination of the components and the loadings: In PCA, the aim is to find vectors that best explain the variance of the data whereas in ICA the kurtosis is in focus. The latent variables are assumed non-Gaussian and mutually independent.

The data analyzed by ICA could originate from many different kinds of application fields, including digital images, document databases, economic indicators and psychometric measurements. In many cases, the measurements are given as a set of parallel signals or time series; the term blind source separation is used to characterize this problem. Typical examples are mixtures of simultaneous speech signals that have been picked up by several microphones, brain waves recorded by multiple sensors, interfering radio signals arriving at a mobile phone, or parallel time series obtained from some industrial process.

In this ICA method follows the same process as PCA's: ICA to the full data panel, GARCH projection, rebuilding of the data, determining the VaR, capital requirement computation.

Again this model includes correlation as well as GARCH estimations however it is an add-on to the previous method due to the following: ICA does not assume the non-correlation of the factors instead it supposes the independence, such that the normality of the data is not a must; on the contrary, non Gaussian factors have an added-value. The difference between PCA and ICA is explained in the literature (cf. [START_REF] Bugli | Comparison between principal component analysis and independent component analysis in electroencephalograms modeling[END_REF]), (cf. Burgos (2013)):

Given a set of multivariate measurements, the purpose is to find a smaller set of variables with less redundancy, which would best represent the data: this is the common goal of PCA and ICA.

However, in PCA the redundancy is measured by correlations between data elements, while in ICA the concept of independence is used, and in ICA the reduction of the number of variables is given less emphasis. (cf. Hyvarinen et al. ( 2001))

The earliest ICA algorithm that we are aware of and one which generated much interest in the field is that proposed by (cf. [START_REF] Herault | Space or time adaptive signal processing by neural network models[END_REF]). Since then, various approaches have been proposed in the literature to implement ICA. These include: minimizing higher order moments (cf. Cardoso (1989)) or higher order cumulants (cf. [START_REF] Cardoso | Blind beamforming for non-Gaussian signals[END_REF]), minimization of mutual information of the outputs or maximization of the output entropy (cf. [START_REF] Bell | An information maximization approach to blind separation and blind deconvolution[END_REF]), minimization of the Kullback-Leibler divergence between the joint and the product of the marginal distributions of the outputs (cf. [START_REF] Amari | A new learning algorithm for blind signal separation[END_REF]).

This work is applied using the fastICA algorithm implemented in a R-pakage. (cf.

III.4. Dynamic Nelson Siegel

The interest rate curve is essential for pricing, hedging and evaluating a portfolio. Various curve fitting spline methods have been introduced (quadratic and cubic splines (McCulloch (1971(McCulloch ( , 1975))), exponential splines (Vasicek and Fong (1982)), Bâ Ȃ Řsplines (Shea (1984))..., these methods were criticized for not being too representative of the economical situations. Nelson and Siegel (1987) and Svensson (1994Svensson ( ,1996) ) therefore suggested parametric curves that are flexible enough to describe the large frame of the financial conditions.

Nelson Siegel method consists of estimating three parameters using the maximum likelihood process or OLS to rebuild the yield curve (cf. [START_REF] Siegel | Long-term behavior of yield curves[END_REF]): it models the slope, curvature and level of the curve with their loadings and movements through time in order to fit an existing curve at each tenor t.

y(τ) = β 1 + β 2 1-e -λτ λτ + β 3 1-e -λτ λτ-e -λτ
The Nelson Siegel model is extensively used by central banks and monetary policy makers (ex: Bank of International Settlements (2005), European Central Bank (2008)).

As a development to the traditional fitting approach, Diebold and Li (2006) introduce the Dynamic Nelson-Siegel (DNS) model by estimating the classical formula with time-varying factors and model them using autoregressive specifications projecting therefore the yield curves by adding dynamism to the parameters: This method shows very encouraging results especially on a long time horizon.

y t (τ) = β 1t + β 2t 1-e -λ t τ λ t τ + β 3t 1-e -λ t τ λ t τ-e -λ t τ
Our chosen approach in this paper is to fit the yield curves using the traditional Nelson Siegel and to project it afterwards: estimating the different yields using the Nelson Siegel function in R,(R-package: YieldCurve),projecting the betas computed using adequate ARIMA processes (based on best-fit approach); maintaining the loadings as calculated historically, rebuilding the projected yields based on diebold and Li's dynamic approach.

Having the projected yields, a new portfolio evaluating could be placed, a value at risk and therfore a capital charge is computed.

Lambda:

The lambda factor in the nelson siegel formula, as provided by Nelson and Siegel (1987), does not have any exact economic meaning. In fact, from the econometric point of view, it is a constant assuring a specific slope of yield curve. In practice, higher values produce a faster decay of yield curve and vice versa. In conclusion, the factor plays a key role especially in fitting the longer end of yield curves.

Nelson and Siegel (1987) fixed λ to simplify the computation allowing usage of the simple and linear ordinary least squares. [START_REF] Diebold | Forecasting the Term structure of government bond yields[END_REF] follow the same logic of exogenously predetermined λ however they suggested interpretation for the parameters as the slope, curvature and level. Based on such assumptions, and interpreting lambda as the maturity at which the loading on the curvature factor achieves its maximum, maximizing this loading would result in a numeric value for this coefficient: 0. In this paper, lambda of the projected yield is also considered as a constant chosen as the mean of the previously estimated lambdas for each specific yield.

IV. Application: bonds portfolios

IV.1. Data used

The data is fetched from Bloomberg: Government yield curves 3 months up till 30 years maturities on a daily basis for: France, Germany, USA and Turkey. For each country, different dates are available, France data starts on 04/30/1998, German on 10/04/1991, US on 11/24/2003 and Turkey on 04/01/2005; all the data ending point is on 05/15/2015.

IV.2. Portfoliosâ Ȃ Ź duration and different capital charge requirements

In order to compare these methods, we build portfolios using either one unique currency or multiple currencies. The following plots represent the different yields for each currency:

Single currency portfolios

We consider four portfolios, for each governmental yield, with the same composition, consisting of 12 zero coupons each: 3 one year ZC, 3 two years ZC, 2 three years ZC, 1 five years ZC, 1 ten years ZC and 2 fifteen years ZC. Sensitivity Based Approach SBA capital requirement is compared below to the portfolio s duration at every date: As previously shown by the three cases equations, the plots above reflect the correlation between the duration of the four chosen portfolios and the capital requirement computed using the SBA method.

It is clear that the coefficient depends on a certain factor reflected by the ris k weight used. Having the same portfolio composition, the factor between the duration and the capital charge is the same. After noting the obvious resemblance between the behavior of the SBA capital requirement and the duration of these single currency portfolios, we proceed in applying the main goal of this paper.

Our aim is to define an equivalent to the SBA capital requirement using a VaR on a given confidence level and time horizon; we proceed as follows: For each chosen methodology, we compute the VaR of the portfolios price (Monte-Carlo simulations basis) and compare these VaRs to the SBA requirement; the interception between these figures will simulate the confidence level and time horizon equivalent to Basel's requirement. We initiate this comparative work by computing the GARCH process value at risk.

GARCH model

For each currency, we have 15 yield curves with different maturities; we compute: ∆i t = i ti t-1 . Building on these differences, we estimate them using an ARCH(1) model; projecting this model one year ahead (252 days) we build our 'future' portfolios. Repeating this process 1000 times, based on a Monte Carlo logic, we conclude the VaR and therefore the capital charge. Note that for all methods the SBA capital charge is computed as a percentage of the initial portfolio, in the other methods capital charge is computed as the relative change between the projected mean and values at risk of the initial portfolio value. Not accounting for the inter-correlation in our GARCH approach, the capital charge is expected to fall mostly below the SBA approach. We can observe the resemblance between the French, German and US market,however a very unstable behavior in the Turkish data.

PCA-GARCH model

Applying the PCA on the 15 maturities of each currency, we reduce the data into two components covering at least 98 % of the data. We project the first differential of these components using an adequate GARCH model then rebuild the entire maturities using the projected factors and the previously computed loadings. Monte Carlo simulations permits the extraction of the VaR at different levels and the capital charge.

ICA-GARCH model

This method is similar to the previous one, however instead of using the principal component approach we used the independent components approach to increase the precision and reduce the assumptions made in the PCA.

DNS-ARIMA model

After estimating the curves using NS model, we projected the beta parameters using the best fitted ARIMA(p,d,q) process. Along with the mean of the historically observed lambda's and the projected beta's, we rebuild the curves, estimate the VaR and compute the capital charge. The results show a quicker convergence in this method. However, in an unstable market such as the Turkish case, we do not observe any convergence in the Nelson Siegel parameter, therefore no projection could be applied.

Multiple currencies portfolios

Denoting the previous portfolios by their government yield we have: port_FRANCE, Port_GERMANY, Port_US and Port_TURKEY. In this section we consider multiple currencies combining the previously mentioned portfolios: Port_FR_GR, Port_FR_US, Port_FR_TRY, Port_FR_GR_US, Port_GR_US_TRY and Port_FR_GR_US_TRY Sensitivity Based Approach In this section, having multiple currencies, the correlation parameters between different curves at different tenors will be added. Note that both France and Germany are held in euros therefore they are part of the same bucket. SBA capital requirement vs duration: Similar remarks could be presented here regarding the parallel movement of the duration and capital requirement of these portfolios. This could be interpreted using the equations in appendix 1,case 3.

FR-GR portfolio

IV.3. Comparing results

In order to reduce the fluctuation, the results were interpolated to a logarithmic function resulting in the following: Based on the single currency portfolios, GARCH model shows a permanent 'low' level of capital requirement compared to Basel and the other methods. This could be explained by the correlations' exclusion between different tenors in this first method.

PCA and ICA methods show a close trend however, the ICA approach is more conservative and demands higher requirements. In all portfolios, DNS converges very rapidly, equaling the SBA on a short time horizon. The results differ as follows for multi-currencies portfolios: In order to decide on the best approach for the capital requirement computation, a backtest analysis is a must. We used the typical 'number of hits' approach (number of violations) based on a scrolling window of 252 dates projecting them 100 notches (daily or weekly) and comparing the results with the actually observed rates. Our findings are summarized in the following table.

Results reject the GARCH model being too unrealistic and raises question marks regarding the DNS model for the daily projection.

Number of violations

V. Conclusion

We have compared in this paper different methods for computing the capital charge of a commercial bank based on a Eurobonds portfolio example and we have explored the performance in an out-of-sample forecasting based on the number of violations. These approaches might be used as internal models compared to Basel's SBA in oder to define a chosen time horizon and confidence level vis a vis the 'standard method'.

The following table summarizes the encountering points (in days) between the SBA and the different other methodologies studied for the ten portfolios. • Except for the Turkish market, GARCH method computes a similar capital requirement as the SBA for a minimum of one year time horizon. • Comparing PCA and ICA we can conclude that the ICA is more restrictive for single currency denoted and mixed portfolios. • PCA gives an eight months, 97.5% adequacy parallel to the four months given by the ICA for FR and GR portfolios. We add a PCA requirement of 4 months horizon for the US portfolio facing a one month limit given by the ICA. • In the mixed portfolios PCA sets (at 97.5%) a limit of one year except when the Turkish lira is involved, ICA shows a lower encounter point of 9 months. • The DNS on the 97.5% gives an adequate capital charge between two and four months. • In the mixed portfolios DNS method on the 97.5% requires a time horizon between four and seven months.

20

Based on the previous, our recommendations are: -> For the Eurozone:

-1 year GARCH capital charge would be equivalent to the SBA for a level of 97.5 %.

-GARCH does not account for inter-maturities correlations therefore an ICA or PCA approach would be more rational: * 8 months PCA 97.5 % on a country level and 1 year when combined. * 3 months ICA 97.5 %on a country level and 8 months when combined. -DNS would inquire an average of 3 months for each country and 6 months for a multi-European portfolio. -> For the US:

-GARCH imposes a 1 month horizon for 97.5% confidence level.

-97.5% PCA for less than 4 months capital charge would do it and a 1 month ICA.

-An average of 2 months DNS results in a close capital charge as the SBA's requirement. -> For the Turkish market:

-TRY is too volatile to be adequately represented by ICA or DNS models: it can be used for very short term : one or two months PCA (97.5%). -> When combining US and Euro markets:

-GARCH results could remain applicable.

-PCA method time horizons' increases significantly to an average of one year.

-ICA method time horizons' increases to 9 months.

-DNS approaches a limit between 5 and 7 months. -> When combining the Turkish lira with any of the US dollar or Euro portfolio: PCA approaches an average of 4 months and ICA an average of 2 months.

The goal of this paper was to provide banks with a tool that explains 'mathematically' Basel's SBA approach in order to fix the time horizon and confidence level of their capital requirement in the trading portfolio. In addition, these models could provide an internal approach with customized coefficients and parameters.

In June 2015, a new consultative document was issued by the BCBS on the 'Interest rate risk in the banking book, presenting new approaches to handle this book's capital charge computation and suggests dividing this amount between the first and second pillar. Incorporating the banking book in the first pillar is a new approach, because that segment was reserved for the trading book. Doing so, a similar methodology to the SBA would be inquired for the banking book. Our next step is to try and construct an internal model that mimics the proposed approach to compute the local parameters for the capital charge computation and interest rate shock scenarios applications.

  0609. Alternatively, in the related literature, Diebold et al. (2006) instead use the value of 0.077. Adding to that, other papers suggest different approaches for computing lambda not related to the aforementioned interpretation(cf. Comisef (2010)) and (cf. Gilli et al. (2012)).

  Figure 1: FRANCE government yield curves

Figure 3 :Figure 4 :

 34 Figure 3: US government yield curves

Figure 5 :Figure 6 :Figure 7 :Figure 8 :

 5678 Figure 5: FR portfolio duration vs SBA CC

Figure 9 :Figure 10 :Figure 12 :

 91012 Figure 9: French GARCH capital charge

Figure 15 :Figure 16 :

 1516 Figure 13: French PCA capital charge

Figure 20 :

 20 Figure 17: French ICA capital charge

Figure 21 :Figure 22 :Figure 23 :

 212223 Figure 21: French DNS capital charge

Figure 24 :Figure 25 :Figure 26 :Figure 27 :Figure 29 :

 2425262729 Figure 24: FR GR duration vs SBA CC

Figure 30 :Figure 31 :Figure 32 :Figure 33 :FRFigure 40 :FRFigure 47 :FRFigure 50 :

 30313233404750 Figure 30: FR-GR GARCH vs SBA CC

FigureFigure 54 :

 54 Figure 53: US capital charge

Figure 60 :

 60 Figure 60: FR-GR-US-TRY CC

Figure 61 :

 61 Figure 61: Encounter days with the SBA capital requirement

Figure 62 :

 62 Figure 62: Color coded encounter dates as months Approaching Basel's method, the results show:• Except for the Turkish market, GARCH method computes a similar capital requirement as the SBA for a minimum of one year time horizon. • Comparing PCA and ICA we can conclude that the ICA is more restrictive for single currency denoted and mixed portfolios. • PCA gives an eight months, 97.5% adequacy parallel to the four months given by the ICA for FR and GR portfolios. We add a PCA requirement of 4 months horizon for the US portfolio facing a one month limit given by the ICA. • In the mixed portfolios PCA sets (at 97.5%) a limit of one year except when the Turkish lira is involved, ICA shows a lower encounter point of 9 months. • The DNS on the 97.5% gives an adequate capital charge between two and four months. • In the mixed portfolios DNS method on the 97.5% requires a time horizon between four and seven months.


Z Hammal

N Eveno

PY A Caplier
email: alice.caplier@lis.inpg.fr

Coulon

Parametric models for facial features segmentation

Keywords: parametric models, deformable model, facial features, segmentation

In this paper, we are dealing with the problem of facial features segmentation (mouth, eyes and eyebrows). A specific parametric model is defined for each deformable feature, each model being able to take into account all the possible deformations. In order to initialize each model, some characteristic points are extracted on each image to be processed (for example, eyes corners, mouth corners and brows corners). In order to fit the model with the contours to be extracted, a gradient flow (of luminance or chrominance) through the estimated contour is maximized because at each point of the searched contour, the gradient (of luminance or chrominance) is normal. The definition of a model associated to each feature offers the possibility to introduce a regularisation constraint. However, the chosen models are flexible enough to produce realistic contours for the mouth, the eyes and the eyebrows. This facial features segmentation is the first step of a set of multi-media applications.

Introduction

Facial features deformations contribute to the communication between humans: for example, lip reading allows to improve the understanding of a noisy vocal message and it is also a support of communication with hard of hearing people.

The aim of our work is the extraction of the contours of permanent facial features (mouth, eyes and eyebrows) with enough accuracy to be able to improve the human-to-human communication through a machine.

Several applications are under consideration:

• The outer contour of lips can be used in a mobile phone application in order to improve the vocal message quality in case of noisy transmission; • This contour is going to be used in a project of phone device for hard of hearing people. This requires speech synthesis from lip shape and motion; • The detected iris contour is going to be used to evaluate the vigilance or interest level of a user by the analysis of his frequency blinking. It will also be used to estimate the gaze direction of a PC user; • All the detected contours (eyes, brows and lips) are used in an automatic system of facial emotion recognition based on video data. All the considered applications require very accurate contours and real time processing.

In this paper, we propose new parametric models well suited for eyes, mouth and brows segmentation. For the initialisation of each chosen model, some characteristic points (eyes and brows corners for example) are extracted. Initial models are then deformed in order to maximize a gradient flow (of luminance and/or chrominance). The originality of our approach is the definition of parametric models able to take into account all the possible deformations of each considered feature. This yields to realistic and accurate segmentation.

Many algorithms have been proposed to solve the problem of lip segmentation. Some methods use lowlevel spatial information such as colour [START_REF] Zhang | Visual Speech Feature Extraction for Improved Speech Recognition[END_REF]. No smoothing constraint or shape constraint is considered so that the extracted contours are often of coarse quality. In [START_REF] Nefian | A couple HMM for Audio-visual Speech Recognition[END_REF], a linear discriminant analysis (LDA) is used to segment lip pixels from skin pixels. This yields to lip contours. Though the discriminant analysis is followed by a smoothing step, the segmentation remains noisy. Snakes [START_REF] Kass | Snakes : Actives Contours Models[END_REF] have been widely used for lip segmentation ([1] [START_REF] Terzopoulos | Analysis and Synthesis of Facial Image Sequences Using Physical and Anatomical Models[END_REF]) because snakes can take into account in a same framework smoothing and elasticity constraints. Snake-based methods yield to interesting results but the main drawback is the tuning of several parameters. Moreover, the quality of the segmentation is dependent of initialization. A priori shape models can be used to obtain smoother contours: the extracted contour belongs to the possible space of the lip shape. For example, Active Shape Models (ASM) have been used [START_REF] Cootes | Active Shape Models: their Training and Application[END_REF]. But this approach requires a very huge learning database in order to be able to take into account all the possible deformations of a speaking mouth. And each image of the learning database has to be calibrated with very high accuracy: face orientation and illumination conditions have to be controlled. One possible solution to avoid learning stage is the use of parametric models [START_REF] Yuille | Feature Extraction from faces using deformable templates[END_REF].

Segmentation of eyes and brows is simpler because such features are less deformable than the mouth. Two kinds of approach exist. First of all a coarse localization of these features based on luminance information is extracted (valley images for example) ( [START_REF] Sobottka | Looking for Faces and Facial Features in Color Images[END_REF], [START_REF] Wang | Facial Feature Extraction and Tracking in Video Sequences[END_REF]). These approaches yield to a very few accurate contours detection for eyes and brows. The second approach introduces models to be related to the searched contours [START_REF] Tian | Dual state Parametric Eye Tracking[END_REF].

In our approach, parametric models are considered. In section 2, chosen parametric models for mouth, eyes and brows are described and justified. In the preprocessing step described in section 3, face illumination variations are removed and iris circular contour is extracted. Section 4 presents the algorithms for the automatic extraction of facial characteristic points used for the initialization of the parametric models. Section 5 describes the deformation model process according to gradient information to make the model coincide with the contours of the processed image. Section 6 gives qualitative and quantitative results in order to evaluated the pertinence of the chosen parametric models and the quality of the segmentation. The accuracy of the extracted contours is compliant with an application of facial emotion recognition.

Model choice

The analysis of face images coming from different databases shows that the models that have been proposed until now for mouth, eyes and brows are too rigid to obtain realistic contours.

Mouth model

Several parametric models have been proposed to model the lip contour. Tian [START_REF] Tian | Robust Lip Tracking by Combining Shape, Color and Motion[END_REF] uses a model made of parabolas. This is very simple but the precision is limited (see Figure 1-a). Others authors propose to model the upper lip contour with 2 parabolas [START_REF] Coianiz | 2D Deformable Models for Visual Speech Analysis[END_REF] or to use quartics [START_REF] Hennecke | Using Deformable Templates to Infer Visual Speech Dynamics[END_REF]. It improves accuracy, but the model is still limited by its rigidity, particularly in the case of asymmetric mouth shape (see Figure 1-a The choice of the right model for lip is a great challenge because lip contour is highly deformable. The use of an a priori model in the segmentation step yields to a smooth contour. But if the chosen model is not suitable, segmentation result will be of bad quality.

We propose a model made of 5 independent curves. Each curve describes a part of the lips boundary. Between Q 2 and Q 4 , the Cupidon's bow is drawn with a broken line and the other parts of the contour are approximated by 4 cubic polynomial curves γ i (see Figure 2). We also consider that each cubic has a null derivative at key points Q 2 , Q 4 or Q 6 . For example, γ 1 has a null derivative at Q 2 .

γ 1 γ 2 γ 4 γ 3 Figure 2: mouth parametric model

Eyes and brows parametric models

The most usual model for eyes is made of parabolas for eyelids and of a circle for iris ( [START_REF] Coianiz | 2D Deformable Models for Visual Speech Analysis[END_REF], [START_REF] Tian | Dual state Parametric Eye Tracking[END_REF]). But the analysis of multiples frames of eyes (on the ORL database [START_REF]The ORL Database of Faces[END_REF]) shows that the contour of the upper eyelid does not always present a vertical symmetry. Figure 3 exhibits an example of non-symmetric eyelid that is not well approximated with a parabola (Figure 3-a) but the contour is well fitted with a Bezier curve (Figure 3-b). On the contrary, lower lip contour presents a vertical symmetry so that a parabola is well suited. Regarding the eyebrows, they are often approximated by two broken lines defined by both corners and a middle point. This is a basic model. We propose the definition of a Bezier curve for each brow.

To summarize, we propose the following parametric model well suited to the multiple possible shapes of eyes and eyebrows (see Figure 4):

• For each eye: a circle for the iris (it could be a semi-circle if the eye is slightly closed); for the lower contour, a parabola defined by 3 points {P 1 , P 2 , P 4 }; for the upper contour, a Bezier curve defined with 3 control points {P 1 , P 2 , P 3 }; in case of closed eye, the considered model is a line defined by P 1 and P 2 . • For each brow: a Bezier curve defined with 3 control points {P 5 , P 6 , P 7 } for the lower contour (only the lower contour is extracted).

Figure 4: parametric model for eye and brow

The parametric model for eye and brow is less flexible than the model for the mouth because eyes and brows are less deformable. In each case, the aim is to propose the simplest well-suited model in order to reduce the complexity. For that reason, no link has been introduced between the relative positions of iris and eyes contours. This kind of link could be considered in a second step of temporal tracking of segmentation.

Pre-processing

Video acquisition conditions

Analyzed sequences are acquired with a Web-cam. Natural lighting conditions (white light) are considered (the aim is to do the acquisition with constraints compliant with the expected applications). The camera is facing the person (minimal admissible size for the face in the image is around 90x90 pixels). In the first frame, the face is supposed to be vertical and the eyes are supposed to be open. In the following frames, the face has to remain close to the vertical axis.

Several algorithms have been proposed for the automatic extraction of face in images (see the survey articles [START_REF] Hjelmäs | Face detection: a survey[END_REF], [START_REF] Yang | Detecting face in images : a survey[END_REF]). Face extraction is beyond the scope of this paper and we use the MPT algorithm [START_REF]Machine Perception Toolbox, face detection algorithm[END_REF] based on the work of Viola and Jones [START_REF] Viola | Robust Real Time Face Detection[END_REF]. This algorithm extracts a square bounding box around the face and gives two points located inside the irises. The facial square dimension is noted H face . With this algorithm, pan and tilt head rotations are possible so long as facial permanent features remain visible. But the MPT algorithm is not very efficient in case of roll head motion: the detection of head that are not close to the vertical axis is not robust.

The case of a face with very long hairs, which are covering eyes and brows, is not considered: the facial features to be segmented are supposed to be visible. Iris contour being the frontier between the dark area of iris and the eye white, it is supposed to be a circle made of points of maximum of luminance gradient. Since the eyes could be slightly closed, the upper part of the iris could be occluded. So for each iris, we are looking for the lower part of the iris circle.

Illumination variations attenuation

With the face bounding box dimension H face , a bounding box (H eye , L eye ) around each eye has been defined with the following relations (see Figure 7 These relations have been learnt on the 400 images of the ORL database [START_REF]The ORL Database of Faces[END_REF]. Each bounding box is centred on the eye point given by the MPT algorithm. These relations are dependent on the algorithm used for face detection. If you change the face detector, a learning step is necessary in order to adapt the dimensions of the eye bounding box with respect to the dimension(s) of the face bounding box. Others face detectors can provide a rectangular bounding box and the limits of this bounding box with respect to the face can be different depending on the method used for the detection.

In each eye bounding box, each semi-circle of iris maximizes the normalized flow of luminance gradient (NFLG):

( ).

p SC

NFLG I p n length SC

∈ = ∇
∑ where I t is the luminance at point p and at time t, is the normal of the boundary at point p and SC is the lower semi-circle. The length of the chosen contour normalizes the NFLG. Several semi-circles scanning the search area of each iris are tested and the semi-circle, which maximizes the NFLG, is selected. Figure 8 shows that the right position of the semicircle SC exhibits a sharp maximum of NFLG in the iris search area. Iris radius is supposed to be known and only the centre position of the searched semi circle is scanning the eye bounding box. The maximization of the NFLG is very fast (compared with a gradient flow slope method for example) and without any parameters tuning. It could be possible to solve the problem of the manual estimation of the iris radius in testing several radius values since iris radius value could be correlated to face dimensions.

( ) n p

Characteristic points extraction

Initial positioning of each model requires the automatic extraction of some characteristic points.

Eyes key points

A tracking process of points with maximum value of luminance gradient is used in order to localize eyes corners.

The centre of eye and brow bounding box coincides with the iris detected centre. This allows an efficient delimitation of search area for eye and brow. Figure 9 gives an illustration of the corners detection method: starting with the points X 1 and X 2 , pixels of maximal gradient of luminance located at two pixels from the vertical of the iris circle limits, a tracking process, to the left direction, of pixels with highest value of luminance gradient yields to the detection of the first corner C 1 . Since the initial point X 1 is located below the corner C 1 , only the three (black) points

1 X • • •     
 located above and on the left of X 1 are tested. The curve between X 1 and C 1 (resp. X 2 and C 2 ) is made of pixels with local maximum of luminance gradient. The tracking stops when the luminance gradient becomes negative since a skin pixel is clearer than an eye corner pixel (see Figure 9).

A similar tracking process to the right direction yields to the detection of the second corner C 2 .

Eyebrows key points

Let S 1 and S 2 be the extremities of each eyebrow.

For each eyebrow, the search area is limited to an area located above the eyes. The abscissas x 1 and x 2 of both brow corners correspond to the left and right zero crossing of the derivative of the quantity (see Brow corners detection is a difficult task especially for the external corner, which is not always well defined on the image to be processed. The proposed method supposes that brows are always visible and that brows are darker than the skin (which is generally the case). Since the results of brow corners detection method depend on the contrast of brows and on noise, initial detected brows corners will be adjusted during the deformation step of the brow model.

Mouth key points

The localization of characteristic points of the mouth is more complex. This detection uses mixed information of luminance and chrominance and a new kind of snake called « jumping snake ».

Skin and Lips Color Analysis

The choice of a color space is crucial because it has a direct influence on the robustness and the accuracy of the segmentation finally achieved. Note that luminance does not contain enough information because of shadows that can occur on face (around the nose, below the inner lip for example). Our goal is to find a color space that enables a good separation of skin and lip pixels. In [START_REF] Zhang | Visual Speech Feature Extraction for Improved Speech Recognition[END_REF] the HSV system is used for the lips detection because hue has a good discriminative power. Even for different talkers, hues of lips and skin are relatively constant and well separated. But hue is generally very noisy because of its "wrap-around" nature (low values of hue lie close to high ones) and its very bad reliability for low saturation pixels.

CIELUV and YCrCb spaces can also be used in face analysis. It been shown that the skin color subspace covers a small area of the (Cr,Cb) or (u,v) planes ( [22][25]). However, the distributions of skin and lip colors often overlap and vary for different speakers. This makes these spaces unsuitable for lip segmentation. In RGB space, skin and lip pixels have quite different components. Figure 12 shows the histograms of each R, G, and B component for skin and lip respectively. For both, red is prevalent. Moreover there is more green than blue in the skin color mixture and for lips both components are almost the same [START_REF] Eveno | A new color transformation for lip segmentation[END_REF]. Skin appears more yellow than lips because the difference between red and green is greater for lips than for skin. In order to distinguish skin pixels from lip pixels, the idea is to exhibit this difference, which can be done by the use of the pseudo-hue proposed in [START_REF] Hulbert | Synthesizing a Color Algorithm from Examples[END_REF] and defined by: ( , ) ( , ) ( , ) ( , )

R x y h x y G x y R x y = +

where R(x,y) and G(x,y) are respectively the red and the green components of the pixel (x,y). Unlike usual hue, the pseudo hue is bijective. It is higher for lips than for skin [START_REF] Eveno | A new color transformation for lip segmentation[END_REF] (cf. Figure 13). Luminance is also a good cue to be taken into account especially when the light comes from above the speaker In this case the top frontier of the upper lip is very well illuminated while the upper lip itself is in the shadow. This difference of illumination is not always totally removed by the preprocessing step.

To combine color and luminance information, we work in the RGB color space and we use the "hybrid edge" ( )

, top R x y introduced in [8]. It is computed as follows: [ ] ( , ) ( , ) ( , ) top N N

R x y h x y I x y = ∇ -

where h N (x,y) and I N (x,y) are respectively the pseudo hue and the luminance of pixel (x,y), nor alized between 0 and 1. m ∇ is the gradient operator. This hybrid edge exhibits much better the top frontier of the mouth than the classic gradients of luminance or pseudo-hue (see [START_REF] Eveno | A new color transformation for lip segmentation[END_REF])

The Jumping Snake algorithm

Active contours, or snakes, have proved their efficiency in many segmentation problems. Since their introduction by Kaas et al. [START_REF] Kass | Snakes : Actives Contours Models[END_REF], many improvements have been proposed in the literature. But none of them has totally removed the two major weak points of the snakes: the choice of parameters and the high dependence on the initial position. The method presented here helps addressing these problems.

To find the upper mouth boundary, we introduce a new kind of active contour that we call "jumping snake". Its convergence is a succession of jumps and growth phases [START_REF] Eveno | Jumping snakes and parametric model for lip segmentation[END_REF]. It is initialized with a seed S 0 that can be located quite far away from the final edge (see Figure 14). The seed is put manually above the mouth and near its vertical symmetry axis. The snake grows from this seed until it reaches a pre-determined number of points. This growth phase is quite similar to the growing snake proposed by Berger and Mohr [START_REF] Berger | Towards Autonomy in Active Contour Models[END_REF], in the sense that the snake is initialized with a single point and is progressively extended to its endpoints. Then, the seed "jumps" to a new position that is closer to the final edge. The process stops when the size of the jump is smaller than one pixel (which requires 5 iterations in average).

During the growth phase, left and right endpoints are added to the snake. They are located at a constant horizontal distance, denoted ∆, from the previous point. Moreover, the search area is restricted to the angular sector inf sup ( , ) θ θ (see Figure 15). The best left and right endpoints, denoted M -(i+1) and M i+1 , are found in this area by maximizing the top R mean flow through the end segments M -(i+1) M -i and M i M i+1 (see Figure 15). These two mean flows can be written as follows: .

i i i i M M top top M M i i i i i i 1 R d n R d n M M M M φ φ + - - + -- + - = = ∫ ∫ - --
where is the vector orthogonal to the segment. The maximizations of

)

π π θ θ = -

). The proposed optimization method being exhaustive, it allows to have an accurate solution very quickly.

When the jumping snake reaches a pre-determined number of points 2N+1, the growth stops and the position of the new seed S 1 is computed. This is the jump phase of the jumping snake algorithm. Let {M -N ,...,M -1 ,S 0 ,M 1 ,...,M N } be the points of the snake and let { }

1 1 N N , , , , , φ φ φ φ … … - -
be the mean flows through the 2N segments. The new seed S 1 has to get closer to high gradient regions, i.e. high mean flow segments. We consider that S 1 is the barycentre of S 0 and the points, which are in the highest gradient regions (the big dots on Figure 14). If {i 1 ,...,i N } are the indices associated to the N highest mean flows, then the vertical position of S 1 can be written as follows:

( )

1 0 1 1 1 2 k k N i k k N S S i k y i y y φ φ = =       = +       ∑ ∑
where y(i k ) is the vertical position of the point M i k .

The horizontal position x S 1 of the seed is kept constant. Then, a new snake grows from this new seed until it reaches the predetermined length and "jumps" again. This growth jump process is repeated until the jumps amplitude becomes smaller than 1 pixel. Typically, 4 or 5 jumps are needed to achieve the jumping snake convergence.

The jumping snake algorithm involves 4 The key points give important cues about the lip shape. They are used as fulcra for the computation of the model. We use six principal key points (see Figure 2): the right and left mouth corners (Q 1 and Q 5 ), the lower central point (Q 6 ) and the three points of the Cupidon's bow (Q 2 , Q 3 and Q 4 ). We also use two secondary points located inside the mouth: Q 7 and Q 8 (see Figure 16). They are used to find the lower central point Q 6 . Moreover, they will be useful in a future work for the inner lip boundary segmentation.

The three upper points are located on the estimated upper lip boundary resulting from the jumping snake algorithm. Q 2 and Q 4 are the highest points on the left and right of the seed. Q 3 is the lowest point of the boundary between Q 2 and Q 4 (see Figure 16).

The points Q 6 , Q 7 and Q 8 are found by analyzing , the 1D gradient of the pseudo hue along the vertical axis passing by Q ( ) y h ∇ 3 (see Figure 16). The pseudo hue is higher for the lips than for skin or teeth and tongue. So, the maximum of below the upper boundary gives the position of Q ( ) y h ∇ The process for model fitting remains the same: each contour is supposed to be made of a set of points with maximum of luminance gradient. The curve to be selected maximizes the NFLG through the contour.

Models fitting 5.1 Eyes and eyebrows

Points (P 1 , P 2 and P 4 ) are detected with enough accuracy so that the lower eyelid parabola has not to be modified again. On the contrary, since the P 3 control point of the upper Bezier curve is initialized at the iris centre, we know that this initial curve has to be adjusted. Point P 3 moves upwards along a vertical axis (points P 1 and P 2 being fixed) until the NFLG through the contour is maximum. For the fitting of the brow model, the same idea is used since brow contour is made of points with local maximum of luminance gradient. The main difference concerns the strategy of points displacement. The three control points P 5 , P 6 and P 7 are moved (along a vertical axis only for P 5 and P 7 , and along all the possible upward directions for P 6 ).

Mouth

The deformation strategy of the cubic curves based mouth model is different because model fitting and mouth corners detection are tightly linked. If a human operator has to find the corners, he implicitly uses the global shape of the mouth. He follows the upper and lower edges, extends them when they are becoming indistinct, and finally put the corner where they intersect. So corners and boundaries are found in a single operation. We propose an algorithm that works the same way.

Basically, a cubic curve is uniquely defined if its four parameters are known. Here, each curve passes by, and has a null derivative on points Q 2 , Q 4 or Q 6 . These considerations bring 2 constraint equations that decrease the number of parameters to be estimated from 4 to 2 for each cubic. So, only two more points of each curve are needed to achieve the fitting. These missing points are chosen in the most reliable parts of the boundary, i.e. near Q 2 , Q 4 or Q 6 . Upper curves missing points have already been found by the jumping snake. On the other hand, only one point (Q 6 ) of the lower boundary is known. To get additional lower points, we make a snake grow from the seed Q 6 . The growth stops after a few points (the white dots in Figure 18). Now that there are enough points for each part of the boundary, it should be possible to compute the curves γ i passing by them and to find the mouth corners where these curves intersect.

In order to make results more robust, we also suppose that the corners (Q 1 , Q 5 ) are known. This consideration brings another constraint equation that decreases the number of parameters to be estimated from 2 to 1 for each curve. Therefore, the least squares minimization is achieved very quickly.

Moreover, the resulting curves are much less sensitive to supplementary chosen points positions. In other words, a given corner position corresponds to a unique and easily computed set of curves. So, the fitting is achieved by finding the corners that give the best curves. Obviously, an exhaustive test over all the pixels of the image would be too long. But, a very simple hypothesis can help reducing the search area to several pixels. We consider that the mouth corners are located in dark areas. So, we look for the minima of luminance for each column between the upper and the lower boundary. It gives the line of minima L mini (see Figure 18). We suppose that the corners are located on this line. A given (right or left) corner corresponds to a unique couple of upper and lower curves (the dashed curves in Figure 18). So, the fitting is achieved by finding the corners that gives the best couple of curves.

To know if a curve fits well to the lip boundary, we use an edge criterion. The model fitting consists in finding the corners, which are going to provide the closest cubic curves to the real contours. Real contours are characterized by a maximal top R flow.

If the upper curves γ 1 and γ 2 fit perfectly to the edge, they are orthogonal to the top R gradient field.

On the other hand, the curves γ 1 and γ 2 have to be orthogonal to the [ ] N h ∇ gradient field. We compute φ top,i and φ low,i , the mean flows through the upper and lower curves respectively: where d n and ds are the vector orthogonal to the segment and the curvilinear abscissa respectively. We consider n possible positions along L mini for each point Q 1 and Q 5 respectively. The best position gives a high φ top,i and a very negative φ low,i . So, on each side we have to maximize φ total , computed as follows:

{ } , , , 1 , ,
k k k total top normalized low normalized k n φ φ φ = - ∈ … with: { } { } { } { } { } { } 1, , , 1, , 1, , min max min

Results

The segmentation method has been firstly tested on a video database acquired at our laboratory. Each video has been acquired with a digital camera and the filmed person was more or less far from the camera. There is no constraint about the lighting conditions (so far as possible as "natural" conditions). The FERET database [START_REF]FERET database[END_REF] (with more than 3500 images) and the Kanade and Cohn database [START_REF] Kanade | [END_REF] have also been used for tests. But for the Kanade and Cohn images, only eyes and brows have been extracted since the data are only luminance data while mouth contour segmentation requires chrominance information.

The choice of a database for testing is a difficult task. In [START_REF] Hjelmäs | Face detection: a survey[END_REF], the MIT and CMU databases are presented. In [START_REF] Yang | Detecting face in images : a survey[END_REF], 8 other face databases are described. Another difficulty is that all these databases have been defined for face recognition and detection. Images are not always adapted to our problem of facial features extraction since some features can be hidden or some faces are too small in the frame in order to proceed to facial features extraction.

Models pertinence and quality segmentation

Figure 19 allows visually evaluating the pertinence and the flexibility of the chosen model for the outer mouth contour. The choice of the right mouth model is a great challenge because mouth is highly deformable. With the proposed model, it is possible to segment very different mouth shapes even in case of a grimace. This is very important with applications such as vocal message understanding improvement in case of noisy transmission. Indeed, the use of visual information in order to increase the intelligibility of a message is efficient only if this information is very accurate (especially, the extracted contours should be very precise). The accuracy of the proposed algorithm is to be related chiefly to the chosen models for mouth, eyes and brows. Indeed, the choice of a model allows putting some regularization constraints on the solution. Figure 20 shows examples of segmentation results for the whole considered facial features. Segmented contours are of good quality even if the subject is wearing spectacles. Spectacles are not a problem since the first extraction concerns the iris semi circle detection and spectacles are in general bigger than the eyes. As far as the brow segmentation is concerned, it is possible that in some cases, the brow contour coincides with the spectacles one. But in such cases, it is not possible to make the difference even visually.

In order to have a quantitative evaluation of the segmentation, a comparison with a manual ground truth has been done. The accuracy is estimated for the key points Q 1 , …, Q 6 of the mouth, for the key points P 1 , P 2 for each eye and for the key points P 5 , P 7 for each brow. For this comparison, 300 frames coming from 8 different sequences have been manually labelled by 5 different persons on the one hand and on the other hand, they have been segmented by using the proposed algorithm. Figure 21 gives a representation of the average error for each point: the radius of each circle is directly proportional to the error rate. Mouth corners are extracted with less accuracy (these points are sometimes very difficult to identify even visually). Concerning the eyes and the brows, the accuracy is lower for the external corners, which are more difficult to characterize than internal corners. The numerical values of accuracy are summarized in Table 1. Errors are relative to the mouth width, eyes width and brows width respectively. For a purpose of comparison, we also give the dispersion obtained during the manual labelling by different persons. Even with a manual labelling, the relative error is non null (cf. Table 1) and this error is higher for the external points of eyes and brows (these points are more difficult to detect even visually).

Q 1 Q 2 Q 3 Q 4 Q 5 Q 6 P 1 P 2 P 5 P 7 P algo 4.5 2.9 2.3 2.4 3.8 4.1 6.1 2.9 7.8 4.9 P han d 1.5 1.1 1.1 1.2 1.6 1.6 2.5 1.7 3.2 1.9 For a more complete quantitative study of the quality of the mouth segmentation, two approaches could be considered: clone lip motion synthesis and speech recognition (both approaches are under study). The accuracy of the segmentation should be related to the considered applications. For example, a good segmentation for facial expression recognition will be the segmentation, which leads to the best classification rates.

Application to emotion recognition

The extracted contours are used in a system of recognition of the six basic universal emotions (joy, surprise, sadness, disgust, anger, fear). From a physiological point of view, facial expressions result from facial features deformations. This approach has been validated by a rate of 60% of good recognition obtained by an experimentation carried out in psychology where 60 judge subjects (30 males and 30 females) had to recognized an expression by only viewing the facial contours.

The classification is based on the study of the temporal evolution of the facial features skeletons. Figure 22 gives examples of emotional skeletons in case of joy and surprise. The considered distances for the fusion process are defined on the same figure .  A mouth opening and an eye wrinkling characterize Joy emotion. A full eye opening and a moving up motion of the brows characterize surprise emotion.

The description of the recognition system using the distances D i in a fusion process based on the belief theory is given in [START_REF] Hammal | Rombaut -Belief Theory Applied to Facial Expressions Classification-3rd International Conference on Advances in Pattern Recognition[END_REF].

About algorithms parameters

The proposed algorithms involve the following parameters:

• For the mouth, 4 parameters inf sup ( , , , N ) θ θ ∆ are used for the jumping snake definition. The values of these parameters are related to the mouth size and to a compromise between accuracy and complexity [START_REF] Eveno | Jumping snakes and parametric model for lip segmentation[END_REF]. The used values for all the presented examples are given in section 4.3.2 • For eyes and brows, the parameter to be manually estimated in the first frame is the iris radius. As far as the processing rate is concerned, the extraction of the contours of mouth, iris and eyes is done at a frame rate of 15 frames per second with a C code implementation on a standard low cost PC and for the processing of QCIF images. The brow segmentation is much slower because the three control points have to be adjusted during the fitting step. A preliminary study done at the laboratory about the implementation on a system on chip of our algorithm for the lips and eyes segmentation shows that real time processing is achievable [START_REF] Mancini | An IIR based 2D adaptive and predictive cache for image processing[END_REF].

We are working on the problem of head pose estimation in order to alleviate the constraint of vertical head. But the pertinence of the proposed parametric models have been shown: the models are enough flexible to describe the considered contours with accuracy.

Conclusion

In this article, parametric models for the purpose of permanent facial features segmentation have been proposed. The extracted contours are enough accurate to be used in an application dedicated to a high level interpretation such as facial expression recognition.

Figure 1

 1 Figure 1: a) model with 2 parabolas; b) model with 3 parabolas; c) model with quartics.

Figure 3 :

 3 Figure 3: a) model with 2 parabolas; b) model with a Bezier curve for the upper eyelid and a parabola for the lower one.

Figure 5 Figure 6

 56 Figure 5: retinal filter

Figure 6 :

 6 Figure 6: left, face with large illumination variations; right, filtered image.

Figure 7 :

 7 Figure 7: Results of face detection and eye bounding boxes construction.

Figure 8 :

 8 Figure 8: Evolution of the NFLG for each tested semi circle in the iris search area: NFLG is maximum when the selected semi circle coincides with the iris in the image.

Figure 9 :

 9 Figure 9: eye corners detection: top, tracking process; bottom, luminance gradient evolution along the X 1 C 1 curve.Points P 1 and P 2 of the model are fitted to both detected corners C 1 and C 2 . Point P 4 of the parabola is aligned with the lowest point of the iris detected semicircle and point P 3 initially coincides with the iris centre point. Figure10presents the result of the initialization of the eye models.

Figure 10 :

 10 Figure 10: initialization of eye models

Figure 11 1

 11 Figure 11): and the

Figure 11 :

 11 Figure 11: a) valley image; b) vertical projection V; c) horizontal projection H.

Figure 12 :

 12 Figure 12: typical R (dots), G (dashed), B (plain) histograms of lip pixels (top) and skin pixels (bottom).

Figure 13 :

 13 Figure 13: pseudo-hue image and histogram for lip in red (or dark) and for skin in green (or bright).

Figure 14 :

 14 Figure 14: Jumping snake initialization and seed jump.1 1

  achieved by a systematic computation over a small set of candidates located in the search area. A method of gradient slope has been rejected since the number of points to be tested is

Figure 15 :

 15 Figure 15: From the seed S 0 , adding left and right endpoints extends the snake. The Rtop mean flows Φ i through each segment have to be maximized.

.

  of the angular sector of possible evolution for each segment of the snake; ∆ the horizontal distance between 2 consecutive points of the snake and N the total number of points of each part of the snake. In this paper, these parameters have been tuned to the For the definition of the angular sector, the condition inf sup θ θ > has to be verified in order to make the jumping snake go towards the mouth. The values of N and ∆ are the outcome of a compromise between accuracy of the contour and convergence speed.

4. 3 . 3

 33 Key points detection.

  requires a face alignment with the vertical so that lips are horizontally aligned.

Figure 16 : 3 ,

 163 Figure 16: The 3 upper points are found on the estimated upper boundary resulting from the jumping snake algorithm (white line). Q 6 , Q 7 and Q 8 are below Q 3 , on extrema of

Figure 17

 17 Figure 17 gives an example of initial models for eye and brow.

Figure 17 :

 17 Figure 17: Initialization of eye and brow contours.

Figure 18

 18 Figure 18 the maximum of k total φ gives the position of the corner along L mini . The dotted curves are the cubic curves associated to the different k tested points along L mini .

  and φ k low are associated to the tested corner number k. φ k top,normalized and φ k low,normalized are their normalized value over the whole tested set. When φ k total is high, the corner position is reliable because the corresponding curves fit well to the lip boundaries. So the boundaries and the corners are found in a single operation. Figure 18 shows the evolution of φ k total for different values of k. The maximum of φ k total gives the position of the corner along L mini .

Figure 19 :

 19 Figure 19: segmented mouth contours in case of a grimace.

Figure 20 :

 20 Figure 20: extracted contours for different persons.

Figure 21 :

 21 Figure 21: Quantitative evaluation of the segmentation accuracy.

Figure 22 :

 22 Figure 22: emotion skeletons for joy (on the left) and surprise (on the right); characteristic distances.

  [1] P.S. Aleksic, J.J Williams, Z. Wu et K. Katsaggelos "Audio-Visual Speech Recognition Using MPEG-4 Compliant Visual Features" EURASIP Journal on Applied Signal Processing, Special Issue on Joint Audio-Visual Speech Processing, September 2002, pp.1213-1227.

  W. Beaudot. The neural information processing in the vertebra retina : a melting pot of ideas for artificial vision. Phd thesis, tirf laboratory, Grenoble, France, 1994.

Table 1 : relative errors (in %) after the automatic extraction (P algo ) and the manual extraction (P hand ) of the key points.

 1