UNIVERSITY OF TRENTO ON THE USE OF SVM FOR ELECTROMAGNETIC SUBSURFACE SENSING. A. Boni, M. Conci, A. Massa, and S. Piffer.

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1 UIVRSITY OF TRTO DIPARTITO DI IGGRIA SCIZA DLL IFORAZIO 3823 Povo Trento (Italy) Via Soarive 4 O TH US OF SV FOR LCTROAGTIC SUBSURFAC SSIG A. Boni. Conci A. assa and S. Piffer January 20 Technical Report # DISI--278

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3 O TH US OF SV FOR LCTROAGTIC SUBSURFAC SSIG A. Boni. Conci A. assa and S. Piffer Departent of Inforation and Counication Technologies University of Trento Via Soarive 4 I Trento Italy. -ail: andrea.assa@ing.unitn.it; Web-page: Abstract In this paper a classification approach for the real-tie identification of occupation areas (instead of the detection of each subsurface object) in sub-surface sensing applications is applied. A suitable SV-based strategy is developed for deterining the probability of occurrence of buried targets and to define a risk ap of the investigation doain. To assess the effectiveness of the proposed approach and to evaluate its robustness selected nuerical results related to a two-diensional geoetry are presented. Introduction In clearing terrains containated or potentially containated by landines and/or unexploded ordnances (UXOs) a quick wide-area surveillance is often required. Such a process is inevitably tie-expensive and it involves coplex acquisition procedures. Consequently high costs should be et. This is one of the ain otivation of the growing research interest in developing unsupervised techniques able to effectively (in ters of tie and resources) repair landine/uxo containated areas. Several solutions have been proposed based on various ethodological approaches (see for instance [] and the references cited therein) which consider different sensor odalities such as ground-sensors or synthetic aperture radars. In such a fraework electroagnetic approaches based on learning-by-exaples (LB) techniques [2][3] have been recently proposed for the on-line (after the training phase perfored once and off-line) detection of subsurface objects. However because of the coplexity of the underlying architecture soe difficulties occur when a larger nuber of unknowns is taken into account. As a consequence LB regression-based approaches turn out to be very effective for the detection of a single (or few) buried object whereas they are not-so-suitable in dealing with the detection of ultiple targets. On the other hand it should be pointed out that the identification of free-areas and the estiation of the concentration of subsurface objects (instead of the localization of each buried scatterer) ight be enough in several situations. Then the goal of a subsurface sensing technique could be oved fro the object detection to the definition of a risk ap. Consequently a classification approach instead of a regression one could be eployed. In this paper such an approach is preliinary investigated and assessed through a nuerical analysis with twodiensional geoetries in noiseless as well as noisy conditions. atheatical Forulation Let us consider a half-space subsurface scenario where the upper region presents the sae characteristics of the vacuu ( ε r = and σ = 0. 0 ) and the lossy subsurface region is described by ε r2 and σ 2. The extension of the investigation doain (where the unknown objects lie) below the surface is D I = { L 2 ( x y) L 2. T transitters ( z -directed electric line sources) located above the air-ground interface radiate an incident field inc = inc ( x y)zˆ. The scattered field scat = scat ( x y)zˆ is collected by a set of R sensors close to the air-ground interface. To define a risk ap let us odel the investigation doain with a two-diensional lattice of. The state χ of the -th cell can be either epty (if any scatterer belongs to the cell) χ = or occupied χ =. Then the proble is recast as to deterine the probability array Q = { q =... defined as Q = Pr{ χ = Γ where q is the ( t) { probability that the -th cell is occupied and Γ = scat xr yr ; r =... R; t =... T. Such a classification proble can be solved by eans of an approach based on a support-vector-achine (SV) starting fro the knowledge of a set of known exaples (i.e. input-output relations {( Γ χ ; =... ) ; n =... called training set). SV-based Classification Approach The proposed SV-based classification approach is forulated as a two-step procedure Step : to deterine a decision function Φˆ that correctly classifies an input pattern ( Γ ) (not-necessarily belonging to the training set); Step 2: to ap the decision function ˆ {( Γ Pr χ = Step : Definition of the Decision Function Φ ) into an a-posteriori probability { Γ

4 At this step the status χ of each cell of the lattice has to be deterined. atheatically such a proble forulates in the definition of a suitable discriinant function Φˆ that separates the two classes χ = and χ =. SV defines a linear decision function corresponding to a hyperplane that axiizes the separating argin between the classes. Such a R ) where { linear data-fitting is carried out in the feature space ℵ ϕ ( Γ ) (different fro the original input space { Γ the original exaples are apped through a non-linear operator ϕ ( ). Consequently the nonlinear SV classifier is defined as ( ( Γ ) ) = w ϕ( Γ ) + b = K Φˆ ϕ () where w and b are the paraeters to be deterined during the training phase. w is a linear cobination of the apped vectors ϕ Γ n w = = n= { α χ ϕ( Γ ) (2) where α 0 n = K = K are the unknown Lagrange ultipliers. oreover fro the Karush-Khun- Tucker conditions at the optiality [4] b turns out to be expressed as follows b = sv being the nuber of patterns In order to deterine sv ( p) ( p) { { ( ) ( ) χ n α ϕ Γ ϕ Γ = = p (3) sv Γ for which α 0 (called support vectors ). = ( ) { α ; n = K ; = K it is necessary the iniization of the following cost function + ( ) 2 w + + n Ω w = + λ ξ λ ξ + ( ) (3) 2 = n= = n= subject to the separability constraints ( ( n ) ) ( n w ϕ Γ ) + b ξ for χ = = K + w ( ) b ( ( n ) ) for ( n ) (4) ϕ Γ + ξ χ = n = K where + ξ are the slack variables ; and indicate the nuber of training exaples for which ( ) ( ) χ = and + χ = respectively; λ + = C = ( ) and λ = C = ( ) [5]; the user-defined hyperparaeter C controls the trade-off between the epirical risk and the odel coplexity to avoid the overfitting. The arising constrained optiization proble (3)-(4) is reforulated in a ore practical dual for axα { Ω Dual ( α ) = ( p) ( p) ( p) [ α Θ( Γ Γ )] n= p= = α χ χ axα (5) 2 α [ 0 ] = = ( ) n subject to = = 0 if n = α χ α λ χ = and α 0 λ + and quadratic function of ( n α ) [ ] otherwise. Since ( α ) Ω is a convex it is solved nuerically by eans of a standard quadratic prograing technique (e.g. the Platt's SO algorith for classification [6]). When the Lagrange ultipliers and b are coputed then Φˆ turns out to be n n n Φˆ ϕ Γ = α χ Θ Γ Γ + { ( ) { ( ) ( ( ) ) = n= sv ( p) ( p) χ α Θ Γ Γ { = n= p= sv () i ( j) i j where Θ Γ Γ = ϕ Γ ϕ Γ is a suitable kernel function. () (6) Dual

5 Step 2: apping of the Decision Function into the A-Posteriori Probability Unlike standard SV classifiers that labels an input pattern according to the following rule [7] χ = sign Φˆ ϕ Γ = K (7) { ( ( )) the proposed approach is aied at defining the a-posteriori probability Pr{ χ = probability is approxiated with a sigoid function Pr { χ = ( Γ ) = + exp { γφˆ ( ϕ( Γ ) ) + δ Γ. Towards this end the a-posteriori = K where γ and δ are estiated according to a fitting process. ore in detail a subset of the input patterns of the training { K ; s = K S set is selected ( s ; = ). (8) Γ χ and the following cost function is defined s S χ + Υ{ γ δ = = = log + s s 2 + exp( γφˆ + δ ) (9) ( s) ( s) χ ( γφ + δ ) exp ˆ log s 2 + exp γφˆ + δ where ( s) ( s) Φˆ = Φˆ ( ϕ ( Γ ). Successively (9) is iniized according to the nuerical procedure proposed in [8]. uerical Results For the experiental validation the following scenario has been considered. The relative perittivity and the conductivity of the hoogeneous subsurface region are ε = 4.0 r2 and σ = [S/] respectively P r (χ= Γ ) (a) (b) Figure. Risk ap for the two-targets scenario: (a) three-diensional and (b) contour level representation The investigation doain is a 2.0λ 2. 0λ region partitioned in a lattice of = 36 square cells. The buried objects odeling UXOs or landines are lossless circular cylinders of diaeter d = λ 6 with a relative perittivity εuxo = 5.0. R = 6 receivers are equally-spaced along an observation line 2.0λ in length and parallel to the air-ground interface d = 0.6λ above the surface. The probing source ( t = ) is located at xt = and y t = 7λ 6. The training is coposed of = 2484 patterns related to two and three-targets configurations. These patterns have been also used during the validation test for defining the a-posteriori fitting odel ( γ = 33 and δ =. 272 ). Concerning the SV structure Gaussian kernel functions were adopted and their paraeters selected according to [9]. Within the nuerical validation the first experient deals with a test set of P = 2484 patterns (related to exaples (2) different fro those of the training phase and concerned with two- and three-scatterers configurations P = 260 and (3) P = 224 ) and noiseless conditions. Figs. (a)-(b) and 2(a)-(b) show the risk aps obtained for two exaples of the test set. The first exaple (Fig. ) refers to a two-targets configuration where the UXOs are located as indicated in Fig. (b).

6 The second exaple (Fig. 2) is related to a three-scatterers configuration. The objects are adjacent and lie at the botto of the investigation doain. As expected when the targets are buried far fro the surface the localization of the dangerous zones is ore difficult. P r (χ= Γ ) (a) (b) Figure 2. Risk ap for the three-targets scenario: (a) three-diensional and (b) contour level representation. The second nuerical experient considers a ore critical scenario where a single target is supposed to be located in the () investigation doain ( P = 296 ). It should be pointed out that such a configuration does not belong to the training set. As an exaple the risk ap for a saple of the test set is shown in Fig. 3. P r (χ= Γ ) (a) (b) Figure 3. Risk ap for the single-target scenario: (a) three-diensional and (b) contour level representation. References [] I Trans. Geosci. Reote Sens. Special Issue on: ew Advances in Subsurface Sensing: Systes odeling and Signal Processing vol. 39 Jun [2] S. Caorsi D. Anguita. Berani A. Boni. Donelli and A. assa A coparative study of and SVbased electroagnetic inverse scattering approaches to on-line detection of buried objects J. Applied Coputat. lectroagnetics Soc. vol. 8 pp [3]. Berani A. Boni S. Caorsi and A. assa An innovative real-tie technique for buried object detection I Trans. Geosci. Reote Sens. vol. 4 pp [4]. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector achines. Cabridge University Press [5] K. orik P. Brockhausen and T. Joachis "Cobining statistical learning with a knowledge-based approach: a case study in intensive care onitoring" Proc. 6th Int. Conf. achine Learning IT Press. 999.

7 [6] J. Platt "Fast training of support vector achines using sequential inial optiization" in Advances in Kernel ethods - Support Vector Learning B. Scholkopf C. J. C. Burges and A. J. Sola (ds.) IT Press 999. [7] K. -R. uller S. ika G. Ratsch K. Tsuda and B. Scholkopf "An introduction to kernel-based learning algoriths" I Trans. eural etworks vol. 2 pp ar [8] J. Platt "Probabilistic outputs for support vector achines and coparison to regularized likelihood ethods" in Advances in Large arging Classifiers A. J. Sola P. Bartlett B. Scholkopf D. Schuurans (ds.) IT Press 999. [9] D. Anguita S. Ridella F. Rivieccio and R. Zunino " Hyperparaeter Design Criteria for Support Vector achines" eurocoputing vol. 55 pp