several complex systems have been recently investigated from the perspective of the ( weighted ) networks that are linking the different elements comprising them .indeed , complex systems are in general made of _ several interacting elements _ and it is rather natural to associate to each element a node and to each interaction a link yielding to a graph .examples include food webs , scientific citations , social networks , communication networks , sexual contacts among individuals , company links in a stock portfolio , the internet and the world wide web .the properties of such graphs have been studied with the aim of catching basic features of the investigated systems . however , the complexity of the system is in general reflected in the associated graph which results in an intricate interweaved and densely connected structure .there is therefore a general need to find methods which are able to single out the key information by filtering such a complex graph into a simpler relevant subgraph .such a filtering is especially essential for correlation - based graphs where , in the absence of any filtering procedure , all links among elements are present .+ in this paper we introduce a new filtering procedure which extracts a representative sub graph with a controlled complexity and maximal information content out of the graph describing the system . to illustrate the method we present a concrete example dealing with 100 stocks belonging to a usa equity portfolio . in the modeling of equity portfoliosa natural starting point is the investigation of cross - correlation among time series of returns of stock pairs .the correlation provides a similarity measure among the behavior of different elements in the system .it was shown by one of the authors that a powerful method to investigate financial systems consists in the extraction of a minimal set of relevant interactions associated with the strongest correlations belonging to the minimum spanning tree ( mst ) .however , the reduction to a minimal skeleton of links is necessarily very drastic in filtering correlation based networks loosing therefore valuable information .the necessity of a less drastic filtering procedure has been already raised in the literature .for example , an extension from trees to more general graphs generated by selecting the most correlated links has been proposed in refs .however , with the method discussed in refs . is highly improbable to obtain a filtered network connecting all elements via some path by retaining a number of links of the same order of the number of elements .+ the method that we present in this paper is based on the key - idea that graphs with different degrees of complexity can be constructed by iteratively linking the most strongly connected nodes under the constraint of generating graphs that can be embedded on a surface of a given genus  .the genus is a topologically invariant property of a surface defined as the largest number of non - isotopic simple closed curves that can be drawn on the surface without separating it , i.e. the number of handles in the surface .we prove that such graphs have the same hierarchical tree associated to the mst but contain a larger amount of information which increases with the genus .we show that , with respect to the mst , the major relative improvement of the information stored in the graph is realized for the planar case when the genus assumes the value .* construction algorithm . * let us first illustrate the method and the associated algorithm to filter significant information out of a given complex system composed by elements where a similarity measure between pairs of elements is defined , e.g. the weight of links in the original network or the correlation coefficient matrix of the system .an ordered list of pair of nodes can be constructed by arranging them in a descending order accordingly with the value of the similarity between element and element .let us first consider the construction algorithm for the mst : _ following the ordered list starting from the couple of elements with larger similarity one adds an edge between element and element if and only if the graph obtained after the edge insertion is still a forest or it is a tree_. a forest is a disconnected graph in which any two elements are connected by at most one path , i.e. a disconnected ensemble of trees . with this procedurethe graph obtained after all links of are considered is the mst .in fact when the last link is included in the graph the forest reduces to a tree .+ in direct analogy with this construction of the mst , we construct graphs by connecting elements with largest similarity under the topological constraint of fixed genus .the construction algorithm for such graphs is : _ following the ordered list starting from the couple of elements with larger similarity one adds an edge between element and element if and only if the resulting graph can still be embedded on a surface of genus after such edge insertion_. this generates _ simple _ , _ undirected _ , _ connected _ graphs embedded on a surface of genus . in the next section we demonstrate that these graphs have the same hierarchical tree of the mst and that they possess relevant additional information associated with the structure of loops and cliques making them natural extensions of the mst .a clique of elements ( -clique ) is a complete subgraph that links all elements .+ a special case is when and the resulting graph is planar , i.e. it can be embedded on the sphere .this graph is the first extension of the mst and we name it planar maximally filtered graph ( pmfg ) .an implementation of the algorithm providing the pmfg written in mathematica is accessible as supplementary information .a basic difference of the pmfg with respect to the mst is the number of links which is in the mst and in the pmfg . on the other hand , in general ,the number of links in a graph g with a fixed genus is at most .therefore , in most practical cases , when and , the relative increase in the number of links that might be included in the graph by increasing its genus is very small .it follows that the pmfg assumes a special status among all the graphs constructed with the introduced algorithm .indeed , it is the simplest and the one providing the most significant additional information with respect to the mst . for this reason we will deserve a special attention to it .it is worth noting that the construction algorithm and the topological constraints on the pmfg force each element to participate to at least a clique of 3 elements . in other words ,the pmfg is a topological triangulation of the sphere. only cliques of 3 and 4 elements are permitted in the pmfg .indeed kuratowski s theorem does not allow cliques with a number of elements larger than 4 in a planar graph .larger cliques can only be present in graphs with genus .the larger the value of the larger is the number of elements of the maximal allowed clique ( specifically  ) . +* hierarchical organization .* we prove the following statement : _ at any step of construction of the mst and graph g of genus if two elements are connected via at least one path in one of the considered graphs then they are connected also in the other one_. to this end we must recall the concept of bridge : a link between two elements is a bridge whenever the elements are disconnected via any path in its absence .it follows from the definition of mst that all links in the mst are bridges . on the other hand , for graphs with a fixed genuswe have the following important property : if a bridge is inserted between two previously unconnected regions of a graph g , characterized by the genus , then the genus of the graph obtained after the insertion is still .this property is straightforwardly proved as a corollary of the miller theorem by noting that the addition of a bridge to a graph leaves unchanged the biconnected components of the graph .the above property implies that if the construction algorithm of g selects a link which is a bridge for the graph at that step of construction , then the link is always added to the graph .+ we now prove the above statement by induction . in the following we indicate as and the graphs constructed by using the similarity measure up to the row of . for the first two steps of construction the statement is true : and graphsare always equal .now suppose the statement is true at the step _m _ of construction , i.e. for and . for the step _m+1 _ only four cases are possible :  \(i ) the new link , connecting the vertices and , is a bridge for the . by the definition of bridgethis implies that the vertices and are not connected via any path in .therefore , by inductive hypothesis , the vertices and are not connected via any path also in and then the new link is a bridge for too . in this caseboth graphs will include the considered link and then the statement is true at the step .\(ii ) the link is a bridge for . by using the same reasoning as in ( i )this implies that the same link must also be a bridge for the due to the inductive hypothesis and both graphs will include the considered link and then the statement is true at the step .  in the remaining two cases we assume the condition that the link between the vertices and is not a bridge for both and is a condition that can be used without loss of generality because if the link is not a bridge for ( or ) then one always concludes that the link is also not a bridge for ( or ) by following the same reasoning of case ( i ) .\(iii ) the link is not a bridge for both and and the genus condition fails . in this casethe link is not included to any of both graphs and the statement is again true at the step .\(iv ) the link under investigation is not a bridge for both and and the genus condition is satisfied . in this case the link and does not .however due to the fact that the link added to is not a bridge the connectivity between pairs of elements in and in rests unchanged in both and and the statement is also verified in this last case .the statement is therefore true and it has an important implication : the fact that the mst is formed only by bridges implies that the mst is always contained in any graph g of genus generated with the construction algorithm presented above and , as a specific case , in the pmfg . moreover , this statement shows even a more important fact : the formation of connected clusters of nodes in coincides with the formation of the same clusters in the . in other words the hierarchical tree associated to graph g coincides with the one of the mst .it is worth noting that the construction algorithm and the associated network properties also hold true in the more general case of weighted networks and non fully - connected networks . in other words the algorithm is general and in the case of a non - connected graph the filtered graph of genus will also be a non - connected graph whereas the equivalent of mst will not be a tree but a forest . +* an illustrative example .* we present here a simple example showing how the pmfg provides additional information with respect to the one contained in the mst and in the associated hierarchical tree .let us consider a simple system composed by 10 elements being characterized by the mst shown in panel a of fig .not all the information about the similarity measure is used to obtain the mst and different similarity matrices can have the same mst and hierarchical tree but end up into different pmfgs .two quite distinct possible pmfgs with the same underlying mst are shown in panel b and c of fig .1 . in this figurethe thicker lines are identifying links belonging to both the mst and the pmfg whereas the thinner lines belong to the pmfg only . by looking just at the mst we observe in both examples of fig .1 b and c two groups , specifically the one formed by vertices ( ) , say cluster , and the other composed by vertices ( ) , say cluster .the two clusters are connected by the link between and .the differences between the two pmfgs can be quantified by a comparison between the link structures of these two planar graphs . in the first planar graph ( panel b of fig .1 ) there are 18 intra - cluster links ( connecting elements to and to ) and 6 inter - cluster links ( connecting elements to ) . in the second planar graph ( panel c of fig .1 ) the number of intra - cluster links is 12 , equal to the number of inter - cluster links .another way to quantify the differences between the two pmfgs is by analyzing the clique structure of the two planar graphs . in the graph of panel b a total of cliques of 4 elements is found , with two of them and being characterized by elements all belonging to the cluster , other two having elements internal to the cluster  and and 3 being composed of inter - cluster elements , and .looking at the cliques of 3 elements we observe 14 intra - cluster cliques and 8 inter - cluster cliques . quite differently , in the graph of panel c we observe just 4 cliques of 4 elements , no one of them being an intra - cluster clique , and just 4 of the 20 cliques of 3 elements are intra - cluster cliques .this analysis shows that the two considered pmfgs refine in a quite different way the information associated with the ( same ) mst .this shows that pmfg provides additional information with respect to the mst and this information is primarily associated with the nature of the observed cliques .* pmfg for 100 us stocks . * in the previous section we have introduced a _ general _ method for constructing graphs of defined genus by using a similarity measure . in the following we constraint ourselves to the case , i.e. to the pmfg and we present an example concerning the filtering of a graph obtained with a correlation based procedure .specifically , we consider pair correlation between daily returns of a set of 100 stocks traded in the new york equity markets in the time period between 1/1995 and 12/1998 . in this casethe measure of similarity is given by the correlation coefficient between stocks .the pmfg for this system is shown in fig .2 . in the figurethe various elements are connected through links with thicker lines indicating links belonging to both the mst and the pmfg . by a comparison between the pmfg and the mst several new details emerge with the most striking difference being that the pmfg allows the existence of loops and cliques as we saw in the above example . in the pmfg we counted cliques of 3 elements out of the possible .the number of cliques of elements is .this number is much smaller than the number of possible cliques of elements present in the fully connected graph which are .the complete lists of cliques with 3 and 4 elements present in the pmfg are accessible as supplementary material .interestingly , these numbers of 3- and 4- elements cliques coincide with the numbers of such cliques attainable when a graph is made by a set of tetrahedra ( 4-cliques ) packed together by sharing a triangular face .the fact that we observe such numbers of cliques can be qualitatively explained .consider elements of a correlation based network , say a , b and c. if a is strongly correlated to b and b is strongly correlated to c then it should also be detected a strong correlation between a and c which makes highly probable the formation of a triangular clique .now , if one of these 3 elements is strongly correlated with a 4th one , say d , then also the other two are likely to have a strong correlation with d generating in this way a 4-clique : a tetrahedron .given the topological constraint of planarity , the next most correlated element can only be connected to maximum 3 of the 4 elements of such tetrahedron .the connection of a new element to three elements of the 4-clique generates another 4-clique which is a new tetrahedron sharing a face with the previous one . by following this reasoning, we expect therefore that the basic structures in the resulting graph are the 4-cliques which during the formation of the pmfg clusterize together locally at similar correlation values and then connect to each other by following the mst as skeleton structure .therefore , if such elements cliques are the ` building blocks ' of the pmfg , then there must be strong relations between their properties and the ones of the system of 100 stocks from which they have been generated .these relations are explored in the following section .+   & + e & 5 & arc & chv & mob & xon & & + b & 4 & bcc & cha & ip & wy & & + f & 6 & axp & bac & jpm & mer & & + t & 8 & csco & intc & msft & sunw & & + h & 2 & bax & bmy & jnj & mrk & & + c & 2 & avp & cl & ko & pg & & + s & 3 & ait & bel & gte & t & & + u & 1 & aep & etr & so & ucm & & +  * financial market properties and 4-cliques structure . *let us first classify each stock accordingly with an economic sector following the classification of the forbes magazine .an analysis on all the 4-cliques in the pmfg reveals a high degree of homogeneity with respect to the economic sectors .indeed , we observe that of the cliques are composed by stocks belonging to the same economic sector ; 22 are composed by 3 stocks belonging to the same sector ; 37 have 2 stocks from the same sector and only 7 have stocks all from different sectors .  in tablei we list the cliques with the largest mean correlation among stocks for each economic sector having at least one clique of four elements .we label the economic sectors as energy ( e ) , basic materials ( b ) , financial ( f ) , technology ( t ) , healthcare ( h ) , consumer non cyclical ( c ) , services ( s ) and utilities ( u ) .the total number of intra - sector 4-cliques ( ) for each sector is given in the second column of the table .it should be noticed that among stocks is different for different sectors .for example the clique with the largest mean correlation is a clique of the energy sector which has . whereas the clique of the sector utilities has the smallest mean correlation with . to better understand the structure of such cliques it is interesting to quantify how much the correlation among the stocks is spread within the clique . in analogy to ref . we compute the quantity inside a clique as the mean value of the _ disparity measure _ ^ 2 ] + e & 8 & & + f & 10 & & + t & 17 & & + b & 11 & & + h & 7 & & + u & 6 & & + s & 13 & & + c & 10 & & + co & 5 & & + cc & 6 & & + tr & 4 & & + cg & 3 & & +  in table iii the connection strength is presented for all the elements belonging to the economic sectors both for 4-cliques and 3-cliques .table iii clearly shows that the connection strength can be quite different across sets of elements .specifically , elements belonging to some economic sectors are strongly connected within themselves whereas others are much less. examples of strong connection are the elements of the energy and financial sectors whereas elements belonging to the conglomerates , consumer cyclical , transportation and capital goods sectors are weakly connected .  in tableiv we list the 7 cliques with all the 4 components belonging to different economic sectors . these 4-cliques provide bridging regions among areas of the pmfg populated by elements belonging to different economic sectors .this interpretation is supported by the fact that in these cliques some of the most connected stocks are present .in fact general electric ( ge ) is present in all the 7 cliques whereas the american international group ( aig ) is present in of them . & + bac ( f ) & bmy ( h ) & ge ( co ) & ko ( c ) & & + bac ( f ) & bmy ( h ) & ge ( co ) & dis ( s ) & & + aig ( f ) & ge ( co ) & nsc(tr ) & wmt ( s ) & & + aig ( f ) & ge ( co ) & nsc(tr ) & vo ( c ) & & + aig ( f ) & ge ( co ) & fdx ( tr ) & vo ( c ) & & + aig ( f ) & bdk ( cc ) & dal ( tr ) & ge ( co ) & & + aig ( f ) & cen ( t ) & gd ( cg ) & ge ( co ) & & +  a joint reading of table iii and table iv shows that economic sectors are not equivalent in the detected pmfg .specifically sectors e , t and b are sectors of elements significantly connected among them but weakly interacting with stocks belonging to different economic sectors .indeed no energy and basic materials stocks and only one technology stock ( cen ) appear in table iv .quite differently the financial sector ( f ) has still elements strongly connected among them but it also participates to all the 7 inter - sector cliques of table iv . in other wordsvarious sets of elements may or may not be clustered among themselves and may or may not be also connected to elements of other sectors . the pmfg is able to extract this information .finally , elements of sectors co and tr are very weakly clustered among them but are often present in cliques of table iv meaning that stocks belonging to these sectors behave like bridges between sectors as f , t , h , s and c. in summary tables iii and iv show how the pmfg is able to quantify the connection strength of elements of the graph via an analysis of the clique structure . an information which is only partially present in the mst and the hierarchical tree .the analyzed correlation structure has a certain degree of statistical uncertainty due to the finite length of time series .the stability of the filtered graphs with respect to such statistical uncertainty has been analyzed by generating surrogated data series using the discrete karhunen - love expansion .the random multivariate gaussian data series are computed starting from a given correlation matrix . for any simulated realizationthe correlation matrix has been calculated .the computed matrices become closer and closer to the reference matrix by increasing the number of records of the simulated time series .we consider the _ empirical _ correlation matrix associated to the system as the reference matrix . for fixed values of the number of records of time series , realizations are simulated and for each of them both the mst and the pmfg have been determined . in fig .3 the percent of non - matching edges in the simulated and in the real data graphs is plotted as a function of the number of records of the simulated time series in a log - log scale .3 shows that the mst is marginally more stable than the pmfg .3 also suggests a power law dependence of the stability of the mst and pmfg with respect to the number of data in the multivariate time series .the significant increase of information gained by the pmfg is therefore fully balancing the marginal decrease of stability for any number of records of the multivariate time series .this is another reason suggesting that the pmfg and the similar graphs characterized by a low value of the genus are the best compromise allowing to consider a graph richer than the mst but characterized by a similar degree of stability with respect to the statistical uncertainty unavoidably associated with graphs modeling complex systems .+ * conclusions * + in summary we have shown that it is possible to determine a family of graphs having the same hierarchical tree associated to the mst but comprising a larger number of links and allowing closed loops . the amount of filtered information with respect to the one present in the mst increases by increasing the genus . a substantial amount of additional filtered information is already obtained in the case for genus which gives the pmfg .a correlation - based investigation of a financial portfolio shows that the method is pretty efficient in filtering relevant information about the connection structure both of the whole system and within each cluster .the example presented is representative of a large class of correlation based clustering investigations .for this example , the stability of mst and pmfg with respect to the statistical uncertainty due to the finite length of time series has been investigated suggesting a power law dependence of the stability with respect to the number of records in time series .the stability of mst turned out to be slightly higher than the one of pmfg .the proposed procedure can be applied to a large number of real and correlation based networks when a filtering of the similarity measure matrix is needed .+ * acknowledgments * + we wish to thank s.t .hyde for fruitful discussions and advices .we acknowledge partial support from research project miur 449/97 `` dinamica di altissima frequenza nei mercati finanziari '' .mt and rnm thank partial funding support from research projects miur - firb rbne01cw3 m , nest - dysonet 12911 eu project .ta and tdm acknowledge partial support from arc discovery projects dp0344004 ( 2003 ) , dp0558183 ( 2005 ) and australian partnership for advanced computing national facilities ( apac ) .onnela , j .-p . , chakraborti , a. , kaski , k. , & kertsz , j. ( 2002 ) eur .j. b * 30 * , 285 - 288 .+ onnela , j .-chakraborti , a. , kaski , k. , kertsz , j. , & kanto , a. ( 2003 ) phys .e * 68 * , 056110 .+ onnela , j .-chakraborti , a. , kaski , k. , & kertsz , j. ( 2003 ) physica a * 324 * , 247 - 252 .barthlemy , m. , barrat , a. , pastor - satorras , r. , & vespignani , a. ( 2004 ) arxiv : cond - mat/0408566 v1 .barrat , a. , barthlemy , m. , pastor - satorras , r. , & vespignani , a. ( 2004 ) proc .usa * 101 * , no .11 , 3747 - 3752 .

Abstract: we introduce a technique to filter out complex data - sets by extracting a subgraph of representative links . such a filtering can be tuned up to any desired level by controlling the genus of the resulting graph . we show that this technique is especially suitable for correlation based graphs giving filtered graphs which preserve the hierarchical organization of the minimum spanning tree but containing a larger amount of information in their internal structure . in particular in the case of planar filtered graphs ( genus equal to 0 ) triangular loops and 4 element cliques are formed . the application of this filtering procedure to 100 stocks in the usa equity markets shows that such loops and cliques have important and significant relations with the market structure and properties .


turbulent combustion in premixed flows is a widely studied topic in both scientific and industrial settings ( see e.g. ) .the interest in the subject is due to an important influence that advection can have on the reaction process : both experimental and theoretical work shows that the propagation speed of the flame can be significantly altered by the fluid flow .specifically , moderately intense levels of turbulence have the tendency to accelerate the flame speed beyond its laminar value .the mechanism and the extent of the flame acceleration depend on the particular regime of burning .the general reason for the enhancement is that the fluid motion distorts the flame front , increasing the reaction area . on the other hand ,if the advection is too strong , it can lead to the flame extinction .the critical strength of advection which leads to quenching depends on the extent of the flame , strength of reaction and diffusion , and properties of the flow .  at this stage, it is unreasonable to expect a complete analytical theory describing the process of combustion in a fluid phase .indeed , detailed modelling of the phenomena involves solving a reaction - diffusion system involving temperature ( or energy ) and concentrations of reactants coupled with compressible navier - stokes equations describing motion of the mixture .therefore , most of the studies in this field which seek analytical conclusions use heuristic reasoning or simplified models , which may approximately describe the system in certain combustion regimes .some of the combustion regimes are relatively well - understood , such as the so - called flamelet regime , where flame thickness is small compared to the fluid velocity scales .the geometric optics approximation where the propagation of the front is ruled by huygens principle is often used as a starting point in the analysis of this regime ( see e.g. ) .our goal here is to study one of the most widely used pde models of combustion , namely the scalar reaction - diffusion equation with passive advection : here is the normalized temperature , , is the fluid velocity , which we assume is incompressible , is thermal diffusivity , and is the typical reaction time . in the absence of fluid velocity eq .( [ eqnt ] ) admits flat propagation front with laminar burning velocity of the order of and characteristic thickness of the order of .the model ( [ eqnt ] ) can be derived from a more complete system under assumptions of constant density and unity lewis number ( the ratio of material and temperature diffusivity ) , as shown , for instance , in  .the equation ( [ eqnt ] ) has a more general applicability than the geometrical optics approximation ; moreover , as we will discuss below , the geometrical optics limit can be obtained from ( [ eqnt ] ) in a certain parameter range .we will consider reaction rates of two types , kpp ( kolmogorov , petrovskii , piskunov ) , and ignition .the kpp type is characterized by the condition that the function is positive and convex on the interval .this reaction type is used often in problems on population dynamics ( see e.g. ) , but is relevant in combustion modelling , for example in some autocatalyctic reactions .a reaction term of ignition type is characterized by the presence of critical ignition temperature , such that the function is identically zero below ignition temperature .this type of reaction term is used widely to model combustion processes ( see e.g. ) , in particular approximating the behavior of arrhenius - type chemical reactions which vanish rapidly as temperature approaches zero .our main goal is to gain insight into the question of how the geometry and the amplitude of the fluid flow influence the combustion process .our study is partly motivated by recent analytical work where rigorous bounds on combustion enhancement and quenching are proved .we test the sharpness of results in , and in addition derive new predictions .we consider two classes of flows .the first is shear flows , a representative of a wider class of flows , called percolating in , which have open streamlines connecting distant regions of the fluid .the second class is cellular flow , where the streamlines are closed and the flow consists of isolated cells . for each class of flows , we study both flame enhancement and quenching .  for flame enhancement study , we consider initial temperature in the form of the laminar front , with in the semi - infinite region behind the front and in the the semi - infinite region ahead of the front .distorted by imposed flow , the flame front propagates as a travelling wave with velocity higher than laminar .the goal of the flame enhancement study is to obtain relations between flame propagation speed and the properties of the flow , especially for the large advection velocities .  in the case of quenching phenomena, we consider initial temperature to be non - zero in a finite region .since quenching can not occur for the kpp - type source term , we use the ignition - type reaction term . as shown by kanel ,there is a critical size of initial hot region below which the flame will be extinguished by diffusion alone , e.g. with no advection , when the temperature drops below the threshold and reaction ceases before the flame establishes a steady travelling wave configuration .when advection is present , the fluid flow stretches the initially hot region so that it can be quenched by diffusion ; hot regions of the size much larger than can be quenched in this manner .our goal has been to understand how the geometry and amplitude of the flow influence the size of the band of the initial hot region that can be quenched .the simulation is set in two space dimensions , in a vertical strip of width with periodic boundary conditions in direction ( fig .[ fgic ] ) . in reactionenhancement studies , the initial temperature was set to in the lower half of the domain and to in the upper half of the domain . in the quenching studiesthe initial temperature was set to in a horizontal band of width in the center of the domain , and to elsewhere .the interfaces between hot and cold fluid were smoothed to match the laminar flame thickness .we consider two types of flows , sinusoidal shear flow with amplitude and wavelength , perpendicular to the initial temperature front(s ) , and cellular flow with amplitude and wavelength , in the quenching simulations , the size of the cell was a fraction of , so that the initial band always contains integer number of cells .  ) , light to the hot fluid ( ) . ]most of the reaction enhancement computations were done using kpp reaction rate in the advection - reaction - diffusion equation ( [ eqnt ] ) , with some of the simulations repeated with ignition type reaction , where represents threshold temperature , below which . in quenching studies we use ignition type reaction ( [ eqignition ] ) with threshold temperature . both reaction rates ( [ eqkpp ] ) and ( [ eqignition ] )were chosen to exactly match , in the absence of advection , laminar burning velocity .the corresponding laminar flame thickness is in both cases of the order of ; for kpp reaction it is several times wider than for ignition .equation ( [ eqnt ] ) with reaction rates ( [ eqkpp ] ) and ( [ eqignition ] ) has been solved using a fourth - order explicit finite difference scheme in space and a third - order adams - bashforth integration in time .the grid size was chosen so as to accurately represent the shear across the reacting region : typically of the order of 12 zones across the flame interface for thin fronts and at least 32 zones per period for thick fronts .the computational domain extended a considerable distance upstream and downstream from the burning front so that boundary effects were negligible . in flame enhancement simulationsthe overall grid was remapped following the propagation of the front , thereby allowing for long integration periods of the order of 1000 reaction times .we found that these long integrations were necessary in order to reproduce correctly the asymptotic behavior of the propagation speed in the case of strong advection .as a measure of the reaction enhancement we use the bulk burning rate the second equality in ( [ br ] ) can be justified by integration by parts .the bulk burning rate coincides with the front velocity in a case where the solution is a travelling wave , but provides a more flexible measure of combustion .physically , can be understood as the total amount of reacted material or the total heat production . in all simulations done for reaction enhancement, approaches an asymptotic value ( for cellular flows one should average in time to arrive at this value ) and we denote this asymptotic value .we remark that in shear and cellular flows equation ( [ eqnt ] ) admits travelling wave - type solutions , called pulsating fronts ( see e.g. ) . a rigorous stability theory for these solutions exists but is not complete ( see for a recent review ) . except when quenching occurs , in our simulations we always observed convergence of the solution to such waves , so the bounds on also provide bounds for the propagation speeds of pulsating fronts .  in the quenching studies, we measure the total amount of burned material per wavelength , which can also be interpreted as the width of the non - perturbed horizontal band with temperature surrounded by the fluid with .this quantity is related to the bulk burning rate per interface , .depending on initial conditions and flow parameters , either approaches an asymptotic value , e.g. , or increases with constant rate , that is ( for cellular flow , in the time - averaged sense ) . in the first casewe say that flame quenches ; the main objective of quenching simulations is to determine under which conditions this happens .we carried out simulations with the sinusoidal shear flow with amplitude flow , and wavelength , , given by eq .( [ eqshear ] ) . in this sectionwe are interested only in reaction enhancement phenomena , and therefore for initial conditions we consider in the semi - infinite domain , and for .we carried out computations for both kpp and ignition type reactions , but did not find significant differences in the qualitative behavior .the numerical results presented in this section are obtained with kpp reaction term ( [ eqkpp ] ) .  of special interestis the dependence of the effective propagation rate on the velocity amplitude , , and wavelength , , which defines the characteristic length scale of the flow ( fig .[ fg1 ] ) . for small amplitudes , ,our results are in agreement with the quadratic law , which goes back to clavin and williams for turbulent flow and has been recently proved rigorously for shear flows in .we did not study this regime in detail since our main interest is in the strong advection case .  ) as a function of the shear flow amplitude for different shear wavelengths . ]  for the amplitudes the results are in good agreement with the linear law where the coefficients and depend on the geometry of the flow . in the situation where the scale of the flow is much larger than the reaction length scale , ,our data agrees with this law has been proposed in for shear flows which vary slowly compared to the typical reaction length and rigorously proved in under similar assumptions . for any fluid flow , the regime closely related to the so - called geometrical optics combustion regime , the limit where reaction time and length scales approach zero . in the framework of the equation ( [ eqnt ] ) this corresponds to the limit while remains constant .indeed , by rescaling equation ( [ eqnt ] ) with a factor in space and time , we find that the bulk burning rate for the original equation ( [ eqnt ] ) is the same as for equation with modified diffusion and reaction ( [ effl ] ) as grows , equation ( [ effl ] ) approaches the geometrical optics limit .  , and .]  quite often , front propagation in the thin front and fast reaction limit is modelled by hamilton - jacobi type equations .one such model is the -equation where the front is defined by a constant level surface of the scalar ( see e.g. ) .the -equation describes propagation of the front according to huygens principle ; that is , the front ( i ) is transported by fluid flow , and ( ii ) propagates normal to itself with the speed .the law can also be understood from the point of view of geometrical optics since it is easily derived from the -equation .recently majda and souganidis pointed out that the -equation does not provide the geometrical optics limit of the reaction diffusion equation ( [ effl ] ) in a precise sense .however the rigorous bounds derived for the true effective equation still give the same prediction for in the case of shear flow .in the situation where becomes comparable to , the coefficient proportionality between velocity amplitude and flame propagation rate is no longer equal to unity .the rigorous lower bound for from takes form , where this bound is in good qualitative agreement with an argument proposed by abel , celani , verni and vulpiani based on the effective diffusivity for the shear flow .it is well known that , if the problem is considered on sufficiently large time and length scales , the effect of the advection of passive diffusive scalar can often be modelled by effective diffusivity .the expression of effective diffusivity in a strong shear flow goes back to taylor , , where and is the typical length scale of the flow . in the presence of reaction , we take since the advection balances with reaction instead of diffusion if .this leads to the qualitative prediction if and if .we obtained good although not perfect agreement with this prediction .this is not surprising given the heuristic derivation of the expression for the effective diffusivity and its possible dependence on more subtle geometric properties of the flow .  .]   and .the individual curves have been rescaled by a factor of in the direction . ]additional understanding of linear dependence can be gained from studying the relationship between the burning enhancement and the structure of the front , in particular level sets of the temperature .assume that in the geometrical optics approximation the front is given by the function ; then for the travelling wave obeying huygens principle and propagating with speed , we have where is profile of the shear , . in the case where is a mean zero flow , this leads to the expression thus , we obtain a well - known fact that the speed of propagation is proportional to the area of the front which in geometrical optics limit coincides with a level set of ( see fig .[ fg2 ] for a picture of level sets in a situation close to geometrical optics ) .it is interesting to test to what extent this relationship remains true in the situations where geometrical optics regime is no longer valid , for example when is comparable to .we found that for large , there is still good agreement between the elongation factor of the level sets of temperature and combustion enhancement ( fig .[ fg2 ] ) .moreover , we found that for large the temperature distribution across the front scales with the shear amplitude ( see fig .[ fg2 ] ) , providing another explanation of the linear dependence of the bulk burning rate on the amplitude of the flow .this scaling behavior can be understood in terms of the approximate self - similarity of equation ( [ eqnt ] ) with respect to the change of variables , , , which gives the only term which now depends on is the one proportional to the second derivative in and it becomes negligible in the limit of large .indeed , equation ( [ nsa ] ) without this term is hypoelliptic , and so addition of the second derivative term does not constitute a singular perturbation .that leads to -independent propagation rate and to linear proportionality for large .we conclude this section by a remark that understanding of the combustion enhancement in a shear flow appears to be useful in some situations where flows with different structure are involved . in particular ,in the reactive boussinesq system the flow consists of vortices moving along with the reaction front .however in the frame moving with the front the effect of such vortical flow is similar to the shear .the prediction for the reaction enhancement in such system based on the results for the shear flows appears to be in agreement with numerically observed behavior .cellular flows have been studied by many authors ( see e.g. ) , since similar fluid motions appear in many important applications ; classical examples are the two - dimensional rolls of the rayleigh - bnard problem and taylor vortices in couette flow .  for cellular flow simulations , we use a velocity field given by eq .( [ eqcell ] ) .the flow is controlled by two parameters , velocity amplitude and wavelength . as in simulations of reaction enhancement by shear flow, we consider initial conditions with in the upper half of the computational domain ( ) , and in the lower half ( ) .most of the results presented in this section were obtained with kpp reaction term ( [ eqkpp ] ) ; the influence of the reaction type on the reaction enhancement is discussed at the end of this section .there are several regimes , which can be classified according to the relations between the characteristic scales present in the problem .there are three characteristic time scales : advection , ; reaction , ; and diffusion , . in this paper, we mostly studied two regimes : and .the first is the regime of strong advection ; in the second regime advection can be very strong as well , but is compensated by large cell size so that the reactive time scale becomes the fastest in the problem .if , or equivalently , , we have diffusive or small cell regime .the remaining regime corresponds to and therefore .hence we have slow advection ; this situation is of less interest to us since under very general conditions , so the effect of the advection on the propagation speed is minor . slow advection geometrical optics ; ( ii )   fast advection geometrical optics ; ( iii )   fast advection with radial burning within cells ; ( iv )   fast advection with uniform burning within cells ; ( v )   limited advection in small cells ; ( vi )   slow advection in small cells . ]   and period ( upper row ) and ( lower row ) .snapshots for the first case were taken with time interval and for the second - with time interval . ]  in the regime our simulations show good agreement with geometrical optics models . in figure [ fg4 ] we show a typical picture of flame in the regime close to geometrical optics .the -equation , ( [ geq ] ) , which is closely related to geometrical optics regime , is invariant under simultaneous rescaling of time and space by the same factor .this suggests that for flames approaching the geometrical optics limit , one should observe this similarity , and indeed we do as shown in fig .the earliest prediction of the front propagation speed in a cellular flow for large within the geometrical optics framework appears to be due to shy , ronney , buckley and yakhot .using heuristic reasoning , they proposed that if one considers front advancing according to huygens principle .the same law is proposed in based on more detailed analysis .to illustrate the origin of this law , let us sketch an argument providing the lower bound for let us look at the propagation of the flame tip along the path abcde on the fig .[ fg3 ] . assuming that at every point of abcde the flame velocity is given by and integrating in time, we obtain a lower bound this lower bound , when doubled , gives a very good fit to our numerical data and is represented by a solid line in fig .[ fg5 ] . as one can see in fig .[ fg4 ] , the tip of the flame follows the path close to abcde , but avoiding corners ; this may account as a factor for the difference in speed compared with the lower bound .our results in the geometrical optics regime agree with results of ; however we stress that our simulation was carried out for the reaction - diffusion equation ( [ eqnt ] ) using the same numerical scheme in all regimes , while uses a different model in geometrical optics limit .we remark that in the possibility of the regime , where and behaves as , was proposed .we could not definitively confirm existence of such a regime due to the closeness of and curves in the range of tested parameters ( fig .[ fg5 ] ) .it should be emphasized that the geometrical optics regime requires not only the thin front assumption , , but also fast reaction in comparison with advection , ; in other words , velocity must be limited by .when this restriction is broken we observe significant decrease in flame propagation speed compared to the geometrical optics prediction ( fig . [ fg5 ] ) .figure [ fg6 ] further illustrates this point , showing that on a logarithmic scale , there is a marked change in the slope of as a function of as increases .when exceeds , we observe power - law , , as proposed by audoly , berestycki and pomeau , and confirmed in in a narrower range of parameters .the measurement of the slope was sufficiently precise to distinguish it from the behavior , a lower bound rigorously proved in .the observed scaling extends to the limit of cell sizes small in comparison with the laminar thickness , .we remark that the laminar front thickness for the kpp reaction is of the order of which is large compared to smallest cell size shown in fig .[ fg6 ] ; limited data available for ( shown in fig .[ fg7 ] ) also confirms the scaling . in the very small cell regime ,the scaling was rigorously proven in using homogenization approach .    ) , dashed line is . ]prediction is based on doubled eq .( [ equlogu ] ) for geometrical optics . ]     and period ( left to right ) ]    we also studied the dependence of on the cell size while is fixed .figure [ fg8 ] shows changes in the structure of the flame with the increase of the cell size from a more diffusive front to a front approaching geometrical optics behavior .the flame propagation speed , normalized by factor , is presented in fig .as increases , we see the transition from the regime , to the geometrical optics , where flame propagation speed is independent of cell size . for the data collapse to a single curve , suggesting the power scaling with , with power changing from for small to for large .the resulting scaling can be summarized as ,   to explain observed the flame propagation speed , let us consider a model based on the effective diffusivity , proposed by audoly , berestycki and pomeau .when velocity is high , , the sharp temperature gradients appear in the narrow boundary layer at cell borders ( fig .[ fg8 ] ) .the thickness of the boundary layer , , is determined by the balance of diffusion and advection , , and is much smaller than , ( this argument is a concise , naive version of the considerations appearing in the derivation of the effective diffusivity in cellular flow , . )  a discrete diffusion equation modelling the original equation ( [ eqnt ] ) has been suggested in : +  \frac{1}{\tau } r(\theta_n).\ ] ] here is the average temperature in the -th cell ; and is effective diffusivity this leads to the propagation rate , and , taking into account eq .( [ eqh ] ) , to the scaling ( [ eqlsm ] ) .we found that the prediction for speed given by ( [ effcell ] ) agrees with numerical simulations only if   for large cell sizes , , the diffusive model ( [ effcell ] ) no longer accounts fully for the flame propagation process .the main objection to the model is that ( [ effcell ] ) assumes temperature is uniform inside the cell , and the reaction term can be estimated at the average cell temperature .however the numerically observed behavior demonstrates at first sharp temperature gradients at the border of the cell , later evolving into flame propagation inside of the cell roughly at the laminar flame speed ( see fig .[ fg8 ] for the structure of the front inside cells ) .indeed , the cellular flow has no efficient mechanism for mixing between the streamlines , and the diffusion time scale in that direction is of the order ( practically not enhanced ) .therefore the combustion process inside the cell takes time of the order of , rather then ( which corresponds to substituting the average temperature into the reaction term ) .  here , we will modify the effective diffusivity model ( [ effcell ] ) to account for slower burning inside large cells .as in the case of small cells , the flame propagation from one cell to another is enhanced because of the high temperature gradient in the boundary layer with width given by expression ( [ eqh ] ) .the heat coming to the cell through the cell boundary is distributed on the scale of ( as opposed to , in the case of small ) .we further notice that due to the fast advection , the temperature is essentially equal along the streamlines inside the cell ( see fig . [ fg8 ] ) , which allows the flame to propagate directly from one boundary layer to another ( fig .[ fg8 ] ) . that allows us to write the discrete diffusion equation similar to ( [ effcell ] ) , butreplacing averaging in the cell by the averaging in the strip of width along the cell border ( -layer ) , +  \frac{1}{\tau } r(\theta_n).\ ] ] here is the average of the temperature in the -layer of the -th cell ; and is effective diffusivity , the temperature in a laminar front varies on the scale and so estimating reaction in a -layer by is justified . equation ( [ efflayer ] ) does not take into account the heat flux from the -layer to the bulk of the cell .however , since this heat flux does not enter the main balance . in other words , the front described by ( [ efflayer ] ) propagates entirely in the -layers , with the rate substituting from eq .( [ eqh ] ) we obtain ,   the time needed to ignite a new cell , e.g. to warm up a layer of the size of the order in that cell , is equal to once the width of a warmed up layer reaches size of the order of , reaction becomes capable of sustaining the temperature. further propagation of the flame corresponds to the basically laminar front movement inside the cell , and takes time .  ) . ]the total bulk burning rate is of the order of times the number of burning cells , which can be estimated as the ratio of the cell burning time , , to the time needed to ignite a cell , .therefore the bulk burning rate is obtained by multiplying the number of burning cells with or essentially by normalizing with a factor of , which agrees with numerically observed scaling ( [ eqllg ] ) .finally , we would like to mention the effect of the reaction rate .similar to the shear flow , we found that the reaction type does not influence asymptotic scaling laws like ( [ eqlsm ] ) or ( [ eqllg ] ) , although there is certainly a difference in the constant factors .however , in cellular flows there is an interesting phenomenon which is present for ignition - type but not kpp reactions .the dependence is always monotone increasing in the kpp case , but it may exhibit a temporary reversal for ignition - type reactions ( fig . [ fgreac ] ) .this effect has been discovered by kagan and sivashinsky and further studied in .we found that this phenomenon is more pronounced when the reaction threshold is closer to unity , in agreement with the arguments of .in this section we address another effect that advection can have on the combustion process quenching .we say that reaction is quenched if the average temperature goes to zero uniformly with time .quenching occurs in the systems with ignition type reaction when , due to diffusion and advection , temperature drops below the ignition threshold everywhere and the integrated reaction rate becomes identically zero .if the size of the region is small enough quenching can be caused by diffusion alone , e.g. without advection , as shown by kanel .kanel considered the one - dimensional reaction - diffusion equation .he found that there exist two critical sizes such that if the initial size of the hot region ( where ) is smaller than reaction quenches , while if the initial size of the hot region is greater than , two fronts form and propagate in opposite directions .two different critical sizes are likely an artifact of the proof ; in our simulations , we always found and will refer to the single critical size in two and three dimensions , the presence of advection may lead to stretching of the initial hot spot , thus making diffusion more efficient at cooling , and consequently , at quenching .  in our numerical simulations we study quenching under the influence of advection , in particular in shear and cellular flows . as in previous sections ,the prescribed flow velocities are defined by eq .( [ eqshear ] ) for shear flow and by eq .( [ eqcell ] ) for cellular flow . for all simulations we used the ignition - type reaction ( [ eqignition ] ) with threshold , since quenching can not occur for the kpp - type source term . as the initial conditions we use a horizontal band of width with temperature above critical ( ) within the band and with temperature below critical ( ) outside the band .  , per wavelength for different values of shear amplitude , measured for and . ]  the typical evolution of the system described above is shown in fig .we found that the temperature distribution in both for shear and cellular flows evolves according to one of the two possible scenarios , depending on the amplitude of the advection velocity . for lower advection velocities ,after an initial transient period , the system develops solution characterized by a wide , steadily growing burned region between two wrinkled fronts propagating in opposite directions .these fronts are exactly as described in the preceding sections with regard to structure , speed , and dependence on the flow properties  and  . for higher advection velocities, the temperature eventually drops below everywhere , after which no burning occurs .we denote by the value of advection velocity which triggers the system between these two scenarios ( further we refer to them as burning and quenching ) , and establish the relation between and the initial conditions and structure of the flow .     in eq .( [ equcrshear ] ) on the wavelength of shear flow .measurements were taken at ]  we measure the critical value of velocity amplitude using the following procedure .for each combination of the initial hot band size and velocity period , we execute a number of simulations for different velocity amplitudes . for each simulation we measured the total amount of burned material per period , defined by eq .( [ eqttotal ] ) , as a function of time ( shown in fig .[ fg11 ] ) .the burning systems ( with higher velocities , where two front are formed ) are characterized in fig .[ fg11 ] by constant , non - zero slopes , corresponding to constant reaction rate .these rates are independent of initial conditions , and are equal to double the burning rate , since there are two fronts .the quenched systems are characterized by evolving to constant .both formation of steady fronts and quenched solutions requires some transition time .the summary of results for a sinusoidal shear flow is given in fig .[ fg12 ] and fig .we found that scales linearly with ( see fig .[ fg12 ] ) , as predicted in , with the coefficient strongly dependent on the wavelength of the advection velocity ( fig .[ fg13 ] ) .shear flow is most effective at quenching in the intermediate range of wavelengths , namely when is of the order of a few reaction lengths .the quenching mechanism for small and for large is different ; one has to distinguish between the ability of the flow to stretch the front over the larger scales and to make the initial hot band uniformly thin .  for small , the rapid spatial variation of the flow velocityis well approximated by effective diffusion .the effective diffusivity for strong shear flow scales as , where .the characteristic length scale for reaction with this renormalized diffusion behaves as and scales linearly with .then gives for small .we remark that in the limit of small the effective diffusivity argument can be justified by a rigorous homogenization procedure .   for a fixed as the size of flat plateau increases . ]  for large , the nature of quenching is related to the appearance of the ( almost ) constant regions in the velocity profile .we observe the behavior for large which can be explained in the following way . in order for quenching to occur, the shear flow should stretch the initial hot region thinner than kanel s critical length ( of the order ) in time less than the reaction time , so that reaction does not have time to compensate cooling by advection .the stretching is least efficient near the tip of the velocity profile . at the tip of a sinusoidal profile ,the difference between flow velocities at two points separated by a distance is .therefore we obtain a sufficient condition for quenching , which leads to   we also examined a degenerate case of shear flows with a plateau in the velocity profile . for such flowsit has been shown in that quenching does not happen as soon as the size of plateau is larger than certain critical size of the order and the size of initial bandwidth exceeds as expected , we found that diverged to infinity as the size of the plateau approached a critical value ( fig .[ fg9 ] ) , in agreement with results of .this phenomenon can be understood in terms of reaction and diffusion alone : in the region where the profile of the velocity is flat , there is no stretching of the initial hot band .if the size of the hot band is roughly larger than kanel s critical size , then reaction can compete with diffusion , there will be no quenching , and eventually propagating fronts will form .   for cellular flows with different periods . ]quenching in cellular flow requires significantly higher advection amplitudes .for relatively small cell sizes where quenching is possible , we find that the critical velocity satisfies ( see figure [ fg14 ] ) .notice that this correlates with the dependence for the speed of advection - enhanced flame .this correlation is not coincidental , and can be explained as follows .the speed up law indicates that the size of the region where reaction happens in a stabilized combustion regime scales as for large if the width of the initial band of hot material is of the order smaller than , advection carries away the energy of the hot material over the larger region faster than reaction is able to compensate the falling temperature .this leads to quenching .we remark that quenching is impossible if the cell size is sufficiently large , , and .the reason is similar to the flat plateau effect in the shear flow .fluid advection does not provide mixing inside cells in the direction perpendicular to the streamlines , and thus if the cell is large enough reaction can sustain itself against diffusion .this result has been proved in .we carried dns calculations of an advected scalar which reacts according to a nonlinear reaction law .we studied combustion enhancement and quenching phenomena in two typical classes of flows , shear and cellular . in a shear flow ,we find linear dependence of the combustion speed on the amplitude of the flow in the strong flow regime .the factor depends on the relationship between the period of the flow and typical reaction length scale is equal to if and tends to zero if the observed behavior is in agreement with recent rigorous and numerical results . in a cellular flowwe studied primarily two regimes characterized by the relationships and between diffusion , reaction and advection time scales .we found that combustion speed in the first regime is close to predictions of models based on geometrical optics limit , in the second regime where large dominates , we found this agrees with the prediction of the effective diffusion model in terms of the power of but has different dependence the cell size we proposed an explanation of the observed behavior with a modified effective diffusion model where enhanced diffusivity is concentrated in the boundary layers .  as opposed to combustion enhancement , quenching may happen if the reaction term is of ignition type and initial temperature is higher than critical in a finite region .if the shear flow velocity profile does not have a plateau of sufficiently large size , or the size of the cells in cellular flow is not too large , then for any initial hot band size there exists such that for quenching takes place . if , two fronts form and propagate with the speed of the developed advection - enhanced front . in the case of shear flow, depends linearly on with a factor .quenching is most efficient for the flows with on the order of a few typical reaction lengths .for the cellular flows , scales as .the results are in good agreement with theoretical arguments .this research is supported in part by the asci flash center at the university of chicago under doe contract b341495 .pc was supported partially by nsf dms-0202531 .ak has been supported by nsf grants dms-0102554 and dms-0129470 and alfred p. sloan fellowship .lr was supported partially by nsf grant dms-0203537 and by an alfred p. sloan research fellowship .p. clavin and f.a .williams , _ effects of molecular - diffusion and of thermal - expansion on the structure and dynamics of premixed flames in turbulent flows of large scale and low intensity , j. fluid mech , * 116 * , 251 ( 1982 ) . _                   a.n .kolmogorov , i.g .petrovskii and n.s .piskunov , _ tude de lquation de la chaleurde matire et son application un problme biologique , bull .univ . mat .* 1 * ( 1937 ) , 1 - 25 .( see pp .105 - 130 for an english transl . ) _                      ya.b .zeldovich , g.i .barenblatt , v.b .librovich and g.m .makhviladze , _ the mathematical theory of combustion and explosions , translated from the russian by donald h. mcneill .consultants bureau [ plenum ] , new york , 1985 . _

Abstract: we perform direct numerical simulations ( dns ) of an advected scalar field which diffuses and reacts according to a nonlinear reaction law . the objective is to study how the bulk burning rate of the reaction is affected by an imposed flow . in particular , we are interested in comparing the numerical results with recently predicted analytical upper and lower bounds . we focus on reaction enhancement and quenching phenomena for two classes of imposed model flows with different geometries : periodic shear flow and cellular flow . we are primarily interested in the fast advection regime . we find that the bulk burning rate in a shear flow satisfies where is the typical flow velocity and is a constant depending on the relationship between the oscillation length scale of the flow and laminar front thickness . for cellular flow , we obtain . we also study flame extinction ( quenching ) for an ignition - type reaction law and compactly supported initial data for the scalar field . we find that in a shear flow the flame of the size can be typically quenched by a flow with amplitude . the constant depends on the geometry of the flow and tends to infinity if the flow profile has a plateau larger than a critical size . in a cellular flow , we find that the advection strength required for quenching is if the cell size is smaller than a critical value .


_ anderson localisation _ is the term used to describe a generic phenomenon , discovered in the late fifties by p. w. anderson , whereby the addition of a certain amount of disorder or randomness in an otherwise deterministic medium causes the waves propagating in the medium to become localised in space . for quantum systems ,the understanding of transport properties requires a thorough study of how the presence of disorder affects the nature of quantum states a question first addressed in an earlier paper of landauer and helland .since that time , the one - dimensional case has been discussed extensively and has led to a better understanding of the physical mechanisms that are responsible for localisation .a remarkable feature of one - dimensional systems is that almost all states become localised as soon as there is _ any _ disorder .this result , first conjectured by mott and twose , was made more rigorous by borland who considered an infinite chain of identical localised potentials separated by regions of zero potential .assuming that the lengths of these regions are independent random variables with the same probability distribution , borland studied the growth rate of the wave function on a semi - infinite chain with prescribed boundary conditions at one end .he then argued , by using a `` matching argument '' , that the positivity of the growth rate implies the exponential localisation of the wave functions .a purely mathematical proof using the properties of transfer matrices was given by matsuda and ishii .these physical arguments can in fact be made completely rigorous .the exponential growth of the solutions of the cauchy problem is a crucial feature of the proof that the spectrum has no absolutely continuous component , and that it is pure - point . if denotes a solution of the cauchy problem ( i.e. a solution of the schrdinger equation on the positive half - line subject to boundary conditions at ) then the quantity is a self - averaging quantity called the _lyapunov exponent _ of the disordered system .a rigorous demonstration of the fact that , under certain hypotheses , also quantifies the exponential decay of the _ eigenfunctions_ and therefore that its reciprocal can serve as a definition of the localisation length appears for the first time in ref .it should be borne in mind , however , that this definition of the localisation length is only useful if certain conditions are fulfilled ; cases where the definition is inappropriate will be considered briefly in [ conclusionsection ] .  in order to illustrate the localisation phenomenon , anderson made use of a model in which the wave function solves a difference equation .the general solution of this `` tight - binding '' model takes the form of a product of random matrices , say , where the are independent and identically - distributed square matrices with a common probability measure . the quantity called the lyapunov exponent of the product of random matrices and , as we shall soon see , is effectively the same as .anderson localisation has been a powerful motivation for the study of products of random matrices . for this reason , the search for precise conditions on the measure that would guarantee the existence and the positivity of the lyapunov exponent has been of particular interest .the main result in this respect is due to furstenberg a result published the same year as borland s paper ; see also oseledec whose work can be considered as an extension to dynamical systems of the work of furstenberg and kesten .further developments of these results , and their application to the study of localisation , are described in the works of ishii , bougerol and lacroix  , carmona and lacroix , lifshits _ , luck , pastur and figotin , crisanti _ et al ._ and the references therein .an important milestone in the development of the theory of localisation was the discovery of the relationship between the lyapunov exponent and the integrated density of states .it turns out that the characteristic function viewed as a complex - valued function on , is _ analytic _ in the upper half of the complex plane .interestingly , this analyticity property of the characteristic function was in fact exploited much earlier by dyson in his famous paper on the dynamics of a disordered chain  .  products of random matrices appear naturally in several other problems related to the physics of disordered systems see the monographs and is still the subject of active research .the concept of lyapunov exponent is a very useful tool for analysing a large class of systems with quenched disorder , such as magnetic systems .a well - known prototype is the ising model in a random magnetic field , where the calculation of the free enegy reduces to analysing an infinite product of matrices ; in this context , the lyapunov exponent is proportional to the free energy per spin .the group from which the matrices in the product ( [ product ] ) are drawn varies not only with the physical context but also with the choice of vector basis . in the analysis of the schrdinger equation in a random potential ,a standard choice is to consider the vector formed by the wave function and its derivative .as we shall see later on , the evolution of this vector is governed by matrices belonging to the group .this is the formulation chosen for example in refs . and in the present article .if , instead , one chooses to focus on the scattering aspects of the problem then the wave function , in a region where , is a combination of incoming and outgoing waves . in this setting , it is natural to consider transfer matrices connecting pairs of complex amplitudes .current conservation then implies that such transfer matrices belong to the group ; see the appendix of ref . for further details , and ref . for a pedagogical presentation and a review of the literature .interestingly , this scattering formulation provides a natural way of generalising the one - dimensional model which is associated with products of matrices to a quasi - one - dimensional model where the matrices in the product are , where is the number of conducting channels .the localisation problem then involves not just one but rather ( counting multiplicity ) lyapunov exponents . since the pioneering work of dorokhov ,whose results were later rediscovered independently by mello , pereyra and kumar  , this topic has attracted a lot of attention owing to its relevance in the description of weakly disordered metallic wires ; see the review  .these early works relied on the so - called _ isotropy assumption _ ,namely that each elementary slice of disordered metal redistributes the current uniformly amongst the conducting channels .this assumption produces a set of lyapunov exponents with the behaviour  ] where ; in the limit of interest here we have and .thus we expect that the contribution to the integrals of the interval  ] , from which we get re - introducing , we indeed get the expected behaviour for .the limit and keeping fixed corresponds to a case where converges to a gaussian white noise of strength a model that has been solved in ref . ( see also ) .therefore we expect that the characteristic function ( [ eq : integralrepresentation ] ) , i.e. eq . ( [ eq : omegageneralcase ] ) , converges in this limit towards the expression close this short review , we point out the limitations of the lyapunov exponent as a means of characterising the localised nature of the spectrum in a disordered system .a first observation is that the definition ( [ gamma ] ) relies on the assumption of a linear increase of , i.e. on the application of the central limit theorem .it is however possible to consider models of disorder such that scales as , leading to the phenomenon of sub - localisation ( for ) or super - localisation ( for ) ; this has been studied in , for example , refs . ( see also references therein ) .however , this limitation is easily overcome by extending the notion of the lyapunov exponent so that it characterises the typical length over which eigenstates are localised .another , more serious , limitation of the usefulness of the lyapunov exponent pertains to the possible existence of extended states in one dimension .this might occur when the disordered potential is correlated as happens for instance in the dimer model .it might occur also for symmetry reasons , as is the case for the supersymmetric hamiltonians studied in refs . and in section [ susysection ] ; there , the lyapunov exponent decays to zero in the limit of zero energy . in this respect , it must first be pointed out that the vanishing of the lyapunov exponent is a necessary but not a sufficient condition for delocalisation ; sometimes it only signals sub - localisation . secondly , several works on disordered supersymmetric quantum mechanics have shown that the lyapunov exponent , when it vanishes at the bottom of the spectrum does not provide the relevant length scale characterising the localisation of eigenstates at low non - zero energies .as mentioned already in section [ susysection ] , when , the lyapunov exponent vanishes like as  .this suggests that low - energy eigenstates are localised on a scale commensurate with .however , the study of the ordered statistics problem , as well as the analysis of other physical quantities such as the averaged green s function at non coinciding points  or the boundary - sensitive averaged density of states , show that eigenstates are localised on a much larger scale , namely .the inability of the lyapunov exponent to capture the localisation property for energies where may be related to the fact that it is defined , via eq .( [ gamma ] ) , in terms of a solution of the cauchy problem that vanishes at just _one _ boundary .this definition may indeed fail to describe the interesting properties of the real eigenstates which are solutions of the schrdinger equation vanishing at the _ two _ boundaries and as these eigenstates become less localised .this work was supported by _ triangle de la physique _ under grant 2011 - 016t-1ddisquarm .let us look for a solution of the differential equation ( [ differentialequation ] ) of the form then y ' + \frac{p\,q}{z ( z-1)(z+1)}\ , y = 0\ , .\label{reducedequation}\ ] ] the solution we require has various representations in terms of special functions .heun s differential equation is  y ' + \frac{\alpha\,\beta z -q}{z ( z-1)(z - a)}\ , y = 0 \label{heunequation}\ ] ] where the parameters satisfy the relation heun s function is the particular solution analytic inside the unit disk : where the satisfy the recurrence relation \,y_{n+1 }  \\ + ( n+\alpha)(n+\beta ) \,y_n = 0 \label{heunrecurrencerelation}\end{aligned}\ ] ] with the starting values comparing equation ( [ reducedequation ] ) with heun s equation ( [ heunequation ] ) , we find the particular solution the recurrence relation for the coefficients of this solution is with the starting values it follows in particular that is strictly positive for . hence , for , where is a normalisation constant .the property ( [ involutionproperty ] ) provides the obvious formula in the interval .set then , by comparing the taylor series , we find that ( 1+z)\ ,  { _ 2}f_1 \left(a+1,b+1;c+1;z^2 \right ) \right \}\ , .  \ ] ] the contiguity relation ( c+1 ) \,{_2}f_1(a+1,b+1;c+1;t ) \nonumber\\  - c(c+1 ) \,{_2}f_1(a , b;c;t ) = 0  \ ] ] leads to the simpler expression the differentiation formula then yields so we obtain the `` compact '' representation where    formula ( 6 ) in , 3.13 , expresses the hypergeometric function in terms of the associated legendre function : ^q\ , \left ( \frac{1+\e^{-\eta}}{1-\e^{-\eta } } \right ) ^{{\frac{p}{4 } - \frac{1}{2}}}\,f_{p , q } \left ( \e^{-\eta } \right ) \ , .\label{legendreformula}\end{aligned}\ ] ]  equation ( 8) in , 7.621 , gives a useful integral representation of the legendre function in terms of a whittaker function : then , by equation ( [ legendreformula ] ) , we obtain ^q\,\frac{1-z}{1+z}\,f_{p , q}'(z )  = \frac{1}{4 \sqrt{\pi } ( { \frac{p}{4 } - \frac{1}{2}}+q ) } \frac{\gamma  ( q+1/2)}{\gamma(2q ) }  \nonumber\\ \times \int_0^\infty \e^{-\frac{x}{4 } \left ( z + 1/z \right ) } m_{{\frac{p}{4 } - \frac{1}{2}},q-\frac{1}{2 } } ( x ) \left [ \frac{q}{xz } - \left ( \frac{1-z}{2 z } \right ) ^2 \right ] \,\d x \ , . \label{intermediateequation}\end{aligned}\ ] ] now ,  = m_{{\frac{p}{4 } - \frac{1}{2}},q-\frac{1}{2 } } ( x ) \left [ \left ( \frac{q}{x } + \frac{1}{2 } \right ) \frac{1}{z } - \frac{1+z^2}{4 z^2 } \right ]\\ = \frac{\frac{p}{4}+q-\frac{1}{2}}{2 q \sqrt{x } } \frac{1}{z } m_{\frac{p}{4},q}(x ) + \frac{1}{z } m_{\frac{p}{4 } - \frac{1}{2},q-\frac{1}{2}}'(x ) - \frac{1+z^2}{4 z^2 } m_{\frac{p}{4 } - \frac{1}{2},q-\frac{1}{2}}(x)\end{aligned}\ ] ] where we have used formulae ( 13.4.28 ) and ( 13.4.32 ) from to obtain the last equality .the proof of theorem [ whittakertheorem ] follows after reporting this in equation ( [ intermediateequation ] ) and using integration by parts for the term involving the derivative of the whittaker function ._ remark . _formula ( 5 ) in , 2.8 , expresses in simple terms : \ , . \ ] ] it then follows from equation ( [ legendreformula ] ) and from the recurrence relation for the associated legendre function ( see formula ( 18 ) in , 3.8 ) that , for , may be expressed in terms of elementary functions .for example , we have \ ] ] and - \left [  z^2- \left ( \frac{p}{2 } - 1 \right ) z + 1 \right ] \right \}\,.\end{aligned}\ ] ] formulae for and are easily deduced .define using theorem [ whittakertheorem ] and changing the order of integration , we find next , formula ( 7 ) in , 6.9 , says hence the integral on the right - hand side is of the same form as that in , 7.625 , formula ( 1) albeit outside the range indicated since in our case and .nevertheless , direct numerical verification indicates that the formula does remain valid and so the normalisation condition yields and so we deduce the formula : for .then , using formula ( [ riccatiformula ] ) for the lyapunov exponent , together with the identity ( [ negativeenergyidentity ] ) and the fact that for , we eventually find the analytic continuation from negative to positive energy consists of replacing by .this completes the proof of corollary [ whittakercorollary ] .100               a. comtet and c. texier , one - dimensional disordered supersymmetric quantum mechanics : a brief survey , in _ supersymmetry and integrable models _ , edited by h. aratyn , t. d. imbo , w .- y .keung and u. sukhatme , lecture notes in physics , vol .313328 , springer , 1998 .( available as cond - mat/9707313 ) .

Abstract: the concept of lyapunov exponent has long occupied a central place in the theory of anderson localisation ; its interest in this particular context is that it provides a reasonable measure of the localisation length . the lyapunov exponent also features prominently in the theory of products of random matrices pioneered by furstenberg . after a brief historical survey , we describe some recent work that exploits the close connections between these topics . we review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case . finally , we point out some limitations of the lyapunov exponent as a means of studying localisation properties .   = 1


video systems have recently gained a lot of attention .many new 3d video systems have been developed . among themsuper multiview television and free viewpoint television can be examples of such novel 3d systems . in the free viewpoint television a user is able to freely choose a position of a virtual camera .the requested view of a scene is generated from dynamic 3d representation of the scene .the most commonly used 3d representation is a multivideo and depth ( mvd ) composed of multiple videos acquired by the set of cameras and accompanied depth maps for each of the views .based on transmitted videos and depth data any view can be easily generated by employing depth - image - base rendering ( dibr ) .recently 3d extension of such standards as avc and hevc that allows for efficient transmission of dynamic 3d scene representation in mvd format has been finalized .depth information in such systems can be acquired either directly by depth cameras , or indirectly by algorithmic depth estimation from recorded videos .commonly depth information is obtained by the conversion from disparity information .although in computer vision , disparity is often treated as synonymous with depth ( distance ) , essentially those terms are the inverse of each other .    disparity is a displacement vector between corresponding fragments ( pixels , blocks ) of two images of the same scene taken from different viewpoints .those two corresponding fragments represent the same fragment of an observed scene but seen from two different viewpoints .stereo correspondence search is an active research topic in computer vision , and one of the basic method of obtaining disparity information .there are many stereo disparity estimation methods known .comprehensive study of stereo disparity estimation methods can be found in , and on the middlebury webpage containing up - to - date benchmark of stereo disparity estimation methods . in the scope of development of multiview systemsstereo correspondence search was extended to multiview correspondence search .  for the sake of simplicity and accuracy, many algorithms assume that images are taken by a rectified set of cameras . consecutively, corresponding fragments of a given image can be found on the same horizontal line in the remaining images .some algorithms use three views ( left , central and right ) as inputs and produce disparity map or depth map for the central view ( fig .[ fig:3viewde ] ) .often when it is not important which of left or right view is referred to , a name `` side view '' is used instead .  during disparity estimation , for a given fragment of the central view , the algorithm searches for the corresponding fragment in the side views that represent the same fragment / portion of the scene .the correspondence search is done on the basis of similarity metric which expresses how probable it is that a certain fragment of one image is the corresponding fragment of the second image .although the metric used is often called similarity , it actually expresses dissimilarity between fragments .there are many similarity metrics known from literature : sum of absolute difference ( sad ) sum of squared difference ( ssd ) , rank , census , cross correlation and other .the correspondence search is often defined as an optimization problem in which for every fragment of the central image the best ( the most similar ) fragment of the side views is selected .this optimization problem maybe expressed in terms of energy function using markov random field ( mrf ) and optimized via one of the optimization algorithms such as belief propagation , dynamic programming , or graph cut .since input videos are captured by multiple cameras with different positions , some parts of the observed scene can be occluded , and thus not visible , within some of the views .disparity estimation for those fragments of a scene is challenging and requires special care .if the the algorithm do not properly taking into account , possible occlusions within the scene , estimated disparity can be wrong .estimated disparity can indicated not truly corresponding fragments . in this paper a novel approach to occlusion handling designed to work in three - view disparity estimation algorithms is proposed .  given three images , center , left and right of the same size , we search for such a displacement for every pixel p of center view ( at coordinates ) that minimize cost function expressing a similarity between pixel p ( or small fragment around the pixel p like block ) and a corespondent pixel p ( small fragment around pixel p ) displaced by in side views ( at positions in left and in right view ) .such displacement is then a disparity of a given pixel p of a center view .  in disparity estimation based on / from three views ( see fig . [fig : occ ] ) , a given point of the scene visible from center view can be visible from both of the side views ( point a ) , or only from one of the side views ( left or right , point b ) , or from neither of them ( point c ) .  if the given fragment of the scene visible from center view is not visible from one or both of the side views , we say that a given fragment of the scene is occluded in side view ( is not visible from that particular side view ) .the simplest method for detecting occluded fragments is cross - checking .cross - checking tests the consistency of estimated disparity value for pixels from center view with those estimated for pixels in left and right views .if the disparity value estimated in each view is different for a correspondent triple of pixels from center , left and right views given pixels are assumed to be occluded .next , the disparity value for occluded pixels are extrapolated from neighboring pixels that are not occluded . in order to perform cross - checking , disparity maps for all of the three viewsare required .estimation of three disparity maps is not always possible . even if the estimation of three disparity maps instead of one is possible it is resource and time consuming .occlusion handling is performed by adding / putting additional constraints , such as ordering constraint or uniqueness constraint to objective function of optimization procedures like graph cut ( gc ) , dynamic programming ( dp ) or belief propagation ( bp ) used to estimate the disparity map .the ordering constraint imposes the same order of corresponding pixels in all views .if a pixel a is on the left of pixel b in the center view , in the side view pixel a that is a corresponding pixel of pixel a must be as well on the left of pixel b , a corresponding pixel of pixel b. in real scenes the ordering constraint can be violated in the case of big perspective change or in case of thin objects . in such cases ordering constraint can introduce errors in estimated disparity maps .the uniqueness constraint imposes the one - to - one correspondence between pixel in center and side views .if a given pixel a of the central view is assigned to a corresponding pixel b in the side view , no other pixel of central view can be assigned to a correspondence with pixel b in side view .this way unique pixel to pixel correspondence is forced across all of the views .there are many disparity estimation algorithms known , that handle occlusion in efficient way .the main drawback of all of those algorithms are additional constraints ( terms ) imposed in optimization procedures with increased complexity and thus execution time of the disparity estimation .another approach to occlusion handling is to change cost term ( eq . [ eqn : cost ] ) composed of similarity metric in optimization algorithms .as we search for corresponding fragment of a central view in both side views simultaneously , there are many ways of defining a function .  commonly it is the sum of similarity metrics between a fragment in the center view and corresponding fragments in left and right views .  because of the occlusions tanimoto proposed to pick just the most similar fragment from either left or right view .the intuition is that the occluded fragment of the images will lead to less similar fragment , thus the minimum of similarity metrics from left and right view is used .  in this paper we propose yet another way to define cost function which takes into account an occlusion possible within the scene .as it was said before , a given fragment of a scene visible from center view can be occluded in one or both side views ( left or / and right ) ( fig .[ fig : occ ] ) .in such a case , searching for a correspondence of a given pixel of center view in this particular side view ( left or right ) is pointless , as the given fragment of the scene is not visible from that particular side view .considering the correspondence with an occluded fragment of an image could cause errors in estimated disparity .therefore the correspondence search should be performed only in side views in which a considered fragment of a center view is not occluded .the cost function should be constructed in such a way that it considers only similarity metrics from not occluded views .if a given fragment is visible in both views , then the cost function should be an average of both similarity metrics , in order to reduce the influence of noise , which is present in all views .we propose to define the cost function in a way that it considers only similarity metrics of fragments from a not occluded view ( either left or right ) ( eq .[ eqn : novelcost ] ) where , expresses whether a given pixel of a center view is not occluded in left and right views respectively . depending on the existence of occlusion in the views , the sum in the denominator of eq .[ eqn : novelcost ] can be 2 if a pixel in not occluded in both views , 1 if it is occluded in one of the side views ( either left or right ) , and 0 if it is occluded in both side views .if a given pixel is occluded in both side views the equation [ eqn : novelcost ] loses its meaning , thus in such a case constant penalty value is used as a cost value .  but why a given fragment ( object a ) of a scene is not visible in a side view ? because in a side view that fragment is occluded by some other part of the scene ( object b ) .object b blocks light rays from object a , so in side view closer object b is visible instead the farther object a.  consider the example on fig .[ fig : occ_detection ] where two points a and b are observed by two cameras ( left and center ) .point b is closer to the cameras and point a is farther .point b is visible in both views ( left and center ) at pixel position and respectively .but due to the occlusion , point a is visible only in center view at pixel position .if there would be no point b , point a should / would be visible in left view at pixel position .the disparity of point b in left view is the difference of the pixel position and and disparity of point a in left view would be ( if the point was / would be visible ) the difference of pixel position and .    the distance to the camera is reciprocal to disparity .so a fragment of an image representing a closer object ( point b ) has bigger disparity than the fragment representing farther object ( point a ) .  for a given pixel of center view at coordinates and considered displacement , corresponding pixel in left view should be at coordinates .so , if we want to check whether a fragment a of a scene is occluded in left view we have to check the disparity ( distance ) assigned to the considered corresponding pixel in left view .if a disparity assigned already to considered corresponding pixel is bigger than the considered displacement t then probably a pixel is not a fragment of the same object a but rather some other closer object b that occludes object a in the left view .based on such a consideration we can create a function assessing whether for a pixel at coordinates and displacement , corresponding pixel is / can be / will be occluded or not in left and right views .     equal 1 means that the corresponding pixel in side view at a given displacement is probably not occluded .the proposed idea is general - it does not impose any particular source of disparity maps for left and for right view .but in general , disparity maps for left and right views are unknown before estimating the disparity for central view .commonly , disparity maps are estimated iteratively with the use of such algorithms like belief propagation or graph cut . in such algorithms , at each iteration of the estimation , algorithm maintains up - to - date / best already estimated disparity map for center view .this disparity map is further refined in the next iteration of the algorithm .  for our occlusion detectionwe propose to use disparity maps of side views created based on the disparity map of center view through depth - image - based rendering ( dibr ) . after each iteration of a disparity estimation algorithm , we create disparity maps of side views ( and ) from the best already estimated disparity map of a center view .this way if the estimation algorithm used assigned already some disparity to some pixel , then pixel can not have such a disparity that the corresponding pixel ( fig .[ fig : occ_detection ] ) is at the same position as corresponding pixel of pixel . in other words fragment b of a scene represented by pixel in center viewshould occlude a fragment a of a scene ( represented by pixel in center view ) seen from left view ..positions of views used for evaluation of quality of estimated disparity maps . [ cols="<,^,^,^",options="header " , ]we have presented a novel approach to occlusion handling in disparity estimation , based on a modification of similarity cost function . proposedapproach has been tested in the three - view disparity estimation scenario . for occlusion detection synthesized disparity maps of left and right views have been used .for well - known multiview video test sequences , the experimental results show that the proposed approach provides virtual view quality improvement of 1.25 db of luminance psnr over the state - of - the - art technique implemented in mpeg depth estimation reference software ( ders ). moreover , direct quality evaluation of estimated disparity reveals that proposed the approach reduces a number of bad pixels by 1.26 p.p .research project was supported by national science centre , poland , according to the decision dec-2012/05/n / st7/1279 .s. y. kim , j. h. cho , and a. oschan , _3d video generation and service based on a tof depth sensor in mpeg-4 multimedia framework _ , ieee trans . on consumer electronics , vol .1730 - 1738 , august 2010 .m. domaski , a. dziembowski , a. kuehn , m. kurc , a. uczak , d. mieloch , j. siast , o. stankiewicz and k. wegner , _ experiments on acquisition and processing of video for free - viewpoint television _ , 3dtv conference 2014 , budapest , hungary , july 2014 .     j. stankowski , k. klimaszewski , o. stankiewicz , k. wegner , m. domaski , _ preprocessing methods used for poznan 3d / ftv test sequences _ ,iso / iec jtc1/sc29/wg11 mpeg 2010/m17174 , doc .m17174 , kyoto , japan , january 2010 .j. stankowski , k. klimaszewski , book : `` image processing and communications challenges 2 '' , chapter : `` application of epipolar rectification algorithm in 3d television '' , advances in intelligent and soft computing : vol .84 , springer - verlag , berlin , 2010 , pp .345 - 352 , isbn : 978 - 3 - 642 - 16294 - 7 .g. egnal and r. wildes , _ detecting binocular halfocclusions : empirical comparisons of five approaches _ , ieee trans . on pattern analysis and machine intelligence , vol .24 no . 8 , pp . 1127 - 1133 , august 2002 .t. liu , p. zhang , and l. luo , _ dense stereo correspondence with contrast context histogram , segmentation - based two - pass aggregation and occlusion handling _ , lecture notes in computer science , vol .449 - 461 , january 2009 .r. ben - ari and n. sochen , _ stereo matching with mumford - shah regularization and occlusion handling _ , ieee trans .on pattern analysis and machine intelligence , vol .2071 - 2084 , november 2010 .d. scharstein and r.  szeliski , _ high - accuracy stereo depth maps using structured light _ , in ieee computer society conference on computer vision andpattern recognition ( cvpr 2003 ) , volume 1 , pages 195 - 202 , madison , wi , june 2003 .g. tech , k. wegner , y. chen , s. yea , _3d - hevc draft text 6 _ joint collaborative team on 3d video coding extension development of itu - t sg 16 wp 3 and iso / iec jtc 1/sc 29/wg 11 doc .jct3v - j1001 , 10th meeting : strasbourg , fr , 1824 october 2014 .annex i _ multiview and depth video coding _ of iso / iec 14496 - 10 , int .standard _ generic coding of audio - visual objects part 10 : advanced video coding _ , 8th ed ., 2013 , also : itu - t rec .h.264 , edition 8.0 , 2013 .s. m. seitz , b. curless , j. diebel , d. scharstein , and r. szeliski , _ a comparison and evaluation of multi - view stereo reconstruction algorithms _ , in proceedings of the ieee computer society conference on computer vision and pattern recognition ( cvpr06 ) , pp .519526 , june 2006 .o. stankiewicz , `` stereoscopic depth map estimation and coding techniques for multiview video systems '' , phd dissertation at poznan university of technology , faculty of electronics and telecommunications , 2014 .

Abstract: the paper presents a novel approach to occlusion handling problem in depth estimation using three views . a solution based on modification of similarity cost function is proposed . during the depth estimation via optimization algorithms like graph cut similarity metric is constantly updated so that only non - occluded fragments in side views are considered . at each iteration of the algorithm non - occluded fragments are detected based on side view virtual depth maps synthesized from the best currently estimated depth map of the center view . then similarity metric is updated for correspondence search only in non - occluded regions of the side views . the experimental results , conducted on well - known 3d video test sequences , have proved that the depth maps estimated with the proposed approach provide about 1.25 db virtual view quality improvement in comparison to the virtual view synthesized based on depth maps generated by the state - of - the - art mpeg depth estimation reference software .   depth estimation , disparity estimation , occlusion handling , mvd , graph cuts , ders , free viewpoint television .