inspired by turing s proposition of pattern formation mechanisms based solely on simple diffusion and reaction mechanisms , there has been a tremendous amount of theoretical and experimental efforts devoted to designing and studying systems that exhibit mechanisms for the selection of spatial structure . theoretically , the most accessible scenario is the instability of an unpatterned , spatially homogeneous state against perturbations with spatial structure .the linearly fastest growing mode then allows for rough predictions of wavenumbers in nonlinear systems . despite the simple theoretical appeal of turing s prediction, it took a long time to support his theoretical predictions experimentally , see e.g. , where sustained , self - organized spatially periodic structures are observed in an open - flow chemical reaction ; , where the roles of the nodal and lefty signal are summarized during the blastula stage ; , where wnt and dkk signaling is discussed in the context of hair follicle spacing ; and , where turing s ideas are quantitatively tested in a cellular chemical system .it s worth noticing that absence of diffusion in one species out of two , does not lead to selection of finite wavenumbers in linear instabilities .however , patterns with well - defined wavenumber laws had experimentally been observed for such systems long before turing s prediction . on the other hand , systems of two speciesdo not exhibit linear instabilities that select temporal oscillations with a finite wavenumber .such spatio - temporal selections through a fastest growing linearly unstable mode , that will be the main theme of the present paper , are possible in reaction - diffusion systems with at least three species , only ; see .our interest here is in a yet simpler system of run - and - tumble dynamics .we think of self - propelled agents moving with the same speed to either right or left .in addition , agents may change orientation , and start moving in the opposite direction .the probability of this tumbling events is assumed to be a pointwise function of the densities of left- and right - moving agents .the resulting tumbling rates can be thought of as encoding probabilities of encounters between left- and right - moving agents , which in turn induce changes of orientation .this caricature picture is motivated by observations of rippling patterns in colonies of myxobacteria . during aggregation and starvationinduced self - organization of myxobacteria an intriguing rippling pattern can be observed within the colony ; see figure [ f : exp ] .the ripple crests are oriented approximately perpendicular to the movement direction of the bacteria .the so - called c - signal , bound to their cell surface , is transmitted upon end - to - end contact of bacteria , and increases the reversal probability of individual bacteria .a number of mathematical models with different assumption have been discussed in the literature in order to decribe this rippling behavior , cf . .all models include active motion and collision - triggered reversal of myxobacteria .continuous , , and discrete approaches , are discussed .several of the models assume a refractory period after collision and/or a reversal triggering internal biochemical clock to play a role .this is not needed in . in cellular states are suggested instead , i.e. the state of excitation ( being able to receive or send the c - signal ) , the state when being in contact with counter - migrating cells , and the state of reversing the motors .although rippling patterns could numerically be observed in all models , a strict argument that the respective assumed mechanisms really do select a wavenumber , as one can see in experiments , could so far only be shown for the model in .as pointed out in any linear stability analysis is not capable to predict a correct wavelength and wavespeed for the models given in . in the limit of weak signaling for the model in a fokker - planck type of equationwas derived by for the reversal - point density , that contains a source term which is absent in the respective limit given by . for small nonlinearitieswave number selection could be found .strengthening the nonlinearity tends to confine and destroy the patterns through a nonequilibrium phase transition , reminiscent of destruction of synchronization in the kuramoto model .+ such an analysis is even less accessible for discrete or individual based models , i.e. leaving the rigorous analysis of the suggested mechanisms for rippling patterns and defined wavelengths in these models still largely open .a further crucial control for the suggested rippling mechanisms are their capability to also reflect mutant , or dilution - experiments , see section [ s:6 ] for a detailed discussion , and the close relation between rippling and aggregation patterns .  tying in with these considerations and since the actual mechanism for motion and tumbling are most definitely quite complex, we think of our caricature example as trying to exhibit the simplest mechanism that can explain the intriguing rippling patterns and also the associated selection of wavenumbers .  to be precise , we start with the model that was analyzed in , where and encode the densities of left- and right - moving bacteria , respectively , is the rate at which left - moving bacteria reverse direction , and , by reflection symmetry , the rate at which right - moving bacteria reverse direction .such systems do arise as special lower dimensional case from structured population dynamics models , cf . , = 0,\ ] ] where denotes a set of internal variables which characterize the state of the considered species , i.e. here the direction of motion .the most distinctive feature of the operator is , that it acts on the density in a local manner . to our knowledge, the classification of pattern formation in such systems is largely open , and one could view our attempts at a precise description of the patterns in the drastically simplified system as a first step towards a systematic description of patterns in the more complex system .  at times , we will also allude to a diffusive regularization of ( [ e : cte ] ) , with , small , encoding brownian fluctuations in addition to directed motion .we mostly think of these systems posed on , but will restrict to periodic boundary conditions when convenient .while most of our results give valid predictions for general systems of the form , we focus in our analysis on the following class of tumbling rates for some .those nonlinearities model a tumbling rate proportional to the number of say right - moving agents , depending in a nonlinear fashion on encounters with oppositely oriented agents . constants encode spontaneous tumbling , linear dependence encodes binary collisions , albeit with zero net effect on the dynamics due to the presence of the reverse tumbling , for .therefore we keep the next simplest term , which encodes triple collisions , and we also allow for a hill - type saturation of the tumbling rate for large densities through the denominator .the interaction of tumbling and transport appears to be surprisingly difficult to characterize .the linear transport equation associated with linear tumbling rates can be converted to a damped wave equation for using the `` kac trick '' , with diffusive long - time dynamics for .for nonlinear tumbling rates , such a simple description does not appear to be possible .one can however find criteria that guarantee global existence of solutions , at least in the presence of small viscosity ; see .in fact , invariance of the domain ^ 2 ] ) is as predicted in the previous section.,title="fig : " ] -component and final profiles of - and -components after perturbing an unstable asymmetric state locally in space .parameter values are ( ) .the initial perturbation is as in with , .note that the -component ( red ) remains almost constant and the spreading of perturbation in a cone (  ] are invariant when for all with , which is true for any when , and for for . therefore decreasing the value of leads to more restrictive invariance conditions hinting at the possibility of large amplitude bursts .in fact , the value of jumps between the two asymmetric branches associated with a fixed -value , respecting mass conservation .as one can immediately infer from figure [ f:1 ] , the value of the second asymmetric equilibrium for given is very large .-component and final profiles of - and -components after perturbing an unstable asymmetric state locally in space .parameter values are , ( ) , initial perturbation as in with , .note that here the -component ( blue ) remains almost constant and the spreading of perturbation in a cone  ] is as predicted in the previous section.,title="fig : " ]  on the other hand , derivatives are larger and resulting wavenumbers smaller ; see figure [ f:5 ] for an illustration of the results .while generally wavenumbers agree well with the prediction , one can notice a `` phase slip '' in space - time plots , that is , the nucleation of a ripple at time which vanishes at time ; we comment on such phenomena in the discussion .a comparison of theoretically predicted wavenumbers and wavenumbers observed in direct simulations is shown in figure [ f:11 ] .the measured wavenumber is typically slightly larger than predicted .we attribute this discrepancy to temporal transients since we observed changes in the wavenumber of magnitude comparable to this discrepancy across the domain , suggesting that a simulation in larger domains over longer time periods would yield more accurate results ; see for instance for an example where such comparisons were performed to very high accuracy and for theoretical results on the slow ( ) convergence of wavenumbers . for lower masses on the upper branch , we noted failure of nucleation , leading to larger ( typically doubled ) wavelengths and hence significant discrepancies to the theoretical predictions ., .,title="fig:",height=288 ] , .,title="fig:",height=288 ] , .,title="fig:",height=288 ]  perturbations at multiple , random locations ( `` shot noise '' ) lead to almost regular patterns , with defects at shot locations ; see figure [ f:6 ] .one can observe the expected crossover to irregular patterns when the distance between locations of perturbations approaches the wavenumber .figure [ f:6 ] also illustrates the absence of a wavenumber selection mechanism for white - noise perturbations .  , we look at localized perturbations at 20 random locations . one can distinguish the selected wavenumber , as well as irregular parts of the pattern associated with the shot locations ; space - time contour plot of -compoent ( left ) and time snap shot ( right ) of - ( blue ) and - component ( red).,title="fig : " ] , we look at localized perturbations at 20 random locations .one can distinguish the selected wavenumber , as well as irregular parts of the pattern associated with the shot locations ; space - time contour plot of -compoent ( left ) and time snap shot ( right ) of - ( blue ) and - component ( red).,title="fig : " ]  finally , we show simulations with white - noise ( random value at grid points ) perturbations of amplitude .as expected , the scale of patterns is controlled by the ( numerical ) diffusion ; see figure [ f:7 ] ., we start from white - noise perturbations , space - time plot of -component ( left ) , final profiles ( right ) of ( red ) and ( blue ) .wavenumbers in the resulting patterns are controlled by numerical diffusion .note the smaller spatial scale compared to figure [ f:6 ] ., title="fig : " ] , we start from white - noise perturbations , space - time plot of -component ( left ) , final profiles ( right ) of ( red ) and ( blue ) .wavenumbers in the resulting patterns are controlled by numerical diffusion .note the smaller spatial scale compared to figure [ f:6 ] ., title="fig : " ]  [ [ invasion - of - symmetric - states . ] ] invasion of symmetric states .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the linear analysis in the section [ s:4 ] predicts instabilities with zero frequency , that is , spreading of disturbances without oscillations .one indeed observes an instability that simply change the state left in the wake , spreading with the characteristic speed . as discussed in section [ s:3 ] , such fronts involve a jump in , say , the -component , with constant across the jump .the resulting state in the wake of the jump is of course asymmetric , and may well be unstable , with small disturbances spreading in an oscillatory fashion .we did observe such a two - stage scenario ; see figure [ f:8 ] .  - and -components ( top row ) ; space - time plot of the total concentration and final profiles of - and -components ( bottom row ) .the symmetric state , is perturbed by a localized perturbation as in figure [ f:5 ] . , title="fig : " ] - and -components ( top row ) ; space - time plot of the total concentration and final profiles of - and -components ( bottom row ) .the symmetric state , is perturbed by a localized perturbation as in figure [ f:5 ] . ,title="fig : " ] + - and -components ( top row ) ; space - time plot of the total concentration and final profiles of - and -components ( bottom row ) .the symmetric state , is perturbed by a localized perturbation as in figure [ f:5 ] . ,title="fig : " ] - and -components ( top row ) ; space - time plot of the total concentration and final profiles of - and -components ( bottom row ) .the symmetric state , is perturbed by a localized perturbation as in figure [ f:5 ] ., title="fig : " ] +  starting from the initial state , , we observe a primary change in the -value , say , to or , which results in secondary instabilities with wavenumbers and , respectively , and predicted wavelengths and .the observed wavenumber is larger : in fact , the instability is related to the instability of the wave train where is piecewise constant with values in .it appears to be difficult to derive more accurate predictions for the resulting wavenumbers .we also explored different types of localized perturbations and observed qualitatively similar dynamics .again , shot noise produces less regular patterns and white noise perturbations of the initial state results in incoherent dynamics without apparent selection of wavelengths .[ [ growth . ] ] growth .+ + + + + + +  the selection of wavenumbers induced by the spatial spreading of disturbances is relevant when initial disturbances are well separated in space .a related scenario is when an unstable state is created through a growth mechanism .we illustrate this here in a prototypical context , again without striving for a systematic exploration .we prepare initial conditions as small random ( white noise ) fluctuations , .we then mimic a growth process , where agents and are `` created '' in the system at spatially localized positions .the system with such a spatio - temporal source term reads we first focus on the simplest case , where , with , for instance .one would expect that the deposition front leads to the formation of a nonlinear solution , with for and for .whenever corresponds to an unstable state , one expects the instability in the wake of this front to result in the creation of spatio - temporal patterns , originating from the front interface .the initial small fluctuations , superimposed on the nonlinear front profile created by the deposition , help initiate the development of the instability at the front interface .figure [ f:9 ] shows the result of a numerical simulation .  - and -components for ( top row ) and final profiles ( bottom row ) with , , , , ; numerical parameters as described in the text at the beginning of section [ s:5].,title="fig : " ] - and -components for ( top row ) and final profiles ( bottom row ) with , , , , ; numerical parameters as described in the text at the beginning of section [ s:5].,title="fig : " ] + - and -components for ( top row ) and final profiles ( bottom row ) with , , , , ; numerical parameters as described in the text at the beginning of section [ s:5].,title="fig : " ] - and -components for ( top row ) and final profiles ( bottom row ) with , , , , ; numerical parameters as described in the text at the beginning of section [ s:5].,title="fig : " ]  deposition at locations , leads to counter - propagating waves in the occupied region .one expects that the selected wavenumber is close to the wavenumber selected by instabilities of the constant state for speeds , but we did not attempt to derive or validate asymptotics , here ; see for such asymptotics in different contexts .figure [ f:10 ] illustrates the resulting patterns in the symmetric case , .-component , of , ( top row ) , and final - and -profiles ( bottom row ) for with from , , , ; numerical parameters as described in the text at the beginning of section [ s:5].,title="fig : " ] -component , of , ( top row ) , and final - and -profiles ( bottom row ) for with from , , , ; numerical parameters as described in the text at the beginning of section [ s:5].,title="fig : " ] + -component , of , ( top row ) , and final - and -profiles ( bottom row ) for with from , , , ; numerical parameters as described in the text at the beginning of section [ s:5].,title="fig : " ] -component , of , ( top row ) , and final - and -profiles ( bottom row ) for with from , , , ; numerical parameters as described in the text at the beginning of section [ s:5].,title="fig : " ]we summarize and discuss our results .[ [ counter - propagating - waves . ] ] counter - propagating waves .+ + + + + + + + + + + + + + + + + + + + + + + + + +  we studied a `` minimal '' model for run - and - tumble dynamics , with nonlinearity confined to tumbling kinetics .our main results here illustrate a dramatic dependence of the selected pattern on the type of perturbation of the initial state ._ white - noise _ perturbations of unstable states result in either oscillations of arbitrary fine scales , or long - wavelength modulations of spatially constant states . localized or _ shot - noise _ perturbations result in spatially periodic waves with well - defined wavenumber .while a distinction between wavenumbers selected as the fastest linearly growing mode ( relevant for white noise perturbations ) and wavenumbers selected through an invasion process ( relevant for shot noise perturbations ) has been noted early on , it is often quantitative , changing the wavenumber slightly . in that regard ,our results are an extreme case , where the nature of the noise enables the selection of wavenumbers .  on the other hand , there are virtually no systems so far , where a mathematical proof of this wavenumber selection process is available .the best mathematical results rely on the construction of invasion fronts that are periodic in a comoving frame , , and proving stability of those in suitably weighted norms ; see for instance .selection of invasion fronts from compactly supported initial data has not been shown in pattern - forming systems .it would be interesting to attempt such a study in the present case , that is , study the existence of invasion fronts with speeds and frequency from the linear calculus given in section [ s:4 ] .in a different direction , the intriguingly simple structure of our systems insinuates that a more direct , hands - on description might be possible . as an example , considering the equation for time - periodic invasion fronts , leads to an evolution equation with `` time '' variable and `` space '' , which is again a coupled transport equation , now with tumbling kinetics .also this equation obviously possesses a large group of transformations that preserve its structure , , , with being invertible .it seems however difficult to reduce the system to a form that is accessible to an explicit analysis .possibly most intriguingly , one would suspect that the _ linear _ coupled transport equation should be amenable to a more direct ( that is , not reliant on fourier transform ) analysis . in particular , one would hope to derive the selected wavenumber ( or ) in a straightforward fashion .[ [ experimental - implication - and - validations . ] ] experimental implication and validations .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  we commented in the introduction on experiments showing rippling patterns with a distinct wavenumber in myxobacterial colonies .while our model is clearly too simple to capture many relevant features , let us shortly discuss whether it can capture an essential feature of the selection mechanism . a semi - quantitative test case are experiments where mutants are introduced into the system that do not release the c - signal .therefore their presence does not induce tumbling of opposite moving bacteria , whereas the mutants themselves do turn upon interaction with counter - migrating wildtype cells .the overall population is always kept constant .experiments show that the higher the density of mutants is in the mixed mutant - wildtype populations , the larger the wavelength of ripples becomes , until the pattern disappears completely when about of bacteria in the population are c - signaling mutants , .the resulting system for unaltered and mutant bacteria is which is in skew - product form , that is , the dynamics do not depend on the -dynamics .in particular , selected wavenumbers depend on the selection through the non - mutant population .fixing the total population in the system , one can therefore view the introduction of mutants as equivalent to a reduction of total mass in a pure non - mutant population .this dilution experiment was considered as test case for the rippling models in   the dependence of wavenumbers on the equilibria and the total mass was shown in figure [ f:11 ] .one notices that wavenumbers are decreasing with mass on the branch with lower mass and are ( mostly ) monotonically decreasing on the branch with higher mass .pictures for smaller values of are somewhat similar but more intricate . in summary , these findings agree with the experimentally observed increase in wavelength as the percentage of mutants in the population is increased as given in figure in , but only within a well defined range of total mass , that is , for masses corresponding to asymmetric equilibria on the lower non - trivial branch . in this respect ,the anomalous decrease of the wavelength with increasing mass is associated with the restabilization and subsequent new destabilization of the asymmetric branch .of course , tuning or possibly altering kinetics further , one can arrange for the lower asymmetric branch to exist for a large range of total masses , and simulataneously push the upper unstable asymmetric branch to very large , potentially unrealistic masses .[ [ modulation . ] ] modulation .+ + + + + + + + + + +  in the marginally stable regime , , , the dispersion relation suggests exponential damping of modes with , such that one would like to expand in terms of long - wavelength modes .one therefore expands the dispersion relation to find near , say , , , a neutral eigenvalue , with eigenvector , and therefore chooses an ansatz we denote the variables corresponding to the diffusive scaling by .then substituting into the equation , expanding in , and projecting onto the kernel , we find at order  a simple diffusion equation .we emphasize that the diffusive effect is generated by the interaction of tumbling and transport , without explicit diffusion in the system !this can be understood somewhat more directly in the interaction of tumbling and transport on the linear level , deriving a damped wave equation for the total mass , as described in the introduction .a description of dynamics in the unstable or marginally stable regime , then appears to be difficult for a variety of reasons .first , since all wavenumbers are simultaneously unstable , an expansion around one particular mode is unlikely to capture dynamics .moreover , expansions around the most unstable mode appear challenging .lastly , resulting patterns are not of small amplitude against the uniform unstable background , such that small amplitude expansions will likely not capture relevant phenomena .[ [ dependence - on - tumbling - rates . ] ] dependence on tumbling rates .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +  our results are for a very specific tumbling rate , only , and one can easily argue for different rates .on the other hand , the linear selection results and the existence of traveling waves depend on the geometric shape of the curves of equilibria , only . from this perspective, it turns out that the class of nonlinearities represented by our specific choice is quite a bit larger .  as a first generalization, one could introduce add a parameter in front of the linear part of the tumbling rate , .figure [ f : tro ] shows that equilibrium configurations are qualitatively similar to the case , studied here .interestingly , the structure of the set of asymmetric equilibria changes qualitatively when nonlinear saturation of tumbling rates is caused by increase of _ total mass _ in the system , , cf .such nonlinearities incorporate a slightly different sensing of also the total population as opposed to only sensing the opposite moving bacteria .one notices that for such systems , the branch of asymmetric equilibria does not re - destabilize ; see figure [ f : tro ] .generally , asymmetric equilibria bifurcate as pde - unstable branches that subsequently stabilize ; the resulting stable branch extends to infinity. moreover , the asymmetric branch appears in a bifurcation from infinity , rather than through the spontaneous emergence of a sub- and a super - critical pitchfork bifurcation at finite mass .   with ( left ) ; , ( center left ) ; , ( center right ) ; , ( right ) ., title="fig:",height=144 ] with ( left ) ; , ( center left ) ; , ( center right ) ; , ( right ) . ,title="fig:",height=144 ] with ( left ) ; , ( center left ) ; , ( center right ) ; , ( right ) . ,title="fig:",height=144 ] with ( left ) ; , ( center left ) ; , ( center right ) ; , ( right ) ., title="fig:",height=137 ]  following , one can generalize further by allowing different power laws for small and large , .a priori bounds based on the comparison principle require  .we notice that for , asymmetric branches of equilibria extend to infinity ; see figure [ f : tro ]. it would be nice to further explore for which parameter regimes solutions blow up in finite time , i.e. reflecting the initiation of fruiting body formation as discussed for the model in .ideally , only small parameter changes in the model would result in the formation of ripples , of aggregates or in the initiation of self - organization .-component resulting from perturbations of asymmetric equilibria through localized perturbations for tumbling kinetics as in figure [ f : tro ] ( from left to right ) .initial mass values were ( left , ) , ( center left , ) , ( center right ) , and ( right , ) . , title="fig:",height=115]-component resulting from perturbations of asymmetric equilibria through localized perturbations for tumbling kinetics as in figure [ f : tro ] ( from left to right ) .initial mass values were ( left , ) , ( center left , ) , ( center right ) , and ( right , ) ., title="fig:",height=115]-component resulting from perturbations of asymmetric equilibria through localized perturbations for tumbling kinetics as in figure [ f : tro ] ( from left to right ) .initial mass values were ( left , ) , ( center left , ) , ( center right ) , and ( right , ) ., title="fig:",height=115]-component resulting from perturbations of asymmetric equilibria through localized perturbations for tumbling kinetics as in figure [ f : tro ] ( from left to right ) .initial mass values were ( left , ) , ( center left , ) , ( center right ) , and ( right , ) ., title="fig:",height=115 ]  in direct simulations , we found little differences when starting with localized perturbations of asymmetric states ; see figure [ f : tro2 ] .we did not see patterning when perturbing symmetric equilibria in the case of saturation by total mass .[ [ coherent - structures - and - stability . ] ] coherent structures and stability .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  striving to put the present analysis on a stronger theoretical footing , one would want to analyze coherent structures and their stability . as a first step, one would study the stability of piecewise constant wave trains using floquet - bloch theory . in a different direction, it would be interesting to analyze the effect of small diffusion on wave trains , fronts , and interfaces , possibly including dispersive terms or even higher - order diffusion .numerical observations from higher - order upwind finite - difference simulations suggest that stability properties of the jump interface can change in subtle ways under diffusive or dispersive regularization . ultimately , the phenomena described here should of coursebe understood in terms of invasion fronts , their existence , and their stability .[ [ wavenumber - selection - through - growth - elsewhere . ] ] wavenumber selection through growth elsewhere .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the phenomena observed here here bear a striking resemblance to wavenumber selection in models for recurrent precipitation , where is non - monotone , for instance , .white noise perturbations of unstable spatially homogeneous equilibria result in spatially decorrelated patterns of arbitrarily fine spatial scale , while shot noise perturbations give rise to patterns with a dominant nonzero spatial wavenumber .structurally , the system is similar to our system as it possesses a reflection symmetry ( which however does not involve the dependent variables ) and a conservation law for the total mass .we also have a plethora of spatially periodic patterns , , analogous to the traveling waves described in section [ s:3 ] , but independent of time .spatially constant equilibria with are unstable but fastest growing wavenumbers are either or as observed in our models .again , invasion processes do select finite , nonzero wavenumbers and observations in direct numerical simulations agree quite well with linear predictions , obtained in a similar fashion to our analysis in section [ s:4 ] . furthermore , selected patterns are discontinuous , stable , yet subject to coarsening when a small diffusivity is added in the second equation .a more subtle similarity is the `` instantaneous coarsening '' observed here : a wave maximum nucleating at the leading edge of the growth process may not develop into a large amplitude peak of a wave train , or merge with a previous peak , as can be seen in figure [ f:5 ] , for the first two peaks or the 13th peak , respectively .this mechanism was demonstrated numerically in and associated with resonances .a more detailed study can be found in for the cahn - hilliard equation .we outline a basic functional analytic setting that allows us to show local existence and uniqueness of solutions , as well as smooth dependence on initial data .previous work in the literature does not quite cover the setting of bounded yet discontinuous functions that we are interested in , here ; see for instance for a linear theory , including boundary conditions .we therefore solve with initial data , a banach space , using the variation - of - constants formula for $ ] , sufficiently small , this system of integral equations defines a contraction mapping principle and yields local existence of solutions in a similar way as the picard - lindelf iteration , provided that controls supremum norms , that is , choosing for instance or .note that solutions will not necessarily be continuous in time for .mimicking the existence proof for odes further , we also obtain continuous dependence on initial data in these spaces for fixed times .u. brner , a. deutsch , and m. br . _ a generalized discrete model linking rippling pattern formation and individual cell reversal statistics in colonies of myxobacteria .. biol . * 3 * ( 2006 ) , 138146 .m. kotzagiannidis , j. peterson , j. redford , a. scheel , and q. wu ._ stable pattern selection through invasion fronts in closed two - species reaction - diffusion systems . _ in rimskokyuroku bessatsu b31 ( 2012 ) , far - from - equilibrium dynamics , eds . t. ogawa , k. ueda , pp 79 - 93 .

Abstract: we study mechanisms for wavenumber selection in a minimal model for run - and - tumble dynamics . we show that nonlinearity in tumbling rates induces the existence of a plethora of traveling- and standing - wave patterns , as well as a subtle selection mechanism for the wavenumbers of spatio - temporally periodic waves . we comment on possible implications for rippling patterns observed in colonies of myxobacteria .   + arnd scheel and angela stevens + _ of minnesota , school of mathematics , 206 church st . s.e . , minneapolis , mn 55455 , usa + mnster , fachbereich mathematik und informatik , einsteinstrasse 62 , 48149 mnster _   * keywords : * traveling wave , standing wave , balance laws , coupled transport equations , invasion fronts , wavenumber selection


a key signal processing operation is the computation of the distance between two signals .this simple operation can suddenly become challenging if the two signals belong to two mistrustful parties who are unwilling to share them .we must now facilitate the continuation of the operations , without exposing their signals to one another .more formally stated , two parties , alice and bob , have two real valued vectors and respectively .the signal processing operation requires the computation of , which may be required by one or both parties .however , at the same time , it is required that none besides alice learns about , and similarly none besides bob is exposed to .a number of methods have been suggested in the literature to solve this problem .these have generally involved the use of secure multi - party computation protocols [ gcs ] or fully homomorphic which enables the computation of distances over encrypted data .although this provides the participants with the desired privacy , they are computationally unattractive. they can increase both the communication and the computational cost of computing the distance by several orders of magnitude .an alternate approach uses _ trusted third parties _ to facilitate the computation . here ,alice and bob interact with a third party , charlie to compute the distances between their signals with information - theoretic privacy .no party gets additional information despite their computational power . under this approach, protocols based on secret sharing using polynomials on finite fields have been proposed to compute the euclidean distance between privately held real - valued vectors . while secure , the precision of the computation depends on how well real numbers are represented in the selected field .moreover , the communication complexity of this kind of protocol is proportional to the dimensionality of the data , which is problematic when alice and bob need to compute a distance between high dimensional vectors in presence of communication constraints ._ even _ if the above concerns were somehow allayed , an additional problem arises from the fundamental definition of the basic problem itself .the outcome of the computation the distance itself reveals information about the signals .the party who obtains this result , which may be either alice or bob , or both of them , learns about the relationship between their signals . in trusted third - party settings , charlie may get to know this distance too. ideally , while this `` leakage '' of information can not be avoided , we would like to limit it .in particular , it is preferable that the third party charlie not be exposed to the information at all .  in this paperwe propose several third - party based protocols for the computation of distances between signals , which address the above issues .we propose to encode the signals through a _modular _ hashing scheme with cryptographic properties that ensure that none of the parties can recover the original signal from the hashes , without the appropriate keys .the hashes have the property that they permit _ limited _ release of the information between vectors the true distance is only revealed if the vectors are sufficiently close and the distance between them lies below a threshold , but is naturally effectively thresholded when the distance is greater , ensuring that even the parties that _ do _ get the distance only get as much as is required for effective computation of the signal processing processes , but not enough to chart out the other person s data .we impose protocols on top of the hashing which ensure that the third party can not gather any information besides the limited distance computed .when the third party may be trusted to know this limited distance then , unlike other cryptographic or secret sharing based schemes , the communication overhead may be made independent of the dimension of the signals , permitting control of the precision of the distance estimate .finally , we also analyze how the protocols may be modified in order to not expose even the limited distance to the third party .  in the next section ( section 2 )we discuss the hashing scheme . in section 3we describe our protocols and in section 4 we explain how they may be extended to hide the distance from the third party ( charlie ) .finally we present our conclusions in section 5 .our proposed approach requires alice and bob to transform their signals into hashes prior to further processing .we now describe the transform . this transformation may be considered as a _ modular _ variant of the -stable locality sensitive hashing ( lsh )  .however , unlike the standard lsh , the introduction of modularity helps to generate _ uninformative hashes _ , i.e. the distribution of the resulting hash does not depend on its input .furthermore , the euclidean distance between two signals may be estimated through comparison of their hashes , provided the distance is shorter than some threshold .   *definition : * let be an integer larger than 1 , be an matrix , ^m ] privately , and sends it to charlie .3 .  bob computes \right) ] is indistinguishable from a vector with uniformly distributed i.i.d components , therefore , the value of should be close to for every pair and .a natural drawback is that the number of parameter to run the protocol , which may cause a computational overhead .in this paper we have presented a random transformation , that given a threshold , can generate hashes from vectors that preserving the euclidean distance between them if it is smaller than .these hashes are uninformative if the random parameters of the function are unknown .  with this kind of transformation , we are able to describe protocols to compute distances privately and efficiently .in fact , the proposed transformation is not only uninformative , it can be see as a transformation which reduces the data dimension and preserves the euclidean distance through the lee distance output space .  although the fact that the distance is preserved until some threshold seems to be a drawback , if both alice and bob know the maximum possible distance between their vectors , then they can set an appropriate value of which lets preserve distances to any desired threshold .we also describe how to hide the distance value from the third party , action that may increase the computational cost of the protocol , as it is expected .we still have many questions .how can we extend these protocols to multiparty versions ?how easy is a reconstruction of knowing and ?we have also trusted the third party to not collude with alice or bob .can this restriction be lifted ?* theorem 2 .* , , even , then we obtain the following relation between the components of and : = \\\frac{k}{4 } -  \frac{2k}{\pi^2 } \sum\limits_{j=1}^\infty \frac{1}{(2j-1)^2 } e^{-2 \left ( \frac{\pi \|{\bf x}_1-{\bf x}_2\|(2j-1)}{\delta k}\right)^2 }  \end{gathered}\ ] ]    + besides , we know that the density function of is + then + but , since both and are positive , we can extend these functions over real line , defining as + similarly we define . then + however, we can see that + where + and \\ &  \\ 1 & \mbox { if } x\in [ -j , j]\\ & \\ -x+j+1 & \mbox { if } x \in [ j , j+1 ] \\ & \\ 0 & \mbox { otherwise } \end{array } \right . } \ ] ] + additionally , using the parseval s theorem we have + where and are the fourier transform of and respectively .but , using the definition of we have + + because + on the other hand , since is a gaussian , it is easy to see + therefore \\ & \qquad \qquad \qquad \cdot e^{-2\left ( \frac{\pi \|{\bf x}_1-{\bf x}_2\| i}{\delta k } \right)^2}\\ & = \frac{k}{\pi^2 } \sum\limits_{i=1}^\infty \frac{\left [ \cos \left ( 2 \pi j \frac{i}{k } \right ) - \cos \left ( 2 \pi ( j+1 ) \frac{i}{k } \right ) \right]}{i^2 }  e^{-2\left ( \frac{\pi \|{\bf x}_1-{\bf x}_2\| i}{\delta k } \right)^2 } \\ & \qquad \qquad + \frac{2j+1}{k}\end{aligned}\ ] ] + finally , noting that if is even , we have and \\ & = \sum_{i=0}^{k/2 } 1- \mathbb{p}\left(d_{lee}(q_{k , a , u}({\bf x}_1),q_{k , a , u}({\bf x}_2))\leq j \right ) \end{aligned}\ ] ] we can prove that = \\\frac{k}{4 } -  \frac{2k}{\pi^2 } \sum\limits_{i=1}^\infty \frac{1}{(2i-1)^2 } e^{-2 \left ( \frac{\pi \|{\bf x}_1-{\bf x}_2\|(2i-1)}{\delta k}\right)^2 }  \end{gathered}\ ] ] +     1 .considering just the first term of the series , we have \leq  \frac{k}{4}-\frac{2k}{\pi^2}e^{-2 \left ( \frac{\pi \|{\bf x}_1-{\bf x}_2\|}{\delta k}\right)^2}\ ] ] + on the other hand , we can notice that then , applying this inequality , the second inequality holds .+     1 .  using the same scheme as in the proof of theorem 2, we can see that = \int_0^\infty f_l(u ) w(u ) du\ ] ] + where is given by and by the function shown in the following plot +  + it is easy to see that + then & \leq & \int_0^\infty f_l(u ) u\ , du\\  & = & \sqrt{\frac{2}{\pi } } \cdot \frac{\|{\bf x}_1 - { \bf x}_2\|}{\delta}\end{aligned}\ ] ] + besides , noting that + + we can write -  \sqrt{\frac{2}{\pi } } \cdot \frac{\|{\bf x}_1 - { \bf x}_2\|}{\delta } \right| = \\\left| \int_0^\infty f_l(u ) \cdot ( w(u)-u ) \,du\right|\end{gathered}\ ] ] + but + + and we have + + hence , setting we have + -  \|{\bf x}_1 - { \bf x}_2\| \bigg| \leq \\ \|{\bf x}_1 - { \bf x}_2\| \cdot e^{- \frac{k^2}{4 \pi \|{\bf x}_1 - { \bf x}_2 \|^2}}\end{gathered}\ ] ] + and defining + + we have the result .+     1 .  to see this is enough to verifythat is increasing in and when tends to infinity tends to 0 .+ to see that increases in we can just compute the derivative and it is easy to verify that is positive . + on the other hand , it is clear that , when tends to infinity , tends to zeros for every value of +     1 .  the hoeffding inequality establish that , if $ ] , and , then in our case , noting that we have + + then , setting + + we can conclude that a sufficient condition is to consider + +    m. pathak and b. raj , `` privacy - preserving speaker verification and identification using gaussian mixture models . ''ieee transactions on audio , speech and language processing , vol 21:2 , pp .397 - 406 , 2013 .j. bringer , h. chabanne and a. patey , `` shade : secure hamming distance computation from oblivious transfer . ''financial cryptography and data security volume 7862 of the series lecture notes in computer science pp 164 - 176 .

Abstract: a critically important component of most signal processing procedures is that of computing the distance between signals . in multi - party processing applications where these signals belong to different parties , this introduces privacy challenges . the signals may themselves be private , and the parties to the computation may not be willing to expose them . solutions proposed to the problem in the literature generally invoke homomorphic encryption schemes , secure multi - party computation , or other cryptographic methods which introduce significant computational complexity into the proceedings , often to the point of making more complex computations requiring repeated computations unfeasible . other solutions invoke third parties , making unrealistic assumptions about their trustworthiness .   in this paper we propose an alternate approach , also based on third party computation , but without assuming as much trust in the third party . individual participants to the computation `` secure '' their data through a proposed secure hashing scheme with shared keys , prior to sharing it with the third party . the hashing ensures that the third party can not recover any information about the individual signals or their statistics , either from analysis of individual computations or their long - term aggregate patterns . we provide theoretical proof of these properties and empirical demonstration of the feasibility of the computation .  secure distance computation , information - theoretic privacy , locality sensitive hashing  .


nowadays , the importance of food and eating goes far beyond a means to survive . often regarded as a social construct, food has become an essential part of modern day life .new jargon has entered our vocabulary as expressions like `` foodie '' , `` food porn '' , or `` food tourism '' , hint at the buzz around the entertainment arising from our culinary experiences . with the rise of social media , and the proliferation ofalways - on always - connected devices , this gobbling revolution is not confined to our kitchens , restaurants , and food stalls , but naturally breaks out on the social web .sharing pictures of one s food has become a growing passion for both tourists and locals  , and dedicated food searching and sharing apps , along with recipe websites and the ubiquitous social presence of celebrity chefs , have all contributed to a thriving culture and passion around food worldwide .  around the world , different cuisines are naturally intertwined with cultures , traditions , passions , and religion of individuals living in different countries and continents .sushi , curry , kebab , pasta , tacos these are just examples of foods conventionally associated with specific countries , as are specific cuisines and ingredients. different dietary habits around the world are also closely related to various health statistics , including cancer incidence  , death rates  , cardiovascular complications  , and obesity  .although there are many common beliefs about cuisines , recipes , and their ingredients , it is still unclear what types of ingredients are unique in / about different countries , what factors make cuisines similar to each other ( e.g. , in terms of ingredients or flavors ) , and how these factors are related to individuals health . with this motivation in mind , in this paper , we set to investigate the way in which ingredients relate to different cuisines and recipes , as well as the geographic and health significances thereof .we use a few datasets , including 157k recipes from over 200 cuisines crawled from yummly , bbc food data , and country health statistics .* overview & contributions . *first , we characterize different cuisines around the world by their ingredients and flavors .then , we train a support vector machine classifier and use deep learning models to predict a cuisine from its ingredients .this also enables us to discover the similarity across different cuisines based on their ingredients e.g. , chinese and japanese while , intuitively , they might be considered different .we look at the diversity of ingredients in recipes from different countries and compare them to geographic and human migration statistics .we also measure the relationship between the nutrition value of the recipes vis - - vis public health statistics such as obesity and diabetes .* paper organization . * the rest of the paper is organized as follows . in section [ sec : data ] , we present the datasets used in our evaluation , then section [ sec : ingredients ] presents an analysis of the diversity of the ingredients around the world , looking at geographic diversity patterns of cuisines and notable ingredients in particular ones . in section [ sec : simil ] , we look at the similarity between the cuisines based on their ingredients and flavors , and use these results to train machine - learning classifiers for ingredient - based cuisine prediction models in section [ prediction ] . in section [ sec :nutrition ] , we correlate the nutrition values of recipes for different countries with their public health statistics . after reviewing related work in section [ sec : related ] , the paper concludes in section [ sec : conclusions ] .our study relies on a number of datasets , namely , a large set of recipes collected from yummly , a list of ingredients compiled by bbc food , and country health statistics . in this section, we describe these datasets in detail .yummly is a website offering recipe recommendations based on the user s taste .it allows users to search for recipes , learning which dishes the user likes and providing them with recipe suggestions .it also provides a user - friendly api , which we use to collect recipes .first , we crawled wikipedia for a list of cuisines , then , in summer 2016 , we queried the yummly api for recipes belonging to each cuisine . in the end , we obtained 157,013 recipes belonging to over 200 different cuisines . due to api restrictions, we limited the number of recipes per to 5,000 .each recipe obtained from the yummly api contains a number of attributes . in our study , we use the following :  1.  * ingredients : * each recipe contains a list of the ingredients that are required to prepare it . since yummly acts as a recipe aggregator from various cooking sites , the ingredients do not always appear with the same wording .in fact , it is very common to see the same ingredient written with different spellings or by using a different terminology .we overcome these issues through a standardization process described in section [ sec : dataset : bbc ] .* flavors : * recipes are identified by six flavors , specifically , saltiness , sourness , sweetness , bitterness , savoriness , and spiciness .these scores are on a range of 0 to 1 .* rating : * users are encouraged to provide a rating , from 1 to 5 , for the recipes that they try . from the yummly api, we retrieve the average review rating for each recipe .* nutrition : * unfortunately , the yummly search api does not directly provide nutritional information for the recipes . as a consequence , we designed a simple web crawler to fetch the corresponding web page for each recipe in our dataset , and extract information on the amount of protein , fat , saturated fat , sodium , fiber , sugar , and carbohydrate of a recipe ( per serving ) , as well as calories .although some ingredients appear in other languages ( e.g. , german , french , etc ) , the recipes presented here are mostly in english ; hence it is possible that some more authentic or niche local recipes might be missing from our dataset .however , considering the number of recipes and a large cut - off threshold introduced later on , we are confident this does not significantly affect our analysis .moreover , authors of the recipes might not represent the entire population , given the fact that they are likely to be tech - savvy .this might introduce a potential bias in the dataset , but at the same time , this potential issue is compensated by its richness in terms of the variety of dishes from different countries available in it as structured information .   bbc food is a part of the bbc website providing information about recipes , ingredients , chefs , cuisines , and other information related to cooking and dishes from all bbc programs . in summer 2016 , we crawled all the ingredients from the bbc food website , collecting about 1,000 ingredients , which we used to organize and standardize the ingredients in the yummly dataset .the standardization process is as follows :  \(i ) we extracted all the 11,000 ingredients from the yummly dataset and performed a preliminary data cleaning , i.e. , removing measurement units ( mass , volume , etc ) , numbers , punctuation marks , and other symbols .\(ii ) due to the multilingualism of the yummly data , we used the google translate api to perform automatic language detection and translation of all the yummly ingredients to english .\(iii ) we used the bbc list of ingredients as a reference , and mapped all possible ingredients from the yummly list to it .\(iv ) as not all ingredients from the yummly list were successfully mapped , we merged the similar ones into groups , and the ingredients in each group were manually mapped to the its representative ingredient .overall , this process yields about 3,000 standardized ingredients .as diet is directly related to the health of individuals , we also set to relate yummly statistics to real - world health data . to this end, we will use the diabetes prevalence estimates from world development indicators by the world bank , the health expenditure as a percentage of total gdp from the world bank , and the obesity prevalence from the world health organization in the countries to which the cuisines are mapped , using the most recent available data , which is from 2014 .in this section , we provide a characterization of the ingredients used in dishes from all over the world .first , we investigate the diversity of ingredients in different countries .next , we define the concept of `` complexity '' of a dish in terms of its ingredients and look at how complexity changes around the world . finally , we discuss a series of case studies of most notable and significant ingredients in some eminent cuisines .   aiming to investigate the diversity of ingredients in dishes of a cuisine , we set to answer the following questions :  1 .how many different unique ingredients are used in total in dishes of each country ? in other words , what is the number of unique ingredients the people of a country have ever used to prepare a culinary dish ?the answer to this question is what we refer to as the _ global diversity_. 2 .how different are the dishes of an individual country relative together in terms of their ingredients combination ?in other words , do different dishes usually share some ingredients or their ingredients are almost different ?the answer to this question is what we call _ local diversity_.  the local and global diversity of ingredients in a country depend on many parameters including the geographical location , climatic conditions , agricultural situation , or even the amount of immigration which directly influences the diversity of culinary cultures .the calculation of the global diversity is performed in two steps . since the number of recipes per different cuisines are variable , we first set a fixed number of 100 recipes per cuisine , discarding cuisines containing fewer number of recipes , and sampling from cuisines containing more number of recipes uniformly at random , to have an equal number of recipes in all cuisines .this results in a final set of 82 different cuisines each containing 100 recipes .we then map the result obtained for each cuisine to its corresponding country . here, some countries are mapped with more than one cuisine . for such countries ,we record the average result over their associated cuisines . to calculate the local diversity , we look at each cuisine as a probability distribution over all standard ingredients . by counting the total number of occurrences of each ingredient in all recipes of a particular cuisine , andthen normalizing the values such that they sum to one , we obtain the ingredient distribution for that cuisine .we then calculate the entropy of these distributions as the local diversity of their corresponding cuisines .the entropy of the ingredient distribution measures the unpredictability of ingredients used in the dishes .therefore , the higher the entropy of the ingredient distribution of a particular cuisine , the more different the ingredients combination of its recipes , and thus the higher the local diversity . to preserve the smoothness of the ingredient distributions, we again keep the 82 cuisines with more than 100 recipes . after calculating the local diversity for each cuisine , we follow the same procedure as for the global diversity to map the cuisine - based results to countries .figure [ fig : div ] shows the local diversity and the global diversity of ingredients for different countries around the world . according to the figure, we can see that the local and global diversities have a meaningful correlation with each other .the countries with high global diversity have also high local diversity , and countries with low global diversity have low local diversity as well .this happens because as the global diversity increases , people will have more options to choose as the ingredients for their foods , so they can prepare relatively different dishes .  another interesting trend observable in figure [ fig : div ] is that countries like the united states and australia which usually accept a high number of immigrants , have a relatively high global diversity .regarding this , we hypothesized that the number of immigrants coming to a country must have an influence on the ingredient diversity of that country .this is mainly due to immigrants bringing their native culinary culture with themselves , which in turn makes the cuisines of their target country richer .to investigate this fact , we collected the net migration data from the world bank which shows the difference between the total number of immigrants and emigrants during a time period .we correlated the global diversity with the average net migration from 1960 to 2016 . to this end , we fitted a polynomial curve to the data points considering the global diversity and the net migration .the result is shown in figure [ fig : div : mig ] . as we expect, the figure indicates that an increase in the net migration would result in an increase in the global diversity of ingredients .[ small , width=0.95 , xlabel = global diversity , ylabel = net migration , x label style = at=(axis description cs:0.5,0.05),anchor = north , y label style = at=(axis description cs:.1,.5),anchor = south , ticklabel style = font= , xmajorgrids , ymajorgrids , max space between ticks=20pt , try min ticks=5 , y tick label style= /pgf / number format/.cd , fixed , fixed zerofill , precision=2 , /tikz/.cd , ] tableresults / diversity / global - migration.txt ;   another interesting concept about the culinary preferences of different countries is the complexity of dishes .the complexity of a dish is simply the number of unique ingredients required to prepare it .accordingly , a cuisine is more complex than another one if its dishes are proportionally more complex than another one s .  formally speaking ,each cuisine is associated with the complexity distribution of its dishes . for a sample cuisine , this distribution , namely ,specifies the probability of a dish from that cuisine to have exactly unique ingredients .this way , the cumulative complexity distribution ( ccd ) will give us an insight about the complexity of dishes in a particular cuisine .[ small , width=0.95 , legend pos = south east , ymin=0,ymax=1 , xmin=0.0,xmax=34 , xmajorgrids , ymajorgrids , xlabel= # ingredients , ylabel = cumulative probability , x label style = at=(axis description cs:0.5,0.05),anchor = north , y label style = at=(axis description cs:.1,.5),anchor = south , ticklabel style = font= , y tick label style= /pgf / number format/.cd , fixed , fixed zerofill , precision=2 , /tikz/.cd , legend entries = norwegian , russian , spanish , tunisian , lao , legend style = font= , ] tableresults / complexity / norwegian.txt ; tableresults / complexity / russian.txt ; tableresults / complexity / spanish.txt ; tableresults / complexity / tunisian.txt ; tableresults / complexity / lao.txt ;  figure [ fig : complexity : cdf ] shows the cumulative complexity distribution ( ccd ) for norwegian , russian , spanish , tunisian , and lao as an example . from the figurewe can see that the ccd for norwegian cuisine grows faster than the others , while for lao , it is relatively slower .this means that lao dishes are more complex than norwegian ones .thus for each cuisine , the area under its ccd is inversely related to its complexity .hence , we use the reciprocal of the area under ccd as a measure of complexity for a cuisine .figure [ fig : complexity ] shows the complexity of dishes for different countries around the world .here we have used the same approach as in section [ sec : ingredients : diversity ] to map the cuisines to countries .we can see from the figure that except for some cases , the complexities are consistent with the diversities .this is due to the fact that as the number of available ingredients increases in a country ( which is the result of global diversity , ) people can leverage more ingredients and prepare more complex dishes .the exceptions here are china and india , two countries with the most population in the world .the complexity of dishes in these countries are relatively high , while their ingredients diversity is low . this can be the result of overpopulation or special culinary culture in these countries . perhaps , these countries had or have good chefs that could cook more complex foods with the available ingredients !specific cuisines are mostly associated with different sets of ingredients due to the geographical locality of the ingredients . in this section ,we study the most notable ingredient associated to some well - known cuisines using our dataset of recipes from yummly .we use the term frequency - inverse document frequency ( tf - idf ) calculation to find notable ingredients in each cuisine . in this approach , each ingredient is considered as an atomic word , and the collection of all the ingredients appeared within a cuisine is considered as a document .a tf - idf calculation leads us to find the weight of different ingredients in the corpus of documents .this way , the importance of each ingredient within each cuisine is specified .figure [ fig : not ] shows the top-50 most notable ingredients for italian , indian , and mexican cuisines as a case study .we also looked at other similar cases , which we do not present here due to space constraints .according to the figure , the bigger the name of an ingredient , the more distinctive it is in its associated cuisine .we verified the soundness of results using google trends service .for example , the term `` mozzarella '' has the highest search frequency in italy .similarly , `` garam masala '' is the most popular food additive in india according to its search volume .how do we determine the similarities between korean and japanese cuisine ?what makes the middle eastern dishes seem similar to one another ? in this section we answer these questions using a number of different methods and data .   at first, we calculate the similarity between different cuisines based on the ingredients used in their recipes . to this end, we convert cuisines into vector space , representing each cuisine as a vector where each element indicates the frequency of an specific ingredient in that cuisine .thereby , for each cuisine we obtain an ingredient - based feature vector which we leverage to calculate the similarity between different cuisines using the following metrics :  1 .* jensen - shannon divergence : * if we normalize each ingredient - based feature vector such that the elements of a vector sum to one , then each vector will represent a probability distribution over standard ingredients .this way , we can use the distance measures proposed for probability distributions like jensen - shannon divergence .the jensen - shannon divergence between two probability distributions and is defined as : \ ] ] where and is the kullback - leibler divergence from to . since the jensen - shannon divergence is a distance measure between 0 and 1 , we take as the similarity measure between two cuisines with their associated ingredient distributions and .we have used jensen - shannon divergence instead of the simpler kullback - leibler divergence because goes to infinity when for an ingredient like , is non - zero while is .this case almost always happens in our data due to the geographical locality of ingredients .therefore , we turned to jensen - shannon divergence which does not have this drawback . 2 .  * tf - idf similarity : * another measure we use to calculate the similarity between two cuisines is the well - known tf - idf . using this measure , we followed the approach described in section [ sec : ingredients : notable ] to calculate the weight of ingredients in each cuisine . this way , each cuisine is represented as a vector where each of its elements indicates the representative power of the corresponding ingredient for that cuisine .the tf - idf similarity between two cuisines is then simply the cosine similarity between their associated tf - idf vectors .  using the above similarity metrics , we calculated all similarities between each pair of cuisines . to assert the smoothness of ingredient distributionsneeded to compute jensen - shannon divergence , we limited our cuisines to those 82 ones having more than 100 recipes .figure [ fig : sim ] illustrates the results for different similarity measures in a graph - based fashion . in these graphs ,each node represents a cuisine and each cuisine is linked to its top-5 most similar cuisines .link weights are proportional to the obtained similarity score between two endpoints .we colored each cuisine node according to the geographical region it resides in .the cuisines are classified into 9 regions which are north america , latin america , africa , western europe , eastern europe , middle east , south asia , east asia , and oceania . to visualize the graph , we have used _ forceatlas _ graph drawing algorithm implemented in gephi tool .this a force - directed algorithm which makes densely connected nodes to be grouped together and thus the communities become revealed .we can see from the figure [ fig : sim ] that using either of the ingredient - based similarity measures , the cuisines which reside in the same region are more similar to themselves and thus are grouped together . for example , we clearly see the clusters formed by eastern and southern asian , middle eastern and african , latin american , and western european cuisines .furthermore , due to the similarity of cultures in europe and north america , and even oceania , it can bee seen that clusters formed by the cuisines of these regions greatly overlap with each other .in addition to the ingredient - based similarity , we calculate the similarity between cuisines in terms of the flavors provided in their recipes .this can help us understand how different cuisines are related to each other based on the taste of their dishes . as we mentioned in section[ sec : dataset : yummly ] , each recipe contains the flavor scores for six different flavors including saltiness , sourness , sweetness , bitterness , savoriness , and spiciness . to calculate the similarity between cuisines based on these flavors , like what we did for ingredient - based similarity, we consider each cuisine as a distribution over different flavors .regarding the fact that different flavors of a recipe are correlated to each other for instance , a dish can hardly be both sweet and spicy simultaneously and due to the continuity of flavor scores , we hypothesize that the flavor scores are sampled from a multivariate gaussian distribution , where each covariate corresponds a particular flavor . considering this assumption ,we fit a multivariate gaussian distribution to each cuisine so that each cuisine become associated by a mean vector representing the average of flavor scores over all of its recipes , and a covariance matrix representing how flavors change relative to each other within that cuisine .  after fitting a multivariate gaussian distribution to each cuisine using maximum likelihood estimation, we use kullback - leibler divergence to measure the distance between the distributions associated to each pair of cuisines . since kullback - leibler divergence is and asymmetric distance measure , for each pair of cuisines with and as their corresponding flavor distributions , we use ^{-1}$ ] as a symmetric similarity measure between them .  .]  figure [ fig : sim : flavor ] shows the result of flavor - based similarity between different cuisines in a graph - based manner .we followed exactly the same steps as in figure [ fig : sim ] to draw the graph , except that we used flavor - based similarity between cuisines .we can see from the figure that even though the flavors are not as much discriminant as ingredients , still we can observe some geographical patterns .for instance the clusters formed by eastern asian , middle eastern , latin american , and northern european cuisines are clear in this case as well .but what is obvious here is the fact that although there is a sense of taste similarity between the dishes from neighboring countries , the flavors are naturally shared all over the world .in this section we address the question of `` _ _ how good we can predict a recipe s cuisine , given its ingredients ? _ _ '' .to answer this question , we use two different classifiers , support vector machine ( svm ) , which is previously used in for the same task , and deep neural network ( dnn ) , which is popular nowadays for classification purposes . to extract a feature vector for each recipe ,we convert it into a boolean bag of words vector , considering each ingredient as an atomic word .therefore , each recipe is represented as a vector with a length equal to the total number of ingredients , which is 3,286 .the labeling of recipes are performed according to the following settings :  1 .* cuisine prediction : * each recipe is labeled to its cuisine .we consider 82 different cuisines having more than 100 recipes as different classes , resulting in about recipes .* region prediction : * each recipe is labeled according to one of the 9 geographical regions where its cuisine belongs to .the regions are considered the same as in section [ sec : simil ] .this results to have about recipes .  for multi - class classification with svm, we use linear kernel with one vs. rest coding . the class imbalance problem is resolved with adjusting the weight of each cuisine inversely proportional to its frequency .the implementation is done using scikit - learn machine learning library in python . for dnn , we use keras deep learning library and create four dense hidden layers and a softmax output layer .each of the first two hidden layers consists of 1000 neurons , and the two last ones each have 500 neurons .dropout regularization is used for all of the hidden layers .we use adadelta with default parameters as the optimizer . for both methodswe take of the data as training set and the remaining as the test set .the prediction performance of both methods are evaluated under accuracy and f - measure .figure [ fig : prediction ] shows the obtained results with both svm and dnn methods .figure [ fig : prediction : cuisine ] illustrates the results for cuisine prediction , while figure [ fig : prediction : region ] shows the results of region prediction task .as we can see in the figure , the dnn model performs about better than svm for cuisine prediction task under accuracy and over better under f - measure . for region prediction task ,since the number of classes are much fewer than cuisine prediction , both methods performed relatively better . in this case , the accuracy and f - measure achieved by the dnn model is about and better relative those achieved by svm , respectively .  in order to get more intuition about the similarity of recipes in different regions , we bring the confusion matrix of the dnn model for region predictions in table [ table : conf ] .each region name is abbreviated in two letters .for example , la means latin american and af means african cuisines . the number of correctly classified recipes are shown in bold and for each class , the greatest number of miss - classifications is shown in red .this table clearly demonstrates that the almost all of the miss - classifications are western european .this is probably due to the huge ethnic composition of western european countries which resulted in the diversity of culinary cultures of that region .the table shows that for some regions like southern and eastern asian , the number of miss - classified recipes are somewhat low relative to the correctly classified ones .this result is as same as in figure [ fig : sim ] in which these regions were almost disconnected from the others . on the other hand , for some cuisines like oceanic , eastern european , and northern america , the number of miss - classifications are relatively high , mostly with western european .figure [ fig : sim ] shows clearly that the cultures in these regions are very similar to each other , mainly due to the common ethnics and history ..confusion matrix for dnn region prediction [ cols="<,<,^,^,^,^,^,^,^,^,^ " , ]recently , public health has been increasingly analyzed through the lens of the web and social media .we refer the reader to for an overview of the recent research in this area .abbar et al . relate food mentions on twitter conversations to the obesity and diabetes rates , using caloric values , and find a high correlation ( coefficient 0.77 ) between caloric values of tweets and obesity values in various states in the us .low - obesity areas of usa have also been shown to be more socially active on instagram ( posting comments and likes ) than those from high - obesity ones by mejova et al . , who present a large - scale analysis of pictures taken at 164k restaurants in the us .ahn et al . study culture - specific ingredient connections , creating a `` flavor network '' from a dataset of about 56k recipes and relating them to the geographical groupings of countries .similar `` flavor - based '' food pairing studies are conducted on cuisines in distinct geographical areas such as india .west et al . mine logs of recipe - related queries to uncover temporal patterns in consumption . using fourier transforms, they show the yearly and weekly periodicity in food `` density '' of the searched recipes , with different trends in southern and northern hemispheres , suggesting a link between food selection and climate . a study of austrian recipe sites by wagner et al . also highlights differences in the recipes of regions which are further apart .zhu et al . conduct a similar study on chinese recipes to investigate the effect of geographical and climatic proximities on ingredients similarity of domestic cuisines .kular et al . create a network of recipes using a dataset of 300 recipes from 15 different countries , and show the network s _ small - world _ and scale - free properties . as opposed to this line of work, we also exploit flavor and nutritional information , alongside health statistics countries to provide a deeper analysis about the dishes , cuisines , culinary cultures , and the impact of food on human life .su et al . investigate underlying connections between cuisines and ingredients via machine learning classification , with an application to predicting the cuisine by looking at recipes .like our analysis , theirs is based on a large - scale data collection of recipes specifically , 226k recipes collected from food.com . however , they only look at classifying cuisines using support vector machine ( svm ) , while we propose a deep neural network architecture to capture the highly non - linear relation of a recipe cuisine and its associated ingredients .the results approve that the proposed deep model outperforms svm by a significant margin in terms of prediction accuracy and f-1 measure .there are major difference between our work and the ones discussed above , in both scale and domain .an important characteristic of our work comes from the size and the quality of the various datasets we used , which enable us to derive first - of - its - kind insight on worldwide cuisines and their relationship to health factors .in addition to the ingredients , we also exploited flavor and nutritional information , alongside health and immigration statistics , allowing us to perform a deeper analysis of the dishes , cuisines , culinary cultures , as well as the impact of food on human life .this paper presented a large - scale study of user - generated recipes on the web , their ingredients , nutrition , similarities across countries , and their relation with country health statistics .our results have multiple implications : we found strong similarities between cuisines in neighboring countries , yet , the diversity of ingredients and flavors varies largely across the continents , mostly affected by net migration trends ._ you are what you eat _ might be a clich , but we did find quantitative evidence of a strong correlation between nutrition information of the recipes ( e.g. , in terms of sugar intake ) and obesity . also , we demonstrated that deep learning can be used to effectively predicting cuisines from ingredients , potentially providing possibility for fine - grained analysis of food and dishes as well as improved recipe recommendations based on individuals profile .our findings indicate that certain ingredients ( e.g. , mozzarella ) uniquely represent a certain cuisine ( e.g. , italian ) and there are strong clusters of ingredients across neighboring countries .this feature eases the prediction of regions ( e.g. , continents ) from the combination of ingredients in a cuisine .moreover , the correlation between the ingredients and health conditions , such as diabetes , is naturally of great importance for public health experts , where behavior nudges or recommendation of similar dishes in flavor and ingredient complexity can be utilized to improve dietary intake  .  in future work , we plan to explore the possibility of recipe recommendation based on regional and personal tastes and user ratings .this is important as a local chinese dish or a distinct flavor combination may be `` alien '' to , e.g. , a western person , but of interest to a japanese individual .we also wish to asses the ability to model flavors with ingredients , and discover ingredients to match a specific flavor palette .finding answers to these questions would provide a better understanding of the composition of flavors and ingredients in popular dishes and provide a better recommendation system for a healthier , tastier , and more diverse experience .

Abstract: as food becomes an important part of modern life , recipes shared on the web are a great indicator of civilizations and culinary attitudes in different countries . similarly , ingredients , flavors , and nutrition information are strong signals of the taste preferences of individuals from various parts of the world . yet , we do not have a thorough understanding of these palate varieties .  in this paper , we present a large - scale study of recipes published on the web and their content , aiming to understand cuisines and culinary habits around the world . using a database of more than 157k recipes from over 200 different cuisines , we analyze ingredients , flavors , and nutritional values which distinguish dishes from different regions , and use this knowledge to assess the predictability of recipes from different cuisines . we then use country health statistics to understand the relation between these factors and health indicators of different nations , such as obesity , diabetes , migration , and health expenditure . our results confirm the strong effects of geographical and cultural similarities on recipes , health indicators , and culinary preferences between countries around the world .


determining three - dimensional information from a single two - dimensional image is a common goal in computer vision .circular features appear frequently in real world objects , the mouth of a cup , the wheels of a car , the iris of a human eye .such circular features ( referred to as 3d circles from here on ) appear as ellipses when imaged if viewed non - frontally .  in this paperwe present a method for matching parts of a two - dimensional scene to existing three - dimensional models using a technique that maps three - dimensional circles to two - dimensional ellipses .additionally , we present a novel method for identifying elliptical regions and the properties thereof in a two - dimensional image . in particular, we aim to recover the pose of motor vehicles in natural scenes .the purpose of this is twofold : first , it allows us to determine the location and facing of a car in a street scene , which has obvious uses in driver assistance systems , and second it allows us to place vehicles in three dimensions from a single image without constructing a depth map of the entire scene .it is important to note that this paper does not present a general - purpose vehicle detection or complete scene reconstruction algorithm , rather it aims to deliver a basis from which such platforms can be built or improved .intelligent driver assist technology is becoming more and more important as our roads become busier . determining vehiclepose in three dimensions allows us to estimate a heading for a visible vehicle , which can alert the driver to approaching vehicles .importantly , our pose detection technique requires only a single frame , allowing it to detect stationary vehicles as potential hazards in case of unexpected motion , unlike many techniques which require a series of images of moving vehicles .  developing a depth map of a scene from a single image is an underconstrained problem .there are approaches based on machine learning , or that make use of the effects of fog , but these either require classification and a large training data set or specific scene conditions .our method is able to place 3d models of imaged vehicles at their locations in the same orientation as in the real world .this has applications for security , as observers will be able to determine the position of a vehicle from surveillance footage , as well as scene reconstruction . unlike the work of hinz et al . , which places simple 3d car models in vehicle locations in aerial photographs , we want to use a specific , detailed 3d model to match to street - level car images .  in order to do this ,we first must isolate ellipses in the image that we can then match to circular areas in a three - dimensional model . for vehicles ,the most applicable choice for these ellipses are the wheels .existing ellipse detection methods , such as the hough transform require a five - dimensional search , which has high computational and memory requirements .there are many recent methods methods which aim to reduce this by using parametrised spaces for the hough transform accumulator , such as .these methods still require edge extraction or other image preprocessing , and often detect significantly more ellipse than we need in this paper . to this end, we present a new method for identifying wheels in real - world vehicle images which incorporates a significant amount of ellipse filtering and parameter estimation as part of the detection process .other feature detection methods are discussed more fully in the text .  while this work is focussed on identifying the pose of cars ,the ellipse to circle mapping process can be applied to any three dimensional object with circular features , as shown in the experimental verification of section [ 2d3dmatch ] .rather than using an existing ellipse detection method based on edge detection , such as those presented in , we propose a method for detecting elliptical objects which allows for significant noise at the edge of the object .the ellipse detection method presented here involves several steps .first , pixels which are brighter than their surroundings are identified by comparing each pixel to the average local intensity . connected components , or blobs ,are then found and labeled .further processing is performed on each of these blobs to fill gaps and then to determine whether the detected object is elliptical . while we use bright image regions here, the method can be easily changed to identify dark regions or regions which adhere to some other color or intensity property .this process is summarised in algorithm 1 and visually in figure [ fig : ellipsealgorithm ] .smooth image with square kernel ( figure [ fig : ellipsealgorithm]b ) normalize image ( figure [ fig : ellipsealgorithm]c ) threshold normalised image ( figure [ fig : ellipsealgorithm]d ) detect and label connected regions ( figure [ fig : ellipsealgorithm]e ) star fill connected regions ( figure [ fig : ellipsealgorithm]g ) get ellipse parameters using eq.([ellipseparams ] ) filter out non - ellipses by comparison to expected ellipse ( figure [ fig : ellipsealgorithm]g ) determine ellipse normal using eq.([ctophi ] ) ( figure [ fig : ellipsealgorithm]g )  in order to identify pixels which are brighter than their surrounding area , a new image which reflects the local mean intensity for each pixel is built .each pixel in this new image contains the mean intensity for a square region centered on that pixel in the original image .this square is set to _b_% of the shortest image side . for the purposes of this paper , a window 10% of the width of the image was found to be effective , but this can depend on the nature of the ellipses that are desired .if this window would extend beyond the edges of the image for a particular pixel , the mean of the closest possible window which is entirely contained within the image is used instead . the intensity integral image used to efficiently calculate this windowed mean value , which was the primary motivation for using a simple box kernel for smoothing .we then subtract this average from the intensity of each pixel in the initial image , and then set a threshold _t _ which defines how much a pixel must exceed the local average to be flagged as ` bright ' . for wheel detection , one standard deviation above the mean of the average image was found to produce good results .this thresholding produces a binary image , with zero corresponding to ` dark ' pixels , and one corresponding to ` bright ' pixels .this thresholding method tends to produce a thick ` edge ' around the ellipse perimeter as seen in figure [ fig : ellipsedetect ] ( ii ) , and frequently contains small gaps corresponding to darker areas of the ellipse surface , corresponding to occlusions , decoration or simply image noise .  to denoise the binary image obtained above , it is first necessary to eliminate small bright regions .simple blob extraction is then performed , labeling each 8-connected region with a unique identifier .  in this ellipse detection method ,objects being investigated must be filled .as it is not desirable to affect the overall shape of a blob , the fill method chosen preserves the perimeter and fills gaps in a radial fashion . in each blob , if a pixel is marked as bright then all pixels on a line between it and the blob center are also marked as bright .we are able to do this efficiently , in linear time , by spiralling in from the edge to the centre .this fill method is used rather than a simple flood fill as blobs frequently do not have fully enclosed edges due to noise or occlusions .each of these filled , denoised blobs is then separately investigated to determine if it is elliptical in nature .see the following section and figure [ fig : ellipsedetect ] ( iii ) for more details .  in order to determine whether a particular blob is elliptical , the extracted blobis compared to an equivalent ellipse .we do this by finding the mean and covariance matrix of each blob .let be the pixels of one .the area , mean and covariance matrix of each blob can then be found as follows : [ ellipseparams ] |w| = _ pw 1 , = _ pwp , + c = 1|w|_pw(p - v)(p - v)^by assuming that the blob is an ellipse , the major and minor axes of the corresponding ellipse can be calculated from the eigenvalues and of the blob covariance matrix , and the center of the corresponding ellipse is given by . from this information , we are able to find the shape of the corresponding ellipse by eq .. this approach is similar to that described in .this step is shown in figure [ fig : ellipsedetect ] ( iii ) .we are able to perform a consistency check at this stage to eliminate some obvious non - ellipses : if is an ellipse , its area is given by as well as .thus if for some small threshold , we can immediately discard this blob .we can then directly compare the remaining blobs to the equivalent ellipse on a per pixel basis by counting the number of pixels in which the blob and its ellipse do not match .as we are interested in the elliptical nature of the blob regardless of size , we divide the number of mismatches by the blob area .blobs with a sufficiently low mismatch can be regarded as ellipses .we found experimentally that a mismatch ratio of 20% identifies human - visible ellipses while discarding non - ellipses .circles in 3d space become ellipses in 2d space after parallel projection .it is possible to exploit this fact to determine the pose of a 3d object containing circular features from a projected image .first , we describe an ellipse in a way that is easily manipulated .let [ ellipsefromparams ] e(v , c ) : = \{p^2 : ( p - v)^c^-1(p - v)4 } describe a solid ellipse with center , where is a symmetric positive definite matrix .it can be shown by integration that is the covariance of the ellipse given by .the eigenvalues of can be computed from the trace and the determinant : [ evalue ] _1/2 = 12c therefore , the lengths of the major and minor axes of the ellipse are [ axes ] a_1=2 a_2=2  in lemma [ circletoellipse ] , we describe how a 3d circle projects to an ellipse in 2d .[ circletoellipse ] let : = \{x^3 : ||x||r & x^v=0 } be a centered circle with radius and unit normal .let be the orthogonal projection of circle along the axis .then and and are related by [ ctophi ] v (  c_x + _ y + _ z  ) =(  c +  + a_2  ) where and are defined in and .the coordinates of describe an ellipse .condition implies . inserting this into yields ||x||^2 &= & x^2+y^2+z^2 = x^2+y^2+(x_x+y_y_z)^2 + & = & ^ 1+_x^2/_z^2_x_y/_z^2 _ x_y/_z^21+_y^2/_z^2 hence the projected circle is an ellipse with covariance [ phitoc ] c & = & r^24 _x_y/_z^21+_y^2/_z^2 ^ -1 + & = & r^24  where we have exploited when inverting the matrix .conversely , given an ellipse with covariance , using , , and , the radius and normal of the underlying 3d circle are given by .there is one hard sign constraint , namely . sincea projected circle is usually only visible from one side of the 3d object , we define as pointing away from the observer into the image .this fixes the sign of , leaving two solutions which can not be disambiguated .these solutions arise from the fact that an ellipse can be projected from a circle facing in two different directions .we give some rough estimates of the accuracy of the reconstructed .in particular we show that reconstruction from a frontal view is somewhat surprisingly harder than from a tilted view .let  ] rotates into . using \end{aligned}\ ] ] leads to the following convenient representation : [ phitovarphi ] a = 2 = , ( b , c , d ) = 2 = 12a(vv ) this representation only breaks down if is exactly antiparallel to .in this case , any orthogonal to will do .subsequently , a rotation around by an arbitrary angle can be applied .hence the complete transformation is as follows :  [ onewheelalign ] eq .projects 3d circle = ( center , axle , radius ) to 2d ellipse = ( center , covariance ) iff q = q ,q = q_2q_1 , q = v - qv , = a_1/r , minor axis is given by and .matrix is given by and and .matrix is given by and with arbitrary  ] representing the shift from car centre to image origin , and the scale factor is a scalar .figure [ fig : fullpageresults ] shows graphically the recovered pose for two artificial images using approximated wheel normals .the pose recovered from exact ellipse normals matched exactly the input pose , and so is not included .the pose returned when ellipse normals are precisely known matches perfectly with the expected pose of the image .however , the estimated ellipse normals produce a pose which does not exactly match that expected .the pose error is relatively small , and it clearly results from inaccuracies in the estimation of ellipse parameters from the wheel detection . even when using estimated parameters , the pose of the vehicle is still close enough to the true pose that it provides a good starting point for techniques which make use of other information about the vehicle to fine - tune the estimation .figure [ fig : fullpageresults ] shows three real - world car images and the estimated pose of a 3d model of these vehicles . as we do not know the true vehicle pose in a real image , we are unable to conduct a quantitative analysis of the recovered pose , and must instead rely on visual inspection .car pose is recovered quite well in terms of scale and position , but there is some misalignment in orientation .this misalignment is caused by a distortion of the detected wheel ellipses , resulting in an imprecise normal recovery and thus a mismatched pose .the recovered model pose is still able to give us significant information about the position of the vehicle .we have presented a method for mapping a two - dimensional ellipse to a three - dimensional circle by estimating the normal of the ellipse in three dimensions .our results show that an ellipse can be matched to a circle in three dimensions to a pose with eight discrete and one continuous ambiguity .we also show that these ambiguities can be reduced by using other information about the object to be matched or the scene it is drawn from . in particular ,vehicle pose can be resolved to a four - fold discrete ambiguity by making use of both of a vehicle s wheels and the fact that a vehicle is not transparent .forcing the vehicle to be upright reduces this to a twofold ambiguity , given that we know which wheel in the image is the rear .we are currently developing a dissimilarity measure that will be able to automatically select the correct pose from these ambiguities by performing a comparison between each returned pose and the input image , but detail on this process is beyond the scope of this paper .it is based on a distance between images which is invariant under lighting conditions and relatively insensitive to the image background .additionally , we described a technique for wheel detection and wheel parameter estimation using a local thresholding technique and blob covariance .our results show that this technique performs well on a variety of real and synthetic vehicle images .r. safaee - rad , k.c .smith , b. benhabib , and i. tchoukanov .application of moment and fourier descriptors to the accurate estimation of elliptical shape parameters . in _ prof . international conf . on acoustics , speech , and signal processing ( icassp91 ) _ , pages 24652468 vol.4 , 1991 .

Abstract: finding the three - dimensional representation of all or a part of a scene from a single two dimensional image is a challenging task . in this paper we propose a method for identifying the pose and location of objects with circular protrusions in three dimensions from a single image and a 3d representation or model of the object of interest . to do this , we present a method for identifying ellipses and their properties quickly and reliably with a novel technique that exploits intensity differences between objects and a geometric technique for matching an ellipse in 2d to a circle in 3d .   we apply these techniques to the specific problem of determining the pose and location of vehicles , particularly cars , from a single image . we have achieved excellent pose recovery performance on artificially generated car images and show promising results on real vehicle images . we also make use of the ellipse detection method to identify car wheels from images , with a very high successful match rate .   computer vision ; image recognition / processing ; ellipse detection ; 3d models ; 2d - ellipse to 3d - circle matching ; single image pose identification ; wheel detection ; 3d vehicle models .