quantum key distribution ( qkd ) is a major communication application of quantum information theory aiming at the development of protocols for establishing secure channels protected by the laws of quantum physics .such channels can then be used to share a secure key for classical symmetrical cryptographic systems . recently , continuous - variable ( cv ) protocols of qkd ( see for review ) were developed and implemented on the basis of squeezed or coherent states .the security of cv qkd protocols in the case of gaussian modulation was then shown against collective attacks in the presence of channel noise , which also implies the security against the most general coherent attacks .cv qkd protocols , however , suffer from various imperfections . the most threatening are the untrusted ( i.e. , being under full control of a potential eavesdropper ) quantum channels , which are inclined to losses due to the attenuation and can add excess noise in the link .such noise can also be detection noise indistinguishable from the effect of the channel . in security analysisit is then supposed that all the channel imperfections are due to the presence on an eavesdropper .it was an important step in the development of cv qkd when with the use of reverse reconciliation it was shown possible to establish asymptotically secure key transmission upon any pure channel loss , while noise remains limiting to the security of the protocols .  however , the insecure quantum channel is not necessarily the single source of information leakage from a qkd protocol .a potential eavesdropper can use imperfections of the trusted ( i.e. , fully controlled by the trusted parties ) devices such as sources and detectors to gain at least partial information on the signal being sent or to control the measurement being performed at the receiver station .the noise , which is present on the trusted sides , can be fully controlled and calibrated by the trusted parties .such noise , however , can still be harmful .it was shown in particular that the preparation noise can already break the security in the reverse reconciliation protocol , but can be suppressed or tolerated in the direct reconciliation scheme .also , the trusted detection noise limits the key rate , but can be partially helpful to make the protocol more robust against noise in the quantum channel .  in the less optimistic scenario the noise or loss on the trusted sides can however be under partial control of an eavesdropper , as depicted in fig .[ scheme](a ) .this is the case of the side channels , which we define as auxiliary channels that have either input or output controlled by a trusted party but output or input , respectively , controlled by an eavesdropper . from this point of view , the side channels differ from the main channel between the sender and receiver . supposedly , any additional information can be used by an attacker to increase the knowledge about the transmitted key .therefore , it is necessary to investigate the influence the side channels can have on security . in the following studywe summarize all possible sources of side information and define them together as the side channels on either the sender or the receiver side of the protocol .one possible way to overcome the negative influence of the side channels is implementing the so - called measurement - device - independent ( mdi ) qkd protocols , which were recently suggested on the basis of cvs , where the trusted detection stations become shielded from a potential eavesdropper .however , the applicability of the device - independent cv qkd protocols is still very limited , particularly in terms of distance .  in the present paperwe study the effect of the side channels in cv qkd protocols with coherent and squeezed states of light .we define the side channels as the imperfections ( signal loss and noise ) on the trusted sides , which are under partial control of an eavesdropper .in particular , we consider ( a ) the leakage from the trusted sender station and ( b ) measurement manipulation by the noise addition in the trusted receiver station .we show the degradation of the key rate and increase of vulnerability to the channel noise in the presence of a side - channel leakage .we also show a security break from the noisy side channel on the detection stage .we suggest methods to compensate for the negative influence of the described side channels . for ( a )we consider the possibility to classically apply an additional correlated signal on the side - channel input , which is under control of a trusted sender party .we show the positive effect of such additional modulation and the possibility to optimize the modulation variance for the given parameters of the protocol .moreover , we show that by applying correlated information encoding and squeezing the input of the side channel , in the case of the squeezed - state protocol , the trusted party is able to completely decouple the side channel from the signal . by decoupling herewe mean decorrelation ( reducing or turning the correlation to zero ) and stopping the leakage of information through the side channel , which completely removes the negative impact of the side channel .for ( b ) we show the possibility to cancel the infused detection noise by monitoring the output of such a noise - infusing side channel .these are the alternative ways of active compensation of the side channels in the gaussian cv qkd protocols with the trusted sender and receiver stations , which keep the advantage of usability of such protocols , including the longer channel distances , compared to the device - independent protocols , and do not involve entanglement or non - gaussian operations and measurements .if for any reason the input of the sender - side leakage or the output of the receiver - side noise infusion are not available for the manipulations or monitoring , respectively , then the negative impact of the side channels shown in the current paper has to be either taken into account in the security analysis or compensated for by the possible use of the mdi schemes .the paper is structured as follows . in sec .[ security ] we define the side channels and recapitulate the methods of cv qkd security analysis being used . in sec .[ prob ] we demonstrate the negative impact of the side channels on the cv qkd security . in sec .[ solve ] we introduce the methods aimed at compensating for the negative effect of the side channels .we summarize in sec .[ concl ] .we study the effect of the side channels on the standard and optimized cv qkd protocols on the basis of the gaussian modulation of squeezed and coherent states , as depicted in fig .[ scheme](a ) .the trusted sending side ( alice ) prepares the signal state ( squeezed or coherent ) with variance ( so that or , respectively ) using the source .alice then applies random gaussian quadrature displacement of variance ( also referred to as the modulation variance ) , so that the overall variance of the modulated states becomes , using the modulator .the prepared state travels through the untrusted channel parametrized by transmittance ( loss ) and excess noise both being under _ full _ control of an eavesdropper ( eve ) .the signal is then detected by the remote receiving party ( bob ) using the homodyne detector h. further , with no loss of generality , we assume that the quadrature is measured by bob . thus , in the standard gaussian cv qkd protocol ( without the side channels ) alice applies displacement to the signal quadrature and sends the state with the quadrature to the channel so that the variances are , and and then . in the standard gaussian cv qkd squeezed - state protocol the signal states are modulated up to the antisqueezing ( variance of the quadrature complementary to a squeezed one ) , so holds , i.e. , the variance of modulation is fixed by squeezing of the signal states .we will also consider the optimized gaussian cv qkd protocols , where modulation is independent of the variance of the signal states and can be freely optimized for a given signal resource and parameters of the setup .the quantum channel transforms the modulated signal such that bob measures the quadrature , where is the quadrature of the vacuum input of the channel loss and is the quadrature of the channel excess noise with the variances and .   and gaussian quadrature displacement applied in the modulator .the untrusted channel is parametrized by transmittance and excess noise .the signal is coupled to the lossy side channel with untrusted output on the sender side and to the noisy side channel with untrusted input of variance on the receiving side .the remote trusted party performs measurement with the homodyne detector h. the _ trusted _ devices and channels are within the dashed boxes .( b ) scheme of the cv qkd with sender - side leakage modeled as coupling of the signal to a vacuum mode on a beam splitter with transmittance .the receiver - side untrusted noise infusion is modeled as coupling to a noisy mode with variance on a beam splitter with transmittance .the _ untrusted _ channels are within the dashed box . ]note that the trusted parties must know the channel parameters to assess the security of the protocols and therefore the channel must be properly estimated .while the issue of the channel estimation was recently studied in the finite - size context , in the present paper we focus on the side - channel effects and assume that the channel parameters are already known to the trusted parties .the channel estimation is still possible in the presence of the side channels because the side - channel parameters ( losses and noise ) can be estimated independently by the local measurements on the trusted sides .this also allows us to consider the protocols based on the preparation and measurement of a single quadrature ( e.g. , ) , while the channel estimation would require additional modulation and measurement in the complementary one .moreover , since the methods of the side - channel compensation suggested below do not change the data ensemble size ( defined by the signaling and detection rate ) , the finite - size effects would not qualitatively change the results of the paper .two types of side channels are considered in our study as shown in fig .[ scheme](a ) .the first one ( further also referred to as the _ type - a _ side channel ) is the sender - side side - channel leakage , when a vacuum mode is coupled to the signal and only the output of the coupling is accessible by an eavesdropper .an eavesdropper eve has no control of the side - channel input and of the strength of the coupling , thus the input state of such a side channel is initially vacuum .eve however receives the side - channel output similarly to non - invasive passive attacks in classical cryptography .the sender ( alice ) , on the contrary , has full control of the input of such a side channel before the coupling .the second type of side channel ( further also referred to as the _ type - b _ side channel ) is untrusted noise addition in the receiver station . in this casean untrusted noise with variance is supposedly prepared by eve and coupled to the signal prior to detection with the output of the coupling being inaccessible to the eavesdropper .the eavesdropper is not able to change the coupling strength . on the contrary , the receiving side ( bob ) is able to control ( e.g. , measure ) the output of the coupling . in both the cases we assume that the trusted parties are not able to directly remove the side channels or change the coupling strengths ( and , respectively ) .these are the two main types of possible semitrusted side channels , while the completely trusted noise is covered by previous research and completely untrusted noise can be attributed to the channel . moreover ,the noise infusion on the sender side ( symmetrical to the type - b side channel on the receiver side that is considered in the present paper ) is equivalent to the additional noise in the untrusted channel . at the same timethe side - channel loss on the receiver side ( symmetrical to the type - a side - channel loss on the sender side that is also considered here ) is equivalent to the additional loss in the untrusted channel .thus , our analysis covers the main possible semitrusted side channels based on the two - mode interaction in the prepare - and - measure cv qkd .we do not consider any specific physical realization of the side channels applying our analysis to the general case of semitrusted side channels based on the linear - optical mode interaction .however , the side channels can be expected in any real implementation of cv qkd in either fiber or free - space channels , where coherent and squeezed states were successfully transferred .the sender - side leakage can take place in particular in the case of imperfect modulation , e.g. , when the signal is mixed with a temporal , spectral , polarization , or spatial mode , which then leaves the sender station .the receiver - side noise infusion may , e.g. , be caused by imperfect light collection from a free - space channel with a background radiation .linear optical crosstalk is well studied in the classical optical communications where it is present in the multiplexed channels and receivers , but was also reported in the quantum communications .linear coupling represented by the beam splitter transformation is generally used to model the interaction of a quantum - optical system with the environment .therefore , in our work we use the typical linear optical interaction and model the mode coupling between the signal and the side channels as the beam splitters [ see fig . [ scheme](b ) ] .the type - a side channel is modeled as coupling to a vacuum mode on a beam splitter with transmittance . on the other side we model the type - b side channel as coupling to a thermal noise mode with variance on a beam splitter with transmittance . in the case of the sender - side leakage ( type - a ) side channel the quadrature that enters the quantum channelis then changed to , where is the quadrature value of the vacuum state on the input of the beam splitter , . in the case of the noise - infusing ( type - b )side channel the output of the quantum channel is changed as , where is the input noise of the type - b side channel with .  in the analysis of the negative impact of the side channels on cv qkd and the methods to compensate for such impact we mainly study the security against collective attacks , which in the asymptotic limit were shown to be no less effective than the most general coherent attacks . in this case eveperforms the optimal collective measurement on the accessible modes after the process of bases reconciliation is completed , implying that eve attaches a separate uncorrelated probe to each transmitted state and keeps probes in a quantum memory until she can gather additional information . to obtain simple insight into the conditions for insecurity of the protocols we also study the security against individual attacks , in which case eveis limited by the individual measurement on the accessible modes .this weaker security analysis allows us to analytically derive the regions of insecurity of the protocols in the presence of side channels since insecurity against individual attacks implies insecurity against the more effective collective attacks .  following the generalization of the cziszr - krner theorem on the quantum measurements performed by devetak and winter ,the protocol is secure if the mutual information between the trusted parties exceeds the information available to eve on the data on the trusted receiver side ( in the case of reverse reconciliation , which is stable against channel loss ) .the security is then described by the positivity of the lower bound on the key rate which in the case of collective attacks reads : where is the postprocessing efficiency and is holevo bound that determines eve s achievable information limit in the case of collective attacks .the efficiency depends on the effectiveness of the data postprocessing algorithms that are being used in the secure key distillation given the particularly low signal - to - noise ratio .we set as an independent parameter and do not consider any particular postprocessing algorithm . in the following analysis we therefore fix the reconciliation efficiency as ,which is realistic taking into account the recent progress in the error - correcting algorithms for the gaussian - distributed data .the holevo bound can be expressed as through the von neumann entropy of the generally multimode state ( including the side channels ) , which is available to eve for the collective measurement described by the respective covariance matrix , and the von neumann entropy of the state available to eve conditioned on the measurement results of the remote trusted party bob and described by the covariance matrix .covariance matrices are the matrices of the second moments of quadratures of the form , where is the quadrature vector of an -th mode . along with the first moments the covariance matrices explicitly describe the gaussian states and are sufficient for the security analysis of the gaussian protocols due to the extremality of the gaussian states .we analyze the security against the collective attacks using the most general purification method , where the equivalent entanglement - based representation of the protocols is used and all the state imperfections corresponding to the side channels and the main channel are attributed to eve .  in the case of individual attacks the upper bound on the information available to an eavesdropper is given by the shannon mutual information instead of the holevo bound and the lower bound on the key rate ( in the optimistic case of perfect postprocessing efficiency ) reads .details of calculations for security analysis in the cases of both individual and collective attacks are given in the appendix , while here we present the main expressions and results .in the next section we study the negative impact of the side channels on cv qkd .let us first consider the type - a side channel .we start by analyzing the region of insecurity of the protocol with respect to the individual attacks and without the untrusted channel noise .the mutual information in this case reads ( see the appendix for details ) while the information available to eve reads [v-\eta_a \eta ( v-1)]}{v}\label{typeamie}\ ] ] and is independent of the signal states ( squeezed or coherent ) .as can be seen the side channel decreases the mutual information between the trusted parties and increases eve s information , therefore limiting the key rate already for individual attacks with pure channel losses .  in the optimal ( given perfect postprocessing ) limit of infinite squeezing and modulation upon pure channel loss ( ) the key rate for the standard gaussian cv qkd protocolcan be written as where for the squeezed - state protocol and for the coherent - state one .the channel transmittance is therefore effectively decreased by the side - channel coupling .thus the presence of the type - a side channel does not break the security , i.e. , the key rate remains positive for any nonzero value of , as one would expect , because the channel remains purely lossy .[ cols="^ " , ]   note that the uncorrelated modulation can be also combined with squeezing on the input of the side channel .this combination in the case of the squeezed - state protocol greatly improves the method of uncorrelated modulation , making it ( provided the modulation is optimized ) almost as effective as the method of optimized correlated modulation combined with squeezing ( on the plots in fig .[ distance ] the corresponding line in the plotted region of parameters overlaps with the black solid line corresponding to the absence of the side channel and the difference corresponding to the limited performance of the method can only be seen for very low values of the key rate , which are irrelevant due to unavoidable finite - size effects ) .   in the case when eve couples an additional noise to the signal prior to the detection at bob s side , the monitoring of the coupling output , which is not accessible to eve , can be used .then , by applying the proper manipulation on the data from the main detector and from the monitoring detector the negative influence of the type - b side channel can also be fully compensated for .we suggest the method of weighted subtraction of data from the main and the monitoring detector and show that the resulting measurement is free from the influence of the type - b side channel . indeed , if the main homodyne detector ( see fig .[ solution ] ) after the noise - infusing side channel measures the quadrature , where is the output of the main quantum channel and is the noise quadrature of the type - b side channel input with , and the monitoring detector measures the quadrature , then the weighted difference ( and similarly for the quadrature ) is free from the influence of the side channel for and . therefore , the additional optimized monitoring on the output of the noise - infusing side channel resulting in the detection of can _ completely _ remove the negative impact of such a side channel . note that any pair of coefficients satisfying fully restores the performance of the protocol leading to , and the linear scaling of the latter observable does not affect the lower bound on the secure key rate .the complete removal of the noise - infusing side channel is possible also with the imperfect detectors .if both the main homodyne detector and the monitoring detector have efficiency and excess noise , which can be modeled by coupling of the signal to the noise mode with the variance on the coupler ( this is the standard model of the imperfect homodyne detector used in the security analysis of cv qkd ) , then the settings and given above also remove from the weighted difference and the variance then reads .that is , the optimal monitoring of the side - channel output then becomes equivalent to the side - channel - free detection of the signal on the same imperfect homodyne detector ( the details of the calculations are given in the appendix ) , which contains only the trusted noise and thus does not lead to the security break in the reverse - reconciliation scheme .  to calculate the security against the collective attacks we consider the scheme using the equivalent interferometric setup , when the residual side channel is coupled to the signal and then detected ( see the appendix for details )this leads to the appropriate transformations of the variances and correlations .the results of calculations are given in fig .[ krbs ] without the side - channel monitoring and with optimal monitoring of the residual side channel .it is evident that the optimal side - channel monitoring restores the performance of the protocol providing exactly the same key rate as in the absence of the side channel .the noise - infusing side channel can therefore be _ completely _ compensated for .simultaneously , the gaussian entanglement between the trusted parties is fully restored even if it was previously broken by the effect of noise infused in the side channel .experimental aspects of noise cancellation by the measurement have been studied in , which demonstrates the feasibility of such a method for cv qkd .note that the result reported here is obtained under different conditions than the previous analysis of the multimode channels , where the auxiliary channels received by bob contain information encoded by alice , i.e. , are parallel to the main quantum channel .   and of the couplers and by the phase shift ) to the noisy side channel with untrusted input of variance on the receiving side and the monitoring of the side - channel output followed by the data manipulation . ]we also consider the side - channel noise infusion based on the generalized interferometric interaction modeled by two couplers with different transmittance values and and a phase shift in one of the arms between the couplers as shown in fig .[ scheme - b - adv ] . in this casethe monitoring of the side - channel output suggested above can fully restore the performance of the protocol only when the phase shift is absent ( i.e. , , see the appendix for the details ) and the optimal coefficients of the data manipulation read and .the setting can be obtained by maximizing the mutual information between the trusted parties and therefore does not require the estimation of and independently .however , when the nonzero phase shift is present and the output of the interferometric coupling contains combinations of and quadratures of the signal and the noise input , simple side - channel monitoring by the homodyne detection in the quadrature and the linear data manipulation are not sufficient to completely restore the performance of the protocol .it can can be used to partly compensate for the negative influence of the type - b side channel , as shown in fig .[ kr - b - adv ] , where the key rate is plotted for the coherent- and squeezed - state protocols with respect to the weighting ( assuming ) , which can maximize the mutual information and , respectively , the key rate .( upper lines ) and the coherent - state protocol ( lower lines ) in the absence of the type - b side channel ( black horizontal dotted lines ) , with no phase shift ( dashed blue lines ) , with ( green dot - dashed lines ) , and ( red solid lines ) .the parameters of the interaction in the presence of the side channel are and , the modulation variance , the channel transmittance is , and the protocols implementation is otherwise perfect . ]  the optimal data manipulation setting in the general case becomes the lengthy function on the parameters of the protocol , including the values of the coupling and the phase shift as well as the signal and modulation variances and the parameters of the channel . in order to improve the decoupling of the side channelan optimal additional phase shift can be applied prior to the control detection ( so that the mutual information is maximized ) or a more general strategy based on the heterodyne detection and subsequent data manipulation could be used and optimized similarly to the elimination of the cross - talk in the channel .we have studied the effect of the side - channel leakage and noise infusion on the trusted sides of the continuous - variable quantum key distribution protocols .the negative effect of the side - channel leakage on the trusted sender side leads to the degradation of the key rate and to the increased sensitivity of the protocol to the channel noise . at the same time, the side - channel noise infusion on the trusted receiver side can completely break the security of the protocols even upon pure channel loss .we suggested and examined the method of additional modulation applied to the side - channel input being under the control of a trusted sending party .we show that if the additional modulation is properly correlated with the main modulation on the signal and squeezing applied on the side - channel input in the case of the squeezed - state protocol , the negative impact of the lossy side channel can be completely removed .alternatively , we show the possibility to compensate for the negative impact of the noisy side channel on the receiver side by introducing the monitoring of the output of the side channel . since both methods work independently by completely removing the side channels , they can be combined in a single protocol . moreover , since the optimal settings for the methods are independent of the channel parameters , the methods can be applied by the trusted parties using only the parameters of their local trusted stations and do not themselves rely on the channel estimation .our result therefore describes the effective and feasible methods of compensating for the quantum side channels in a continuous - 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656 ( 1948 ) .f. grosshans , n. j. cerf , j. wenger , r. tualle - brouri , and p. grangier , quantum inf .* 3 * , 535 ( 2003 ) .here we provide detailed calculations for the security analysis of the above - described gaussian continuous - variable quantum key distribution protocols with side channels .the scheme of the protocols is given in fig .[ scheme ] .as mentioned , the channel is parametrized by transmittance ( loss ) and excess noise , while side channels are parametrized by coupling ( for the sender - side type - a lossy side channel ) and by coupling and excess noise ( for the receiver - side type - b noise - infusing side channel ) .the protocols in the prepare - and - measure ( pm ) setting are based on the preparation of a signal state ( coherent or squeezed ) characterized by the quadrature values and which are gaussian distributed around zero with variances , , where is generally the squeezed variance which in the case of coherent states is saturated by . here and in the following , with no loss of generality , we assume that the states are squeezed and measured in the quadrature . the results for the quadrature squeezing and measurement are obtained by replacing and vice versa .the signal is modulated by applying the displacement and randomly chosen from a gaussian distribution centered around zero with variance so that the resulting quadrature becomes . here and in the followingthe equivalent expressions apply to the quadrature since the main quantum channel and the side channels are assumed to be phase insensitive ( which is valid for typical optical channels such as optical fiber or free - space links ) . nowif the channel is present then quadrature values after the channel are given by , where is the quadrature value of the vacuum state coupled to the signal to describe the loss , and is the quadrature value of the excess noise , .if the side - channel loss is present at the sender side ( type - a side channel ) , then the signal is coupled to the vacuum input of the side channel , which is modeled by a beam splitter with transmittance , which is the side - channel loss . as it was mentioned , the quadrature that enters the quantum channelis then changed to , where is the quadrature value of the vacuum state on the input of the beam splitter , .if the noise - infusing side channel is present , then , as mentioned in the main text , the output of the quantum channel is further modified to , where is the input noise of the type - b side channel with . knowing the transformation of the quadrature values we can obtain the variances and correlations between the quadratures andderive the covariance matrices describing the states shared between the trusted parties alice and bob and available to an eavesdropper eve , which are then used in the security analysis below .   as mentioned , we estimate the security of the protocols in the presence of the side channels and upon additional manipulations aimed to remove the side channels , as the value and positivity of the lower bound on the key rate , which in the case of collective attacks ( when eve is able to collectively measure her probe states after interaction with the signal ) and reverse reconciliation reads , where is the postprocessing efficiency that takes into account the amount of data that trusted parties lose due to imperfections of the error correction algorithms , is the mutual information between the trusted parties , and is the holevo bound , giving the upper bound on the information that is available to eve . in the case of individual attacks ,when eve is limited by the individual measurement on her probe states , the holevo bound is replaced by the classical shannon information between eve and bob .   in order to calculate the classical ( shannon ) mutual information we use the expression for shannon entropies in the case of gaussian continuous distributions  where and are two zero - mean gaussian random variables with variances and , respectively , and is the conditional variance with the correlation ( covariance ) between and .note that ( [ ixy ] ) is symmetrical with respect to and . in the case of the gaussian protocols considered in the paper ,the variables are the quadratures displacements being introduced by modulation and the quadrature values measured on the remote side of the channel and by a potential eavesdropper are all gaussian distributed .this allows us to calculate the mutual information and upper bound the information leakage in the case of individual attacks .the calculation of the mutual information is straightforward . following the expression ( [ ixy ] ) , we can derive the mutual information between alice and bob as , where is the variance of the data imposed by alice by displacement ( typically equivalent to ) , while conditional variance involves correlation , i.e. , the covariance between the data kept by alice and the data measured by bob , and the variance of bob s measurement results ( which is if the type - b side channel is present ) .the calculation of eve s information , , in the case of individual attacks , is similar .it requires knowing the variances of the modes that are available to eve for the individual measurements and correlations with the measurement results on the side of bob ; these will be derived in the particular cases below .   in the case of collective attacks , as mentioned , the information , which is available to eve , is bounded by the holevo quantity , which is the capacity of a bosonic channel between eve and bob .it is calculated as , the difference of the von neumann ( quantum ) entropies of the state of the modes , which are available to eve for a collective measurement described by the covariance matrix , and of the same state conditioned on the measurement results of bob .  in the general case ,when the excess noise is present in the channel and/or in the type - b side channel , we use the purification method , i.e. , we assume that an eavesdropper eve can purify the state shared by the trusted parties , so , where ab is generally a multimode initially pure state shared between the trusted parties in which all the impurity is assumed to be caused by eve s collective attack .after bob s projective measurement of one of the quadratures similar equivalence holds for the conditioned states : .thus the holevo bound in eq .( [ security2 ] ) is expressed as .the entropy is determined from the symplectic eigenvalues of the -mode covariance matrix as \:,\label{eq : von_neumann}\ ] ] where is the bosonic entropic function  the subtrahend in the expression for the holevo bound is the entropy similarly determined by the symplectic eigenvalues of the respective conditional covariance matrix after bob s projective measurement ( with no loss of generality , we assume measurement of the quadrature ) : ^{mp}\sigma_{b|a}^t\ : , \label{conditional_covm}\ ] ] where is the correlation matrix between mode b and the rest of the trusted modes , , where denotes a diagonal matrix , and is the moore - penrose pseudoinverse of the matrix .the purification is typically based on introducing the entangled [ also referred to as einstein - podolsky - rosen ( epr ) ] sources , which are the two - mode vacuum states described by the covariance matrices of the form where is the variance of each of the two modes , is the 2x2 unity matrix and .it is assumed that alice is performing a homodyne ( in the or quadrature ) or heterodyne ( in the and quadratures simultaneously using two homodyne detectors on the signal , split on a balanced beam splitter ) measurement on one of the modes , which conditionally prepares the squeezed state with variance or coherent state in the other mode , respectively .the unmeasured mode is then being sent through the channel and the side channels .such a scheme is then equivalent to a pm scheme based on squeezed or coherent states with or , respectively ( depending on the homodyne or heterodyne measurement applied by alice ) and .the mode interactions in the side channels and the main channel based on the liner coupling are taken into account in the covariance matrices using the input - output relations for the quadrature vectors of interacting modes 1 and 2 in the form where stands for the transmittance of a coupling beam splitter .such transformations lead to changes of variances and covariances that form the resulting covariance matrices .the lower bound on the key rate secure against collective attacks is then calculated numerically using [ eq : von_neumann ] and [ eq : bosonicef ] . in the case when modulation is independent of the signal squeezing the more general entanglement - based scheme is used instead of the standard epr - based purification described above .the purification method allows us to analyze the security of the protocols in the conditions of untrusted noise by estimating the lower bound on the secure key rate and in particular to study the region of insecurity where the lower bound turns to zero .further , we describe the theoretical purification schemes used to calculate the holevo bound in the particular cases .note that the purification schemes give also the same mutual information as in the pm versions of the protocols .we also cross - check our results using the entangling cloner collective attack being the particular purification of the channel noise by an epr source possessed by eve , which is also widely used in cv qkd security analysis as a typical collective attack ( see , e.g. , ) .the results obtained using the entangling cloner exactly confirm our calculations based on the purification models .   in the case of the type - a side channel ,the variance of alice s data is unchanged and remains and the correlation between alice and bob is scaled by the channel and the side channel so that .the variance of the state measured by bob in the quadrature after the side channel and the main noisy and lossy channel + 1 ] [ which also changes the mutual information ( [ typeamie ] ) ] , is unchanged by the channel noise , and .the correlations are , , and . from this the conditional variance }\ ] ] can be obtained and used to calculate the key rate secure against the individual attacks in a noisy channel .  in the case of collective attacks in a noisy channel and no additional manipulation on the side - channel inputthe security is calculated through the 4x4 covariance matrix \mathbb{i } \\\end{array } \right ) , \label{covm1}\ ] ] which describes the state shared between the trusted parties in the epr - based version of the protocols .the conditional matrix after the measurement at bob in particular contains separately from and reads from these two matrices the security of the protocol can be analyzed for the case of collective attacks .we do not provide the explicit expressions for the multimode covariance matrices in the further analysis since they are too lengthy ; thowever , they can be directly obtained using the input - output relations [ inout ] and the details of the purification schemes given below .further , we present the purification schemes for the different methods of the side - channel decoupling as well as the changes of the variances and correlations of measured data upon the additional manipulations on the side channels .  first ,if the uncorrelated noise is added to the input of the side channel , it is modeled as the coupling of one of the modes of the epr source n [ see fig .[ scapur12 ] , ( a ) ] to the side - channel input using a strongly unbalanced beam splitter with transmittance ( for mode ) .the variance of the source is respectively set to ; this way the noise is added losslessly .the state of the modes abcd contains all the purified trusted noise and the holevo bound for the standard gaussian protocols is then calculated following the purification method as .if the known modulation is applied to the side - channel input , then the purification is based on a similar scheme but the second mode of the epr source n is coupled to the mode a , measured by alice . this waythe displacement that is applied to the input of the side channel is also added to the displacement measured by alice [ see fig .[ scapur12 ] , ( b ) ] .alice s data in this case have the variance , while the correlation with bob after the side channel and the main channel is given by .bob s measured variance is + 1 ] . from thisthe expression for the mutual information can be obtained in the general case .  for the optimal and with the side - channel input substituted by the squeezed state with variances and ( in the case of the squeezed - state protocol ) it is easy to see that the main signal mode and the side - channel output described by and , respectively become completely uncorrelated , i.e. , .therefore , the side channel becomes decoupled from the main signal . for the calculations of the holevo bound the equivalent scheme depicted in fig .[ scheme - corr ] must be purified .this is done by introducing the epr source mn ( see fig . [ scapur3 ] ) with variance  ] , which in the regime of and gives , corresponding to the modulation of the signal state with variance in both the quadratures with only one value being kept .our purification scheme ( see fig .[ scapur3 ] ) is therefore equivalent to the pm one ( shown in fig .[ scheme - corr ] ) , providing ( in the limits and ) the same variances and correlations , and resulting in the same conditional states .moreover , the developed scheme allows purification of practically any pm scheme being more adjustable than the standard epr - based approach .the asymmetrical modulation can be introduced by the general preparation of the state using two different orthogonally squeezed states , however , such an extension was not needed in the tasks of the present paper .  in the purification scheme the state of the modes is pure , while the channel noise and loss introduce impurity to the state .the mode is not relevant in the scheme since it is uncorrelated from the rest of the setup due to the equality of the variances of modes and prior to the side - channel coupling ( it is shown on the scheme only for explanatory purposes ). then the holevo bound is calculated as .   in the case of the squeezed - state protocol as depicted in fig .[ scheme - corr ] .the source produces an infinitely squeezed state , the entangled source epr : mn provides the modulation of the signal , and the trusted parties perform homodyne detection on their respective modes a and b. ]   in the case of the type - b side channel the variance of alice s data remains and the correlation between alice and bob is scaled by the channel and the side channel so that .the variance of the state measured by bob in the quadrature after the main noisy and lossy channel and the side channel is +(1-\eta_b)v_n ] , where is the main signal mode before the side - channel interaction and is the side - channel input prior to interaction .it is easy to see that upon the resulting quadrature of mode contains no side - channel noise due to the destructive interference and the negative effect of the side channel is removed completely .note that if the reflection of the mode on the beam splitter would be absent and the mode would be directly coupled to the mode on the beam splitter with transmittance , the equivalent measurement that removes the type - b side channel would be on the mode upon .the described scheme allows calculations using the purification method simply as since the side - channel output monitoring emulated by the interferometric setup does not change the purity of the states .the performance of the protocol thus becomes equivalent to the one of the protocol without the type - b side channel , which is confirmed in the case of collective attacks in a noisy channel . in the case of the generalized preparation( when modulation is independent of the signal state variance ) we apply a similar scheme but replace the epv : v source with the generalized entangled state preparation as described in .  in the case of the interferometric - type interaction between the signal and the type - b side channel , as shown in fig .[ kr - b - adv ] , the mode transformations during the interactions become more complex and read  +x_{scb}\big[\sqrt{\eta_{b2}(1-\eta_{b1})}+ { } } \nonumber \\ & { } + \cos{\phi}\sqrt{\eta_{b1}(1-\eta_{b2})}\big]-p_b\sin{\phi}\sqrt{(1-\eta_{b1})(1-\eta_{b2})}+p_{scb}\sin{\phi}\sqrt{\eta_{b1}(1-\eta_{b2})}\end{aligned}\ ] ]  and + x_{scb}\big[\cos{\phi}\sqrt{\eta_{b1}\eta_{b2}}- { } } \nonumber \\ & { } -\sqrt{(1-\eta_{b1})(1-\eta_{b2})}\big]- p_b\sin{\phi}\sqrt{\eta_{b2}(1-\eta_{b1})}+ p_{scb}\sin{\phi}\sqrt{\eta_{b1}\eta_{b2}}\end{aligned}\ ] ]  now involving the contributions from the quadratures and of the signal and side - channel noise modes respectively , which is caused by the phase shift in the interaction .this prevents the complete decoupling of the type - b side channel by simple manipulation on the homodyne measurement results in the form , as illustrated in fig [ kr - b - adv ] ( plotted based on the numerical calculations using the equivalent transmittance in the purification - based scheme ) .the complete decoupling in such a case is possible only when and the cross - quadrature terms are absent .

Abstract: the role of the side channels in the continuous - variable quantum key distribution is studied . it is shown how the information leakage through a side channel from the trusted sender station increases the vulnerability of the protocols to the eavesdropping in the main quantum communication channel . moreover , the untrusted noise infusion by an eavesdropper on the trusted receiving side breaks the security even for a purely attenuating main quantum channel . as a method to compensate for the effect of the side - channel leakage on the sender side , we suggest several types of manipulations on the side - channel input . it is shown that by applying the modulated coherent light on the input of the side channel that is optimally correlated to the modulation on the main signal and optionally , introducing additional squeezing in the case of the squeezed - state protocol , the negative influence of the lossy side channel on the sender side can be completely removed . for the trusted receiving side , the method of optimal monitoring of the residual noise from the side - channel noise infusion is suggested and shown to be able to completely eliminate the presence of the noisy side channel . we therefore prove that the side - channel effects can be completely removed using feasible operations if the trusted parties access the respective parts of the side channels .


fractal theory has presented a growing interest in many applied areas in the last decades , for instance , in medicine , physics , computer science , engineering , among many other fields .most of these applications employ the fractal dimension to describe objects that be classified or simply described in some manner .fractal dimension provides important information about the object . while in a fractal, the dimension measures the `` fractality '' , in a real - world object , it expresses the spatial occupation of the structure .more practically , this implies that fractal dimension is capable of capturing important physical and visual attributes , like , or the repetition of geometrical patterns .  despite its importance as a powerful descriptor ,fractal dimension is still limited in the representation of more complex structures .this is true mainly in the analysis of real world objects , when the level of `` fractality '' varies along the same object .the literature shows some approaches to solve this problem , , and .this work is focused on the fractal descriptors solution .  here, we propose a novel fractal descriptor based on the triangular prism fractal dimension and apply it to the discrimination and retrieval of texture images . in this approach, the fractal dimension is estimated different scales of observation and a weight parameter is introduced as an exponent in the sum of the areas , changing the influence of each scale in the final result and ensuring a more complete and flexible description of the image .the proposed technique is tested over two well - known .the results of the classification and retrieval of such data sets are compared to other classical and recent texture analysis methods .the results confirm that the proposed descriptor is a valuable tool for image analysis tasks .texture analysis is a paradigm where the image is described in terms of statistical patterns formed by spatial arrangements of pixel intensities .the first known systematic study on this topic was carried out by haralick and his co - occurrence matrices . since then , a large number of methods on texture analysis have been proposed in the literature . among the most successful approaches one can mention local binary patterns , bag - of - features , scale - invariant feature transform , spatial pyramid matching , invariants of scattering transforms , fast features invariant to rotation and scale of texture , and others .  during the last decades , another branch of methods that have presented interesting results in texture analysis , especially on natural images , are those based on fractal geometry , particularly multifractals , multiscale fractal dimension and fractal descriptors . in this context, this work proposes the study and application of fractal descriptors based on the estimative of the fractal dimension using a tessellation of triangular prisms .our proposal has some particular characteristics that distinguish it from other approaches in the literature .first , rather than pre - selecting preferable regions in the image as in , here all pixels and scales are equally important _ a priori _ , which simplifies the modelling and interpretation of the texture descriptors . another difference from methods such as those in is that image invariances are not treated explicitly , although the underlying model and multi - scale process ensures that such effect is attenuated in practice .this is confirmed here in the experimental analysis and avoids the use of cumbersome strategies when in many cases invariances are not a critical issue or even when , for example , a rotated texture should be interpreted as a different object .finally , an important distinction should be done from approaches such as those in where direct relations are established based on te pixel values .here there is a complete and well - defined physical model behind the statistics extracted from the image , causing it to be more precise in most cases and more robust to deformations usually found in natural structures .fractal geometry many diverse areas .this is motivated mainly by the flexibility of fractal theory in objects , which can not be represented through conventional euclidean geometry .fractal theory also presents a concise and powerful framework to describe and identify a natural object , based primarily on the fractal dimension concept . roughly speaking, fractal dimension measures the complexity of a structure . in this case, complexity is related to the property of presenting details at different of observation . in this way ,fractal dimension is of particular importance because it is strongly related to fundamental physical and visual features of the object , , , , others .the following section describes in a few words some important aspects of fractal geometry theory and its application to texture analysis .fractal dimension is formally defined as being the hausdorff - besicovitch dimension of a geometrical set of points that composes the fractal .  in order to define the hausdorff - besicovitch dimension , be defined : where  the behavior of against , observes that jumps from to at a particular real non - negative value of .this value is the or fractal dimension .although the above definition is consistent and be applied to any set of points immersed in the euclidean space , it to be difficult or even impracticable in many situations where the fractal dimension of an object has to be estimated . with the aim of simplifying the computation in such situations , an approximate discrete version of the dimension , known as similarity dimension , can be defined by : where is the number of with length used to cover the fractal object . actually , is a metric that can be generalized to a large sort of measures , both in spatial and frequency domain .this gives rise to a lot of methods for estimation of fractal dimension , like bouligand - minkowski , box - counting , fourier , etc .the triangular prism dimension employed in this work is an example of method derived from the similarity dimension .this method for the estimation of fractal dimension , proposed in , is based on the relation between the surface area of a triangular tessellation of the gray - level map and the dimension of the base of each triangle .the image is divided into a grid of squares with side - length . for each square, a triangular prism is constructed using the pixel intensities in each corner as the heights and a central point whose height is given by the average of the corner heights .thus in a gray - level image , be the pixel intensities delimiting the grid square , such that : the center of the prism has height given by the simple average :   figure [ fig : tri ] depicts a scheme of the prism construction .the set of prisms composes a tessellation surface , such that the total area of this surface can be computed by using some geometrical procedures .thus for each prism , the of each face , , and is given by : : the total area of each triangular prism is provided by : where the total area of the surface is computed by summing the area of each prism in the grid with step : where is the set of points in the grid with step .this procedure is repeated for a range of values of and in each step the area is estimated .the fractal dimension is extracted from the log - log relation between and , such that , , where is the slope of a straight line fit to the curve .fractal descriptors are an extension of fractal dimension concept . actually , although fractal dimension is it is still insufficient to represent .we fractals the same fractal dimension despite their completely different .such situation is when we deal with objects from the real world , which are not real fractals . in these structures ,we find different levels of `` fractality '' according to the or even to the spatial region analyzed . in this contextwe need a tool capable of the object in all its extension .fractal descriptors a solution to fill this gap , possible a richer analysis of .figure [ fig : desc ] illustrates two distinct texture images whose fractal dimensions ( fd ) are similar .the fd estimation is not enough distinguish the images .for the same images , figure [ fig : desc ] shows the normalized fractal descriptor curves , which demonstrates capable of discriminating the textures in a straightforward manner .essentially , the purpose of fractal descriptors is to estimate the fractal dimension under different scales , providing information of different patterns and arrangements present in the structure . a natural candidate to allowthis analysis is the power - law intrinsic to the fractal dimension . from the we can write : where is any measure , related to the spatial or frequency distribution of the object and is the scale parameter .the power - law relation may be stated in a quite simple fashion :   fractal descriptors consist in extracting features from the following function :   the function may be used in different manners directly , as in , or after a multiscale transform or by the application of principal component analysis or still functional data analysis in , among many other possibilities .this work proposes a novel fractal descriptor based on the fractal dimension estimated by triangular prisms . in this case, the fractality function corresponds to the area function .following the general idea of fractal descriptors , the proposed features are provided by ., to improve the ability of identifying multiscale patterns along the texture , the proposed method introduces an exponent weight to the area sum , which now has the following expression : where is defined as in equation [ eq : area ] .  for image analysis purposes , a range of values is chosen empirically and the following values are taken into account to compose the descriptors :   finally , since the above expression generates a too much large set of features , a reduction of dimensionality is necessary .such procedure is carried out by a karhunen - love ( kl ) transform .thus let be the feature matrix , that is , a matrix each line corresponds to the vector of a sample and each column is a descriptor .the covariance matrix is defined by : where is the number of descriptors in , is the column of and is the average of the vector . in the following a second matrix defined as having in each column the eigenvectors of , sorted according to decreasing values of eigenvalues .finally , the kl transform of is obtained by now , is the new matrix of features .it has the same dimensions of and each line in contains the final descriptors of the respective sample .based on the kl transform theory , the first descriptors are the most meaningful for the analysis . here , the descriptors are considered in this order and , as it will be described in results section , the number of such descriptors is varied between 1 and a predefined maximum , to find out the best configuration empirically .figure [ fig : method ] summarizes the described steps in a visual diagram .the proposed method combines two types of information to provide a rich and reliable image descriptor .the first is the complexity , measured by the fractal dimension .such property is strongly correlated with physical characteristics of the object .the second one is the multiscale analysis that is accomplished by the parameter . using the curve of ensures that the complexity information is expressed along a range of scales .finally , the exponent provides an empirical weight for each scale , making the proposed descriptors more flexible to address the description of so diverse objects .all this combination yields a method capable of identifying and discriminating objects even in severe situations as when there is high variability among elements of a same class and/or high similarity among samples of different classes .figure [ fig : descriptor ] visually illustrates the discrimination of two classes of textures by the proposed method .      to better understand how and why tringular prisms work on the image, we should look at its mathematical interpretation .the classical analysis of the relation between triangular prisms and fractal theory is that adopted in .it relies on the idea of extending the `` walking - dividers '' approach to a two - dimensional manifold .here we propose a second interpretation based on the theory of fractional brownian motion ( fbm ) .our first objective is to establish the relation between the prism areas and the pixel gray levels in the image .we illustrate the computation in face .as can be written as a function of we can rewrite and : we can also plug the definition of into the area : replacing , and with their respective representation in terms of , , and we end up with the following expression : ^ 2 }     } { 4}\\      = \frac{\epsilon}{4 } \sqrt{1.25a^2 + 1.25b^2 + 0.25c^2 + 0.25d^2 - 1.5ab - 0.5ac - 0.5ad - 0.5bc - 0.5bd + 0.5cd + \epsilon^2}\\     = \frac{\epsilon}{8 } \sqrt{5a^2 + 5b^2 + c^2 + d^2 - 6ab - 2ac - 2ad - 2bc - 2bd + 2cd + 4\epsilon^2}\\      \end{array}\ ] ] rearranging terms we can write the above expression as the expression enclosed by the square root contains three squared terms and the leading one is .the points in the image where the intensities are and are separated by a distance .therefore the distribution of this term is classically related to the fbm process .an fbm process is a gaussian non - stationary stochastic process , with mean zero and variance , whose covariance statistics satisfies where is the expected value and is a parameter named hurst exponent .another way of defining the same process is to write   seminal works on fractal geometry in images , such as those of mandelbrot and pentland , already demonstrated that fractal characteristics in images reflect fractality in the physical process originating the pictured object .pentland even carried out a survey to discover that such relation can be perceptually confirmed by human beings .since then , a number of studies have been based on this assumption and fbm has been considered the canonical representation for such physical processes giving rise to fractal images .  in this context , the expression ( [ eq : var ] ) has been paramount .if the image obeys the statistics in ( [ eq : var ] ) with an acceptable -value the image can be analyzed as a statistical fractal .furthermore , as demonstrated in , the fbm has fractal dimension ( in the sense of hausdorff - besicovitch ) equals to with probability .  in practiceit is usual to verify the statistics ( [ eq : var ] ) in an image by checking the curve of  where stands for the average and and are points separated by .this is in essence what is expressed by the term in ( [ eq : trifbm ] ) .the second squared term ( ) can be seen as a correction taking into account the influence of the neighbor pixels in the fbm covariance .similarly , the parameter appears as a weight expressing the scale size .we can conclude from this that the triangular prisms are capable of representing two classical viewpoints in fractal geometry , to know , a geometrical interpretation such as `` walking - dividers '' and a statistical approach derived from fbm . back to the context of fractal descriptors , the statistics ( [ eq : var ] )does not to fit perfectly the image .rather what we expect to find is how the image is more or less close to a fractal in each scale . andthis analysis highly depends on the richness of the measure provided by the method employed to estimate the fractal dimension . in this way ,a method with a more complete description of the fractal process , both in a geometrical and a statistical sense , tends to be more appropriate than other classical methods in the literature , hence explaining the success achieved by the proposed approach .now we turn our attention to the exponent in ( [ eq : desc ] ) .although a generalized version of the multinomial theorem with real exponents exists , it is not straightforward to be interpreted . therefore we opted for employing maclaurin series up to the second order .the general expression for a real multi - variable function is : we will replace the first term in the summation ( [ eq : desc ] ) by and all the others by resulting in the function .plugging the power function derivatives : whereas the first summation term is related to the mean of the following ones relate to the correlation between groups of variables .the exponent gives weight to these terms and the higher the exponent the larger the influence of the correlation within larger groups of variables .correlation is a well - established statistics in texture images as it is related to haralick second - order statistics and quantify such property in terms of the prism areas and indirectly on the local fractality of the image .altogether the combination of an accurate fractal analysis with an intrinsic correlation is what allows for the analysis proposed here to be powerful even in the most challenging scenarios such as when we have high variability among images of a same group or when images from different groups can be confused by more traditional approaches in the literature , where the modelling may be not sufficiently solid and flexible at the same time neither metrics employed can express the parameters of the underlying model with the necessary accuracy .the following pseudo - code contains the algorithm for the triangular prism descriptors .essentially it is not too much than a mere translation of the mathematical expressions in section [ sec : tri ] . a simplified version written in matlab/octavecan also be found at ;   proposed method is tested over two data sets , that is , brodatz and vistex .brodatz is a gray - level texture database composed by 111 texture images divided each one into 16 windows with 128 pixels without overlapping .vistex is composed by 54 color texture images , each one divided into 16 128 windows .the color images are converted into gray - level ones before the descriptor analysis . in both experiments ,each entire image is a class and each window corresponds to a sample .the classification is performed by a linear discriminant analysis ( lda ) method .the classification of each is achieved by inputting the proposed descriptors and other well - known texture descriptors to the classifier , using a 10-fold scheme , where one tenth part of the database is used for testing and the remaining samples for training .the classification results are compared with other state - of - the - art and classical texture descriptors in the literature , that is , local binary patterns ( lbp ) , gabor - wavelets , multifractal spectrum , gray level co - occurrence matrix ( glcm ) and fourier descriptors .the proposed method depends on an exponential parameter that can be to find the most suitable descriptors for each type of image . in this way , to find the best parameters is a fundamental task to ensure a robust solution . in real situations , these variables can be empirically evaluated over the training set .figure [ fig : exponent](a ) shows the precision in the classification of brodatz data set when using exponents ranging between -1 and 2 . outside this rangethere is no significant gain .based on this plot , the triangular descriptors with exponents between 1 and 1.8 were concatenated and summarized through pca to provide the texture descriptors used the following experiments .descriptors are used for each exponent .  [ cols="^,^ " , ]   the proposed method differs from other approaches to texture analysis in the literature in that the physical process underlying the generation of the image is taken into account to a high extent , as we demonstrated its relation with fractional brownian processes .these processes are known to be widely present in nature and its association with the way that our brain perceives materials ans scenarios around us is well established .parallel to this , the triangular prisms also encompass a geometrical analysis similar to `` walking - dividers '' , intimately related to the primitive idea of fractals and also commonly observed in natural structures . in itself , such complementary and dual viewpoint of fractality is an important contribution and novelty with regards to other fractal approaches such as multifractals , bouligand - minkowski , and others .the consideration of the physical process also showed to be more advantageous than simply quantifying relations between neighbor pixels without accounting for the semantic involved in the represented object , as in lbp and glcm descriptors .even though the interpretation of these approaches can be considered more straightforward , the lack of a more realistic model makes them insufficient in more complex cases with larger and more heterogeneous databases as those presented here .finally , we also compare to other methods where the image is observed beyond the simple pixel values , such as in fourier and gabor descriptors .these descriptors , however , are ultimately based on linear filtering and are not the ideal solution to explain the non - linearities present in many natural images , mostly caused by the chaotic behavior associated to fbm processes .generally speaking , the great performance of triangular fractal descriptors in the presented results confirm what is expected from the fractal analysis of natural texture images . indeed, fractal geometry constitutes a reliable tool to model those structures , which can not be well - described in an euclidean framework .this analysis is especially complete here as two fractal viewpoints ( statistical and geometrical ) are merged to provide an even more accurate description of the image .furthermore , fractal descriptors approach enhances the conventional fractal geometry analysis by extracting relevant information of complexity under different scales , by the multiscale analysis implicit when the dimension is estimated for different values of . in this way , these descriptors are capable of providing features representing precise and rich measures of the spatial structure of complex textures like those in benchmark data , as used here , as well as in practical application involving the identification and discrimination of intricate patterns in objects and scenarios represented in a digital image .this work proposed a novel gray - level texture descriptor based on a particular method of fractal dimension estimation , named triangular prism method .the descriptors were obtained by combining values of area of a triangular tessellation of the texture , using different steps for the tessellation grid as well as different values for an exponential parameter used as a weight for the area within each cell in the grid .these descriptors were compared to classical and state - of - the - art texture descriptors in a task of classification of two well - known benchmark texture datasets , e.g. , brodatz and vistex .the classification results illustrated that the novel approach is a valuable descriptor , achieving remarkable results in image classification and retrieval tasks and suggesting its application to a wide range of problems comprising tasks of pattern recognition in digital images .o. m. bruno gratefully acknowledges the financial support of cnpq ( national council for scientific and technological development , brazil ) ( grant # 307797/2014 - 7 and grant # 484312/2013 - 8 ) and fapesp ( the state of so paulo research foundation ) ( grant # 14/08026 - 1 ) .j. b. florindo gratefully acknowledges the financial support of fapesp proc .2013/22205 - 3 and 2012/19143 - 3 .s. lazebnik , c. schmid , and j. ponce . beyond bags of features : spatial pyramid matching for recognizing natural scene categories . in _ proceedings of the 2006 ieee computer society conference on computer vision and pattern recognition - volume 2 _ , cvpr 06 , pages 21692178 , washington , dc , usa , 2006 .ieee computer society .e. t. m. manoel , l. da fontoura costa , j. streicher , and g. b. mller .multiscale fractal characterization of three - dimensional gene expression data . in _sibgrapi _ , pages 269274 .ieee computer society , 2002 .r. o. plotze , j. g. padua , m. falvo , m. l. c. vieira , g. c. x. oliveira , and o. m. bruno .leaf shape analysis by the multiscale minkowski fractal dimension , a new morphometric method : a study in passiflora l. ( passifloraceae ) ., 83(3):287301 , 2005 .

Abstract: this work presents a novel descriptor for texture images based on fractal geometry and its application to image analysis . the descriptors are provided by estimating the triangular prism fractal dimension under different scales a weight exponential parameter , followed by reduction karhunen - love transform . the efficiency of the proposed descriptors is tested on two well - known texture data sets , that is , brodatz and vistex , both for classification and image retrieval . the novel method is also tested concerning invariances in situations when the textures are rotated or affected by gaussian noise . the obtained results outperform other classical and state - of - the - art descriptors in the literature and the power of the triangular descriptors in these tasks , suggesting use in practical applications of image analysis based on texture features .   carlos institute of physics , university of so paulo , po box 369 , 13560 - 970 , so carlos , sp , brazil . + scientific computing group - http://scg.ifsc.usp.br + of mathematics , statistics and scientific computing - university of campinas + rua srgio buarque de holanda , 651 , cidade universitria `` zeferino vaz '' - distr . baro geraldo , cep 13083 - 859 , campinas , sp , brasil  * _ keywords _ * pattern recognition , texture analysis , fractal descriptors , triangular prism


the modeling of interactions between ultrasonic waves and cracks is of great interest in many fields of applied mechanics . when the wavelengths are much larger than the width of the cracks , the latter are usually replaced by zero - thickness interfaces with appropriate jump conditions .linear models for crack - face interactions have been widely developed .however , these models do not prevent the non - physical penetration of crack faces .in addition , laboratory experiments have shown that ultrasonic methods based on linear models often fail to detect partially closed cracks .a well - known nonlinear model for cracks is the unilateral contact model .a more realistic hyperbolic model accounting for the finite compressibility of crack faces under normal loading conditions has been presented for applications in engineering and geomechanical contexts .the well - posedness and the numerical modeling of the latter model in the context of 1-d linear elastodynamics was previously studied in .the generation of harmonics and the distortion of simulated scattered velocity and stress waves were also observed .subsequent numerical experiments have brought to light an interesting property . in the case of a sinusoidal incident wave , the mean values of the elastic displacements around a crackwere found to be discontinuous .the simulations conducted also indicated that the mean dilatation of the crack increases with the amplitude of the forcing levels .this purely nonlinear phenomenon ( the linear models predict no dilatation ) is of physical interest : experimenters can measure the dilatation and use the data obtained to deduce the properties of the crack .a preliminary theoretical study based on the use of a perturbation method was presented in . an analytical expression for the dilatationwas obtained in that study and successfully compared with the numerical results . however , this expression gives only a local estimate , valid in the particular case of the hyperbolic model subjected to very low sinusoidal forcing levels .the aim of the present study is to provide a theoretical analysis which can be applied to any range of forcing levels and to a larger number of contact models .the present paper is organized as follows :  * in section 2 , the physical configuration is described .the family of strictly increasing convex jump conditions dealt with in this study is introduced .numerical simulations of the process of interest are presented ; * in section 3 , the problem is reduced to a nonautonomous differential equation . up to section 6 , only the case of sinusoidal forcing , which allows a complete understanding of the phenomena involved , is studied ; * in section 4 , preliminary results on the differential equation are presented .classical tools for dynamical systems are used : poincar map , phase portrait , lower and upper solutions ; * in section 5 , the main qualitative results of this study are presented .the mean , maximum and minimum aperture of the crack are bounded , and local estimates for small forcing are also proved ; * in section 6 , some of the previous results are extended to non - monochromatic periodic forcing conditions .the increase of the mean dilatation with the forcing parameter is proved in theorem [ theoremybargene ] .the most general quantitative result of the paper is given in equation ( [ ybarlocalgene ] ) ; * in section 7 , some conclusions are drawn about the physical observables .some future perspectives are also suggested .and with rough contact surfaces , under constant static stress .static ( left ) and dynamic ( right ) case , with incident ( i ) , reflected ( r ) and transmitted ( t ) waves . ]we consider the case of a single crack with rough faces separating two media and , which are both linearly elastic and isotropic , taking to denote the density and to denote the elastic speed of the compressional waves .these parameters are piecewise constant and may be discontinuous around the crack : if , if the media are subject to a constant static stress . at rest , the distance between the planes of average height is ( figure [ figcrack ] , left ) .elastic compressional waves are emitted by a singular source of stress at in , where is a median plane of the actual flaw surface .the wave impacting gives rise to reflected ( in ) and transmitted ( in ) compressional waves .these perturbations in and are described by the 1-d elastodynamic equations where denotes the causal stress source , is the elastic velocity , is the elastic displacement , and is the elastic stress perturbation around .the dynamic stresses induced by the elastic waves affect the thickness of the crack ( figure [ figcrack ] , right ) . the constraint \geq\xi_0-d>0 \label{penetration}\ ] ] must be satisfied , where =u^+-u^- ] .  [ cols="^,^ " , ]   let }y(t)\ ] ] be the minimum value of the -periodic solution .the following properties are satisfied by :  for all , we obtain the bounds :   at small forcing levels , we also obtain the local estimate   at infinite forcing levels , the crack is completely closed :   [ theoremymin ]  . proposition [ propositionty ] shows that is negative . as in the proof of proposition [ propositiony0 ], the existence of the lower nonporous fence means that .  .as in the proofs of proposition [ propositiony0 ] and theorem [ theoremymax ] , the local estimate when follows from a second - order taylor expansion of and from ( [ zeps0 ] ) .  .first we consider the property used in the proof of theorem [ theoremybar ] .evolution of in ( [ ode_y ] ) and the definition of , in proposition [ propositionty ] give & = & \displaystyle -\left(\int_0^{t_{y_1}}|f(y(\tau))|\,d\tau+\int_{t_{y_2}}^t|f(y(\tau))|\,d\tau\right)+\int_{t_{y_1}}^{t_{y_2}}|f(y(\tau))|\,d\tau , \end{array}\ ] ] hence secondly , let us consider the ode \displaystyle s(t_{y_1})=0 . \end{array } \right .\label{proofyminode}\ ] ] straightforward calculations and property ( iii ) of corollary [ corollaryty ] give with ( [ zeps0 ] ) .comparison between ( [ ode_y ] ) and ( [ proofyminode ] ) shows that is a lower solution of when :  ] and given in ( [ ode_y ] ) , must tend towards -1 when , to make blow up , which concludes the proof .+ the evolution of stated in theorem [ theoremymin ] is shown in figure [ figymin ] . at moderate forcing levels ( a ) , one can not distinguish between and its lower bound .at higher forcing levels ( b ) , one observes that is above its lower bound , as stated in theorem [ theoremymin ] .although no rigorous proof has been obtained so far , the numerical simulations indicate that decreases strictly as increases .most of the results obtained in sections [ secpreres ] and [ secquality ] were based on the simple analytical expression of sinusoidal forcing . in the case of more general forcing ,the key properties stated in proposition [ propositionty ] and corollary [ corollaryty ] are lost , which makes it impossible to obtain estimates such as theorems [ theoremymax ] and [ theoremymin ] .however , some properties are maintained if two assumptions are made about in ( [ odeorig ] ) : -periodicity , and a null mean value .the latter assumption is physically meaningfull : as deduced from ( [ vinc ] ) , a non - null mean value of results in an amplitude of that increases linearly with .since the techniques required are the same as those used in the above sections , the derivations will be shortened . setting the parameters and the fourier decomposition of the source as follows }\left|s(t/\beta)\right| , \quad a=\frac{\textstyle 2\,v_0}{\textstyle \beta\,d},\quad y=\frac{\textstyle [ u(\alpha,\,t)]}{\textstyle d},\quad f(y)=-{\cal f}(y),\\ [ 10pt ] \displaystyle g(t)=\frac{\textstyle 1}{\textstyle 2\,\rho_0\,c_0 ^ 2\,v_0}\,\left|s(t/\beta)\right|=\sum_{n=1}^{\infty } \left(a_n\,\sin n\omega t+b_n\,\cos n\omega t\right ) ,  \quad t=\beta\,t_0,\quad \omega=\frac{\textstyle 2\,\pi}{\textstyle t } , \label{adimgene } \end{array}\ ] ] we obtain the model problem \displaystyle f:\,]-1,\,+\infty[\rightarrow ] f_{\min},\,+\infty[,\quad \lim_{y\rightarrow-1}f(y)=+\infty,\quad -\infty\leq f_{\min}<0,\\ [ 6pt ] \displaystyle f(0)=0,\quad f^{'}(0)=-1,\quad f^{'}(y)<0<f^{''}(y),\\ [ 6pt ] \displaystyle g(t+t)=g(t),\quad \overline{g}=0,\quad \left|g(t)\right|\leq 1,\quad \max_{t\in [ 0,\,t]}\left|g(t)\right|=1,\quad 0\leq a<+\infty,\\ [ 6pt ] \displaystyle y(0)=y_0\in]-1,\,+\infty[. \end{array } \right .\label{ode_ygene}\ ] ] the isocline of zero slope of ( [ ode_ygene ] ) is    the following result generalizes the proposition [ propositionyperiod ] : since the same notations and processes are used , the proof is only sketched here .there is a unique -periodic solution of ( [ ode_ygene ] ) .this solution is asymptotically stable .[ propositionygene ]  three cases can be distinguished .this case is identical to case 1 in proposition [ propositionyperiod ] . . based on the funnel s theorem, the compact set \ ] ] is invariant under the flow of ( [ ode_ygene ] ) , and hence .a fixed point argument shows the existence of a -periodic solution .equations ( [ proofyperiodunique ] ) and ( [ proofyperioda0 ] ) still hold : the uniqueness and attractivity of are straightforward consequences of .the isocline ( [ isoclineygene ] ) is not defined when . in these cases ,the invariant set deduced from the isocline is the non - compact set , which makes it impossible to apply the fixed point argument . to build a compact invariant set , we take , which means that . as long as , integration gives lower and upper solutions of  from the -periodicity of , it follows that it is then only necessary to bound to establish that on  ] .-periodicity of and yield .the solution is therefore sought as a fourier series with null mean value injecting this series in the ode satisfied by and using ( [ adimgene ] ) provides now , parseval s formula leads to the last equality in ( [ zeps0gene ] ) .the next result extends the theorem [ theoremybar ] .it shows that a positive jump in the mean elastic displacement still occurs .the mean value of the -periodic solution in ( [ ode_ygene ] ) is positive and increases strictly with the forcing amplitude : at small forcing levels , the following local estimate holds where and are the fourier coefficients of the source ( [ adimgene ] ) .[ theoremybargene ]  based on the -periodicity and the null mean value of , the proof of theorem [ theoremybar ] can be straightforwardly extended to prove . to prove the second inequality, we proceed in three steps .first , as stated in the proof of proposition [ propositionygene ] , null forcing gives , and hence .secondly , lemma [ lemmazeps0gene ] gives .thirdly , the -periodic function satisfies ( [ ode_w ] ) .the sign of and in ( [ ode_ygene ] ) show that the isocline of zero slope ( [ isoclinew ] ) is positive or null : occurs only at points where vanishes .the proof of proposition [ propositionw ] therefore still holds here , yielding at all  ] and {\min} ] and {\min} ] , the source , and the physical parameters of and are known , then the second equation in ( [ result1 ] ) provides a straightforward mean of determining .note that the stiffness , which is involved in via the proposition [ propositionode ] , is classically measured using acoustic methods .it then suffices to measure } ] implies }=t_0^a - r_0^a+\frac{\textstyle v_0}{\textstyle \omega}. \label{conclueq1}\ ] ] on the other hand , the causality of the source and the continuity of the stress ( [ jcsigma ] ) mean that solving ( [ conclueq1 ] ) and ( [ conclueq2 ] ) gives }&=&\displaystyle -\frac{\textstyle \rho_0\,c_0+\rho_1\,c_1}{\textstyle \rho_1\,c_1}\,r_0^a+\frac{\textstyle \rho_1\,c_1-\rho_0\,c_0}{\textstyle \rho_1\,c_1}\,\frac{\textstyle v_0}{\textstyle \omega},\\ [ 10pt ] & = & \displaystyle \frac{\textstyle \rho_0\,c_0+\rho_1\,c_1}{\textstyle \rho_0\,c_0}\,t_0^a+2\,\frac{\textstyle v_0}{\textstyle \omega}. \end{array } \label{conclueq3}\ ] ] purely acoustic measurements of or therefore give }$ ] , from which the nonlinear crack parameter can be easily determined .the monotonicity of in ( [ fconcave ] ) - or equivalently in ( [ adimentionalize ] ) and ( [ adimgene ] ) - was the key ingredient used here to prove the properties of the crack : the strict increasing of - or the strict decreasing of - proves the uniqueness and attractivity of the periodic solution .in addition , the strict concavity of - or the strict convexity of - shows that the mean value of the solution increases with the forcing parameter . relaxing these hypotheses may lead to more complex situations , and require more sophisticated tools .one thinks for instance to the case of hysteretic models of interfaces .this paper was focused on the normal finite compressibility of a crack .coupling with shear stress is also an important topic to examine .lastly , further studies are also needed on networks of nonlinear cracks . these configurations frequently occur in applications .direct numerical simulations have been carried out in , but a rigorous mathematical analysis is still required .we are grateful to didier ferrand , martine pithioux and benjamin ricaud for their useful comments on the manuscript .many thanks also to jessica blanc for her careful reading .

Abstract: the interactions between linear elastic waves and a nonlinear crack with finite compressibility are studied in the one - dimensional context . numerical studies on a hyperbolic model of contact with sinusoidal forcing have shown that the mean values of the scattered elastic displacements are discontinuous across the crack . the mean dilatation of the crack also increases with the amplitude of the forcing levels . the aim of the present theoretical study is to analyse these nonlinear processes under a larger range of nonlinear jump conditions . for this purpose , the problem is reduced to a nonlinear differential equation . the dependence of the periodic solution on the forcing amplitude is quantified under sinusoidal forcing conditions . bounds for the mean , maximum and minimum values of the solution are presented . lastly , periodic forcing with a null mean value is addressed . in that case , a result about the mean dilatation of the crack is obtained .  elastic wave scattering , contact acoustic nonlinearity , nonlinear jump conditions , periodic solutions of differential equations , dependence of solutions on parameters  34c11 , 35b30 , 37c60 , 74j20