a recent development in radiotherapy treatment planning is to use an optimization process to obtain a treatment plan .to do so one needs to formalize the clinical objectives in a so called object function whose optimum should correspond to an optimal treatment plan .many current treatment planning systems rely on a quasi - newton algorithm to perform the optimization . in order to guarantee that this process indeed converges to an optimum the object functionshould at least be twice continuously differentiable .this last property is the main subject of this study .as it turns out the object functions used in radiotherapy do not always have this property so that convergence of the optimization process can not directly be guaranteed in those cases .in section [ sec : optbg ] we present some background for the use of optimization in radiotherapy treatment planning .the object functions currently used are built from so called dose - volume and eud constraints . by dose - volume constraints ,the dose - volume histogram enters the object function .it is the finite differentiability of the former that causes the insufficient differentiability of the latter . in section[ sec : objdiff ] this relationship is explained in more detail .a example illustrates how finite differentiability of the object function influences the optimization process .the practical consequences thereof are the subject of section [ sec : conc ] . where section [ sec : objdiff ] is kept informal , section [ sec : proof ] provides formal statements and proofs of the differentiability results .this last section does not contain new arguments but serves as a support for the exposition .in radiotherapy treatment planning a careful choice of values for treatment parameters is necessary to obtain a dose distribution in the patient that meets criteria for eradicating a malignant tumour . at the same timeit should meet criteria so that healthy tissues and organs are not critically damaged . due to the fact that dose is deposited in every irradiated part of the patient , these criteria are conflicting in almost every clinical situation .experienced planners are usually very good at intuitively balancing such conflicting goals , still there is a need for more objective and quantifiable methods to devise and evaluate treatment plans .this has led to the construction of object functions , depending on treatment parameters , which formalize the objectives of dose on tumours and healthy tissues , see . by optimizing this object function with respect to treatment parameters , one hopes to get an optimal treatment plan .another , very different , reason to use optimization in treatment planning is the emergence of imrt where the sheer number of treatment parameters simply precludes manual planning .even if we know that a function has a unique optimum , there is no simple recipe to find it .therefore we have to resort to numerical , iterative methods to solve optimization problems , which , given a starting point , generate a sequence converging to an optimum .the latter can be guaranteed only if certain conditions on the object function are met .it seems , however , that if these conditions are considered at all , it is taken for granted that they are fulfilled .many treatment planning systems use a quasi - newton method , like the lbfgs algorithm , to find an optimum of the object function , see for example or as a general reference .one of the conditions guaranteeing that this algorithm yields a sequence locally converging to an optimum is that the object function is twice continuously differentiable .the object functions currently used in treatment planning systems are limited to so called _ eud constraints _ and _ dose - volume constraints _ , or combinations of both , see for example , but also see section [ sec : objrt ] remark [ rem : compro ] .our results show that the former do satisfy the differentiability condition .the latter however , do not , that is not for all values of the treatment parameters .   the goal of radiotherapy treatment planning is to devise a treatment plan achieving a high _ tumour control probability _( tcp ) combined with a low _ normal tissue complication probability _ ( ntcp ) . a prioriboth tcp and ntcp are unrelated to treatment parameters . even relating them to criteria on the dose distribution in the patientis a non - trivial task . to simplify mattersone does not consider the 3-dimensional dose distribution , but a derived quantity namely the dose - volume histogram for each relevant region .the latter are tumours and healthy organs for which the dose - volume histogram tells which part of the volume receives at least a certain dose , see equation ( [ eq : volfun ] ) below .data from previously treated patients are used to relate tcp and ntcp to dose - volume histograms , see . at this pointfurther simplifications are necessary , because tcp and ntcp can only be related to certain points of dose - volume histograms or even only to averages . generally for tumoursthis leads to dose - volume constraints , that is a minimum and a maximum dose for volumes and respectively , or in a more concise notation and . in case of healthy tissues the situation is much more complicated , in practice one uses a mixture of dose - volume constraints and eud constraints , see .criteria for the dose distribution or the derived dose - volume histograms are related to treatment parameters , thus the treatment planning objectives can be expressed with an object function of these treatment parameters .now the object function used in treatment planning is a weighted sum of local object functions , one for each dose - volume constraint or eud constraint .let us consider a tumour with a minimum and a maximum dose - volume constraint .furthermore is the dose - volume histogram of the tumour for a given dose distribution and treatment parameters , that is is the ( relative ) volume of the tumour , region , receiving at least dose  in this expression is the volume of the region and is the dose distribution .one constructs what we will call _ local object functions_. for example the local object function for a minimum does - volume constraint for a specific volume is a function which increases the more the actual relative volume at is below and decreases or even becomes zero if the actual relative volume at is above .similarly a maximum dose object function is constructed such that it increases when the relative volume at is above and decreases or becomes zero otherwise .thus the object function for the tumour alone would be the weighted sum of two local object functions , usually of the following form where and are weights and although the object function is a function of the treatment parameters it does not depend on them directly , but only via the dose - volume histograms .the latter depend on the dose distribution which directly depends on the treatment parameters .the eud constraints are in a similar way expressed in the _ equivalent uniform dose_. a quantity used to measure the biological effect of radiation especially for healthy tissues , see .we denote the equivalent uniform dose by , here it is a function of treatment parameters ^{1/\alpha}.\ ] ]  * remarks *  1 .  [ rem : compro ] the word _ constraint _ in this context is somewhat misleading because a dose - volume nor an eud constraint is used as a constraint in the sense of constrained optimization .if they were , many if not most optimization problems in treatment planning would have no solution .it is a common clinical situation that a tumour is located adjacent to a critical organ . for several reasons the region , called ptv , on which the treatment dose is prescribed is larger than the tumour .see for a systematic description of xtv s , where x = g , c , i or p. therefore the critical organ is possibly partly located in the ptv .suppose for example that one third of the volume of the critical organ is located inside the ptv .furthermore suppose that the minimal dose - volume constraint on the ptv is 60 gy ( gy is unit of dose ) and the maximum average dose constraint on the organ is 20 gy . since the dose distribution varies continuously in space and its gradient is bounded , this constrained optimization problem will have no solution .due to the anatomy a minimum dose of 60 gy on the ptv implies an average dose well above 20 gy on the organ and vice versa .a weighted sum of local object functions is a way to find a clinically acceptable compromise for this situation . to avoid the term _ constraint_ some authors use the term _ soft constraint _ , but in my opinion this term is superfluous because we already have the notion of object function .nevertheless we adhere to the word constraint , in order to stay in line with the existing literature .[ rem : eud ] the equivalent uniform dose , as defined in equation ( [ eq : eud ] ) , can be regarded as a generalized average , in fact for it is the standard average of the dose distribution over the region .formally it is identical to the -norm of the dose distribution restricted to a region of the patient . but is called and it has a biological interpretation .one associates a value of to each tissue type , but there is no agreement yet in the literature about these values .[ rem : dvh ] the dose - volume histogram refers to the graph of a function that is usually given the same name .this function , defined for each in equation ( [ eq : volfun ] ) , can be regarded as a volume function in the sense of .let us fix for the moment .the dose - volume histogram of a region is a function whose value is the volume of receiving at least dose . in other wordsit gives the volume of a region enclosed by two level sets of the dose distribution , namely those where and the where ( this last level set will be a point ) .[ rem : treatpar ] any quantity that influences the dose distribution may be considered as a _ treatment parameter_. usually treatment parameters are _ beam intensities _ and _ beam angles _ , that is parameters of the treatment unit[ rem : planpar ] the object function in equation ( [ eq : objfun ] ) also depends on , , , , and .such parameters might be called _ planning parameters_. here we will keep them fixed .the numerical optimization process based on a quasi - newton method converges locally to an optimum if the object function is at least twice continuously differentiable . in the sequelwe will simply say differentiable . in current radiotherapy practicethe object function is a function of treatment parameters , but only indirectly via dose - volume constraints and or eud constraints . herewe will consider their differentiability properties .let us start with object functions only containing eud constraints .then is a weighted sum of local object functions involving only like in equations ( [ eq : objfun ] ) and ( [ eq : locobjfun ] ) . in this casethe object function is differentiable .here we have to assume that the dose distribution is differentiable .  however , when the object function contains dose - volume constraints it is not always differentiable .this is due to the finite differentiability of the dose - volume histogram at critical values of the dose distribution , even when the latter is differentiable . since it is not immediately obvious how the differentiability of the dose - volume histogram as a function of dose affects the differentiability of the object function as a function of treatment parameters , we present the following explanation . essentially , the treatment parameters enter the argument of the volume function because in the presence of parameters , the critical values are parameter dependent .when treatment parameters vary , the dose distribution varies and in particular its critical values . thus a critical value may pass through .suppose is a critical point for ( a critical point can always be translated to ) .it can be shown , see section [ sec : proof ] , that for values of near and near , can be transformed to a new function . where is a independent local standard of with critical point at zero and critical value zero and is the critical value of for near .  using the previous we will show in section [ sec : proof ] that can be split into a global and a local part .the global part depends differentiably on .let us consider the local part in more detail .on a neighbourhood of we have the function is proportional to the volume in the expression above and the proportionality factor is a differentiable function of . the last line in the formula above shows the volume function for a region enclosed by level sets of . according to function is not twice continuously differentiable at critical values of .thus is not twice continuously differentiable at values of for which this , in turn , means that the object function is not twice continuously differentiable at values of such that is a critical value of .  * remarks *  1 .  [ rem : gnotsmo ] the function in equation ( [ eq : locobjfun ] ) is not differentiable at .this alone would already imply that the object function in ( [ eq : objfun ] ) is not differentiable .however we will assume that we are in a situation like in remark [ rem : compro ] where we are looking for a compromise .this means we stay away from points where is not differentiable .[ rem : codim ] note that equation ( [ eq : hyper ] ) defines a hyper surface in parameter space . it will be -dimensional in most of its points , therefore we need to vary only one parameter to pass through .[ rem : dosesmo ] throughout we assume that the dose distribution is a differentiable function .if it were less differentiable , the volume function would also be less differentiable for both critical and regular values .the aim of this section is to provide two examples .one shows a differentiable object function , the other shows an object function that fails to be twice continuously differentiable at a certain value of the treatment parameters , here we take . in the first casethe newton iteration converges to an optimum , wheras in the second case the newton iteration does not converge if one of the iterates becomes . to avoid lengthy calculations obscuring the points we wish to illustrate we will make several simplifications .( 100,25)(0,0 ) ( 48 , 0 ) ( 70 , 19) ( 140,21) ( 70 , 10) ( 140,10) ( 70 , 15) ( 140,15) ( 75 , -3) ( 140,-3)  let be a dose distribution on the region .the object function consists of a single maximum dose - volume constraint so .the difference between the to examples will be the dose distribution .suppose for in a sufficiently large neighbourhood of the dose distribution is a function with a single maximum and for a similar neighbourhood of it is a function with two maxima for example : here and are different parameter values . in figure [ fig : distroos ] we draw the functions and .for both dose distributions we call the value at the global maximum and is the value of the other local maximum of . both and on the treatment parameters .let us assume that always .furthermore we assume that .  for near dose distribution has only one critical value namely so for , is a differentiable function of which we call . in the second case , where is near , we will be particularly interested in . for near but , is again a differentiable function . at , can be split in a local and a global part , as in the previous section , where the global part is differentiable .we assume for simplicity that the global part is again .the local part is essentially a function of the following form where is a local standard form of the dose distribution , is a constant .since the volume function is not differentiable at , the object function ( that we will construct shortlye hereafter ) is not differentiable at values of for which .the latter defines a hyper surface in parameter space of codimension one , which means that we need to vary only one parameter to cross .therefore we choose new parameters and the first one we take to be .only this parameter will be relevant for the example , we call it again , but now .thus for the second dose distribution we finally have the following volume function :   now we construct the object function .first we scale such that .then the object function for the first dose distribution is the object function for the second dose distribution is given by almost the same formula , but is replaced by the volume function in equation ( [ eq : volfun2 ] ) .the function must satisfy several conditions so that it is consistent with the dose distributions we chose . first must be differentiable and decreasing , furthermore since we assumed that .a possible choice is .  in the first optimization problemthe object function is differentiable so there are no difficulties in applying the newton algortihm . however , in the second optimization problem the object function is not twice continuously differentiable at .recall that the 1-dimensional newton iteration for is .fixed points of correspond to extremal points of if is continuous and nonzero at .since the limit from the left of the second derivative of is unbounded , the newton iteration has a fixed point at which is not related to an minimum or maximum of . butthis spurious fixed point is not stable for the newton iteration . only if the next iterates will also be zero .therefore it does not lead to convergence to a non extremal point of in general . although it may slow down the convergence of the newton iterates . for a graphical explanation see figure [ fig : graphs ] .( 100,35)(0,0 ) ( 0 , 15 ) ( 68 , 16) ( 117,31) ( 95 , 33) ( 107,33) results show that the radiotherapy treatment planning optimization process based on a quasi - newton method locally converges to an optimum of the object function for most values of the treatment parameters .only when the object function contains a dose - volume constraint , convergence can not be guaranteed .the reason is that the object function is not sufficiently differentiable for values of the treatment parameters such that is a critical value of the dose distribution .although for all other values of , the object function is differentiable . the question remains whether this leads to practical consequences for optimization in treatment planning .let us take a closer look at the values of the treatment parameters for which the object function is not sufficiently differentiable .in parameter space these values of are determined by the equation , see section [ sec : objdiff ] , expressing that is a critical value of the dose distribution .geometrically this equation defines a hyper surface in parameter space . roughly speaking is -dimensional so it has measure zero .this means that given a starting point for optimization , the newton iteration scheme generates a sequence of points that will lie on with probability zero .the points on are spurious fixed points of the newton iteration like in the example of the previous section .the reason is that we can generalize this example .first all types of extreme points of the dose distribution , namely minima , saddles and maxima lead to the same type of non - differentiability of the volume function .second , also in higher dimensions the second derivative of the object function becomes unbounded at points of insufficient differentiability .third , the spurious fixed points of the newton iteration scheme are unstable .our final conclusion is that in practice the convergence of the optimization process is most likely only slowed down near .in order to make a statement about the differentiability of the object function , we need a few definitions .let be an open set of _ treatment parameters_. let be a domain in representing the patient and let represent a tumour region or a healthy organ . for each , the _ dose distribution _is represented by a function ( here we do not consider as a distribution in the sense of generalized functions ) . a functionis called _ smooth _ if it is infinitely many times continuously differentiable .we now assume that is a smooth function of position and moreover smoothly dependent on the parameters ._ critical points _ of are points in where the gradient of vanishes , other points are called _ regular points_. the value takes at a critical point is called a _ critical value_. if the level set contains no critical points , is called a _regular value_. we now state our results on the differentiability of object functions containing eud constraints and dose - volume constraints respectively .   [ the : diff2 ] let the object function be given by , where is a smooth function and is a volume function for dose distribution .then is smooth with respect to provided that is not a critical value of . if is a critical value of , then is not twice continuously differentiable at .the proof of theorem [ the : diff1 ] is rather straightforward , but the proof of theorem [ the : diff2 ] is more involved .since in the setting of theorem [ the : diff2 ] the differentiability of the object function is completely determined by the volume function , the proof is about the latter only .the theorem contains two statements which we here state and prove separately .the proofs heavily rely on the results in , but due to the presence of parameters we have to make several adjustments .let us start with some definitions .   1 .let be a positive smooth function for all and moreover depends smoothly on .for each , vanishes at infinity in the following sense , for each there is a compact such that for all , .we assume that critical points of are non - degenerate , that is if is a critical point of then .2 .  also is a smooth function with non - degenerate critical points .although we consider and as different functions , their values are taken to be identical : for all and .we denote the level sets of by .the level sets of are denoted by .* proof of theorem [ the : diff1 ] . *the function is a positive and smooth function , smoothly depending on parameters .then ^{\alpha}$ ] for is also smooth in both variables an parameters .now it immediately follows that as defined in equation ( [ eq : eud ] ) is again a smooth function of .     let us begin with the nice situation where is a regular value of , so that smoothly depends on . in the proof of proposition[ pro : diff ] we use a property of the level sets of .[ lem : ndiffeo ] let and be as defined above and furthermore be a regular value of .then an open neighbourhood of exists , such that is diffeomorphic to for all and the diffeomorphism depends smoothly on .  * proof of proposition [ pro : diff ] . * the volume is given by an integral over a region bounded by . since , according to lemma [ lem : ndiffeo ] , the latter depends smoothly on at , the integral and thus depends smoothly on at .   * proof of lemma [ lem : ndiffeo ] .* let be a shorthand for .note that . since is a regular value , contains no critical points of .thus the intersection of and is transversal .then a neighbourhood of exists such that for all , the intersection of and is transversal .thus for all , is a regular level set , consequently there are no critical points of in .suppose and then an index exists such that .now the gradient flow of the height function on is a regular flow on thus defining a diffeomorphism from to .  let us now turn to the case where is a critical value of .then there is at least one critical point in , but for the sake of simplicity we assume that is unique . in the proof of proposition [ pro : finidiff ]we will use a similar construction as in without providing all details .* proof of proposition [ pro : finidiff ] .* let us assume that is the unique critical point of on the critical level set .since by assumption is a non - degenerate critical point we can apply the implicit function theorem to conclude that an open neighbourhood of exists such that the map and is a critical point of is smooth .now we define a new function , which is a smooth function of and .moreover has a critical point at with critical value .then we have . following the construction in we split the latter region in a partnot containing the point and a remaining small part including .the volume of the first part depends smoothly on using the same arguments proving proposition [ pro : diff ] .since is the sum of both volumes , the differentiability of is determined by that of the volume of the second part .let us consider this second part contained in a small neighbourhood of .we may assume that is small enough to put into the standard form of . in order to keep notation simplewe denote the standard form again by , but now we can write . the actual standard form is , with .note that no longer depends on , it has a critical point at and the critical value is .this form can be obtained by a smooth change of coordinates , even smoothly depending on . hereagain we need the non - degeneracy of critical point of . in the following we use the results of . as a function of ,the volume function is only finitely differentiable with respect to at .now the volume of the second part is proportional to .the proportionality factor is of no concern because it originates from smooth coordinate changes and depends smoothly on .thus we obtain that the differentiability of with respect to is determined by that of .the latter is only finitely differentiable with respect to at those values of where . or , put differently , where is a critical value of .   1.  the statements and proofs in this section are valid for any dimension .therefore we have not specialized them to the case . in this respectfinitely differentiable means less than two times continuously differentiable .2 .  the critical points of a parameter family of functions will generally be degenerate for certain parameter values .however we excluded this possibility because the volume function will be even less differentiable at a critical value when the critical point on the level set of this value is degenerate .99 a. brahme , treatment optimization using physical and radiobiological objective functions , in : `` radiation therapy physics '' , alfred r. smith ( ed . ) pp .209 - 246 , springer berlin , 1995 .byrd , p. lu , j. nocedal , c. zhu , a limited memory algorithm for bound constrained optimization , siam j. sci .* 16(5 ) * ( 1995 ) pp .1190 - 1208 .a. eisbruch , r.k .ten haken , h.m .kim , l.h .marsh , j.a .ship , dose , volume and function relationships in parotid salivary glands following conformal and intensity - modulated irradiation of head and neck cancer , international journal of radiation oncology biology and physics * 45(3 ) * ( 1999 ) pp .577 - 587 .i. hoveijn , differentiability of the volume of a region enclosed by level sets , journal of mathematical analysis and applications , * 348 * ( 2008 ) pp .530 - 539 .icru , prescribing , recording and reporting photon beam therapy ( supplement to icru report 50 ) , icru report 62 , international commission on radiation units and measurements , 1999 .a. niemierko , reporting and analyzing dose distributions : a concept of equivalent uniform dose , medical physics * 24(1 ) * ( 1997 ) pp .103 - 110 .j. stoer , r. bulirsch , introduction to numerical analysis , springer 1983 .c. thielke , t. bortfeld , a. niemierko , from physical dose constraints to equivalent uniform dose constraints in inverse radiotherapy planning , med .* 30(9 ) * ( 2003 ) pp .2332 - 2339. q. wu , r. mohan , algorithms and functionality of an intensity modulated radiotherapy optimization system , med .phys . * 27(4 ) * ( 2000 ) pp .701 - 711 .

Abstract: the radiotherapy treatment planning optimization process based on a quasi - newton algorithm with an object function containing dose - volume constraints is not guaranteed to converge when the dose value in the dose - volume constraint is a critical value of the dose distribution . this is caused by finite differentiability of the dose - volume histogram at such values . a closer look near such values reveals that convergence is most likely not at stake , but it might be slowed down .


gamma - ray bursts ( grb ) are bursts of gamma - rays that arguably signal the birth of a black hole somewhere in the universe . based on the duration and spectrum , two classes of burstshave been observed  : those that last less than two seconds and have on the average hard spectra ( short grbs ) , and those that last longer than two seconds and are spectrally softer ( long grbs ) . the exact nature of the grb progenitors is unknown , although it is possible that long grbs come from the collapse of massive , rapidly rotating stars  and short grbs result from the merger of compact objects  .regardless of the progenitor system , accretion onto the resulting compact object is thought to create a highly relativistic jet .the prompt gamma - ray emission from the grbs may arises from the internal shocks due to collisions of faster shells with slower ones ejected earlier by the central engine .the subsequent softer multi - wavelength emission , referred to as the afterglow , may be due to the collision of the fireball with the extra - stellar material  .our understanding of grbs progressed very rapidly after the detection of multi - wavelength afterglows .well localized , favorably positioned grbs get fairly good multi - wavelength afterglow coverage .currently , the leading grb afterglow detection mission is which detects 90100 grbs annually .most of the grbs get observed by various instruments around the world because of its rapid arc - minute localization capability .compared to , gamma - ray burst monitor ( gbm ) detects about 250 burst per year but with poor localization . the error circle ( ec ) of gbm detected burstsis too large for a single telescope to observe effectively .the typical statistical uncertainty of the gbm burst location is about 3.3 degrees .however , when combined with the systematic uncertainty of 3.8 degrees  , the total burst location uncertainty is 5.0 degrees ( i.e. , 5.0 degree error radius ) .naturally , a brighter burst will have a smaller gbm ec than a weaker burst .  even though the localization is poor , gbm detected bursts have very good timing and spectral information including crucial measurements ( is the peak energy of the grb spectrum ) . if there is a method to localize gbm detected grbs to a few arc - seconds uncertainty , then large telescopes can do deeper follow up observations to determine the redshift of the burst and also potentially identify any emerging supernova . in addition , can also slew quickly to the gbm burst in order to observe the x - ray afterglow and obtain its light curve in x - ray wavelengths .based on observations about 60% of grbs have optical counterparts  .these optical counterparts are detected by various observatories with r magnitudes ranging from 14 to 22 within few hours after the burst  .thus , it is reasonable to assume about 60% of the gbm detected grbs also have optical counterparts with similar brightness distribution . if we were able to cover the entire gbm ec within about 24 hours after the burst it is conceivable that we would be able to find optical afterglows of grbs per year , which is more than the total number of burst detects per year .  due to the small energy range ( 15 - 150 kev ) of the burst alert telescope ( bat ), measurements alone can not constrain the of all bat detected bursts  . in contrast , due to the wide energy range ( 8 kev - 40 mev ) of gbm , all grbs detected by gbm have fairly good measurements . hence, addition of possibly another bursts per year with good and redshift measurements may allow us to explore the validity of various grb luminosity relations and to conduct detailed grb hubble diagram studies .we have investigated the feasibility of using a system to do coordinated monitoring of the bat field - of - view ( fov ) for prompt optical emission from grbs  .the study showed that with the current instrumentation , performing such a coordinated monitoring is not practical mainly due to the bat s very large fov .however , a similar coordinated observing campaign can be used to find the optical afterglow of gbm detected bursts .the gbm ec is much smaller than the bat fov and observers do not need to continuously monitor the field to detect the optical afterglow .this enables a given observatory to perform multiple observations inside the gbm ec and thereby increase the chance of a afterglow detection .the basic proposal is to design a system to facilitate scanning of the gbm ec for optical emission from grb afterglows .this observing program will be specially aimed at amateur astronomers around the world .proliferation of amateur telescopes with high quality ccd cameras has opened a new avenue to study optical emission from grbs .the basic objective of the system is to coordinate a significant number of ground based telescopes to scan different patches of the gbm ec in order to find the location of the optical afterglow . unlike the gcn system  which sends notices to large number of recipients ,this system will send customized targeted messages to individual registered telescopes .these individual messages will be sent via email or socket connections and they will have one or more assigned pointing locations for each telescope .the target telescopes can be either robotic or non - robotic .the selection of various patches in the gbm ec will be done based on the number of available telescopes , individual telescopes physical location , field of view ( fov ) and sensitivity .it is also reasonable to assume that these telescopes can observe multiple patches of the gbm ec , which will increase the chances of detection significantly .some of the important impacts of the proposed project are :  1 .the project will significantly enhance the value of gbm as a grb discovery instrument .2 .  potentially increase the number of burst with good timing , spectral and redshift measurements .the project will allow and attract the participation of amateur astronomers and their telescopes .      in order to investigate the feasibility of the project, we performed a monte carlo simulation to study the probability of detection of optical afterglows from gbm grbs .we assumed that gbm detects about 150 grbs with optical afterglows ( total rate is 250 per year ) per year distributed isotropically in the sky and throughout the year .we have distributed telescopes in such a way that they roughly trace the major cities in the world .then for each burst we tracked the path of the sun and selected a set of telescopes away from the sun and within few hours from the burst location to scan the gbm ec .we also calculated the illuminated fraction of the moon s disk ( ) at the time of each burst .the probability , , of finding a grb was estimated by here is the total number of independent attempts to observe , with probability of success , . in this case is equal to the product of number of available telescopes and number of patches each telescope can observe .we calculated the probability of success , i.e. , the probability of detecting a given burst afterglow per observation using the following equation . note that typically the one gbm ec is .we repeated this procedure for every simulated burst , while changing the total number of telescopes participating in the program , the fov of telescopes , and total number of patches a given telescope can cover .the results of our simulation are shown in figure [ sim_results01 ] and figure [ sim_results02 ] .for these particular simulations we have assumed that all the telescopes have the same fov and all telescopes can cover some constant number of patches in the gbm ec .furthermore , we assumed that these telescopes will be able to observe assigned sky patches within a few hours after the burst .hence , in the simulation we used only telescopes which are within a few hours ( 6 hours ) of the burst location .the number of participating telescopes in the observing program was varied from 20 to 100 .figure [ sim_results01 ] shows the number of localized bursts ( total number of bursts ) as a function of the fov of participating telescopes .here we have fixed the number of participating telescopes to 25 .various curves correspond to different number of patches that each telescopes can observe .figure [ sim_results02 ] shows the number of localized bursts as a function of number of participating telescopes in the program assuming that each telescope can observe 20 patches .the six curves shown in the plot correspond to various field of views . according to the simulation , with 60 telescopes participatingit is possible to detect about 40 grb optical afterglows per year using telescopes with fov of .this value is close to the value of a typical fov of an amateur telescope .it is also interesting to note that if we have about 10 telescopes participating with fov of , then it is possible to detect about 25 grb afterglows per year . in order to put these values into perspective we point out that thus far , no one has managed to observe optical afterglow of a grb based only on a gbm localization ( the gbm has been detecting grbs for more than three years ) .it is also worth noting that not all the gbm locations will have a statistical error of 3.3 degrees . about a third of the grbswill have a statistical error less than this value .about 10% will have error of 1 degree or less .in such cases the gbm ec radius will be less than 5 degrees and may be as small as 4 degrees . obviously , for those cases we have a much higher chance of detecting the afterglow .  a schematic block diagram of a potential software system is shown in the figure [ diagram ]the system has two components : 1 ) a scheduling system that will assign various observing patches to participating telescopes , and 2 ) a online system that will let observers to upload their images and search for candidate transients .the algorithms in the scheduling system will check the gbm ec observability of each participating telescope and assign them to different parts of the gbm ec . in doing thisthe algorithm will consider individual telescopes fov , sensitivity and local weather conditions .in addition , it will also assign more than one patch for each astronomer .on average an amateur astronomer may receive about 40 notices per year .the exposure time for each patch depends on many factors such as aperture , seeing , type of ccd camera etc and typically may vary from 1 to 30 mins .the probability of success depends on the telescope configuration ( fov , sensitivity ) , local weather and sky conditions , and the number of patches observed .however , every amateur observer who submits an observation to the system will get credit for their effort by being a co - author of the subsequent gcn notice that results from a successful detection .the online system is envisioned to have a web interface where the participants can submit their observations .it will also have online tools that will compare the submitted observations with existing catalogs and search for the optical afterglow of the grb .if one of the observations has a positive detection then the system will initiate a follow up observation to establish whether the candidate source is fading .if the candidate is found to be fading ( telltale signature of a grb afterglow ) then the magnitudes of the two images will be determined and a gcn circular will be sent .we investigate the feasibility of implementing a system that will coordinate ground based telescopes ( both amateur and professional ) to scan the gbm ec in order to localize gbm bursts .unlike the gcn system , proposed system will send individual customized messages to telescopes to observe certain patches in the gbm ec .the scientific objective of the system is by localizing gbm detected burst , we will be able to increase the number of gbm bursts with mutil - wavelength followups potentially with redshifts measurements .these measurements are scientifically very important because there are hints that gbm bursts may represent significantly different burst population . based on our simulation, we can detect about 25 grb afterglows per year using just 10 telescopes with field - of - view . with more telescopes participating in the program, we should be able to detect many more afterglows and study a potentially interesting burst population that is currently inaccessible to the grb community .

Abstract: we investigate the feasibility of implementing a system that will coordinate ground - based optical telescopes to cover the fermi gbm error circle ( ec ) . the aim of the system is to localize gbm detected grbs and facilitate multi - wavelength follow - up from space and ground . this system will optimize the observing locations in the gbm ec based on individual telescope location , field of view ( fov ) and sensitivity . the proposed system will coordinate gbm ec scanning by professional as well as amateur astronomers around the world . the results of a monte carlo simulation to investigate the feasibility of the project are presented .


spatial modulation ( sm ) is emerging as a promising multi - antenna modulation scheme ( see and the references therein ) .sm uses multiple antenna elements but only one radio frequency ( rf ) chain at the transmitter .the single - rf chain feature of sm brings in several advantages including reduced rf hardware complexity , size and cost , no spatial interference , and simple detection at the receiver , while offering the capacity and diversity benefits of multi - antenna systems . in sm , only one antenna element is activated in a given channel use and a qam / psk symbol is sent on the activated antenna ; the remaining antenna elements remain silent .the index of the active antenna element also conveys information bits .it has been shown that sm can achieve better performance than spatial multiplexing ( smp ) under certain conditions .generalized spatial modulation ( gsm ) is a generalization of sm , where the transmitter uses multiple ( ) antenna elements and more than one ( ) rf chain . among the available antenna elements are activated in a given channel use , and qam / psk symbols are sent simultaneously on the active antennas .the indices of the active antennas also convey information bits .both sm and smp can be seen as special cases of gsm with and , respectively .it has been shown that for a given modulation alphabet and , there exists an optimum that maximizes the spectral efficiency .further , for the same spectral efficiency , gsm can perform better than both sm and smp , .this is illustrated in fig .[ fig1 ] which shows the bit error rate ( ber ) performance of maximum likelihood ( ml ) decoding for different configurations of mimo systems each with and same spectral efficiency of 8 bits per channel use ( bpcu ) .we can see that the ( 8,8,2)-gsm mimo system outperforms both the ( 8,8,1)-system ( i.e. , sm system ) and ( 8,8,8-system ( i.e. , smp system ) .this superior performance of gsm - mimo motivates us to study its capacity limits .while most studies on sm / gsm in the literature so far have focused mainly on performance analysis and receiver algorithms , capacity of sm / gsm systems remains to be studied .the need for capacity analysis of sm has also been highlighted in . in ,the authors have obtained the capacity of sm for miso systems through simulation . however , an analytical characterization of the capacity of sm / gsm has not been explored .our contribution in this paper addresses this gap .in particular , we derive lower and upper bounds on the capacity of gsm .another contribution is the proposal of low complexity encoding ( i.e. , bits - to - gsm signal mapping ) and detection methods suited for gsm - mimo systems with large number of antennas and rf chains .the proposed encoding makes use of combinadic representations in combinatorial number system , and the detection makes use of layered message passing .  the rest of the paper is organized as follows .section [ sec2 ] introduces the gsm - mimo system model .gsm - mimo capacity bounds are derived in section [ sec3 ] .the proposed combinadics based gsm encoding scheme and the layered message passing based gsm detection scheme are presented in section [ sec5 ] .conclusions are presented in section [ sec6 ] .consider a ( )-gsm mimo system with antennas and rf chains at the transmitter ( ) , and antennas at the receiver .figure [ sysdia ] shows the gsm transmitter .an switch connects the rf chains to the transmit antennas . in each channel use , out of transmit antennas are chosen and activated .the remaining antennas remain silent .the selection of the antennas to activate in a channel use is done based solely on information bits ( not based on csit ) .therefore , the indices of the active antennas convey information bits per channel use . on the active antennas , modulation symbols ( one on each active antenna ) from a modulation alphabet are transmitted .this conveys additional information bits .therefore , the total number of bits conveyed in a channel use in gsm is given by    let denote the gsm signal set , which is the set of all possible gsm signal vectors that can be transmitted .therefore , if is the signal vector transmitted by the gsm transmitter , then .for example , for , , and bpsk , i.e. , , a gsm signal set can be as follows :    let denote the vector of modulation symbols that are to be transmitted in a channel use over the chosen antennas , i.e. , .* definition : * define a matrix of size as the _ antenna activation pattern matrix_. the matrix represents a particular choice of antennas from the available antennas , such that the gsm signal vector .the matrix is a sparse matrix consisting only of s and s , with exactly one non - zero entry in every column and one non - zero entry in rows .if are the indices of the chosen antennas , then is constructed as where denotes the element in the row and column of .for example , in a system with and , to activate antennas 1 , 3 , 6 and 8 , the matrix is given by .\label{aap1}\ ] ] note that the indices of the non - zero rows in matrix give the support of the gsm signal vector . out of the antenna activation choices , only are needed for signaling .let denote this set of all allowed antenna activation pattern matrices , where and .let .now , is given by   note that any gsm signal vector , , can be represented as with some , and , , and .conversely , given any and , there exists a gsm signal vector such that . since and are chosen by two independent information bit sequences , and are independent .that is , .thus , the above system model of gsm - mimo decouples the choice of active antennas and the transmitted modulation symbols into two independent random variables , namely and .this makes the gsm - mimo system analysis simpler .the received signal vector ^t ] is the noise vector whose entries are modeled as complex gaussian with zero mean and variance . since is obtained based only on information bits , and are independent . since , and are independent .also , and are independent .so , and are independent .so , the mutual information between and in gsm is given by .the capacity of a spatially multiplexed mimo channel with channel state information at the receiver is  \big\ } , \label{cap}\ ] ] where is the covariance matrix of the transmit signal vector with elements from gaussian codebook . for i.i.d .modulation symbols and total power , the capacity expression becomes \big\}. \label{capsmp}\ ] ] here , we are interested in the capacity of gsm - mimo , where the symbols sent on the active antennas are from gaussian codebook .the capacity of gsm - mimo can be written as ) , \label{gsmcap}\end{aligned}\ ] ] where denotes the differential entropy . to compute , we need to evaluate , which requires the knowledge of the distribution of . from ( [ sysmodel ] ), we can see that the distribution of is a gaussian mixture given by where \nonumber\\ & = & { { \bf h}}{{\bf a}}_i{{\mathbb e}}_{{{\bf s}}}({{\bf s}}{{\bf s}}^h){{\bf a}}_i^h{{\bf h}}^h +\sigma^2{{\bf i}}_{m}. \label{covar}\end{aligned}\ ] ] if all the possible antenna activation patterns are equally likely , then .when the transmitted modulation symbols are independent of each other , , now , ( [ ydist ] ) becomes the differential entropy of is given by it is difficult to get a closed - form solution to the above expression .hence , we try to bound it above and below by the following techniques ._ lower bound 1 , _ : since is a convex function , by jensen s inequality , \geq-\log{{\mathbb e}}[p(y)].\ ] ] hence , the differential entropy can be lower bounded as \nonumber \\ & \geq&-\log_2\int p^2({{\bf y}})d{{\bf y}}\nonumber \\ & = & -\log_2\int \big(\frac{1}{l}\sum_{i=1}^l{{\mathcal n}}({\bf 0},{{\bf \phi}}_i)\big)^2 d{{\bf y}}\nonumber \\ & = & -\log_2\bigg\ { \frac{1}{l^2\pi^{m } } \sum_{i=1}^l\sum_{j=1}^l\frac{\det(({{\bf \phi}}_i^{-1}+{{\bf \phi}}_j^{-1})^{-1})}{\det({{\bf \phi}}_i)\det({{\bf \phi}}_j)}\nonumber \\ & & \hspace{2cm}\underbrace{\int \frac{e^{-{{\bf y}}^h({{\bf \phi}}_i^{-1}+{{\bf \phi}}_j^{-1}){{\bf y}}}}{\pi^{m}\det(({{\bf \phi}}_i^{-1}+{{\bf \phi}}_j^{-1})^{-1 } ) } d{{\bf y}}}_{= 1 } \bigg\}\nonumber \\ & = & -\log_2\bigg\ { \frac{1}{l^2\pi^{m } } \sum_{i=1}^l\sum_{j=1}^l\frac{1}{\det({{\bf \phi}}_i+{{\bf \phi}}_j ) } \bigg\ } \label{l1}\\ & { \triangleq } & l_1.\nonumber\end{aligned}\ ] ] from ( [ gsmcap ] ) and ( [ l1 ] ) , a lower bound on can be obtained as   _ lower bound 2 , _ : since differential entropy is a concave function , we can write from the above equation , can be lower bounded as   based on the two lower bounds and , a refined lower bound on gsm - mimo capacity is given by   _ upper bound 1 , _ : by the property of entropy , we can write further , we also have using the above properties , an upper bound for can be written as now , an upper bound on can be written as   _ upper bound 2 , _ : here , we approximate the probability distribution of to a gaussian distribution .this leads to an upper bound because the entropy of any random variable is bounded above by the entropy of a gaussian random variable with the same mean and variance .the mean and covariance of are given by {{\bf h}}^h + \sigma^2{{\bf i}}_{m}\nonumber \\ & = & { { \bf h}}\bigg(\frac{1}{l}\sum_{i=1}^l{{\bf a}}_i\big(\frac{\sigma^2_x}{r}{{\bf i}}_{r}\big){{\bfa}}_i^h\bigg){{\bf h}}^h + \sigma^2{{\bf i}}_{m}\nonumber \\ & = & \frac{\sigma^2_x}{rl}{{\bf h}}\big(\sum_{i=1}^l{{\bf d}}_i\big){{\bf h}}^h + \sigma^2{{\bf i}}_{m } , \label{upb1d1}\end{aligned}\ ] ] where . let be the set of active antenna indices that corresponds to the antenna activation pattern matrix .it can then be seen that is a diagonal matrix , such that where is the element in the row and column of .assuming that all activation patterns are allowed , i.e. , , the number of times any particular antenna will be active among the activation patterns is .therefore , and ( [ upb1d1 ] ) becomes now , an upper bound on the gsm - mimo capacity is given by \big\ } , \label{ub2}\end{aligned}\ ] ] which is the same as the capacity of a spatially multiplexed mimo system .based on the two upper bounds and , a refined upper bound on gsm capacity can be obtained as   , and ,width=336,height=249 ]  , , and varying .,width=336,height=264 ]  [ capfig2 ]  _ numerical results _ : we evaluated the lower and upper bounds on the gsm - mimo capacity for different system configurations .figure [ capfig1 ] shows the lower and upper bounds for gsm - mimo systems with , , .we see that the lower bound and upper bound are tighter at low snrs .whereas , the lower bound and upper bound are tighter at high snrs . it can be noted from the figure that the lower and upper bounds are very close at low snrs ; so , in this regime , the gsm - mimo capacity is almost same as that of a spatially multiplexed mimo system with the same and . in fig .[ capfig2 ] , we compare the bounds on gsm - mimo capacity with the true gsm - mimo capacity obtained through simulation .we consider gsm - mimo with , , snr db , and varying . in the top figure of fig .[ capfig2 ] , we observe that , at snr = 2 db , the gap between the upper and lower bounds is bps / hz for and bps / hz for . also , in the bottom figure of fig .[ capfig2 ] , at snr = 32 db , the gap is bps / hz for and bps / hz for .it is known that large - scale mimo systems provide a variety of advantages . however , for large values of in gsm - mimo , the encoding and decoding complexities become prohibitively large . in this section ,we propose low - complexity methods to encode and detect gsm signals that scale well for large .   in gsm - mimo , bits are used to choose an activation pattern matrix from , and bits are used to generate from . while the mapping of bits to modulation symbols in is straight - forward , the mapping of bits to a choice of activation pattern is not . a table or a map of bit sequence to activation patterns has to be maintained both at the transmitter and receiver . for large values of , the size of this map can become prohibitively large . for example , if , then .implementation of an encoding map of this size is impractical . to alleviate this problem , we use _ combinadic _ representations in combinatorial number system .  *definition * : the combinadic of a number  ] .obtain .2 .  find the combinadic of .3 .  construct matrix such that the indices of the non - zero rows of are given by the combinadic of .an example of the encoding map for based of the above procedure is illustrated in table [ tab1 ] .the demapping procedure is listed below .  1 .obtain the combinadic corresponding to the decoded antenna activation pattern matrix .2 .  compute from this combinadic as .3 .  demap to get .the above low complexity mapping / demapping procedures allow practical implementation of gsm encoding for large ..[tab1 ] encoding map for gsm system with . [ cols="^,^,^,^ " , ]   _ example : _ let . then , .+ _ mapping _ : let  ] .this gives the corresponding bit sequence of the detected antenna activity pattern at the receiver .the maximum a posteriori probability ( map ) detection rule for gsm - mimo is given by note that , .therefore , the exact computation of ( [ map ] ) requires exponential complexity in . to address this problem ,here we propose a low complexity layered message passing ( lamp ) algorithm which gives an approximate solution to ( [ map ] ) . in the proposed lamp algorithm , we define four sets of nodes and exchange messages between them in layers .there are two layers of message exchanges , one corresponds to the detection of modulation symbols and the other corresponds to the detection of the antenna activation pattern . in order to describe this algorithm, we define a new variable as follows .* definition * : a variable is called the antenna activity indicator if if the antenna is active , else .that is , , where is the indicator function .  therefore , .note that , which we call as the gsm system constraint .let  ] ) , csr detection in for the following large - scale gsm - mimo system configurations : ( ) -gsm , 4-qam , , bpcu , ( ) -gsm , 4-qam , , bpcu , and ( ) -gsm , 4-qam , , bpcu .it can be seen that the proposed lamp algorithm performs better and its performance improves as the dimensionality of the gsm signal increases .  in fig .[ sim3 ] , we plot the average snr required by the lamp detection algorithm to achieve a target ber of as the number of receive antennas is varied .we consider two gsm mimo systems here , ( ) , 4-qam , bpcu , and ( ) , 4-qam , bpcu . from fig .[ sim3 ] , it can be seen that as increases the snr required to achieve the target ber reduces .it can also be observed that the lamp detection algorithm performs well even when the system is underdetermined , i.e. , .this is because the lamp detection algorithm exploits the sparse structure of the gsm transmit signals to efficiently perform detection .   as the number of receive antennas is varied for , and gsm mimo systems with 4-qam.,width=336,height=249 ]we studied gsm mimo capacity and low complexity methods for gsm encoding and detection .we made the following key contributions in this paper : 1 ) we derived lower and upper bounds on the gsm mimo capacity , 2 ) we proposed an efficient gsm encoding method using combinadics for large number of transmit antennas and rf chains , and 3 ) we also proposed a message passing based low - complexity detection algorithm suited for large - scale gsm mimo systems .1 m. di renzo , h. haas , a. ghrayeb , s. sugiura , and l. hanzo , `` spatial modulation for generalized mimo : challenges , opportunities and implementation , '' _ proceedings of the ieee _ , vol .1 , jan . 2014 .j. wang , s. jia , and j. song , `` generalised spatial modulation system with multiple active transmit antennas and low complexity detection scheme , '' _ ieee trans .wireless commun .1605 - 1615 , apr .2012 .t. lakshmi narasimhan , p. raviteja , and a. chockalingam , `` generalized spatial modulation in large - scale multiuser mimo systems , '' _ ieee trans .wireless commun .14 , no . 7 , pp. 3764 - 3779 , jul .

Abstract: generalized spatial modulation ( gsm ) uses antenna elements but fewer radio frequency ( rf ) chains ( ) at the transmitter . spatial modulation and spatial multiplexing are special cases of gsm with and , respectively . in gsm , apart from conveying information bits through modulation symbols , information bits are also conveyed through the indices of the active transmit antennas . in this paper , we derive lower and upper bounds on the the capacity of a ( )-gsm mimo system , where is the number of receive antennas . further , we propose a computationally efficient gsm encoding ( i.e. , bits - to - signal mapping ) method and a message passing based low - complexity detection algorithm suited for large - scale gsm - mimo systems .   _ * keywords * _


magnetoencephalography ( meg ) is a noninvasive neurophysiological technique that measures the magnetic field generated by neural activity of the brain using a collection of sensors outside the scalp [ ] .when information is being processed at some regions of the brain , small currents will flow in the neural system , producing a small electric field , which in turn produces an orthogonally oriented small magnetic field according to maxwell s equations .the meg inverse problem refers to recovering neural activity by means of measurements of the magnetic field .the neural activities are usually represented by magnetic dipoles , which are closed circulations of electric currents , that is , loops with some constant current flowing through .each dipole has a position , an orientation , and a magnitude .the inverse problem then becomes determining the position , orientation , and magnitude ( or amplitude ) of the dipoles .one challenge of the meg inverse problem is that it does not have a unique solution and so it is ill - posed [ ; ; ] . as early as in the 19th century, von helmholtz demonstrated theoretically that general inverse problems , such as those aiming at identifying the sources of electromagnetic fields outside a volume conductor , have an infinite number of solutions [ ] .hence , to derive a practically meaningful solution from the infinitely many mathematically correct solutions , one has to introduce constraints to the solution and/or use prior knowledge about the brain activity .existing approaches for the meg inverse problem can be grouped into two major classes that differ in how they impose constraints on the source signals . within the first class ,the dipole fitting [ ; ; ; ] and scanning methods [ ; ; ; ; ; ] assume that there exist a limited number of dipoles as point sources of the magnetic field in the brain . by constraining the number of sources ,the locations of these dipoles are estimated by least squares fitting [ ] or iterative computing [ ] .dipole orientations and amplitudes can be effectively estimated within these locations . however , estimating the source locations involves solving a difficult nonlinear optimization problem which has multiple local optima [ ] .our proposed method belongs to the second class , which contains various imaging methods .different from the first class , imaging methods assume that there are a large number of potential dipole locations evenly distributed all over the cortex . by dividing the cortical region into a fine grid and attaching a dipole at each grid , imaging methods model the orientations and magnitudes for all the potential dipoles simultaneously .dipoles with nonzero magnitudes are identified as the source dipoles .imaging methods are based on the theory that the primary sources can be represented as linear combinations of neuron activities [ ] .one can express the inverse problem using a linear model where is an matrix containing meg time courses measured by sensors at time points , recording the amplitudes of the magnetic field . without loss of generality, it is assumed that the measurements for each time course are sampled at the same evenly - spaced time points .the known design matrix links the source signals to the sensor measurements , and is computed using a boundary element model prior to application of model ( [ eqdata ] ) [ ] .the matrix represents the unknown dipole activities in the form of unobservable source time courses .the matrix contains some additive noise .the amplitudes and orientations of the signal for each dipole at a time point can be decomposed into three components in the coordinate system .therefore , represents the total number of the dipole components , and it is three times that of the number of grid cells . in a typical meg study , is usually from a few hundred to a few thousand , is a few hundred , but is as large as over 10,000 , and so .  defining the matrix frobenius norm as . to recover  , one can solve a penalized least squares problem where is a penalty function that promotes certain desirable properties on .  in the literature of meg source reconstruction ,spatial focality and temporal smoothness are two valid assumptions .that is , the source signals are smooth in time , and only a small number of compact areas are responsible for the recordings [ ] .many of the imaging methods focus on either the first assumption or the second .earlier methods using the smoothness assumption usually adopt the -norm penalty , for certain weighting matrix .the simplest such method is the minimum norm estimate ( mne ) [ ] which uses .the loreta methods [ ; ] set to be the discrete spatial laplacian operator .two advantages of the -penalty based methods are the computational efficiency and the less - spiky property in the time domain . nevertheless , the -penalty lowers the spatial resolution and causes the well - known `` blurring effect '' in the spatial domain .utilizing the -penalty , the focuss method [ ] reduces the blurring effect by reinforcing the strong signals while weakening the weak ones using an iterative algorithm to update .however , it is noticed that focuss is very sensitive to noise [ ] .many hierarchical bayesian approaches induce the temporal smoothness by employing smoothing priors which penalize discontinuities [ see , e.g. , ; ; ] .an alternative penalty is the -norm , , which promotes the focality of the recovered signals .related work includes the minimum current estimate ( mce ) [ ; ; ] and the sparse source imaging method [ ] .in contrast to the -penalty , the -penalty causes `` spiky '' discontinuities of the recovered signals in both temporal and spatial domains .bayesian methods developed by , , take into account the spatial focality by employing anatomic sparse priors .however , these methods have similar problems as methods based on the -penalty .  to prevent the spiky property from the -penalty and the blurry property from the -penalty , some -norm methods with and have been introduced [ ; ] . however , the optimization problems associated with -penalties are more difficult to solve than with and penalties .more recently , some spatio - temporal regularization methods have been proposed , which take into account both the smoothness and focality properties by combining basis representation with penalization .the -regularization discussed by first projects onto a temporal basis and then imposes the -penalty on the spatial domain and the -penalty on the temporal domain .the event sparse penalty procedure [ ] divides the brain surface into several patches based on its anatomic features and uses temporal basis functions to represent source time courses within each patch .one drawback of both methods is that it is not straightforward to choose the basis .both methods have some shortcomings .the former makes the assumption that the source temporal basis can be extracted perfectly from the meg recordings .the latter utilizes comprehensive prior information of the experiment task and the brain geometry .in addition , the use of basis representation can potentially cause information loss , since information orthogonal to the basis set can not be recovered after the projection to the basis set is done .the goal of this paper is to develop an innovative two - way regularization method ( twr ) for solving the meg inverse problem that promotes both spatial focality and temporal smoothness of the reconstructed signals .the proposed method is a two - stage procedure .the first stage produces a raw estimate of using a fast minimum norm algorithm .the second stage refines the raw estimate in a penalized least squares matrix decomposition framework . a sparsity - inducing penalty and a roughness penalty are employed to encourage spatial focality and temporal smoothness , respectively .the proposed twr has three major advantages over the existing methods .first , twr regularizes in both spatial and temporal domains , and simultaneously takes into account both focality and smoothness properties .hence , it should be superior to one - way regularization methods ( e.g. , mne and mce ) .second , unlike some aforementioned spatio - temporal methods , twr does not rely on the choice of basis functions . hence, it avoids the information loss due to basis approximation .third , the two - stage procedure is computationally efficient .the advantages of our method are well illustrated in the empirical studies , which show clearly that twr outperforms one - way regularization methods that focus either on the focality or the smoothness alone , and some existing two - way spatio - temporal methods as well .two - way regularization techniques for matrix reconstruction have been studied in other contexts . a two - way regularized singular value decomposition for analyzing two - way functional data that imposes separate roughness penalties on the two domains . and develop sparse singular value decomposition methods that impose separate sparsity - inducing penalties on the two domains .however , to the best of our knowledge , a two - way regularization with the sparsity penalty on one domain and the roughness penalty on the other domain of the data matrix has not appeared in the literature .this paper provides a novel application of the two - way regularization method in solving the highly ill - posed meg inverse problem , where different types of penalties are naturally used to meet the dual requirements of spatial focality and temporal smoothness on the unknown source signals .the rest of the paper is organized as follows .section [ secmethod ] presents the details of the twr methodology including the computational algorithm . through a synthetic example, section [ secsim ] shows advantages of the twr over some existing methods for solving the meg inverse problem .section [ secmeg ] applies the twr to a real - world meg source reconstruction problem .section [ seccon ] concludes the paper with some discussion about an alternative one - step approach and related complications .we propose a two - way regularization ( twr ) method to regularize the recovered signals in both spatial and temporal domains .we adopt a penalized least squares formulation that uses suitable penalty functions to ensure the spatial focality and the temporal smoothness of the recovered signals .twr is implemented in a two - stage procedure where the first stage produces a rough estimate of the source signals and the second stage refines the initial rough estimate using regularization .   the goal of stage 1 is to obtain a rough estimate of the location and the shape of the source signals . at this stage ,source information in the data is retained as much as possible .it is natural to obtain such a rough estimate by solving the following minimization problem : note that the forward operator contains the information of positions and orientations of the dipoles , and how they are represented at the sensor level .this information can be decomposed by applying a singular value decomposition ( svd ) to , that is , , where is an orthogonal matrix and is a thin ( since ) orthonomal matrix , such that and .then the objective function in the optimization problem ( [ eqmin ] ) becomes   let and .the minimization problem ( [ eqmin ] ) is equivalent to let and be the row of and the row of , respectively . since  is a diagonal matrix , the minimization problem ( [ eqminraw ] ) can be obtained by separately solving for each , where is the diagonal element in .this problem has a unique solution .then the estimated matrix with in the row can be obtained .thus , a rough estimate of can be obtained by solving note that is , is , and is . since and , equation ( [ eqchat ] ) does not have a unique solution for .any solution of ( [ eqchat ] ) can be written as , where is a orthonormal matrix whose columns are orthogonal to the columns of and is a matrix .we pick the minimum norm solution , which is .in fact , solves the following optimization problem : we can see this by noticing that and the equality holds when is a matrix of zeros .we call this the raw estimate . note that the raw estimate can only recover information that lies in the column space of , and thus any information orthogonal to the columns of the columns of are the right singular vectors of , the column space of is equivalent to the row space of .this information loss can also be understood by viewing the forward operator as a filter that maps the source to the space of the observations , , and so the information in the columns of that is orthogonal to the rows of can not be recovered .since all imaging methods are based on model ( [ eqdata ] ) , information loss is a common problem to these methods .this is the limitation of the meg technology .fortunately , according to our experience , most important information still remains in many real - world applications , as we will see in our real data example .note that the methods that require basis representation may cause additional information loss , since any information in the columns of that is orthogonal to the basis chosen will also be lost .it is obvious that the raw estimate , , can be noisy .the purpose of stage 2 is to polish the raw estimate by incorporating the smoothness and focality assumptions .the polished solution from this stage is denoted as .as we will see in the simulation study in section [ secsim ] , the shapes of the time courses in the rows of are noisy but usually follow the shapes of the true curves , and may suggest a broader range of active regions . in a penalized least squares framework, we apply a roughness penalty to smooth the recovered time courses and apply a sparsity - inducing penalty to refine the active regions .  in order to apply two penalty functions to , we first use the two - way structure of the raw estimate and decompose it into spatial - only and temporal - only components .specifically , we write as where the matrix contains only the temporal features of  , and can be treated as the spatial coefficients .when , the decomposition ( [ eqbapp ] ) suggests a reduced - rank representation of . our empirical studies, however , suggest that any reduced - rank representation would lead to information loss and thus the full rank model is needed in practice .we shall focus on the full rank model ( ) for the rest of the paper . for identification purposes ,we require that is an orthogonal matrix , that is , .note that when the full rank model is used , the reconstruction error of using to represent , , is exactly zero .we propose to introduce focality and smoothness requirements on and respectively at the cost of allowing some errors in reconstructing .in particular , we consider the following penalized least squares problem : where and are appropriate penalty functions , and and  are the corresponding penalty parameters .to ensure the spatial focality of the recovered source signals , we employ a sparsity - inducing penalty on so that the estimated is a sparse matrix , that is , a large proportion of its entries are zero. note that if a row of has all zero entries , then the corresponding row of has all zero entries , indicating no signal or an inactive location .although other choices are possible , we use the penalty to serve our purpose . on the other hand , to induce smoothness to the time course of the recovered source signals , we apply a roughness penalty to the columns of so that each column of is a smooth function of time .let represent a generic vector representing a column of .one choice of the roughness penalty is the squared second order difference penalty , defined as .this penalty is a quadratic form and can be written as for a nonnegative definite roughness penalty matrix ,  .the overall penalty on is the summation of the penalty on each column , .using the penalties defined above , the penalized least squares problem ( [ eqpen - ls ] ) becomes    we propose an iterative algorithm to solve ( [ eqpen ] ) that alternates the optimization with respect to and .the algorithm starts with setting the initial to be the orthonormal matrix of the right singular vectors from the svd of .that is , let , where and are orthonormal matrices , and we set the initial .  _fix , update ._ when is fixed as , the roughness penalty term in the objective function ( [ eqpen ] ) is irrelevant to the optimization of .thus , updating reduces to solving the problem this is similar to one step of the iterative algorithm for the sparse principal component analysis as formulated by .express as a summation of a serial of rank - one terms where and are the column of and , respectively .since simultaneous extracting of all the rank - one terms is computationally expensive , we propose to obtain them sequentially .for the first rank - one term , we solve for fixed  this problem has a closed - from solution which is given below . for the sake of notational simplicity, we drop the subscripts for now and express the objective function of ( [ eqvecg ] ) as \\[-8pt ] & & \qquad = \sum^p_{i=1 }  \biggl\ { a_i^2\sum^s_{l=1}\hat{g}_l^2 - 2a_i\sum ^s_{l=1}\hat{b}_{il}\hat{g}_l+\sum^s_{l=1}b^2_{il}+\mu_1|a_i| \biggr\ } , \nonumber\end{aligned}\ ] ] where is the element in , and , , are the elements of the vector .the minimization of ( [ eqseq ] ) is equivalent to independently solving  optimization problems according to lemma 2 of , the minimizer of each objective function in ( [ eqsoft ] ) is the soft thresholding rule where , and .the -vector that minimizes ( [ eqseq ] ) is .  after the first rank - one term is obtained , we find the second rank - one term by solving the following minimization problem , while fixing : this is the same problem as ( [ eqvecg ] ) except that the in ( [ eqvecg ] ) is replaced by the residual from the rank - one approximation .the rest of the rank - one terms , , , can be obtained sequentially in a similar manner by using the residuals from the lower - rank approximations .  _ fix , update . _ when is fixed as , the penalty term in ( [ eqpen ] ) becomes constant and thus is irrelevant to the optimization with respect to  .the update of  then solves the following problem : since directly solving this problem is complicated , we solve for the columns of sequentially . to obtain the first column of , we solve the problem which has the solution .let be the eigen - decomposition .then to obtain an update of , we solve the problem which has the solution \\[-8pt ] & = & { { \mathbf{p}}}(\hat{\mathbf{a}}^t_2\hat{\mathbf{a}}_2{{\mathbf{i}}}+ \mu_2 { \boldsymbol{\lambda}})^{-1 } { { \mathbf{p}}}^t\hat{{\mathbf{b}}}_{\mathrm{res},1}^t \hat{\mathbf{a}}_2 , \nonumber\end{aligned}\ ] ] where again is the residual from the rank - one approximation .the rest of , , can be obtained similarly using the residuals from the corresponding lower rank approximations .when all columns of are obtained , we orthonormalize the columns of by taking the qr decomposition of and assigning the matrix to .  the iterative twr procedure , including stage 1 and stage 2 , is summarized in algorithm [ algtwr ] .we consider the algorithm has converged if the frobenius norm of the relative difference between the current solution and the previous solution is smaller than a prespecified threshold value . in our implementation , we declare convergence when . based on our empirical studies ,only a few iterations are needed to reach convergence ; 15 iterations are usually sufficient for our numerical examples in sections [ secsim ] and [ secmeg ] .there are two tuning parameters in the twr algorithm : the focality parameter , , and the roughness penalty parameter ,  .the choice of and can be done using the cross - validation ( cv ) techniques and the generalized cross - validation , respectively .  to select , we can utilize the _ leave - one - out _ cv that minimizes the leave - one - out cv score defined as where is the row of corresponding to the time course , is the row of , and and are the estimates of and using all observations except the time course . however , practical application of the cv has some difficulties .the gradient - based optimization is not feasible for minimizing the cv score since it is not a smooth function of , a consequence of using the penalty .in addition , direct computation of the cv score is costly because of the usual large scale of the real problem . in a typical meg study, is over 200 , is a few hundred , and can be over 15,000 . in order to reduce the computational cost , we propose to use the -fold cross - validation .specifically , we divide the rows of and into about equally sized parts and leave out one part each time for validation , and use the rest of the parts for estimating and .the -fold cv score is defined as where contains the part of the rows of , contains the corresponding rows of , and and are the estimates of and using all observations except the part of time courses that are left out for validation .we used in our implementation .to further speed up the algorithm , we restrict our search only in a moderate - sized set of discrete candidate values for .such restrictive search is satisfactory , since we find that the results are usually not very sensitive to mild changes of ( see sections [ secsim ] and [ secmeg ] ) and thus fine tuning of is not necessary .we used 10 different values evenly - spaced between 0 and 1 for in our simulations and the real - world meg example ; the search range may need to be changed for different problems .  to select , note that given , the update of columns of can be obtained by solving separate penalized regression problems . for the column ,the regression has as the input , where , as the output , and the hat matrix of the regression is , according to equation ( [ eqg2 ] ) .theoretically , can take different values for different s , but we decide to use a common for all the s based on computational efficiency consideration .the advantage of this strategy is that there is only one optimization problem to solve for choosing the tuning parameter when updating  .then , the overall gcv criterion is the average of all individual gcv criteria : where , and .the gcv optimization is nested in the iterations because it is defined conditioning on the current value of  .since the gcv criterion is a smooth function of , the optimization can be done using a combination of golden section search and successive parabolic interpolation [ ] .   by separately setting one of the penalty parameters in ( [ eqpen ] ) to be zero ,one can obtain two different one - way regularization methods : towr and sowr , as explained below .these two one - way regularization methods will be used as a comparison to twr to demonstrate the need for two - way regularization .  letting leads to a method that emphasizes temporal smoothness of the recovered signals , which is referred to as towr ( temporal one - way regularization ) , and is related to the functional pca [ ] .the corresponding optimization problem becomes a modified version of algorithm [ algtwr ] can be applied for computation , with the `` update '' step in the algorithm simplified to .letting leads to a method that encourages spatial sparsity of the recovered signals , which is referred to as sowr ( spatial one - way regularization ) and is related to the sparse principal component analysis of . in this case , the optimization problem ( [ eqpen ] ) reduces to again , a modified version of algorithm [ algtwr ] is applicable , but with the `` update  '' step simplified to .in this section we illustrate the proposed twr method using a synthetic example that mimics human brain activities .both the source and the forward operator are created based on real - world meg studies . generated the forward operator , , from a  subject head boundary element model using the mne software ( available at : http://www.nmr.mgh.harvard.edu/martinos/userinfo/data/sofmne.php ) .the matrix is a matrix , corresponding to a meg device with 248 valid channels . to mimic real - world scenarios and ensure enough difficulty of the problem, we located two source areas on the left and the right hemispheres , respectively .the sources were generated from two sine - exponential functions [ ] and are shown in figure [ figbsim](a ) .the black solid and the red dashed curves are source signals located at the left motor and the right visual cortical areas , respectively .as we can see , the sources reach their energy peaks at 25 ms and 58 ms , respectively .the synthetic meg time courses were generated using equation ( [ eqdata ] ) and were obtained using a sampling frequency 355 hz with a duration of 200 seconds [ see figure [ figbsim](b ) ] . by mimicking the real meg data after preprocessing ,that is , denoising and smoothing , the signal - to - noise ratio , , is set to be 5db .we compare twr with eight different methods that can be put into two categories as given below .* one - way regularization : * * the -based mne method [ ] * * the -based mce method [ ] * * sowr ( i.e. , spatial sparsity only ) * * towr ( i.e. , temporal smoothness only ) * two - way regularization : * * the method proposed by * * mne ( i.e. , obtaining the mne solution as stage 1 and then applying sowr ) * * mce ( i.e. , obtaining the mce solution as stage 1 and then applying towr ) * * mne ( i.e. , obtaining the mne solution as stage 1 and then applying stage 2 of twr )  we put mne+sowr in the two - way regularization category because the  penalty in mne puts constraints on both domains , and sowr puts the  penalty only on the spatial domain . as a result, the temporal domain is regularized by the penalty , while the spatial domain is regularized first by the penalty and then by the penalty .similarly , mce+towr is also categorized as a two - way regularization method .mne+sowr and mce+towr can be considered as two alternative ways for two - way regularization and are suggested by a reviewer .mne+twr , also suggested by a reviewer , is a slight modification of twr , replacing the first stage of twr by mne .its inclusion in comparison helps us study the effect of using a different stage 1 estimator on the performance of twr .we implemented all the methods in r , and the tuning parameters are selected using either cv or gcv .three comparison criteria are utilized : the overall mean squared error ( mse ) , the standardized distance between the energy peak of the estimated source and the energy peak of the true source , and the computation time .the overall mse is defined as where and are the true and recovered source matrices , respectively .  .7d3.6d3.7d9.4@ * method * & & & & + mne & 544.0 ( 9.0 ) & 50.2 ( 7.3 ) & 42.9 ( 5.9 ) & 4371 ( 4.3 ) + mce & 903.7 ( 8.9 ) & 337.1 ( 6.4 ) & 156.1 ( 11.4 ) & 1545 ( 3.0 ) + towr & 407.9 ( 8.9 ) & 40.2 ( 5.8 ) & 39.6 ( 4.3 ) & 1841 ( 3.4 ) +sowr & 153.2 ( 7.7 ) & 19.3 ( 4.6 ) & 13.9 ( 3.9 ) & 1798 ( 3.6 ) + twr & & & & 1872 ( 3.5 ) + & 44.3 ( 7.1 ) & 31.0 ( 6.1 ) & 17.8 ( 2.3 ) & 40872 ( 8.8 ) + mne+sowr & 187.3 ( 8.8 ) & 27.9 ( 6.8 ) & 14.5 ( 3.1 ) & 5998 ( 3.9 ) + mce+towr & 912.7 ( 10.9 ) & 343.8 ( 6.2 ) & 145.2 ( 12.7 ) & 3321 ( 3.8 ) + mne+twr & 28.6 ( 7.2 ) & 16.9 ( 4.3)&10.7 ( 3.9 ) & 6201 ( 3.1 ) +  the energy of the dipole at time point is defined as , where , are the amplitude components for the dipole at the time point in the cartesian coordinate system .the energy of the reconstructed source can be defined similarly . the standardized distance between the estimated and the true energy peak at time point defined as where is the total number of dipoles , are the coordinates of the location for the maximum source energy at time point , and are the coordinates for the maximum estimated source energy at the corresponding time point . in this simulation example , there are two peak times , 25 ms and 58 ms , so we are interested in and .the simulation was conducted 100 times with the noise term in model ( [ eqdata ] ) newly generated for each run .the criteria described in the previous subsection ( i.e. , mse , , , computation time ) were evaluated for each simulation run , and the mean and standard error of the criterion values across the 100 runs were calculated .the numerical results are shown in table [ tblcompare ] .several interesting observations can be made from the table .twr is the best method in the sense of having the smallest mse and the shortest distances between the true and the estimated peaks . among the four one - way regularization methods , sowr and towroutperform the classical mne and mce methods , and towr outperforms sowr .the fact that twr outperforms the four one - way regularization methods justifies our proposal of using two - way regularization .the method is the third most accurate method , but its computation time is more than 21 times as large as that of twr .mne+sowr and mce+towr are less satisfactory , demonstrating the importance of the first stage .mne+sowr is not better than sowr because the penalty of mne does not smooth the temporal domain .the performance of mce+towr is similar to mce and is not better than towr because mce does not recover well important information at the first stage , and hence towr based on mce is inaccurate .note that the reported computation time for twr , sowr and towr are based on fixed 15 iterations in order to make the calculation of the average computation time meaningful .such report is conservative because these algorithms usually converge rapidly and fewer iterations ( usually less than 10 ) are enough to obtain considerably good accuracy .figures [ fig25 ] and [ fig58 ] show the 3-d brain mapping by different methods at 25 ms and 58 ms for a randomly selected simulation run .twr performs the best among the nine methods in detecting the true source locations even though it misses some small regions .it is able to identify the majority parts of both source locations , and its solutions are focal .solutions from sowr and mne+sowr are more scattered than twr .mne and towr produce even more diffuse solutions .mce misses the main parts of both active areas and so does mce+towr , and they are the least satisfactory methods .the method recovers some of the activity , but the solution is overly focal .the plot of mne+twr is very similar to that of twr , so it is not presented here to save space .direct comparison of results of twr and towr clearly demonstrate the positive effect of using regularization in the spatial domain .method also identifies the active area but the solution is too focal .sowr and mne+sowr produce more scattering solutions than twr .mne and towr detected active areas are diffuse .mce and mce+towr misidentify the active region . ]   identify the major active area and the solution is focal .sowr and mne+sowr produce more scattering solutions than twr .mne and towr detected active areas are diffuse .mce and mce+towr misidentify the active region . ]figures [ figsource1 ] and [ figsource2 ] show the true and the recovered time courses by the nine methods for an arbitrarily chosen single dipole component in the two active areas , respectively , for a randomly selected simulation run .each subfigure shows the true time course and the estimated time course by one method .as one can see , the methods considering the temporal smoothness reconstruct the shape of the source time course well .twr , towr , , mce+towr and mne+twr all produce smooth time courses .twr recovers the most energy of the source , while mce+towr recovers the least .mne+twr tends to overshrink the amplitude of the time course because mne overshrinks the amplitude .the methods without considering the roughness regularization in the temporal domain result in noisy time courses even though some methods can recover the general trend . in figure [ figsource2](b ), mce does not capture the major peaks of the signal , and , consequently , mce+towr [ figure [ figsource2](h ) ] , which relies on the solution of mce , does not recover any signal activity either .direct comparison of results of twr and sowr clearly demonstrate the positive effect of using regularization in the time domain .   and mne+twr recover the shape of the time course reasonably well and the solutions are smooth .but mcr+towr , mne+twr and overshrink the amplitude .mne , mce , sowr and mne+sowr estimate the general trend reasonably well , but the estimated time courses are too noisy .twr gives the best result .]  and mne+twr recover the shape of the time course reasonably well and the solutions are smooth .but mne+twr overshrinks the amplitude .mne , sowr and mne+sowr estimate the general trend reasonably well , but the estimated time courses are too noisy .mce and mce+towr do not recover the shape of the time course .twr gives the best result . ]the selection of the focality parameter and the roughness penalty parameter was conducted using the method presented in section [ sectuning ] .figure [ figtune](a ) and ( b ) shows the cv and gcv scores for twr as functions of and , respectively .the optimal values of the tuning parameters are and .figure [ figtune](c ) shows the sparsity level of the reconstructed source matrix , , for twr as a function of the number of iterations when the tuning parameters are set at the selected values .the sparsity level for a matrix is defined as the number of zero entries over the total number of entries . herethe total number of entries for is . from this figure, we observe that the sparsity of levels off rather rapidly and stays steadily at about 0.996 , a fairly high sparsity level .in fact , this sparsity level matches closely the true level in the simulation setup : the number of true source dipoles is 20 , and so the total number of active source components is 60 after considering orientations .thus , the true sparsity level is .   and and the sparsity level as a function of the number of iterations .the optimal and are around 0.33 and 5.9 , respectively .the sparsity measure levels off at around 0.996 . ]in this section we demonstrate the proposed method using a human meg data set obtained from the center for clinical neurosciences at the university of texas health science center at houston .the study subject is a 44-year - old female patient with grade three left frontal astrocytoma who underwent the meg test as part of the presurgical evaluation .the patient underwent a somatosensory task which is designed to noninvasively identify the somatosensory areas of the patient .we choose this study because of the clinical usefulness of the somatosensory task in presurgical mapping .data collection was done with a whole - head neuromagnetometer containing 248 first - order axial gradiometers . during the meg somatosensory session ,558 repeated stimulations were delivered to the patient s right lower lip through a pneumatically driven soft plastic diaphragm .each stimulation lasted 40 ms with 450 ms epoch duration ( including a prestimulus baseline of 100 ms ) and an interstimulus interval randomized between 0.5 s and 0.6 s. we removed the offset and averaged the 558 epochs to obtain the final event - related magnetic field response .then a bad channel was removed .the meg device recorded 228 time points in each epoch .the measurement matrix is , where is the number of valid meg channels and is the number of recorded data points per epoch .the forward operator was obtained using the mne software with .the measured meg recordings from the 247 valid channels are plotted in figure [ figmeg](a ) . among the 228 time points ,there are two peaks at time points 85 and 99 , corresponding to the activation of the primary somatosensory area contralateral to the stimuli , as expected by clinical experiences and brain anatomic theories .  nine methods , mne , mce , twr , towr , sowr , mne+twr , mne+sowr , mce+towr and ,were applied to solve the meg inverse problem .figure [ figmeg](b ) shows the reconstructed time courses for an arbitrary source location by different methods .as we can see , twr , sowr and towr , are satisfactory in terms of estimating the shape of the source time course and capturing the peak features at time points 85 and 99 .but sowr produces a noisy time course .mne and mne+sowr overshrink the magnitudes in addition to producing a noisy time course .mne+twr recovers the shape of the time course but underestimates the amplitude .the method does not distinguish the two peaks .mce only identifies the first peak but misses the second one .mce+towr does not capture any activity because it smoothes the spikes caused by mce and hence is the least satisfactory method .figure [ figmegmaps ] shows the side views of the brain mapping at time point 85 by different methods .as we can see , the somatosensory area was correctly identified by twr , which matches the clinical expectation . as with the synthetic example ,towr and mne produce diffuse solutions , leading to false positives around the somatosensory area .sowr produces a scattering solution and so does mne+sowr .mne+twr and also identify some activity in the frontal lobe . solutions from mce and mce+towrare too focal and do not cover the somatosensory area .identify some activity in the frontal lobe in addition to the somatosensory area .solutions from mne , sowr , towr and mne+sowr are too diffuse to be satisfactory .both mce and mce+towr miss the activity in the somatosensory area . ]figure [ figtunmeg](a ) shows the cv error as a function of .the cv error was minimized when the sparsity parameter , , is about 0.44 .figure [ figtunmeg](b ) displays the gcv error as a function of .it shows that the optimal is about 59.5 . the sparsity level as a function of the number of iterationsis shown in figure [ figtunmeg](c ) . as we can see , the sparsity level increases at first and then levels off rapidly , indicating the algorithm converges fast .the optimal sparsity level was about 0.999 .   and and the sparsity level as a function of the number of iterations .the optimal and are around 0.44 and 59.5 , respectively .the sparsity measure levels off at around 0.999 . ]twr solves the meg inverse problem by using two - way penalties that promote both the temporal smoothness and the spatial focality of the solution .we developed a computational efficient two - stage procedure for implementing twr .we also considered a one - stage approach that tries to recover the source signal matrix by solving the optimal matrices and can be obtained by alternating optimization . when fixing as , the optimal can be obtained as in algorithm [ algtwr ] , as described in section [ secalg ] . when fixing as , the problem ( [ eqpen-1stage])becomes +\mu_1|{{\mathbf{a}}}|\ } \\ & & \qquad = \min_{{{\mathbf{a}}}}\{\|{{\mathbf{y}}}\hat{{\mathbf{g}}}-{{\mathbf{x}}}{{\mathbf{a}}}\|_f^2+\mu_1|{{\mathbf{a}}}|\},\nonumber \ ] ] which is equivalent to different problems , one for each column of , namely , where is the column of the matrix .each of these problems is a standard lasso regression problem [ ] with over 10,000 variables . although efficient computational algorithms exist for the lasso regression , the fact that the lasso problem needs to be solved a few hundred times during each iteration of updating makes this approach computationally unattractive .developing a scalable algorithm for the one - stage approach is an important issue for its practical application and remains an interesting research topic .the authors thank the editor , the associate editor and two referees for their comments , which helped improve the scope and presentation of the manuscript .the authors thank the meg lab at the university of texas health science center houston for providing the data . in particular , thanks are due to professors andrew papanicolaou and eduardo castillo for their suggestions and comments .

Abstract: the meg inverse problem refers to the reconstruction of the neural activity of the brain from magnetoencephalography ( meg ) measurements . we propose a two - way regularization ( twr ) method to solve the meg inverse problem under the assumptions that only a small number of locations in space are responsible for the measured signals ( focality ) , and each source time course is smooth in time ( smoothness ) . the focality and smoothness of the reconstructed signals are ensured respectively by imposing a sparsity - inducing penalty and a roughness penalty in the data fitting criterion . a two - stage algorithm is developed for fast computation , where a raw estimate of the source time course is obtained in the first stage and then refined in the second stage by the two - way regularization . the proposed method is shown to be effective on both synthetic and real - world examples .  ,  ,   +  .