every setting for a measurement on a quantum system can be completely specified by an orthonormal basis of the system s hilbert space .identical systems can be independently prepared in a ( pure ) state such that , every time , we get a definite outcome when a system is measured in a setting .if we change to a physically distinct setting , then we observe sometimes one and sometimes other multiple outcomes .in other words , there the probability is one for an outcome in -setting , whereas none of the probabilities is one in -setting . of course ,in any setting , all the probabilities are nonnegative numbers that sum up to one . apart from that ,the probability vectors ( distributions ) and with the two settings and , respectively must follow certain constraints , called _ quantum constraints _ ( qcs ) , together .  historically , such qcs are expressed in terms of uncertainty relations ( urs ) by taking hermitian operators rather than orthonormal bases .an ur is an inequality , , between two real - valued functions : uncertainty measure and its lower bound . in 1927 , heisenberg introduced the first ur for the position and momentum operators .robertson generalized the heisenberg s relation for an arbitrary pair of operators .both employed the standard deviation as a measure of uncertainty . in robertson s ur ,the lower bound is a function of state .deutsch criticized it and introduced a new ur for a finite - dimensional state space , by taking entropy as a measure of uncertainty .he achieved a state independent .later , a better lower bound was conjectured and then proved .review articles  summarize further developments in the field .  throughout the article, we are considering -level quantum systems ( qudits ) and projective measurements .our primary objective is to study a set of combined - probability vectors , called _ combined - probability space _ , where every vector respects certain , if not all , qcs . here the elemental qcs are the triangle inequalities ( tis ) between _ quantum angles _ , and the ( un)certainty relations emerge from them . as an angle between a pair of kets called quantum angle is a metric over the set of all pure states , we own tis . in sec .[ sec : ps - c ] , we present the combined space that is a compact convex subset of the -dimensional real vector space .thanks to the krein - milman theorem ( see theorem  and appendix a.3 in ) , every compact convex subset of can be generated by the convex combinations of its extreme points . as a principal result, we provide a family of _ parametric curves _ in sec .[ sec : ps - c ] , which represents all the extreme points of the combined space .an uncertainty measure should be a concave function on the combined - probability space argued in the beginning of sec .[ sec : um - ur ] .the concavity of ensures that its global minimum will occur at the parametric curves ( extreme points ) of the space ( see theorem  and appendix a.3 in ) .hence , one can exploit these curves to obtain a ur , rather easily , for her or his liking of and , of course , for general measurement settings and .  in sec .[ sec : um - ur ] , we choose a concave , thus uncertainty , measure .a significance of our choice lies in the fact that is again a concave function on every parametric curve ( that is , as a function of the parameter ) . therefore its absolute minimum will occur nowhere but at the endpoint(s ) of these curves .a simple three - step procedure is delivered to find the lower bound for an arbitrary pair of settings and for a finite .one can employ an ordinary computer to run the procedure . besides , is presented in analytic forms for and in the case of mutually unbiased bases ( mubs ) .if a suitable concave function can be a measure of the uncertainty , then an appropriate convex function will be a measure of certainty . in sec .[ sec : other - um ] , we pick some other concave and convex functions and present the ( un)certainty relations for by exploiting the same parametric curves .we conclude the article with sec .[ sec : conc - out ] .the appendices are kept for certain technical details and proofs : the tis are derived in appendix [ sec : der - q - const ] .it is manifested in appendix [ sec : compact - convex - w ] that the combined space is a compact convex set .the parametric curves are explicitly obtained in appendix [ sec : extreme - points - w ] with the help of appendix [ sec : inter - results ] .in quantum theory , observables are represented by hermitian operators .if such an operator is degenerate , then it possesses more than one eigenbases , where some of them can represent physically different measurement setups . hence ,` measurement in an orthonormal basis ' of the underlying hilbert space is rather well defined than ` a measurement of an operator ' ( see chapter 7 in ) .in fact , measurement in a basis measure all the operators whose eigenbasis is .moreover , deutsch pointed out that a measure of uncertainty for a discrete observable must not depend on its eigenvalues , but on its eigenbasis .with all these considerations , we choose orthonormal bases instead of hermitian operators to specify different projective measurements for a qudit .we begin with two orthonormal bases of a -dimensional hilbert space to depict the two measurement settings and , respectively .our ( un)certainty relations are applicable in the following experimental scheme . a similar scenario peres used in his book  at page 93 to interpret the position - momentum ur . in proposal , clearly , the two measurements have no influence whatsoever on each other .  throughout the text, we assume is a pure quantum state so that we can associate angles  and tis  with the state vector .although every ( un)certainty relation presented in this paper as it is applicable for every qudit s state [ see the text around ] .the state provides two probability distributions for the two measurement settings [ given in ] by the born rule : are the probabilities of getting outcome in the -setting and outcome in the -setting , respectively .next , we present quantum angles : are the angles between and and between and , respectively . in the entire article, we consider only the principal values }]and the probabilities that lie in } ] and } ] , so applying it on both sides of ti  and using , , , and , we attain after a rearrangement of terms . as both sides in are nonnegative functions of the probabilities , squaring and further simplification lead to for every .all those pairs that obey qc  for every build the combined - probability space for the two measurement bases in . in the case of ,even if we consider all tis given in for each , they do not capture the full qcs for a general pair of settings .therefore , one can still find some that corresponds to no quantum state .nevertheless , our analysis relies on the following fact : _ every that does not belong to can not be obtained from a quantum state , thus it is discarded . _ to investigate a space contains all those , and only those , pairs that originate from the quantum states is not the aim of this paper .however , it is not tough to realize that for ; in general , .  note that is a _ proper _ subset of . to provethis one can show : only one out of the two extreme points specified by and , where  can belong to .recall that if and only if then the point described by belongs to , otherwise will be violated . secondly, if then , and can not be obeyed by the other point ; hence that lies outside of .the space is held by the conditions  and a compact and convex subset of [ for a proof , see appendix [ sec : compact - convex - w ] ] .every point of such a set can be written as a convex combination of its extreme points due to the krein - milman theorem ( see theorem  and appendix a.3 in ) .we begin our journey from an interior point of in appendix [ sec : interior - w ] and arrive at its extreme points at the end of appendix [ sec : extreme - w ] .there it is concluded that the set of all extreme points of comes from a family of parametric curves .one can skip all those technical details and start constructing the parametric curves straight from the conclusion : the first step is to pick a set of angles from a single column or row of the matrix given in .such a set is called -set , and .for instance , we pick the top angles from the first column .then we associate tes with the -set as by taking , where the subscript 1 reflects the selected column .  next , with  and, we assign probabilities to the angles : and .they create the probability vectors one can observe that serves as a vector - valued function of a single real parameter , thus it exhibits a parametric curve .since the curve is associated with an -set and all its points obey tes , we call it an -parametric curve.  a part of the curve , identified by the upper and lower limits , lies in and represents its extreme points because can not be written into a convex combination of other points of . in appendix[ sec : limit - beta-1 ] , we realize that the two limits are fixed by [ see ] .equations  and are like eq . , whose roots are stated in .always the root with + sign delivers the correct limit [ for justifications , see the last paragraph in appendix [ sec : limit - beta-1 ] ] .if one chooses an -set from a row of , say , then the -parametric curve is constructed as now the parameter is } ] functions , each of a single variable [ for example , see ] . then the smallest minimum will be the lower bound in a urthis task can be easily completed with a regular computer . in the next two sections ,we discuss certain concave as well as convex functions on .if quantifies the uncertainty about the outcomes when a qudit is measured in the basis of then should be a concave function of .it is because mixing probability distributions , and as with } ] .now we define our combined uncertainty measure on the convex set , rather than .sum of two concave functions is concave , so is also a concave function .a mixed quantum state is a convex combination of pure states , the probabilities are linear functions of the state ( , ) , and is a compact and convex set . as a result , every associated with any ( pure or mixed ) quantum state lies in . and , because is a concave function on , our ur given in applies to every state for a qudit .this is also true in the case of other ( un)certainty relations presented in sec .[ sec : other - um ] , because there also we have either a concave or a convex function . by the way, one can check that if then the born rule  reduces to .the range of and our ur are presented as is the global minimum that will occur at the -parametric curves [ given in sec .[ sec : ps - c ] ] . whereas , gains its absolute maximum only at the point identified by for all .it is called the _ center _ of , which represents the uniform distribution for both the settings .now recall from sec .[ sec : ps - c ] that an extreme point of , describe by , belongs to if and only if .only in such a situation that does not necessarily require both the bases and to be the same in any way we have the _ trivial _ lower bound and thus the ur . a similar statement is made by deutsch in . for ,the trivial case is possible if and only if the two measurement settings are ( physically ) the same .a nontrivial lower bound materializes when the settings are completely different , that is when for every .so the following analysis is obviously for the nontrivial cases .  to find the lower bound  and to establish the ur , we write the functional form which of acquires on an -parametric curve specified by . to show that of is a concave function of , we present +     \frac{\partial^2\,\sqrt{p_s}}{{\partial\beta_1}^2}\,,\quad\\ \\ \label{d - sqrtps / d - beta } & & \frac{\partial^2\sqrt{p_s}}{{\partial\beta_1}^2}=  -\,\frac{1}{4\,p_s^{3/2 } }  \left(\frac{\partial\,p_s}{\partial\beta_1}\right)^2  + \frac{1}{2\sqrt{p_s}}\,\frac{\partial^2\,p_s}{{\partial\beta_1}^2}\ , ,  \quad\mbox{and}\\ \label{d - ps / d - beta }  & & \frac{\partial^2\,p_s}{{\partial\beta_1}^2}=  -2\left[\,2\,p_s+(m-2)\,\right ] .\end{aligned}\ ] ] with these derivatives , one can clearly see for .whereas , for , one can directly realize .this proves that is a ( strictly ) concave function on every parametric curve .therefore , its global minimum will always be at the endpoints of the curves .endpoints of an -parametric curve are identified by the two limits on a parameter [ see as well as ] .it is manifested in appendix [ sec : limit - beta-1 ] that , to compute a limit , we always have to solve an equation such as ; which carries m number of angles from a column or a row of [ given in ] .note that we use small letter ` ' when we construct a parametric curve with an -set [ see sec .[ sec : ps - c ] ] and use capital letter ` m ' when we compute a limit with an m - set . essentially , one needs to follow a three - step procedure to compute a limit and then the value of [ defined in , see also ] at the corresponding endpoint of a curve : the equation in step 2 is like eq .  that is solved in appendix [ sec : limit - beta-1 ] , and every time we take the solution  with + sign .one can observe that and therefore are solely determined by the -set picked in step 1 .after repeating the three - step procedure for every -set and for each , we collect a set of values for all the endpoints. then , the smallest value in this set will be [ defined by ] , and thus we own our ur [ presented in ] . since every is determined by the entries in -matrix , the lower bound only on the measurement bases in is _ independent _ of a quantum state .besides , to compute , we can employ an ordinary computer , which repeats the three steps of by taking \ ] ] number of -sets one by one .in fact , } ] , and it is a concave function on with the hassian matrix composed of the second - order derivatives where . considering the same function for the -setting , that is , one can formulate a combined uncertainty measure by the sum and then produce an entropy ur .such urs are reviewed in . for ,tight entropy urs are presented in ( see also ) .we can choose as another ( un)certainty measure .one can prove that the hassian matrix with entries = \frac{\partial^2 u_\gamma}{\partial p_l\partial p_k}\,,\ ] ] , is a negative and positive semidefinite matrix for and , respectively .it confirms that is a concave ( uncertainty ) and convex ( certainty ) measure when and , respectively .a similar observation is made in .in fact , our uncertainty measure of  is with the exponent .furthermore , the range of is } ] if . when , for every due to eq . , thus is not a genuine ( un)certainty measure .like before , one can establish a ( un)certainty relation with the sum . for , in the case of , we obtain as a tight certainty relation ; which is also given in . due to , one can immediately derive from the ur .where of reaches its absolute minimum ( uncertainty ) on , there the function achieves its global maximum ( certainty ) the certainty measure  hits its absolute minimum 1 at the center of [ depicted by the star in figs . [fig : u , pi/6 ] and [ fig : std , pi/3 ] ] .lastly , we pick the function that defines a norm on if we replace with . since every follows , the modulus sign is not shown in .every norm is a convex function , so can be considered as a certainty measure on ; } ] , and the probabilities belong to the interval } ] . applying square root to both sides of the inequality ,we gain } ] , both and are nonnegative numbers , hence there is no need to use the modulus on either side of the above inequality . asthe function is a strictly decreasing function and for } ] , will be a true upper bound on only if it is smaller than or equal to .hence , we can further improve as latexmath:[\[\label{qc_beta - alpha - imp }  taking the right - hand side inequality and applying the cosine function that decreases monotonically on } ] .thanks to the convexity of , the combination  belongs to and meets all the demands .  now we demonstrate that the components and of respect inequality : \\ \label{pq-3 } & \leq&r_{ij}+1 + 2\sqrt{r_{ij}}\;\sqrt{1-\lambda p'_i-(1-\lambda)p''_i}\nonumber\\ & & \sqrt{1-\lambda q'_j-(1-\lambda)q''_j}\\ \label{pq-4 } & = & r_{ij}+1 + 2\sqrt{r_{ij}}\sqrt{(1-p_i)(1-q_j)}\;.\end{aligned}\ ] ] we have equality  due to the convex combination , and then we acquire inequality  by employing and .the next inequality  is attributed to the concavity of a real - valued function defined on \times[0,1]} ] . now we consider and since the difference between angles } ] .one can also perceive as a ti .next we are going to validate a result that is applied in appendix [ sec : extreme - points - w ] . let us designate and by and , respectively , and write just like .one can show that the sum due to and .clearly because } ] . as a net result, , the last term in turns out to be a nonnegative function , and thus we achieve .it completes a proof of .  if and then evidently we have the equality of .now let us prove the converse under the requirements of .if then the last term in must vanish , which occurs provided the sum in  attains its upper bound or . the case arises when and , and happens when and .both these cases come under is when the sum in  reaches its upper bound which materialize if and only if and ; it validates .  similar to we have and to plus we have appendix [ sec : compact - convex - w ] , we demonstrate that the combined - probability space is a compact convex set in . according to the krein - milman theorem ( see theorem  and appendix a.3 in ) , every point of such a setcan be decomposed into a convex combination of its extreme points . in this appendix ,starting from an arbitrary interior point of , we move toward its extreme points .a point that obeys each of the constraints , , and with _ strict _ inequality , is called an _ interior _ point of . in certain cases , such as and ,there exist no interior point only extreme points , then the following analysis is not needed . however , for , there is always an interior point : with , one can show that the center specified by forall  is an interior point when .we begin our journey from a general but fixed interior point along a straight line , which is the locus of points , where obey the linear equation and .one can acknowledge that two points on this line differ from each other only in the first two coordinates , hence are the only variables here .in , the inequality saturates for and becomes strict due to when .since we never want to move outside of the combined space , we only consider those points on the line that lie in . from sec .[ sec : ps - c ] recall that a point of lies in if and only if it meets all the requirements  , and if it also satisfies all the tis  only then it belongs to .so a point on the line , defined by , is contained in if and only if   with and , one can derive as per and , we can attach angles and with and , correspondingly .if these angles comply with only then .observe that the other demands for to be in for and are automatically met , because and are also parts of the interior point .considering the suprema we can convert all the conditions in into two throughout the paper , in the subscripts of angles , capital letters are used to highlight a supremum .a supremum , say , can not be a negative number : implies for every by the definition . which leads to for each by the relations , , , and , and then to the contradiction .furthermore , if and only if for every  .so , both suprema  and lie in } ] , we can translate the constraints  as and then as by the way , inequalities  and impose stronger restrictions than , , and .since follows with eq . , all the restrictions , , and can be put together as one can witness that these bounds on depend on the chosen interior point . inshort , only those that fulfill the requirements and belong to the combined space .  from the interior point , we can travel on the line in two directions : where increases and where decreases . while moving we pass four points of that are presented in table [ tab : endpoints ] .when we proceed in the direction where increases , then we reach first either or .it all depends on the minimum value in .the point that we reach first belongs to . whereas the other point, then , fails to satisfy , and thus it lies outside of .while moving in the other direction , where decreases , we encounter first either or . depending on the maximum value in one of be in , other will be out of , ( unless both these points are the same ) .all the above possibilities are communicated through table [ tab : endpoints - cond ] . forany , only two of these possibilities can and will materialize , thus contains only a duo of ( distinct ) points from table [ tab : endpoints ] . in table[ tab : endpoint - pairs ] , we present every such duo .in fact , the interior point can be expressed as a convex combination of points of the one duo that lies in . for each duo , is presented in table [ tab : endpoint - pairs ] .  by varying from 0 to 1 in the combination, one can generate the line segment from to . recall that the line is described by .if belong to the combined space , then obviously the whole segment will be in thanks to its convexity .the line segments connecting with ( provided ) and connecting with  remain outside of .therefore , these two duos are not listed in table [ tab : endpoint - pairs ] .  in this part , it is shown that every interior point in can be decomposed as a convex combination of _ boundary _ points of , which are decomposed in the next part .note that the subsequent analysis is for . in the case of , , andtable [ tab : endpoints ] already carries the extreme points of .in fact , for , we only need and , because contains and if and only if and , respectively .   the boundary of is made of regions , where a region is characterized by equality in one of the constraints , , and : for . a point from table [ tab : endpoints ] , provided it is in , called a boundary point because it belongs to one of the regions . to reveal that the boundary points of can be decomposed into certain convex combinations ,let us suppose that the duo belongs to and analyze first and then .of course , an identical treatment can be delivered in the case of other duos from table [ tab : endpoint - pairs ] .now we start from and travel within the region along a new set of points by changing according to where . repeating the procedure similar to appendix [ sec : interior - w ], here we have which is like .the supremum is defined by and if and only if respects and follows with , then a new .analogous to tables [ tab : endpoints][tab : endpoint - pairs ] , here we compose tables [ tab : endpoints - p3][tab : endpoint - pairs - p3 ] , in that order . table [ tab : endpoints - p3 ] holds a collection of four points . table [ tab : endpoints - cond - p3 ] has the conditions that decide whether a point of table [ tab : endpoints - p3 ] is in or out of .table [ tab : endpoint - pairs - p3 ] supplies all possible couples of points from table [ tab : endpoints - p3]out of which one belongs to , that one is determined by .the line segment connecting the one couple carries and completely occupies in the region .now we are going to focus on .let us proceed from by altering only of another new vector with respect to note that , and identifies a straight line , a segment of which is contained in the region .in addition to , if agrees to only then the new vector . like tables [ tab : endpoints ] and [ tab : endpoints - p3 ] ,here we assemble table [ tab : endpoints - p1 ] of four points using the four bounds in .due to  and from appendix [ sec : inter - results ] , we have these inequalities are strict because a requirements in , , can not be met since is caused by . now taking with , one can deduce that the vectors and of table [ tab : endpoints - p1 ] can not belong to unless and , respectively .this fact is recorded in table [ tab : endpoints - cond - p1 ] with some other conditions , together they tell when a point of table [ tab : endpoints - p1 ] will be in or out of the region .a duo , out of the four listed in table [ tab : endpoint - pairs - p1 ] , resides in and expresses through a convex combination . as tables [ tab : endpoints][tab : endpoint - pairs ] are linked with the interior point and tables [ tab : endpoints - p3][tab : endpoint - pairs - p3 ] are attached to , tables [ tab : endpoints - p1][tab : endpoint - pairs - p1 ] are associated with .tables [ tab : endpoints ] , [ tab : endpoints - p3 ] , and [ tab : endpoints - p1 ] carry the boundary points of , , and , respectively .                      in the above parts , it is demonstrated that every interior point can be decomposed into a convex combination of the boundary points of , which can further be decomposed into convex combinations of the boundary points of regions . continuing this decomposition process , we reach at a point , where since every of is a supremum , [ see the explanation below ] and due to , we deduce that   the point , designated by , satisfies and number of _ equality _ constraints of type  and , respectively . if of follows then , where is a supremum like , , , and .one can check that points in table [ tab : endpoints ] for and in tables [ tab : endpoints - p3 ] as well as [ tab : endpoints - p1]provided and  are like ; remember that due to .furthermore , one can easily recognize in each of these points .then , one can see through table [ tab : endpoints - cond ] , [ tab : endpoints - cond - p3 ] , and [ tab : endpoints - cond - p1 ] that one of the two inequalities in  is required for a point to be in .the other inequality is automatically obeyed due to and the conditions appeared in the earlier decompositions .  if we start our journey from a point , where then we will arrive at the point , where [ for * 0 * , see ] .this point represents an extreme point of and a special case of and . in the case , the supremum that is possible if and only if , means , for every .indeed , it is so [ see ] . in all other cases , for every [ see the limits  on of ] , and can be decomposed further by adopting the same procedure as before .  without loss of generality ,let us suppose for the subsequent analysis . herewe begin with , where and .one can acknowledge that represents all those points , including , that fall on the straight line characterized by .  if stays on the line with , which follows then .here are suprema , and the angles are related to the components of through and [ see also and ] .the constraints  look alike and .identical to tables [ tab : endpoints ] , [ tab : endpoints - p3 ] , and [ tab : endpoints - p1 ] , we enter a list of four points in table [ tab : endpoints - q ] , where the points are drawn from the four bounds on given in .now , to establish criteria for a point of table [ tab : endpoints - q ] to be in or out of , we are going to address the two cases individually [ see eq .  for and the range  of ] .let us first take the case : whatever the suprema  and are , we have to demonstrate this , we consider , the cases with can be handled likewise . for , we have ( where ) due to . if associated with the supremum  is 1 , then by taking we can validate the strict inequality thanks to .if , we can do the same by now considering . in a similar fashion , we can establish the other inequality .we draw the following inferences from inequalities and . implies that the maximum and the minimum values in are 0 and , respectively .consequently , the points and of table [ tab : endpoints - q ] never , whereas and always , belong to in the case .moreover , can be broken into the convex combination , where [ see table [ tab : endpoint - pairs - q ] ] .next , it is not difficult to realize that both and can be decomposed further and further until we arrive at a point , where in the decomposition process one will encounter inequalities , such as and , that can be tacked like the above .for , a point defined by and is an extreme point of , because it can not be written into a convex combination of other points of .furthermore , is a vector - valued function of since -angles are fixed by once the measurement settings are selected in .let us now turn to the case , where according to , since supremum  is a nonnegative number , can either be or 1 here .it is due to when and , because then and every .similarly , related to the supremum  can either be or 1 here .when or or both , we encounter situation similar to the case : when then due to we have one can perceive that and are analogues to and , respectively .the inequalities in  suggest that is the minimum value in .therefore , without exception lies in , if then .identically , for , always , and belongs to only when it is .  when and only then and can be in without being equal to and , respectively [ see table [ tab : endpoints - cond - q ] ] . with table [ tab : endpoints - cond - q ] , for the case , one can find out whether or not a duplet of points from table [ tab : endpoints - q ] lies in .all such duplets are gathered in table [ tab : endpoint - pairs - q ] , which reveals that the point can be split into a convex combination .as before , we can break the points of table [ tab : endpoints - q ] further and further until we reach extreme points of .  in the case, the decomposition process leads to if of obeys then the point stated by and belongs to .it is an extreme point of in the case .one can also realize that both there and are functions of by noticing in with .in fact , the extreme point identified by and in the case  can also be represented with these and of and by taking , which make it as an endpoint of the parametric curve . in conclusion , we realize the _ structure _ of extreme points of :    we start with the -parametric curve identified by . according to , a part of the curve that lies in represents its extreme points .this part is specified by the upper and lower limits of . to compute these limits , here , we only need to consider when and then , and when and , then .so one can easily perceive that the points fulfill rest of the requirements  as well as to be in .  for in or in , the tiis always obeyed : due to with , , and one can sequentially go through the steps  , and the left - hand side inequality in is a consequence of . since and obey , they certainly follow the ti as every .  if we decrease then decreases , and reaches its lower limit when the inequality , for , gets saturated .it means that is a solution of the equation and thus of [ by and ] .if we increase then and  decrease , and attains its upper limit as soon as one of the inequalities  and gets saturated .using , , and [ owing to ] , these inequalities can be expressed as now we need to investigate the two cases , and listed in , separately for .   in the case , when we increase then the inequality , rather than , gets saturated first . hence, is now a solution of one can justify these statements by proving where . as is a root of eq ., is a root of equations , , and are of the form where angles the -set taken from the first column of matrix [ given in ] .always , we must choose the root of eq .that respects for every .furthermore , as we add more angles from the first column to the -set , the number of nonnegative terms increases on the left - hand side of eq . .then of smaller value will satisfy eq . .so , by comparing eqs .  andin this way , we can certify the left - hand side inequality in .whereas , after a simplification , the right - hand side inequality turns into , which is true as every .in fact , eq .where two angles are taken from the first column of also like eq . .basically , one needs to solve equation such as where angles are picked from a row or a column of get a limit and then an endpoint of an -parametric curve .when then m can only be 2 [ see and ] .and , when then m can either be or [ see and ] .  to solve eq .  for , we transform it into calling by the relations and , we can write eq .  as the two roots of eq.  are which only depend on the -set associated with eq . .we pick the root  with + sign due to the following reasons .first , for , we have equation such as , and its root in corresponds to the + sign solution [ see also with ] .second , for , is the only permissible solution of eq . .it is because angles are not random real numbers , they follow .when , [ see and ] , and always the solution  with + sign offers .third reason , for a pair of mubs , where every is the same , one can directly solve eq . .for every -set , we get the same [ see in ] , which corresponds to ^ 2}{d\,\textsc{m}}\ ] ] that is clearly the root  with + sign .          i. bialynicki - birula and .rudnicki , _ entropic uncertainty relations in quantum physics _ , in statistical complexity : applications in electronic structure " , edited by k. d. sen ( springer , netherlands , 2011 ) , pp .134 ; e - print arxiv:1001.4668 [ quant - ph ] ( 2011 ) .

Abstract: the born rule provides a probability vector ( distribution ) with a quantum state for a measurement setting . for two settings , we have a pair of vectors from the same quantum state . each pair forms a combined - probability vector that obeys certain quantum constraints . such a restricted set of combined vectors , titled combined - probability space , is presented here for a -level quantum system ( qudit ) . the combined space turns out a compact convex subset of a euclidean space , and all its extreme points come from a family of parametric curves . considering a suitable concave function on the combined space to estimate the uncertainty , we deliver an uncertainty relation by finding its global minimum at the curves for a qudit . if one chooses an appropriate concave ( or convex ) function , then there is no need to search for the absolute minimum ( maximum ) on the whole space , it will be at the parametric curves . so these curves are quite useful for establishing an uncertainty ( or a certainty ) relation for a general pair of settings . in the paper , we also present a few other ( un)certainty relations for a qubit .


microfluidic devices have found applications in various scientific and industrial processes .a typical example is their integration as important components of chemical and biological sensors . a micromixer is a microfluidic device used for effective mixing of different fluid constituents .it can be used to efficiently mix for example a variety of bio - reactants such as bacteria cells , large dna molecules , enzymes and proteins in portable integrated microsystems with minimum energy consumption .it is also used in mixing of solutions in chemical reactions ,  sequencing of nucleic acids or drug solution dilution .hence , the design of practical and efficient micromixers is a major research topic in microfluidics ,  especially in the development of micro total analysis systems . over the yearsvarious methods of efficient mixing have been developed and many of those have been successfully applied in industry .  the small length scales of the micromixers has a negative impact on mixing as it results in laminar flows inside the channels . in this flow regime , mixing is influenced mainly by the process of molecular inter - diffusion . experiments with channels with complex surface topology including grooved walls have revealed that microscale mixing is enhanced by `` chaotic advection '' , a process which was first reviewed by aref in 1984 .  he describes how mixing is still possible even at low reynolds number by repeated stretching and folding of fluid elements . if properly applied , this mechanism causes the interfacial area between the fluids to increase exponentially , which can then lead to an enhanced intermaterial transport , hence mixing .a comprehensive mathematical description of the exponential growth of interfacial surfaces can be found in the book by ottino . mixers were designed in the following years , which utilize the principle of `` chaotic advection '' . however , it is important to note that the term `` chaotic '' is used strictly in a lagrangian sense .  depending on the working principle , micromixers can be categorized into two important types : if energy from an external source is used to drive the mixing process , then it is termed as `` active mixer '' .these external energy sources can be acoustic bubble induced vibrations , periodic variation of the flow rate , piezoelectric vibrating membranes , valves , etc .the external sources are often moving components such as micropumps and they require advanced fabrication steps . the second category of micromixers is based on restructuring the flow profile using static but sophisticated mixer geometries .these are termed as `` passive mixer '' .passive micromixers have the advantage of simple fabrication , easy operation and no elements which can generate heat .the absence of heating is an important factor for applications to biological studies where temperature is a sensitive parameter .the mixing length and mixing time are defined as the distance and time span the fluid constituents have to flow inside the mixer in order to obtain a homogeneous mixture .an effective micromixer should reduce the mixing length and mixing time substantially in order to achieve rapid mixing . a common practice to design passive micromixersis to create alternating thin fluid lamellae .these result in an interfacial area that increases with the number of lamellae rendering the diffusion process more effective and hence allowing fast mixing . there are many examples of bi - lamellation  and multi - lamellation  in the literature on micromixers . however , the drawback of such devices is that the number of lamellae is generally limited due to the negative impact on the applied pressure drop caused by the required microstructures inside the channel .other examples of passive micromixers include the twisted pipe mixer ,  the superfocus micromixer , where several jets are made to collide at the focal point of the jets and the three dimensional serpentine model .  the recently developed so - called `` chaotic micromixer '' has gained substantial interest in the literature since it overcomes some of the drawbacks of conventional mixers based on muti - lamellation .such a device consists of microstructured objects such as `` herringbones '' ( see fig .[ fig_1 ] ) . the staggered herringbone mixer ( shm )was introduced as one of the first experimental implementations of chaotic micromixers in 2002 by stroock et al .  the half cycles of the shm consist of grooves with two arms which are asymmetric and unequal in length .these arms are inclined at an angle of to the wall and against each other , while the pattern interchanges every half cycle of the herringbone .the peculiar arrangement of the herringbone structure enhances the mixing process by `` chaotic advection '' where the interfacial area between the fluids grows exponentially in time another important advantage over mixers using the concept of multi - lamellation .the shm was used by several authors for studies related to mixing and analysis of mixing quality . to characterize the mixing quality of the shm , aubin et al . implemented particle tracking techniques using the variance of tracer dispersion .li and chen report on an optimization of the shm using the standard deviation of particle concentrations .as in the current paper , they used the lattice boltzmann ( lb ) method to model the flow field . further , the shm was used by kirtland et al . to study the mass transfer to reactive boundaries from three - dimensional micro - channels .  for the development of micromixers it is important to have reliable tools at hand to quantify their performance .efficiency and mixing quality have been studied by various methods in the past .these include the analysis of the probability density function of the flow profiles , studying the stretching of the flow field , the poincar section analysis , or the intensity of segregation as introduced by danckwerts in 1952 . in this paper an alternative numerical procedure is presented which is tailored for the optimization of chaotic micromixers .it is based on lb simulations to describe the flow inside complex mixer geometries together with a measurement of finite time lyapunov exponents ( ftle ) as obtained from trajectories of massless tracer particles immersed in the flow .the lb method can easily handle flows in complex geometries , which makes this method convenient for flows in advanced microstructures such as the micromixers the current paper focuses on .the lyapunov exponent provides a quantitative measure of long term average growth rates of small initial flow perturbations and thus allows a quantification of the efficiency of chaotic advection . since the systems of interest are finite and simulations are limited to a finite time span , the proposed method utilizes wolf s method to calculate the ftle . while lb simulations of flows in micromixers and ftle computations to quantify the degree of chaotic advection have been reported in the literature before , their combination for 3d performance quantification of realistic chaotic micromixers has to our knowledge not been published before .the numerical scheme has the potential to assist an experimental optimization since geometrical parameters or fluid properties can easily be changed without requiring a new experiment . to demonstrate its applicability ,the scheme is applied to evaluate the parameters of the staggered herringbone mixer that lead to improved performance .  in the following sections the lattice boltzmann method which is applied to simulate the fluid flow and the wolf s algorithm from which the finite time lyapunov exponents are obtained are described .finally , the numerical results and conclusions are given .the lattice boltzmann method is used to describe the fluid flow .the lb method is a simplified approach to solve the boltzmann equation in discrete space , time and with a limited set of discrete velocities . the boltzmann equation , given as represents the evolution of the velocity distribution function by molecular transport and binary intermolecular collisions . represents the distribution of velocities in continuous position and velocity space , and respectively . in the lb approach the position at which is defined ,is restricted to a discrete set of points on a regular lattice with lattice constant , implying that space is discretized .the velocity is restricted to a set of velocities implying that velocity is discretized along specific directions . denotes the discrete time step .the model we adopt is a d3q19 model which is a 3 dimensional model with 19 different velocity directions , . the right hand side of the above equation represents the collision operator which is simplified to the linear bhatnagar - gross - krook ( bgk ) form . in a discretized form the bgk operator is written as here , is the reciprocal of the relaxation time of the system controlling the relaxation towards the maxwell - boltzmann equilibrium distribution . by considering small velocities and constant temperature ,a discretized third order taylor expansion of the above equilibrium distribution function can be written as where denotes the equilibrium distribution function corresponding to the velocity vector , are the lattice weights , is the density , a reference density , and is the speed of sound . is the equilibrium velocity of the fluid , which is shifted from the mean velocity by an amount under the influence of a constant acceleration .the evolution of the lb process takes place in two steps : the collision step where the velocities are redistributed along the directions of the lattice and the propagation step by which they are displaced along these directions .these discrete lb steps are implemented by the equation which gives the dynamic evolution of the distribution function and is referred to as a discretized boltzmann kinetic equation .the macroscopic fluid density is given by and the macroscopic fluid velocity in the presence of external forcing is given by  it can be shown by a chapman - enskog expansion that the macroscopic fields and from the above equations fulfill the navier stokes equation in the low mach number limit and for isothermal systems . in order to simulate a fluid flow through microchannels , periodic boundary conditions are implemented along the direction ( see fig . [ fig_1 ] ) and no - slip bounce back boundary conditions are imposed at the channel walls .the experimental setup of stroock et al .consists of a number of repeated cycles ( ) consisting of two asymmetric half cycles each . by simulating one full cycle only and applying periodic boundary conditions we can reduce the required computational effort substantially .if the system is in steady state , we do not expect any substantial differences in the flow field in different cycles .therefore , this approach is valid .if , however , the experimental system would not be periodic , such a simplification would not be allowed and the full geometry would have to be simulated .we simulate a fluid which is hydrodynamically similar to water , flowing inside a shm with a cross section of .the length of the channel is , but can be varied in order to always accommodate a full cycle of the herringbone structure . for computational efficiencywe have chosen a lattice resolution of resulting in a fixed cross section of and system lengths of the order of .such a relatively low resolution is sufficient to properly resolve the flow field as can be observed from fig .[ profile ] , where the velocity field perpendicular to the flow is shown at positions within different half cycles ( a , b ) .[ profile]c ) shows the profile in flow direction .while the first two subfigures nicely demonstrate the swirling motion of the flow , the third one depicts how the flow penetrates between the grooves and stays mostly unaffected close to the upper boundary of the channel .previously , we have shown that a well resolved velocity field with less than 4 percent error as compared to the analytical solution can be obtained even for a resolution of 6 - 8 lattice nodes in the case of a 3d rectangular poiseuille flow . we further demonstrated that flow over random rough surfaces can be well resolved even if the smallest obstacles are only described by 2 - 4 lattice units . in the lb method , the kinematic viscosity is related to the discrete time step through the expression . since chosen to be to minimize artefacts due to the mid - grid bounce back boundary conditions and the simulated fluid has the kinematic viscosity of water , , this implies for the current choice of that and . when the magnitude of is along the direction , the average steady state velocity of the system is which corresponds to a subsonic flow .the reynolds number of the flow is , where is the characteristic length of the channel .one set of simulations is obtained for being which corresponds to .  when the flow simulation has reached its steady state , massless and non - interacting tracer particlesare introduced at fluid nodes in the plane and then their velocities are integrated at each time step . to calculate the ftle from particle trajectories a group of five particles forms four pairs , with every 5th particle placed at the center and the remaining ones being placed at the four nearest off - diagonal neighboring lb lattice sites .the particle at the center traces a fiducial orbit . with this arrangementwe are able to follow 800 particle pairs by using only 1000 particles .the trajectories are obtained by integrating the vector equation of motion where denotes the position vector of an individual tracer particle .the velocity is obtained from the discrete lb velocity field through a trilinear interpolation scheme .ll a ) & b ) +  &  +  +  `` chaotic '' systems in general have the important feature that two nearby trajectories diverge exponentially in time . the rate at which these trajectories diverge can be related to the ability of the flow field to create conditions for chaotic mixing . the lyapunov exponent is a possible measure of the performance of a micromixer as it is related to the rate of stretching of the fluid elements .it is defined by where is the distance between two trajectories at time which evolve from an initial separation . gives the value of the lyapunov exponent as tends to infinity .due to the finite size of any microfluidic system it is not possible to implement this definition in a simulation code to study the performance of a micromixer also , when two trajectories separate from each other , this definition does not allow to understand the ongoing stretching and folding dynamics of the flow .a quantitative measure of the mixer performance based on the lyapunov exponent can be obtained by using the ftle instead of the previous expression . the ftle takes the dynamical process more completely into account and provides a numerically implementable scheme for quantification of the performance of mixers .the ftle is defined as  where is any particular instant of time and is a finite time after which the ftle is measured . the same process is repeated over times , where is a large number denoting the number of times the ftle is evaluated from trajectories of particle pairs .amon et al . named the ftle as finite time lagrangian lyapunov exponent . the convergence of the average ftle to the lyapunov exponent for large is discussed in the paper by tang and boozer ,    wolf et al . suggested a method to calculate the ftle from a set of experimental data which is well applicable to our simulations . the key idea of the method is to monitor the distance between particles forming a pair and to renormalize it by moving back one of the two particles towards the other one in case they have separated more than a given threshold distance . the ftle is then computed from the sum of individual lyapunov exponents measured between replacements .the method has been verified on systems with known lyapunov spectra and exact results have been achieved . following wolf s approach , we implement the following equation to quantify the mixer performance on the basis of the average ftle as where is the time when a ftle is evaluated , and are the distance between particle pairs at time step and , respectively . is the number of time steps which a pair of particles take until the next replacement and its magnitude could be different for each measure . is the total number of replacements made until time when is evaluated .if has a positive and non - zero value the particles separate from each other at an exponential rate .these particle pairs are initially very close to each other and evolve in time .if the separation between the pair is greater than a maximum distance , the distance between the particles is re - adjusted to the initial distance .for the implementation of the scheme , for every particle pair one of the trajectories is chosen as the fiducial path , while the position of the other particle is replaced if the distance becomes larger than the threshold value .this is schematically represented in fig .[ fig_2 ] .the choice of the maximum distance is based on a variation of it together with a comparison of the obtained ftle : if it is chosen too large , many tracer particles hit the channel walls and do not separate any further resulting in a too low value of the ftle . if it is chosen too small , the tracers do not have a sufficient amount of time to separate sufficiently so that an exponential increase of the distance can not be detected . the chosen value has been found to be a good compromise between the two extreme cases .in order to avoid errors due to orientation one of the particles of a pair is placed along the line of separation . in case a replacement point can not be found due to a wall node present at the location , a nearby fluid node is selected as the replacement point .if even such points can not be found since all surrounding nodes are surface nodes , the replacement is postponed until a suitable replacement is possible .  a schematic representation of wolf s method . is the distance between a particle pair at an arbitrary time .if the distance is greater than a maximum distance then one of the particles is replaced near to the other particle along the line of separation . ]the following section presents how ftle can be utilized for an optimization strategy for chaotic micromixers . as an example , the influence of different parameters which directly affect the performance of the shm is evaluated .these are the ratio of the height of the grooves to the height of the channel , the ratio of the horizontal length of the long arm to the channel width , the ratio of distance between the grooves to the length of the channel and the number of grooves per half cycle .the width of the grooves is kept fixed at for all simulations .[ fig_1 ] provides a pictorial representation of these parameters .the performance of the shm is studied by varying four geometrical parameters . while keeping all other parameters fixed ( .089 , .2 , =10 ) , the width fraction ( )is varied within the range of 0.5 and 0.82 and the distance fraction ( ) from 0.04 to 0.11 .then , the number of grooves per half cycle ( ) is varied from 2 to 10 and the height fraction ( ) from 0.125 to 0.343 .the optimization of the shm as presented here is meant to demonstrate the feasibility of the algorithm only since several optimization studies of the shm are already available in the literature .therefore our optimization is reduced to a limited variation of the four dimensional parameter space .this surely leads to a local optimum of the geometrical parameters , but it is not assured that the global optimum has been found . however , as shown below , our parameters corespond to the optimal ones found by other groups suggesting that our local optimum coincides with the global optimum .  fig .[ fig_3 ] depicts simulated for different width fractions , , and .the reynolds number is kept fixed at and is obtained from tracing the trajectories of 1000 particles . from fig .[ fig_3 ] it can be observed that in each case fluctuates before finally converging to a particular value after time steps .all further simulations are run until the ftle have thoroughly converged .the effect of the geometry can be measured by comparing the average of the converged ftle which is denoted by .the error bars in figs .[ fig_4 ] to [ fig_8 ] are given by the standard deviation of the data from the point where it has converged .the average ftle for 1,000 particles is shown versus the time steps for .after time steps , is converged and the average of this converged value is denoted by .the value of depends on the width fraction as it is further investigated below . ]when one of the parameters is varied it is taken care that the other parameters remain unchanged , because we are interested in the dependence of the individual parameters on the performance of the shm .[ fig_4 ] shows the variance of and as such the performance of the shm with respect to the parameter for two different reynolds numbers , and 1.3 .larger values are difficult to obtain due to the limits of the lb method or would include a substantial increase in computing time .due to the symmetry of the mixer geometry , only values for are shown .the datasets peak at , which implies that the degree of chaotic advection is maximized for this particular value of the width fraction . according to the units chosen above this corresponds to .the curves for the different reynolds numbers depict that changing the driving force of the fluid does influence the absolute value of , but has no influence on the general shape of the curve .similar studies of the  dependence for other geometrical parameters and various different driving forces confirm this behavior .our findings are consistent with the original experimental work of stroock et al . as well as 2d numerical optimizations by stroock and mcgraw .  the latter study how the so - called heterogeneity factor of a dye concentration varies with the number of cycles in the mixer . in both publications it is found that generates a maximum swirling motion of the fluid .the variation of the maximum averaged finite time lyapunov exponent with different width fraction , for two different reynolds numbers .the data indicates that the maximum can be obtained for a width fraction of .error bars are given by the standard deviation of the mean value of . ]  for the next set of simulations is fixed at the optimized value of and the distance fraction is varied from 0.04 to 0.11 . since it is observed from the previous results that the optimal value is independent of the reynolds number , the following simulations are performed at a constant .the average value of converged ftle for different distance fractions is shown in fig .[ fig_6 ] .it can be observed that after a moderate increase of with , the curve has a sharp peak at , which corresponds to a value of for the current choice of .afterwards , decreases in a similar fashion as for small , but still at higher absolute values .a possible explanation is as follows : when the grooves are very close to each other , they create `` dead spaces '' , i.e. regions in the micro - channel where the fluid gets trapped and can not move freely . with increasing the distance between the grooves , the transverse component of the velocity increases , hence `` chaotic advection '' is enhanced resulting a large value for . on the other hand ,if the distance between grooves is too large , the mixer behaves like a plain channel without any chaotic advection component .the maximum in fig .[ fig_6 ] is then given by the optimal interplay of these two effects .the variation of versus the distance fraction for and .the ftle rises with the increase of until it reaches a distinct peak .after that the data shows a decrease , indicating that the optimized performance of the micromixer is at a groove distance .the error bars are given by the standard deviation from the mean . ]after having optimized the values for and , the number of grooves per half - cycle is varied from 2 to 10 .it can be understood from fig .[ fig_7 ] that a variation of has the largest impact on the performance of the mixer as compared to or . for the current setup , by variation of it is possible to change the value of by a factor of 2.3 as compared to 1.2 for and 1.3 for .[ fig_7 ] clearly shows that a staggered herringbone mixer with performs best .the existence of a maximum performance in dependence on the number of grooves can be explained by the fact that interplay between advection in flow direction and the swirling motion needs to be optimized for a well performing mixer .if the number of grooves is too small , the flow field is not sufficiently rotated when flowing through a half cycle .the change of direction of the swirling motion at the beginning of the following half cycle does not result in a relevant distortion of the trajectories then .if the number of grooves is too large , however , the fluid might perform one or more full rotations and come back to its original position before entering the next half cycle .an optimized value for therefore depends on the ratio of optimum rotation to the frequency at which the distortions at the end of a half cycle occur .similar to the work presented in the current paper , li and chen performed lb simulations and used tracers to follow the flow field . they , however , quantify mixing by computing the standard deviation of the local tracer concentration and conclude that shm with or perform best . even though this result is in agreement with our finding, the ftle analysis clearly shows that the channel with performs better than the one with .the variation of with the number of grooves per half cycle ( ) is shown .it can be observed that the shm with performs best .error bars indicate the standard deviation from the mean values . ]the final parameter to be varied is the ratio of the half depth of the grooves to the height of the channel .[ fig_8 ] shows the average value of the converged lyapunov exponents for different between 0.125 and 0.343 . after a strong increase of , the data shows a maximum at .for the units chosen above this corresponds to a groove depth of . for larger value of decreases again .our result is similar to the original experimental analysis of stroock et al . again , an argument can be found for the existence of an optimal value for the ratio between groove depth and system height : if the grooves are too shallow , they are not able to generate the swirling motion required for chaotic advection . on the contrary , if the grooves are too deep the flow is not able to fully penetrate the grooves .further , the volume between grooves and top surface becomes so small that the swirling motion can not develop anymore .the variation of the average converged ftle with the height fraction ( ) .the data indicates that the maximum ftle can be obtained for .error bars indicate the standard deviation from the mean values . ]  in this section we have demonstrated that a numerical scheme based on the lb method for the flow in complex mixer geometries together with wolf s method to calculate ftle from trajectories of passive tracers are a powerful tool for the quantification of chaotic mixing . without loosing generality ,a limited optimization study was performed for the particular example of the shm .the optimal parameters , , and have been found which is in good agreement with known experimental and numerical literature data .mixing at the microscale can be efficient if a large interface between fluids is provided .this can be obtained by passive chaotic micromixers utilizing repeated stretching and folding of the fluid interfaces .the performance of such mixers depends on the rate at which `` chaotic advection '' of the fluid takes place . for the development of efficient chaotic micromixersit is mandatory to understand the underlying transport processes as well as their dependence on the geometric structure of the microfluidic device . in this paperwe have demonstrated an efficient numerical scheme which allows the quantification of the performance of a micromixer .the scheme is based on a lb solver to describe the time dependent flow field in complex mixer geometries combined with wolf s method to compute ftle from passive tracer trajectories .we have demonstrated the applicability of the quantification method by applying it to find an optimal geometrical configuration of the shm , but the scheme should be generally applicable to any chaotic mixer . by performing a systematic variation of the relevant geometrical parameters we obtained a set of optimal valueswhich is consistent with literature data published by others .however , those data was obtained from experiments or numerical simulations not taking the chaotic nature of the tracer trajectories into account .the method presented here , however allows a quantification and optimization of the mixing performance by investigation of the underlying flow profiles . further work could include the optimization of other chaotic micromixer geometries and the introduction of multiple fluids to the problem . for the latter ,the lattice boltzmann method offers a number of possibilities to simulate multicomponent flows and is thus a well suited candidate .the authors thank f. janoschek , f. raischel , g.j.f .van heijst , and m. pattantys - brahm for fruitful discussions .this work was financed within the dfg priority program `` nano- and microfluidics '' , the dfg collaborative research center 716 , and by the nwo / stw vidi grant of j. harting .we thank the jlich supercomputing center and the scientific supercomputing center , karlsruhe for providing the computing time and technical support for the presented work .m. a. burns , c. h. mastrangelo , t. s. sammaraco , f. p. man , j. r. webster , b. n. johnson , b. foerster , d. jones , y. fields , a. r. kaiser and d. t burke , proc. national acad .usa * 68 * , 5556 ( 1996 ) .

Abstract: chaotic micromixers such as the staggered herringbone mixer developed by stroock et al .  allow efficient mixing of fluids even at low reynolds number by repeated stretching and folding of the fluid interfaces . the ability of the fluid to mix well depends on the rate at which `` chaotic advection '' occurs in the mixer . an optimization of mixer geometries is a non trivial task which is often performed by time consuming and expensive trial and error experiments . in this paper an algorithm is presented that applies the concept of finite - time lyapunov exponents to obtain a quantitative measure of the chaotic advection of the flow and hence the performance of micromixers . by performing lattice boltzmann simulations of the flow inside a mixer geometry , introducing massless and non - interacting tracer particles and following their trajectories the finite time lyapunov exponents can be calculated . the applicability of the method is demonstrated by a comparison of the improved geometrical structure of the staggered herringbone mixer with available literature data .


random set theory has recently been proposed as a mathematically elegant framework for bayesian multi - object filtering .research within this theoretical framework has resulted in new multi - object filtering algorithms , such as the probability density hypothesis ( phd ) filter , cardinalised phd filter , and multi - bernoulli filter .the main feature of multi - object filters is that they estimate sequentially the number of objects in the surveillance volume of interest ( the so - called cardinality ) and their states in the state space .formulation of multi - object _ trackers _ from random - set based multi - object _ filters _ has attracted a lot of interest recently , see e.g. .the output of a tracker is a set of _ tracks _ , that is , labeled temporal sequences of state estimates associated with the same object .( a )    ( b )  in this paper we adopt the `` tracking - by - detection '' approach to pedestrian tracking , which has become very popular in computer vision due to its applicability to moving un - calibrated cameras .typically pedestrian detections are obtained using the histogram of oriented gradients ( hog ) detector , trained using either head images ( for head detections ) or body images ( for body detections ) .an example of head and body detections is shown in fig.[f:1].(a ) .head and body detections are unreliable in the sense that : ( 1 ) not all pedestrians are detected in every frame ; ( 2 ) the chance of false detections is quite real , with the spatial density of false detections typically non - uniform .this is evident in fig.[f:1].(a ) .if the state vector of each object ( pedestrian ) contains the position ( e.g. the head centroid ) , but not the size of the object , then both the head and body detections are instances of _ imprecise _ measurements : they represent rectangular regions ( two - dimensional intervals ) within which the true object is possibly located . as such, they can be modelled as random closed sets ( rather than random variables ) .the first tracking algorithm considered in the paper is designed specifically for imprecise measurements : it represents a multi - object tracker built from the bernoulli filter described in .we will refer to this tracker as algorithm 1 .an imprecise measurement ( e.g. an interval in the measurement space ) can always be converted to a precise but random measurement ( e.g. a point in the measurement space which is affected by additive noise ) .the two remaining algorithms considered in the paper assume precise random detections of heads / bodies for pedestrian tracking .algorithm 2 is the same as algorithm 1 , but using the precise random ( gaussian ) measurements model .algorithm 3 is based on the cardinalised phd filter with data association .evaluation of multi - object tracking performance has been one of the main stumbling blocks in advancing the scientific field of target tracking .a large number of evaluation measures have been proposed , both in the general context ( e.g. ) and specifically for video surveillance ( e.g. ) . at presentthere is no consensus in sight on the preferred common approach . in this paper tracking errorwill be measured using the recently formulated optimal sub - pattern assignment ( ospa ) metric for tracks ( or ospa - t ) .the ospa - t metric has an important advantage over all above mentioned performance metrics : it is a mathematically rigorous metric ( it satisfies the axioms of a metric ) for measuring the distance between two sets of tracks ( i.e. between the ground truth and the tracker output ) .ospa - t is also consistent with intuition , as discussed in .the remainder of the paper is organised as follows .section [ s:2 ] describes the performance evaluation framework : the video dataset , the method of pre - processing detections and the ospa - t metric .section [ s:3 ] reviews the three random - set based tracking algorithms .section [ s:4 ] presents the experimental results under various conditions , while the conclusions of this study are drawn in section [ s:5 ] .the video dataset and the ( hand labelled ) ground truth are downloaded from the website .the dataset is a video of an occasionally busy town centre street ( the average number of pedestrians visible at any time is 16 ) .the video was recorded in high definition ( pixels at fps ) .only the first 4500 frames of the video are used in the performance evaluation .frame number 320 of the dataset is shown in fig.[f:1 ] .head detection and pedestrian body detection algorithms were applied to each frame in the video sequence .the fasthog gpu implementation  of the histogram of oriented gradients ( hog ) detection algorithm  was used for both detectors .the hog detector applies a sliding window to the image at a regular grid of locations and scales , and classifies each sub - window as containing or not containing an object ( head or pedestrian ) .classification is performed using a pre - trained linear support vector machine , the input to which is a set of block - wise histograms of image gradient orientation .a classification threshold of 0.75 was used for both detectors .sliding window detectors tend to give multiple detections for one object due to their tolerance to shift and scale , so a post - processing step groups overlapping detections .the head and pedestrian ( whole body ) detections have some complementary characteristics .the detector is only partially tolerant to occlusions , so the head detector tends to have a higher probability of detection since heads are generally more visible than whole bodies in surveillance video .however pedestrian textures are more distinctive than head textures , so the head detector tends to have a higher false alarm rate , picking up on round - ish background objects such as clocks and signs .the pedestrian detector is more able to detect people at a distance where the head becomes too small in the image .head and body detections are treated as if they are independent .each tracker can then be regarded as a centralised multi - source fusion node , where one source of detections is the head detector while the other is the body detector .the rectangles corresponding to body detections are converted to head - like detections as follows .suppose a body detection rectangle is specified by its upper - left corner , width and height .then for its corresponding head - like detection , the upper - left corner coordinates are computed as : and , while the width and height are and , respectively .traditional multi - object tracking performance measures describe various aspects of tracking performance , such as _ timeliness _( e.g. track initiation delay , track overshoot ) , _ track accuracy _( e.g. position , heading , velocity error ) , _ track continuity _( e.g. track fragmentation , track labelling swaps ) and _ false tracks _( their count and duration ) .these measures are based on heuristic , and it is unclear how to combine them into a single score because they are correlated .ospa - t is defined as a theoretically rigorous distance measure on the space of finite sets of tracks , and it has been proven that it satisfies the axioms of a metric .the computation of ospa - t is described in table [ aa:1 ] .suppose we are given two sets of tracks , the ground truth tracks and estimated tracks .a track , , is defined as a temporal sequence where each , , is either an empty set ( if track does not exist at time ) or a singleton whose element is .here is the track label and is its state at time .the labels of ground truth tracks are by convention adopted to be .the first step in the computation of ospa - t is to label the estimated tracks ( steps 3,4 and 5 in table [ aa:1 ] ) .this first involves finding the best assignment of estimated tracks to ground truth tracks .an assignment is a mapping , for .this is typically carried out using a two - dimensional assignment algorithm , such as the auction or munkres algorithm .if for an estimated track exists a true track such that , then track is assigned label .estimated tracks which remain unassigned according to are given labels different from all true track labels ( i.e. integers greater than ) .then , for each time step , the ospa distance between the two labeled sets : is computed .the set represents the set of existing ground truth labeled track states at time ; similarly is the set of existing estimated labeled track states at time .the ospa distance between these two labeled sets is computed as : ^{1/p }  \label{e : ospa - t}\end{aligned}\ ] ] where , and  * is the _ cut - off distance _ between two tracks at , with being the cut - off parameter ; * is the _ base distance _ between two tracks at ; * represents the set of permutations of length with elements taken from ; * is the ospa metric order parameter .  for the case ,the definition is .if both and are empty sets ( i.e. ) , the distance is zero .  ' '' ''   label the estimated tracks for , label = i ] compute the distance for form the labeled sets at : + -ground truth : + -estimated : compute the ospa distance between and   ' '' ''  [ aa:1 ]  the base distance is defined as : where : is the base distance order parameter ; is the localisation base distance , typically adopted as the -norm : ; is the labeling error , adopted as :  ] is the complement of the kroneker delta , that is =0 ] otherwise .parameter  ] , where is the position ( in pixels ) of the pedestrian head centroid and is its velocity vector ( in pixels / s ) . the number of objects from frame to frame varies .the random finite set of _ head _ detections at frame is denoted .accordingly , the random set of _ head - like _ body detections ( see the explanation in the last sentence of sec.[s:11 ] ) is .algorithms 1 and 2 are based on the multi - sensor bernoulli filter , where the `` sensors '' are the two types of pedestrian head detections .separate and independent bernoulli filters are run for each target .target interactions are taken care of by the appropriately increased clutter intensity , as in .this multi - object tracking algorithm has been described in some detail in .the difference between algorithms 1 and 2 is in the model of the single - object likelihood function .let , for , be a detection resulting from an object ( i.e. a pedestrian head ) in the state .a head detection is a rectangle , thus is specified by a tuple , where determines its upper - left corner , while and are the width and height , respectively .the single - object likelihood function used in algorithm 1 treats the detection as an imprecise measurement and is defined as in : where is the gaussian cumulative distribution function with mean and covariance  ] . if , then ( [ e : glf ] ) simply states that if is inside the rectangle , and zero otherwise .the algorithm is applied to the video dataset using and .algorithms 2 and 3 first convert the rectangular detection into a point measurement ^\intercal ] . then the single - object likelihood function of is adopted as : where is the gaussian probability density function with mean and covariance .algorithm 3 is based on the cardinalised phd ( cphd ) filter , but with additional logic to deal with track labeling .the key idea of is to form the clusters of targets , and to apply the cphd filter update to each cluster separately .the update uses every available detection ( measurement ) to calculate the _ weight _ of the track - to - measurement association .the weight of no - measurement association is also computed .finally these weights are used to form an association matrix which is solved using a two - dimensional assignment algorithm ( e.g. auction , munkres ) . at last each predicted track is updated with the measurement which has been assigned to it by the assignment algorithm . since we have at our disposal two types of detections ( are head detections , and are head - like body detections ) , the update step in algorithm 3 is applied twice , first using and then using .although this is not an optimal approach , it has been suggested as a reasonable approximation ..a summary of the contesting tracking algorithms [ cols="^,^,^",options="header " , ]   [ t : algs ]  all three algorithms used the same clutter maps ( one map for heads , the other for body detections ) .the probability of detection was set to and .a short summary of algorithms is given in table [ t : algs ] .the localisation base distance of the ospa - t error only takes into account the positional error ( i.e. neglecting the velocity error ) .fig.[f : ospat ] shows the resulting ospa - t error for the three random - set based tracking algorithms .the parameters of the ospa - t metric used in evaluation : , and .identical head detections and body - to - head converted detections , from every frame , have been used by all three algorithms .fig.[f : ospat ] also shows , as a guideline , the ospa - t error of the benfold - reid ( br ) algorithm , whose tracking results are available online .we point out that the comparison between the br algorithm and the three random - set based trackers is not fair because the br algorithm is a smoother ( operates as a batch algorithm over a sliding window of image frames ) and does not use body / head detections in every frame . from fig.[f :ospat ] one can observe that ranking of the algorithms according to ospa - t varies with time .for example , from frame number 800 to 1100 , the br algorithm is far superior than the random - set based trackers , but the opposite is true from frame 1400 to 1600 . in order to obtain an overall ranking , the time averaged ospa - t error has be computed : its value for algorithms 1 , 2 , 3 and the br algorithm is 45.2 , 42.8 , 40.7 and 40.4 , respectively .the conclusion is that the most accurate of the three random - set tracking algorithms is algorithm 3 .furthermore , it appears that the imprecise measurement model is not justified in the adopted context : the transformation of head and body - to - head rectangles ( imprecise detections ) into random precise measurement points provides better tracking results .this can be explained by the nature of head and body - to - head rectangular detections ; it has been observed that if a detection is not false , then its rectangular centre is a very accurate estimate of the centre of a pedestrian head .thus the likelihood ( [ e : glf ] ) , which is based on the interpretation that the true head centroid is somewhere inside the rectangle , appears to be too cautious and consequently does not use the full information content of a measurement .we repeated the ospa - t error computations for ( no penalty for the labeling error ) .this case corresponds to the original ospa error proposed in . the obtained time averaged ospa - t error for algorithms 1 , 2 , 3 and the benfold - reid algorithm in this case were 34.1 , 29.5 , 27.4 , and 30.2 , respectively .again algorithm 3 performs the best among the random - set based trackers , and even outperforms the benfold - reid algorithm .this result reveals that the major problem with algorithm 3 is the lack of track consistency ( too many broken tracks ) , which by adopting is not penalised .track consistency can be improved by smoothing over multiple image frames ( to be considered in the future work ) .head and body detection algorithms are very computationally intensive and consequently in real - time applications it may not be possible to provide them at every image frame .next we compare the ospa - t error performance of algorithm 3 for the situations where head and body detections are available for : ( 1 ) every frame , ( 2 ) every 2nd frame , ( 3 ) every 4th frame and ( 4 ) every 8th frame .the results are shown in fig.[f:3 ] .we note that the error performance does not change dramatically with the reduced frequency of head and body detections .the time averaged ospa - t error for the four cases are : 40.7 , 37.9 , 39.2 , and 42.7 .somewhat surprisingly , using body / head detections every 2nd and every 4th frame , reduces the number of false tracks and overall improves the accuracy .only when body / head detections become available only every 8th frame , some of the true tracks start to be missing occasionally and consequently the ospa - t error performance deteriorates .the paper presented a framework for performance evaluation of multi - object trackers .the framework is illustrated in the context of video tracking by comparison of three random - set based pedestrian tracking algorithms , using a video data set of a busy town centre .the multi - object tracking error was evaluated using the `` ospa for tracks '' ( ospa - t ) metric .the ospa - t metric has an important property that it satisfies the axioms of a metric .the mathematical proof the triangle inequality axiom is presented in the paper .  the results of performance evaluation indicate that the cphd based tracker of performs the best .although this is a single - frame recursive algorithm , its performance is comparable to that of ( which operates over a sliding window of frames ) .future work will consider a smoothing version of the algorithm in since a delay by a few frames in reporting the tracks is tolerable and has the potential to further improve the tracking accuracy .n. t. pham , w. huang , and s. h. ong , `` tracking multiple objects using probability hypothesis density filter and color measurements , '' in _ proc .multimedia and expo _ , july 2007 , pp .1511 1514 .m. d. breitenstein , f. reichlin , b. leibe , e. koller - meier , and l. v. gool , `` robust tracking - by - detection using a detector confidence particle filter , '' in _ proc .computer vision ( iccv )_ , 2009 .a. gning , b. ristic , and l. mihaylova , `` bernoulli particle / box - particle filters for detection and tracking in the presence of triple measurement uncertainty , '' _ ieee trans .signal processing _ , 2012 ,( in print ) .b. e. fridling and o. e. drummond , `` performance evlauation methods for multiple target tracking algorithms , '' in _ proc .spie , signal and data processing of small targets _ , vol . 1481 , 1991 , pp . 371383 .u. o. o. active vision group , dept .engineering science , `` town centre dataset , '' http://www.robots.ox.ac.uk/activevision/research /projects/2009bbenfold_headpose/ project.html#datasets , accessed dec . 2011 .b. ristic , b .-n . vo , and d. clark , `` performance evaluation of multi - target tracking using the ospa metric , '' in _ proc .information fusion_.1em plus 0.5em minus 0.4emedinburgh , uk : isif , july 2010 .

Abstract: the paper evaluates the error performance of three random finite set based multi - object trackers in the context of pedestrian video tracking . the evaluation is carried out using a publicly available video dataset of 4500 frames ( town centre street ) for which the ground truth is available . the input to all pedestrian tracking algorithms is an identical set of head and body detections , obtained using the histogram of oriented gradients ( hog ) detector . the tracking error is measured using the recently proposed ospa metric for tracks , adopted as the only known mathematically rigorous metric for measuring the distance between two sets of tracks . a comparative analysis is presented under various conditions .