the no - signaling principle , one of the no - go theorems in quantum theory , asserts that information can not be sent faster than light .in fact , the fundamental theorem can be utilized to derive some limitations on performance of quantum operations .one possible way to make use of the no - signaling condition is to incorporate a quantum operation to a communication scenario , in which the operation should be limited such that any information can not be sent faster than light .it was shown that quantum cloning can be incorporated with a communication scenario . by the no - signaling principle , an upper bound to the optimal cloning fidelitywas obtained in the universal quantum cloning , and it turns out that the bound coincides with the optimal cloning fidelity .it was also shown that quantum state discrimination can be incorporated with a communication scenario , to which the no - signaling principle could be applied .upper bounds to the optimal ones were obtained in cases of unambiguous discrimination , minimum - error ( two - states ) discrimination , and maximum confidence state discrimination .it was shown that the upper bounds are tight , so the optimal bounds are immediately implied by the no - signaling principle .note that postulates of quantum theory , such as positivity of quantum operations and measurement postulates , are assumed throughout .the present article contributes to a general and novel connection between the no - signaling principle and the minimum - error state discrimination among multiple quantum states . from the no - signaling constraint ,we derive an explicit formula that non - signaling probability distributions have to fulfill , by which a bound to the minimum - error state discrimination among quantum states can be obtained .we show that the bound coincides with the optimal ones for some cases where the minimum - error state discrimination is known , such as pure or mixed symmetric qubit states .note that the bound can be compared with other known bounds , for instance in .the importance of this work is twofold . on one hand , in the side of applications ,our work provides a systematic way to compute an upper bound to the success probability in the minimum - error state discrimination .although quantum state discrimination is one of basic operations to estimate and characterize capabilities of tasks in quantum information theory , the optimal minimum - error discrimination is known for few cases such as two - state or symmetric states discrimination . here , our method gives a bound to the minimum - error state discrimination among high - dimensional states for which the optimal discrimination is not known yet .  on the other hand , in the fundamental point of view, our work strengthens the quantitative connection between two no - go theorems , the no - signaling and no perfect state estimation .first , from two facts that i ) the no perfect cloning holds for all non - signaling theories having bell violations and ii ) quantum theory is non - local , it is clear that the no - signaling principle in quantum theory immediately implies that perfect quantum cloning is not possible . to be quantitative ,it was shown that the optimal cloning fidelity can be obtained by the no - signaling principle .moreover , quantum cloning generally converges to the optimal state estimation in the asymptotic limit .however , the connection between the state estimation and the no - signaling is only qualitative so far through the above mentioned connection : the no - signaling principle implies that the perfect cloning is not possible , and therefore the state estimation can not be perfectly done either . along these lines, the quantitative connection has been shown in the minimum - error state discrimination between two pure states and , generally , two mixed ones .  in this work , we provide a general upper ( lower ) bound to success ( error ) probability in discriminating among quantum states by the no - signaling principle . in sec.[main ] , we derive the bound using the no - signaling constraint . in sec.[qubit ] , the bound is shown to be tight for cases where the minimum - error state discrimination is known . finally , an example of discriminating among high - dimensional states is provided in sec.[qudit ] .let us first introduce the communication scenario to which the minimum - error state discrimination can be incorporated .two parties , alice and bob , are separated by a distance such that local actions can not affect the other side .suppose that they share copies of entangled states |_ab = _n |u_n_a |v_n_b , [ ab]where and , finite dimensional hilbert spaces , and .now , alice applies general measurement , positive - operator - valued - measure ( povm ) , a complete measurement , i.e. . here, let the choice of povm , denoted by the value , be the message that alice wishes to send to bob .the measurement by alice , say , prepares the decomposition on bob s side ,  _ b , k = _ j p_k^j _ b , k^j , [ bobs ] where  ] . since alice s measurement is complete for each , bob s ensembles are all identical , _ b , k = _ b , l , k l , [ con ] while they are in general different decompositions . moreover , alice can generate any decomposition as long as the ensemble is identical , as it is known by gisin - hughston - jozsa - wootters theorem .it is clear that , since the ensembles are identical , bob never knows which measurement has been applied by alice .the no - signaling principle would be then violated if bob could discriminate among the different decompositions of his ensemble without any further communication with alice . inwhat follows , using the no - signaling constraint , we derive the condition that has to be fulfilled in the state discrimination .suppose that bob applies a detector to make a guess of which measurement has been applied by alice .note that we do not know all properties of the detector but the probability distributions .in fact , the detector can be thought of being in a black box from which one can only know the input - output list .the detector of bob works as follows .once alice applies measurement on her system to send a message , bob s detector gives an outcome with some probability , from which he makes a guess that has been applied .since they are probabilities , for each the normalization condition holds _k d(k|a_l ) = 1.[unit]then , the probability that the detector tells the correct result is , , where is the probability that alice applies measurement . throughout the paper , we restrict to the case where is the same for all , i.e. a measurement for some is applied randomly . if there are possible measurements by alice , for all .the no - signaling principle can be imposed as follows .suppose that the detector works too well such that the success probability is better than random , _k d(k|a_k ) > .[ns ] together with eqs . ( [ unit ] ) and ( [ ns ] ) ,k[d(k|a_k ) - d(k|a_l ) ] > 0 .[ new ] clearly , it holds for at least a single value that that is , there exists a value , alice s encoding , to which bob s detector responds better than other ones . this immediately implies that a faster - than - light communication is possible in the following way .if alice wants to send a message ( ) , she repeatedly performs her measurements such that the -th ( -th ) decomposition is generated at bob s site .bob repeatedly performs his measurements. then bob can discriminate between the two cases by observing how frequently his detector gives the -th outcome , by eq .( [ a ] ) .thus bob can decode a one bit message .we then have the following proposition .+ * _ proposition 1 .* _ suppose that two parties alice and bob sharing copies of entangled states in eq .( [ ab ] ) , alice prepares different ensemble decompositions on bob s side by applying complete measurement to her systems with the equal probability , where .let be the probability that a detector of bob answers the -th decomposition when alice has applied measurement .then , the no - signaling principle constrains the success probability , _ k d(k| a_k ) 1 .[ nos]note that neither dimensions nor particular structures of given states are assumed ._   now we apply the condition in eq .( [ nos ] ) obtained by the no - signaling constraint to minimum - error discrimination among states in .the idea is to construct the identical ensemble having different decompositions such that each decomposition contains .the condition in eq .( [ nos ] ) then gives a bound to the minimum - error discrimination among states in .suppose that , as it is shown in eq .( [ bobs ] ) by alice s measurement , each state is contained in the ensemble decomposition with probability _ b , k = p_k _ k + _ kk q_k _ k. [ decom]note that in the above we are not interested in the state , which only helps to construct the identical ensemble . because the ensembles corresponding to are identical ( see eq .( [ con ] ) ) , we have p_k _ k + _ kk q_k _k = p_l _ l + _ lll. [ con2 ] let us now consider bob s detector designed to discriminate among states in .the detector can also be thought of being in a black box from which bob only knows the input - output list .let denote the probability that bob s detector gives an outcome ( meaning that bob guesses a state is given ) when is actually given .it follows from eq .( [ decom ] ) that p_k p(k|_k ) d(k|a_k ) , [ dp ] since bob s detection consists of the contributions both by the state with probability and by other states in the ensemble such as . from eq .( [ dp ] ) and eq .( [ nos ] ) in the proposition , the no - signaling condition implies that .we can then summarize in the following corollary .+ _ * corollary 1 . *suppose that a set of states for is given with a priori probabilities , to be discriminated between . if one can construct a set of identical ensembles where each ensemble is decomposed such that is contained in the ensemble with probability ( see eqs .( [ con ] ) and ( [ decom ] ) ) , then it must be fulfilled by the no - signaling constraint that _ k p_k p(k|_k ) 1 .[ bd ] here is the probability that bob s detector answers the -th state when is prepared . _ + note that the corollary can be directly applied to the success probability in the state discrimination .let us now explain how to apply the corollary to the minimum - error state discrimination among multiple quantum states , .first , one has to construct an identical ensemble in different decompositions , ( i.e. for all ) , such that each decomposition contains one of the states to be discriminated among , _ b,0 & = & p_0 _ 0 + + _ b,1 & = & p_1 _ 1 + + & & + _ b , n-1 & = & p_n-1 _ n-1 + . [ ensem ] it is then straightforward to apply the corollary to obtain a bound to state discrimination .the goal here is to minimize the error , or equivalently to maximize the success probability .the success probability is , , on average since we are considering the case where each state to be discriminated is generated with equal probabilities , .now let us assume that different decompositions , denoted by , of an identical ensemble are given .the success probability is then constrained from the corollary .therefore , the bound to the success probability for a given decomposition is ,  p_s^c = _ _ k p_k p(k|_k ) 1 \ {  _ k p(k|_k)}.[pd ] note that the no - signaling principle must be fulfilled for any ensemble decompositions under consideration .hence the success probability in the state discrimination is upper bounded as follows , p_s _ c p_s^c , [ res ] where the minimization is taken over all decompositions that can be constructed in eq .( [ ensem ] ) .note also that , given a decomposition , the success probability in eq .( [ pd ] ) gives a general upper bound to the success probability in the minimum - error state discrimination .for the particular case of the ensemble decomposition where for all , the bound to state discrimination is simplified , i.e. the success probability is upper bounded , p_s ( np)^-1 [ form].in this case , although the decomposition may not be the optimal one , no maximization step is required .therefore , constructing the identical ensembles immediately provides an upper bound to the success probability .note that the discrimination bound in eq .( [ res ] ) assumes neither dimensions of given states nor any structure among states to be discriminated .it only assumes that an identical ensemble can be in different decompositions ( see eq .( [ con2 ] ) ) .the upper bound to the success probability can be compared with known bounds shown in refs . . in the next section, our bound is shown to be tight for symmetric qubit states for which the minimum - error state discrimination is known .we will explicitly compute the upper bound to success probability from eq .( [ form ] ) , and compare with the optimal bound .we consider minimum - error discrimination among symmetric qubit ( pure and mixed ) states for which the optimal measurement is known .we first apply the bound in eq .( [ res ] ) to symmetric qubit states . here , the symmetric states are characterized by a unitary operator that satisfies , , and u_ju^ = _ j+1 , j=0 , 1 , , n-1 .[ sym]they are also called as geometrically uniform states . for qubit states ,the unitary operation is a rotation operator with respect to an axis in the bloch sphere .it will be shown that , remarkably , bounds obtained by the no - signaling principle turn out to be tight .to apply the corollary to state discrimination , one has to first construct identical ensembles .let us first start with the fact that a qubit state can be expressed as , ( ) = ( i + ) /2 , [ state ] where , and for pure states , see fig ( [ fig1 ] ) . symmetric qubit states that we are to discriminate among can be parameterized as follows : _ j & = & ( i + ( _ j ) ) /2 ,  j = 0,,n-1 , + ( _ j ) & = & || ( j , j , ) , [ nqstate ] with the unitary operation , denoted by , that rotates a state to by with respect to the -axis . for each state in ( [ nqstate ] ) , we introduce the state lying on the half plane of the bloch sphere , _ j & = & ( i + ( _ j ) ) /2 , j=0,,n-1 [ delta ] + ( _ j ) & = & ( -sgn ( ) j , - sgn ( ) j,0 ) .then , the identical ensembles can then be constructed such that their bloch vectors point to the same point in the -axis of the bloch sphere : _ b , i & = & _ b , j , ij + _ b , i & = & p_i + ( 1-p ) _i , [ ben]which gives the same in all decompositions p = .[ p ]  we now apply eq .( [ res ] ) to obtain a bound to the minimum - error state discrimination . since each state is contained in the ensemble with the equal probability in ( [ ben ] ) and ( [ p ] ) , the bound in ( [ form ] ) can be applied , p_s ( 1+|| || ) .[ sp ] this can also be expressed in terms of a lower bound to the minimum error , p_e 1- .[ ep ] the bound above works for both pure ( ) and mixed ( ) states for any number .note that the optimization over ensemble decompositions , defined in ( [ res ] ) , is not performed here and a particular ensemble decomposition constructed in ( [ ben ] ) is considered . in the next subsection , we show that the bound in eq .( [ ep ] ) is in fact tight , meaning that the decomposition is also optimal for states in ( [ nqstate ] ) .we briefly review the optimal minimum - error discrimination among symmetric states . in a dimensional hilbert space , symmetric statescan be parameterized as follows , |_j =c_k e^ j(k-1 ) |k , [ sys]with orthonormal basis . note that each state is normalized , i.e. . the optimal measurement that minimizes the error probability is known , m_j = |_j _ j| ,  |_j = _k=1^d e^ j ( k-1 ) |k .[ ms ] it is therefore straightforward to compute the success probability , p_s & = & _ ip ( i  |  |_i_i| ) = ( _ k|c_k| ) ^2 .[ mine]for ( i.e. qubits ) , symmetric ( pure ) states can be parameterized as , .the success probability in discriminating among those states is then , p_s = ( 1 + || ) , which coincides with eq .( [ sp ] ) for pure states ( i.e. , ) .this also means that the decomposition taken in ( [ ben ] ) is optimal such that the tight bound can be obtained .the optimal discrimination among symmetric mixed states was analyzed by chou and hsu in ref . , where the optimal povms are constructed by solving the following equations , _k = v^k _ 2 |_0_0 | _ 2^ ( v^)^k , _ 2 = _c_|| where are eigenvectors of , is chosen such that is real and non - zero for all , and . applying the above to the symmetric qubit states , it is straightforward to compute the optimal povms , which turn out to be the same as the case of pure states .it turns out that the minimum - error in state discrimination coincides with the success probability in eq .( [ sp ] ) .this also shows that the decomposition taken in ( [ ben ] ) is optimal such that the tight bound can be obtained .+ * _ example . _* we revisit the example shown in ref . as an application of the upper bound in eq .( [ res ] ) , the minimum - error discrimination among three symmetric mixed states : _ 0 & = & ( i + ( -_z ) ) , + _ 1 & = & v _ 0 v^ ,  _ 2 = v^2 _ 0 ( v^)^2 [ est ] with the operator that rotates a state by about the axis in the bloch sphere , v = (  ccc & - + & +  ) . in ref. ,the minimum - error was shown to be after optimization of the measurement basis .now we explicitly derive the same value of minimum error , by plugging the following parameters , the purity of the state in eq .( [ est ] ) , , the angle given , and , into eq .( [ ep ] ) : p_e = 1 - = .5556 .hence , it is shown that the minimum - error can be obtained by the no - signaling principle .one can also consider other known bounds for the states in the example above . in ref . , known bounds are extensively compared with one another , and it turns out that the bound to the minimum - error , in , seems to be a relatively good one , l_4 = 1 - _ k(_k + _ jk ( _ j_j - _ k_k)_+ ) , [ l4]where are _ a priori _ probabilities , and denotes the positive part in the spectral decomposition of an operator .one can consider the bound for the states in ( [ est ] ) assuming that _ a priori _ probabilities are equal .as it was shown in , the bound coincides with the optimal one if and only if , for some and all , and are orthogonal .however , this condition is not fulfilled in the states in ( [ est ] ) .in fact , for those states , it holds that for all .therefore , the bound is l_4 = 1 - ( + + ) .4742 .in addition , the bound proposed in ref . , denoted by in ref . , is even worse as .it is remarkable that the result in eq .( [ res ] ) does not assume particular dimensions of quantum states to be discriminated among .therefore the bound in eq .( [ res ] ) can be applied as long as an identical ensemble having different decompositions is constructed . when applying the conditions ( [ res ] ) and ( [ form ] ) to derive a bound to the minimum - error state discrimination , one has to first construct the identical ensembles fulfilling the constraint in eq .( [ ensem ] ) .this is in fact non - trivial . for the cases of qubits , as it is shown in sec .[ qubit ] , the state space naturally has the picture , the bloch sphere , based on which one can construct the identical ensembles .this , however , can not be generalized to high dimensional systems due to the lack of representation as useful as the one in the two - dimensional case .  in what follows , we show an example of high dimensional quantum states to which the conditions in eqs .( [ res ] ) and ( [ form ] ) can be applied .the example generalizes the picture of the bloch sphere to high dimensional systems .  _ states ; _ for quantum states of spin- particles , the basis are where .generators for rotations are , , , and satisfying the commutation relation , = i _ j_. [ com ] note that the generators can be written explicitly using the following elements , j , ( m+1 ) |  ( j_1 + i j_2 ) |  j , m & = & + j , ( m+1 ) |  ( j_1 - i j_2 ) |  j , m & = & + j , m^ |  j_3 |  j , m & = & m _ m^,m .for instance , for it holds that where and are pauli matrices .a rotation operator can be represented with as , that rotates a spin- state by with respect to the -axis .based on the rotation picture , identical ensembles of and can be constructed as follows .the states for that we wish to discriminate between are : _ 0 & = & ( i_2j+1 + _ 0 ) ,  _ 0 = ( - , 0 , ) , + _ k & = & u_k _ 0 u_k^ ,  u_k = ( -i_k j_3),[rhok ] for some angles for each .we also need the following states , _ 0 & = & ( i_2j+1 + _ 0 ) ,  _ 0 = ( , 0 , ) , [ si ] + _ k & = & u_k _ 0 u_k^ ,  u_k = ( -i_k j_3),[sigma]where the angles are given in eq .( [ rhok ] ) . in the above , and supposed to be given such that and are non - negative . in the matrix representation ,the operators and are sparse matrices that can be easily diagonalized .the ensembles identical for all are thus constructed , _ b , k = p _ k + ( 1-p ) _ k , with , which points out to the direction .note that the parameter from states in ( [ si ] ) is given only to construct the identical ensembles .then , the formula in eq .( [ form ] ) can be applied and immediately gives the bound , p_s _ .[ fbd ] it is thus shown by the no - signaling principle that the success probability in discrimination between states in eq .( [ rhok ] ) is upper bounded .  * _ example ._ * let us consider a particular case of the states in ( [ rhok ] ) that three states of spin- systems are given with equal probabilities .then , let us take the following representation of generators , j_1 = (  ccc 0 & 1 & 0 + 1 & 0 & 1 + 0 & 1 & 0 +  ) ,  j_3 = (  ccc 1 & 0 & 0 + 0 & 0 & 0 + 0 & 0 & -1 +  ) .then , and can be expressed by the rotation  ] .therefore , the upper bound to the success probability from the no - signaling principle is , as it is shown in ( [ fbd ] ) , p_s ( 1 + ) .[fbd1 ]  let us now compare the bound above with the success probability in ref. .as it was discussed , the bound is optimal if and only if for some and all , and are orthogonal .however , this condition is not fulfilled in the states in ( [ rhok ] ) .in fact , for those states , it holds that for all . since , we obtain 1- l_4 & = & ( 1 + ( _ 1,_2 ) ) , + ( _ 1,_2 ) & = & ( ( ) + ( ) ) .[ l4s]note that for all and .hence , it is shown that for three states in ( [ rhok ] ) the bound is closer to the minimum - error than the one in ( [ fbd1 ] ) .it is also clear that the bound in ( [ fbd ] ) is not tight in general .finally , let us comment on why the bound in ( [ fbd ] ) is not tight in contrast to the case of qubit states . for symmetric qubit states we have shown in sec .[ qubit ] that the tight bound can be obtained from the no - signaling condition .in fact , for the bound in ( [ sp ] ) to be tight , it is necessary that the states in to construct the identical ensemble in ( [ ben ] ) are pure .note that this was also mentioned in ref . in the context of generalized probability theories .now , for the states , however , it is not fulfilled that the states in ( [ si ] ) are pure , since = ( 3 + 4\beta^{2})/9 < 1 ] .nevertheless , it can not be excluded that a bound from the no - signaling principle is tight for or high - dimensional states unless one may consider the problem of finding the optimal ensemble decomposition as it is defined in ( [ res ] ) .we have derived an upper ( lower ) bound to the success ( error ) probability in the state discrimination by the no - signaling principle .the upper bound can be compared with known bounds to the success probability , e.g. refs . , that are obtained by applying some inequalities to the minimum - error formula .the bound we provided here depends on neither dimensions nor particular structures of given quantum states . what is required to apply the bound in eq .( [ res ] ) is to construct an identical ensemble that can have different ensemble decompositions such that each decomposition consists of each of quantum states to be discriminated among , as it is shown in eq .( [ ensem ] ) . as long as the ensemble satisfies the constraint, one can apply the bound in eq .( [ res ] ) to a set of quantum states .it was also shown that the bound coincides with that of the minimum - error state discrimination for known cases such as symmetric ( pure and mixed ) qubit states .we derived a bound to minimum - error multiple state discrimination in high dimensions for which the optimal state discrimination is not known yet .our results strengthen the quantitative connections among the no - go theorems , the no - cloning , the no perfect state estimation , and the no - signaling principle .we thank a. acn and t. cubitt for useful discussions and r. renner for discussions and comments on a connection to ref .we also thank j. tyson for useful comments and references .this study was financially supported by chonnam national university 2009 , the korea research foundation grant funded by the korean government under the contract number , krf-2008 - 313-c00185 , and the it r program of mke / iita ( 2008-f-035 - 01 ) .                         in the case that the probability is obtained by measuring quantum states , the probability can be expressed with povms , that discriminates among , , where each concludes that the state is detected : $ ] .the value can be computed by taking the two identical ensembles and in ( [ ben ] ) with .it follows from the fact that , that .one can then apply the trace norm , denoted by , and obtain the following relation p = .[ pp]note that the trace norm is defined as , , or equivalently , where are eigenvalues of .it is then explicitly computed in ( [ p ] ) .

Abstract: we provide a bound on the minimum error when discriminating among quantum states , using the no - signaling principle . the bound is general in that it depends on neither dimensions nor specific structures of given quantum states to be discriminated among . we show that the bound is tight for the minimum - error state discrimination between symmetric ( both pure and mixed ) qubit states . moreover , the bound can be applied to a set of quantum states for which the minimum - error state discrimination is not known yet . finally , our results strengthen the quantitative connection between two no - go theorems , the no - signaling principle and the no perfect state estimation .


x - ray powder diffraction ( xrpd ) technique is nowadays a well known tool to study crystalline properties , which provide important information for applications in fields such as nanotechnology . all the applications benefit from a reliable pre - processing aimed at enhancing the quality of xrpd data .the pre - processing procedure consists of four steps : denoising , background suppression , peak fitting and signal deblurring , also known as line - broadening ( see fig .1 ) . many techniques can be applied to remove noise from xrpd data ( see ref . and references therein ) .the step of background suppression is needed to emphasize the peak features of the sample .traditional techniques are : young s polynomials , chebyshev approximation and linear interpolation .peak fitting is a very challenging step to extract information about the properties of polycrystalline powder .the final stage in xrpd data pre - processing is the signal deblurring aiming at disentangling the instrumental line - broadening out of data . on this topicwe spot a recent review in ref . . in this paper we propose a new approach for two of the above pre - processing steps of xrpd data .the signal background is removed by means of a procedure based on morphological mathematics .the peak shape profile fitting is carried out by using a subspace - based parameter estimation method called hankel lanczos singular value decomposition ( hlsvd ) technique .the main advantage of this approach is twofold : while both background suppression and peak shape profile fitting are model independent , the signal deblurring procedure , employed thereafter , crucially benefits from the model independence of the previous two steps . indeed , our signal deblurring steps over the major drawback of most of the xrpd analysis tools available in the literature : the peak overlapping problem , namely the difficulty in singling the peak out of the full xrpd profile .this problem is mainly due to the model - induced bias in background / peak profile reconstruction . as to the denoisingwe use a wavelet based filter , a popular tool available in several software packages .real xrpd intensity profiles of ceria samples are used to test the performances .four different raw datasets were used in pairs . for each pair ,one dataset was collected on the annealed ceria specimen ( representing the instrumental broadening ) and the other was collected on the broadened sample .the selected pairs are those measured at the university of birmingham ( a high resolution x - ray laboratory ) and at the national synchrotron light source ( nsls x3b1 ) .the structure of the paper is as follows .section [ sec : method ] is devoted to the description of the proposed method and of the results .conclusions are drawn in section [ sec : conclusions ] .two different raw datasets were downloaded for the ceria sample : the instrumental standard representing the instrumental broadening and the ceria xrpd pattern of the broadened sample .our strategy in disentangling the profile broadening out of the experimental sample relies on a three step procedure which is sketched in the sequel .the noise was removed by applying wavelet transforms to the full xrpd spectrum and subtracted prior to the background suppression .  among the many applications of wavelets ,signal denoising has been deeply investigated and the wavelet filter can be considered as the state of art on this subject . the discrete wavelet transform ( dwt ) is a linear operator which modifies the data vector in a similar way as the discrete fourier transform ( dft ) . in both cases ,the transform , given by a matrix acting on the input -vector data , is invertible .the matrix entries are combinations of basis functions ( the familiar sines and cosines in the case of dft ) .tables 1 - 2 summarize the main properties of some wavelet bases .an interesting property of wavelet basis is the localization in both space and frequency domain .this means that they have a finite support or a decay in both domains .regularity is another important property of wavelet basis .regularity means that the derivative exists _ almost everywhere _ ( see table 1 ) . in the case of daubechies wavelet, regularity depends on the order ( see table 2 ) .a wavelet is defined by particular set of numbers , called wavelet filter coefficients .they are determined by imposing the constraints of vanishing moments , ( ) , see table 1 , and orthogonality .the coefficients characterize a low - pass filter while the coefficients results in a high - pass filter .coefficients give the entries of a matrix , which iteratively applies to the -data vector thus resulting in the -vector of detail coefficients . the whole procedure described aboveis the wavelet transform ( see for further details ) . generally speaking ,the denoising procedure involves three steps .the basic version of the procedure is the following :  * calculate the wavelet transform of xrpd profile and sort the components of the output vector by increasing frequency .this shall result in -vector containing the xrpd profile average coefficient and a set of detail coefficients .* noise thresholding , calculated on the highest frequency detail coefficient of the wavelet spectrum .* signal reconstruction by using the average coefficient and thresholded detail coefficients .  in this paperwe choose the daubechies wavelet basis with .we address the reader to daubechies for further details .the background was determined by means of morphological transforms for the full xrpd spectrum and subtracted prior to the signal deblurring .the morphological mathematics is based on the language of set theory .considered a discrete binary image and a structuring element , the four basic morphological mathematical operations on by are : where denotes the translation of by , namely . the value of each pixel in the output image is based on a comparison of the corresponding pixel in the input image with its neighbours , whose number and location is given by the structuring element . generally , dilation expands image objects , whereas erosion shrinks them . in practice , dilation and erosionare employed in pairs .opening is the erosion of an image followed by the dilation of the eroded image , and closing is the dilation of an image followed by the erosion of the dilated image .opening eliminates sharp peaks smaller then the structuring element while the closing fills in the small holes and gaps .the binary morphological operations of dilation , erosion , opening and closing can be extended to grey - scale images .let and be the domains of the gray - scale image and the grey - scale structuring element respectively .the grey - scale dilation and erosion can be computed by where and . for such images ,the minimum and maximum values are computed within neighbourhood represented by the structuring element ( see for details ) .  in our background suppression procedurethe xrpd pattern is reshaped and padded into a 2-d image .a disk with a radius of three pixels is used as structuring element both for erosion and for dilation . as to the erosion ( dilation ) , pixels beyond the image borderare assigned the maximum ( minimum ) value afforded by the data type .the morphological opening removes small objects from the image while preserving the shape and size of larger objects in the image .the overall result is a peak smearing effect while the background intensity remains unalterated .restoring the original 1-d pattern provides the xrpd spectrum background .figure 2 summarizes the whole procedure .we compared our findings to the traditional interpolation method and we found a satisfactory agreement ( the percentage difference between the background computed by traditional techniques and our finding is below 3 % ) . up to our knowledge, this technique has never been applied to xrpd spectrum background suppression and it provides a reliable and user independent estimate of it .the main problem in analysing an xrpd spectrum is the peak search , since the exact position is crucial in the extraction of the relevant microstructural information .had the peak well defined , its shape would be straighforwardly achieved ( for instance by a high resolution interpolation / fit by means of a model ) . unfortunately the data resolution is rarely high enough to reach the goal of a well profiled peak . in that respect several methods have been devised so far to accomplish the peak fitting by means of gaussian , lorentzian , voigt , pseudo - voigt , pearson vii and other models .the main drawback of the aforementioned methods cited above is the dependence of results on the model used in the fit procedure itself and the poor description of the real peak profile shape ( for instance asymmetry ) .here we use hlsvd method to the purpose .the main advantage of this method is the flexibility since the number of parameters is not fixed and it can be chosen to achieve a more satisfactory agreement between the model and the real peak profile shape .the hlsvd method works as follows .let us model the xrpd intensity samples collected at angles , as the sum of exponentially damped complex sinusoids \ ; , \label{eq : model}\ ] ] is the amplitude , the phase , the damping factor and the frequency of the sinusoid , , with the number of damped sinusoids .the data points defined in ( [ eq : model ] ) are arranged into a hankel matrix , , , with ( ) .the svd of the hankel matrix is computed as , where , , , and are orthogonal matrices and the superscript h denotes the hermitian conjugate .the lanczos bidiagonalization algorithm with partial reorthogonalization is used to compute svd .this algorithm , based on fft , computes the two matrix - vector products which are performed at each step of the lanczos procedure in rather than in in order to obtain the `` signal '' subspace , the matrix is truncated to a matrix of rank  , where , , and are defined by taking the first columns of and , and the upper - left matrix of , respectively . as subsequent step ,the least - squares solution of the following over - determined set of equations is computed , where and are derived from by deleting its first and last row , respectively .the eigenvalues of matrix are used to estimate the frequencies and damping factors of the model damped sinusoids from the relationship , \label{eq : eigen}\ ] ] with .values so obtained are inserted into the model equation ( [ eq : model ] ) which yields the set of equations \ ; , \label{eq : model_eigen}\ ] ] with .the least - squares solution of ( [ eq : model_eigen ] ) provides the amplitude and phase estimates of the model sinusoids which are used in the next step .the xrpd pattern has to be corrected for the instrumental broadening .several methods have been devised so far to deal with this problem . among themwe quote the stokes method and the bayesian approach .the main drawbacks of these methods stem from the difficulty in evaluating the background level , mainly due to peak overlapping .the xrpd pattern plugged in the deblurring algorithm is noise - background free since it has been already pre - processed by the wavelets filter and the morphological operator .the technique proposed in this paper is a modified version of the one presented in ref .a blurred or degraded xrpd pattern can be approximately described by a volterra equation , where is the blurred xrpd pattern and is the distortion operator due to several causes , namely the point spread function ( psf ) , and is an additive noise , introduced in the xrpd acquisition , that corrupts the signal . strictly speaking in an xrpd experimental setup we deal with a poissonian noise which is a multiplicative noise .however the poisson distribution function resembles the gauss one provided a sufficiently large statistics in photons counting .as to the deblurring procedure we implement the damped lucy richardson algorithm .this function performs multiple iterations , using optimization techniques and poisson statistics . in our approachthe psf is the raw dataset downloaded for the ceria sample - the instrumental standard - resembling the instrument profile .the algorithm maximizes the likelihood that the resulting image , when convolved with the psf , is an instance of the blurred image , assuming poisson noise statistics .this function can be effective when the psf is known but the knowledge about the additive noise in the image is poor .richardson algorithm introduces several adaptations to the original maximum likelihood algorithm that addresses complex image restoration tasks . by using these adaptations ,the effect of noise amplification on image restoration can be reduced , nonuniform image quality can be accounted for ( e.g. , bad pixels , flat - field variation ) and the restored image resolution can be improved by subsampling .  due to the denoising / background suppression ,the original volterra equation is readily simplified : , where , and have been already defined . was the inverse explicitly known, we would solve the former equation at a glance .unfortunately this is not the case and the solution has to be approximated as follows . , and are positive and the psf can not change the norm , _ i.e. _ . thus : where ; the solution can be found by an iterative procedure : , where the initial guess xrpd spectrum is uniform . as already stressed in ref . , the main drawbacks in applying such an algorithm to the single peak deconvolution are the noise amplification and the peak fitting bias .noise amplification is dramatically reduced by both the denoising procedure and the small ( some five ) number of iterations used in the algorithm .moreover the damp in the algorithm specifies the threshold level for the deviation of the resulting image from the original image , below which damping occurs . for pixels that deviate in the vicinity of their original values ,iterations are suppressed .  as to the peak fitting bias , unlike the balzar approach, our procedure uses the instrumental standard pattern ( with no overlapping ) as the psf to deconvolve the full xrpd pattern and then we extract the deconvoluted / deblurred xrpd pattern in the same range of the psf used for the deconvolution itself .the rationale of this choice relies on the fact that while the psf peaks have no overlap , this is not the case for the broadened sample peaks and , thus , the peak ranges can be defined starting on the annealed sample rather than the broadened one .moreover the discrete fourier transform ( dft ) , used by the deblurring functions , assumes that the frequency pattern of an image is periodic .this assumption creates a high - frequency drop - off at the edges of an overlapping peaks cluster .this high - frequency drop - off can create an effect called boundary related ringing in deblurred images , that is a systematic error affecting any further investigation on the physical meaning of the deconvolved spectrum . to reduce ringing, our full pattern deconvolution , as described above , resembles an edgetaper function .it removes the high - frequency drop - off at the edge of an image by blurring the entire image and then replacing the center pixels of the blurred image with the original one . in this way , the edges of the image taper off to a lower frequency .in this paper we presented a new approach for background removal and peak fitting of xrpd profiles .such operations are crucial in the line - broadening analysis of x - ray diffraction profiles , an important pre - processing step in the investigation of the crystal powder samples by means of xrpd data .backgroud suppression relies on the use of morphological mathematics while peak fitting is carried out by means of hlsvd technique .in order to enhance the signal - to - noise ratio of xrpd profiles a wavelet based filter is preliminarly applied to xrpd data .the output of the proposed precedure is supplied to a damped lucy richardson algorithm for deblurring .the main advantage of this approach is twofold : background suppression and peak shape profile fitting are model independent .real xrpd intensity profiles of ceria samples are used to test performances .results show that the background estimate is in agreement with that computed by traditional interpolation methods with a percentage difference below 3% .further , the output xrpd profile has narrower peaks , located at the same position and with the same shape as the original one .it is worth noting that once the deblurred , noise - background free xrpd spectrum is convoluted back to the psf and added to the removed noise and background signals , the resulting xrpd pattern resembles the original one .1 cervellino , a. , giannini , c. , guagliardi , a. & ladisa , m. ( 2005 ) , _ nanoparticle size distribution estimation by full - pattern diffraction analysis _ , 1em plus 0.5em minus 0.4emsubmitted to physical review b ; [ arxiv : cond - mat/0502583 ] .cervellino , a. , giannini , c. , guagliardi , a. & ladisa , m. ( 2005 ) , _ disentangling instrumental broadening _ ,1em plus 0.5em minus 0.4emsubmitted to journal of applied crystallography ; [ arxiv : cond - mat/0502582 ] .ladisa , m. , lamura , a. , laudadio , t. , and nico , g. ( 2005 ) , _ application of the hlsvd technique to the filtering of x - ray diffraction data _, 1em plus 0.5em minus 0.4emsubmitted to signal processing ; [ arxiv : math.sp/0502539 ] .jenkis , r. , and snyder , r.l .( 1996 ) , _ introduction to x - ray powder diffractometry _ ,1em plus 0.5em minus 0.4emnew york : wiley .rams , e. , penton , a. , martinez garcia , j & fuess , h. ( 2005 ) , _ the use of analytical peak profile functions to fit diffraction data of planar faulted layer crystals _ , 1em plus 0.5em minus 0.4emcryst ., * 40 * , 1/2 , 166 - 176 .balzar , d. , audebrand , n. , daymond , m.r . , fitch , a. , hewat , a. , langford , j. i. , le bail , a. , louer , d. , masson , o. , mccowan , c.n . , popa , n.c ., stephens , p.w . &toby , b.h . (2004 ) , 1em plus 0.5em minus 0.4emjournal of applied crystallography , * 37 * , 911 - 924 .laudadio , t. , mastronardi , n. , vanhamme , l. , van hecke , p. and van huffel , s. ( 2002 ) , _ improved lanczos algorithms for blackbox mrs data quantitation _ ,1em plus 0.5em minus 0.4emjournal of magnetic resonance , * 157 * , 292 - 297 .data are available at the urls : www.du.edu/%7ebalzar/s-s_rr.htm , www.ccp14.ac.uk , www.boulder.nist.gov/div853/balzar .press , w.h . , teukolsky , s.a . ,vetterling , w.t . andflannery , b.p .( 1992 ) , _ numerical recipes in c _, 1em plus 0.5em minus 0.4emcambridge university press .daubechies , i. ( 1992 ) , _ ten lectures on wavelets _ , 1em plus 0.5em minus 0.4emsiam .serra , j. ( 1994 ) _ morphological filtering : an overview_,1em plus 0.5em minus 0.4emsignal processing , * 38 * , 3 - 11 .stokes , a.r .( 1948 ) , 1em plus 0.5em minus 0.4emproc .( london ) , * 61 * , 382 - 391 .lucy , l.b .( 1972 ) _ an iterative technique for the rectification of observed distributions_,1em plus 0.5em minus 0.4emthe astronomical journal , * 8 * , 243 - 246 .richardson , w.h .( 1972 ) , 1em plus 0.5em minus 0.4emjournal of optical society of america a , * 62 * , 55 - 59 .biemond , j. , lagendijk , r.l . ,mersereau , r.m .( 1990 ) , _ iterative methods for image deblurring _ , proceedings of the ieee , * 78 * , 5 , 856 - 883 .

Abstract: precise knowledge of x - ray diffraction profile shape is crucial in the investigation of the properties of matter in crystals powder . line - broadening analysis is a pre - processing step in most of the full powder pattern fitting softwares . final result of line - broadening analysis strongly depends on preliminary three steps : noise filtering , removal of background signal and peak fitting . in this work a new model independent procedure for two of the aforementioned steps ( background suppression and peak fitting ) is presented . the former is dealt with by using morphological mathematics , while the latter relies on the hankel lanczos singular value decomposition technique . real x - ray powder diffraction ( xrpd ) intensity profiles of ceria samples are used to test the performance of the proposed procedure . results show the robustness of this approach and its capability of efficiently improving the disentangling of instrumental broadening . these features make the proposed approach an interesting and user - friendly tool for the pre - processing of xrpd data .   hankel lanczos singular value decomposition ( hlsvd ) , morphological filtering , x - ray powder diffraction


pose estimation is a fundamental component of many computer vision applications , ranging from robotic vision to intelligent image analysis . in general ,pose estimation refers to the process of obtaining the location and orientation of an object .however , the accuracy and nature of the pose estimate required varies from application to application .certain applications require the estimation of the full 3d pose of an object , while other applications require only a subset of the pose parameters .the 2d-3d registration problem in particular is concerned with estimating the pose parameters that describe a 3d object model within a given 2d scene .an image / photograph of a known object can be analysed in greater detail if a 3d model of the object can be registered over it , to be used as a ground truth . as an example , consider the case of automatically analysing a damaged car using a photograph .the focus of this work is to develop a method to estimate the pose of a known 3d object model in a given 2d image , with an emphasis on estimating the pose of cars .we have the following objectives in mind .* use only a single , static image limited to a single view * work with any unknown camera ( without prior camera calibration ) * avoid user interaction * avoid prior training / learning * work under varying and unknown lighting conditions * estimate the full 3d pose of the object ( not a partial pose as in an overhead view of traffic or machine parts along a conveyor belt )  a 3d pose estimation method with these properties would also be useful in remote sensing , automated scene recognition and computer graphics , as it allows for additional information to be extracted without the need for human involvement .many methods , including point correspondence based methods , implicit shape model based methods and image gradient based methods , have been developed to solve the pose estimation problem .however , the methods identified in the literature do not satisfy the objectives mentioned above , hence the necessity of our novel method .a more detailed review of existing pose estimation methods ranging over the past 30 years is presented in section [ secrelatedwork ] .this paper presents a method which registers a known 3d model onto a given 2d photo containing the modelled object while satisfying the objectives outlined above .it does this by measuring the closeness of the projected 3d model to the 2d photo on a pixel ( rather than feature ) basis .background and unknown lighting conditions of the photo are major complications , which prevent using a naive image difference like the absolute or square loss as a measure of fit .  a major contribution of this paper is the novel `` distance '' measure in section [ secmatching3dto2d ] that does neither depend on the lighting of the real scene in the photo nor on choosing an appropriate lighting in the rendering of the 3d model , hence does not require knowledge of the lighting .technically , we derive in section [ seclossfunction ] a loss function for vector - valued pixel attributes ( of different modality ) that is invariant under linear transformations of the attributes .  to analyse the nature of the developed loss function , we have applied it to a series of test cases of varying complexity , as detailed in section [ seclosslandscape ] .these test cases indicate that for our target application the loss function is well behaved and can be optimised using a standard optimisation method to find an accurate pose match .as presented in section [ secoptim ] , we achieve good pose recovery results in both artificial and real world test cases using this optimisation scheme . in these optimisation tests , negative influence of the background is attenuated by clipping the photo to the projection of the 3d model when calculating the loss .technical aspects of the optimisation and loss calculation methods are discussed in section [ sectech ] .model based object recognition has received considerable attention in computer vision circles .a survey by chin and dyer shows that model based object recognition algorithms generally fall into 3 categories , based on the type of object representation used - namely 2d representations , 2.5d representations or 3d representations .representations store the information of a particular 2d view of an object ( a characteristic view ) as a model and use this information to identify the object from a 2d image .global feature methods have been used by gleason and algin to identify objects like spanners and nuts on a conveyor belt .such methods use features such as the area , perimeter , number of holes visible and other global features to model the object . structural features like boundary segments have been used by perkins to detect machine parts using 2d models .a relational graph method has been used by yachida and tsuji to match objects to a 2d model using graph matching techniques .these 2d representation - based algorithms require prior training of the system using a ` show by example ' method .approaches are also viewer centred , where the object is known to occur in a particular view .they differ from the 2d approach as the model stores additional information such as intrinsic image parameters and surface - orientation maps .the work done by poje and delp explain the use of intrinsic scene parameters in the form of range ( depth ) maps and needle ( local surface orientation ) maps .shape from shading and photometric stereo are some other examples of the use of the 2.5d approach used for the recognition of industrial parts .a range of techniques for such 2d/2.5d representations are described by forsythe and ponce , by posing the object recognition problem as a correspondence problem .these methods obtain a hypothesis based on the correspondences of a few matching points in the image and the model .the hypothesis is validated against the remaining known points .approaches are utilised in situations where the object of interest can appear in a scene from multiple viewing angles .common 3d representation approaches can be either an ` exact representation ' or a ` multi - view feature representation ' .the latter method uses a composite model consisting of 2d/2.5d models for a limited set of views .multi - view feature representation is used along with the concept of generalised cylinders by brooks and binford to detect different types of industrial motors in the so called acronym system .the models used in the exact representation method , on the contrary , contain an exact representation of the complete 3d object .hence a 2d projection of the object can be created for any desired view .unfortunately , this method is often considered too costly in terms of processing time .the 2d and 2.5d representations are insufficient for general purpose applications .for example , a vehicle may be photographed from an arbitrary view in order to indicate the damaged parts .similarly , the 3d multi - view feature representation is also not suitable , as we are not able to limit the pose of the vehicle to a small finite set of views .therefore , pose identification has to be done using an exact 3d model .little work has been done to date on identifying the pose of an exact 3d model from a single 2d image .huttenlocher and ullman use a 3d model that contains the locations of edges .the edges / contours identified in the 2d image are matched against the edges in the 3d model to calculate the pose of the object .the method has been implemented for simple 3d objects . however, this method will not work well on objects with rounded surfaces without clearly identifiable edges .recent work by arie - nachimson and ronen basri makes use of ` implicit shape models ' to recognise 3d objects from 2d images .the model consists of a set of learned features , their 3d locations and the views in which they are visible .the learning process is further refined using factorisation methods .the pose estimation consists of evaluating the transformations of the features that give the best match .a typical model requires around 65 images to be trained .there are many different types of cars in use and new car models are manufactured quite frequently . therefore , any methodology that requires training car models would be laborious and time consuming .hence , a system that does not require such training is preferred for the problem at hand .gray scale image gradients have been used to estimate the 3d pose in traffic video footage from a stationary camera by kollnig and nagel .the method compares image gradients instead of simple edge segments , for better performance .image gradients from projected polyhedral models are compared against image gradients in video images .the pose is formulated using 3 degrees of freedom ; 2 for position and 1 for angular orientation .tan and baker use image gradients and a hough transform based algorithm for estimating vehicle pose in traffic scenes , once more describing the pose via 3 degrees of freedom . pose estimation using 3 degrees of freedom is adequate for traffic image sequences , where the camera position remains fixed with respect to the ground plane .this approach does not provide a full pose estimate required for a general purpose application .work done by and later by attempt to simultaneously solve the pose and point correspondence problems .the success of these methods are affected by the quality of the features extracted from the object , which is non - trivial with objects like cars .our method on the contrary , does not depend on feature extraction .can be used to represent a distance between two data sets , and hence give a measure of their similarity .therefore , distance metrics can be used to measure similarity between different 2d images , as well as 2d images and projections of a 3d model . a basic distancemetric would be the _ euclidian distance _ or the 2-norm .however , this has the disadvantage of being dependant on the scale of measurement .the _ mahalanobis distance _ on the other hand , is a scale - invariant distance measure .it is defined as ||x - y||_c^-1 for random vectors and with a covariance matrix of .the mahalanobis distance will reduce to the euclidean distance when the covariance matrix is the identity matrix ( ) .the mahalanobis distance is used by xing et al . for clustering .it is also used by deriche and faugeras to match line segments in a sequence of time varying images .we describe our approach of matching 3d models to 2d photos in this section using a novel illumination - invariant loss function .a detailed derivation of the loss is provided in section [ seclossfunction ] .assume we want to match a 3d model ( ) to a 2d photo ( ) or vice versa .more precisely , we have a 3d model ( e.g. as a triangulated textured surface ) and we want to find a projection for which the rendered 2d image has the same perspective as the 2d photo .as long as we do not know the lighting conditions of , we can not expect to be close to , even for the correct .indeed , if the light in came from the right , but the light shines on from the left , may be close to the negative of .formally , let be the set of ( integer ) pixel coordinates , and be a pixel coordinate .alternatively a smaller region of interest may be used for instead of as explained in section [ clipbg ] .let be a photo with real pixel attributes , and a projection of a 3d object to a 2d image with real pixel attributes .the attributes may be colours , local texture features , surface normals , or else . in the followingwe consider the case of grey - level photos ( ) , and for reasons that will become clear , use surface normals and brightness of the ( projected ) 3d model .a simple lambertian reflection model is not realistic enough to result in a zero loss on real photos , even at the correct pose .nevertheless ( we believe and experimentally confirm that ) it results in a minimum at the correct pose , which is sufficient for matching purposes .we use phong shading without specular reflection for this purpose .let be the global ambient / diffuse light intensities of the 3d scene , and be the ( global ) unit vector in the direction of the light source ( or their weighted sum in case of multiple sources ) . for reasons to become clear later, we introduce an extra illumination offset ( which is 0 in the phong model ) . for each surface point , let be the ambient / diffuse reflection constants ( intrinsic surface brightness ) and be the unit ( interpolated ) surface `` normal '' vector. then the apparent intensity of the corresponding point in the projection is i(p ) = k_a(p ) i_a + k_d(p)(^v(p))i_d + i_0 am_(p ) + b the last expression is the same as the first , just written in a more covariant form : are the known surface ( dependent ) parameters , and are the four ( unknown ) global illumination constants , and .since is linear in and , any rendering is a simple global linear function of .this model remains exact even for multiple light sources and can easily be generalised to color models and color photos .we measure the closeness of the projected 3d model to the 2d photo by some distance , e.g. square or absolute or mahalanobis .we do not want to assume any extra knowledge like the lighting conditions under which the photo has been taken , which rules out a direct use of .ideally we want a `` distance '' between and that is independent of and is zero if and only if there exists a lighting condition such that and coincide .indeed , this is possible , if ( rather than defining as some -dependent rendered projection of ) we use -independent brightness and normals as pixel features as defined above , and define a linearly invariant distance as follows : let |f : = _ pp f(p ) |m_:=_pp m_(p)^4 be the average attribute values of photo and projection , and c_f m _ : = 1|p|_pp(f(p)-|f)(m_(p)-|m_)^^14 be the cross - covariance matrix between and and similarly and the covariance matrices and . with this notationwe can define the following distance or loss function between and : [ deflil ] ( ) : = \{n , m } - obviously this expression is independent of . in the next sectionwe show that it is invariant under regular linear transformation of the pixel / attribute values of and and zero if and only if there is a perfect linear transformation of the pixel values from to .this makes it unnecessary to know the exact surface reflection constants of the object ( ). we will actually derive ( ) = d_(f , am_+b ) this implies that is zero if and only if there is a lighting under which and coincide , which we desired .a detailed derivation of the loss function is given in this section .although together with section [ secmatching3dto2d ] , this is a main novel contribution of this paper , it may be skipped over by the more application - oriented reader without affecting the continuity of the rest of the paper .using the notation of the previous section , we measure the similarity of photo and projected 3d model ( returning to general ) by some loss : [ defloss ] ( ) : = d(f , m _ ) : = 1|p| d(f(p),m_(p ) ) where is a distance measure between corresponding pixels of the two images to be determined below .a very simple , but as discussed in section [ secrelatedwork ] for our purpose unsuitable , choice in case of would be the square loss .it is convenient to introduce the following probability notation : let be uniformly distributed in , i.e. =|p|^{-1} ] , but this norm is ( only ) invariant under component scaling .to get invariance under general linear transformations , we have to `` divide '' by the covariance matrix c_yy : = , |y:=the mahalanobis norm ( cf .section [ secrelatedwork ] ) ||y||_c_yy^-1 ^ 2 : = y^c_yy^-1 y is invariant under linear _ homogenous _ transformations , as can be seen from  we have used .the following distance is hence invariant under _ any _ non - singular linear transformation of and any non - singular ( incl .non - homogenous ) linear transformation of : [ defdinv ] d(x , y ) : = _ a^mn_b^m  since the norm is quadratic in and , the minimisation can be performed explicitly , yielding [ abmin ] b = b_min:=|y - a_min|x a = a_min:=c_yxc_xx^-1 , c_xy : = ( x , y ) = ,   .inserting back into and rearranging terms gives d(x , y ) = = m - .this explicit expression shows that is also nearly symmetric in and .the trace is symmetric but is not . for comparisons ,e.g. for minimising w.r.t. , the constant does not matter . since the trace can assume all and only values in the interval $ ], it is natural to symmetrize by \{d(x , y),d(y , x ) } = \{n , m } - returning to original notation , this expression coincides with the loss .it is hard to visualize this loss , even for and , but the special case is instructive , for which the expression reduces to d(x , y)=1-^2(x , y ) , ( x , y ) = ( x , y)_x_y is the correlation between and . the larger the ( positive or _ negative _ ) correlation , the more similar the images and the smaller the loss .for instance , a photo has maximal correlation with its negative .in this section , we explore the nature of the loss function derived in section [ seclossfunction ] for real and artificial photographs , together with a pose representation specific to vehicles .it is important to select a pose representation that suitably describes the 3d model that is being matched .careful selection of pose parameters can enhance the ability of the optimisation to find the best match , and can allow object detection or coarse alignment methods , such as that presented in to specify a starting pose for the optimisation .we use the following pose representation for 3d car models , temporarily neglecting the effects of perspective projection : [ eqn : posex ] : = ( _ x , _ y , _ x , _ y , _ x , _ y ) is the visible rear wheel center of the car in the 2d projection . is the vector between corresponding rear and front wheel centres of the car in the 2d projection . is a unit vector in the direction of the rear wheel axle of the 3d car model .therefore , and need not be explicitly included in the pose representation .this representation is illustrated in figure [ fig : poserepresentationx ] .to understand the behaviour of the loss function , we have generated loss landscapes for artificial images of 3d models . to produce these landscapes ,an artificial photograph was generated by projecting the 3d model at a known pose with phong shading .we then vary the pose parameters , two at a time about and find the value of the loss function between this altered projection and the `` photograph '' taken at .these loss values are recorded , allowing us to visualise the behaviour of the loss function by observing surface and contour plots of these values .the unaltered pose values should project an image identical to the input photograph , giving a loss of zero according to the loss function derived in section [ seclossfunction ] , with a higher loss exhibited at other poses .the variation of the loss with respect to a pair of pose parameters is shown in figure [ fig : f1surf ] .it can be seen from these loss landscapes that the loss has a clear minimum at the initial pose .the loss values increase as these pose parameters deviate away from , up to . from this data , we are able to see that the minimum corresponding to can be considered a global minimum for all practical purposes .the shape of the surface plots was similar for all other parameter pairs , indicating that the full 6d landscape of the loss function should similarly have a global minimum at the initial pose , allowing us to find this point using standard optimisation techniques , as demonstrated in section [ secoptim ] .the landscape of the loss function was analysed for real photographs by varying the pose parameters of the model about a pose obtained by manually matching the 3d car model to a real photograph .the variation was plotted by taking a pair of pose parameters at a time over the entire set of pose parameters . a loss landscape obtained by varying and for a real photographis shown in figure [ fig : losslandscaperealphoto ] .the variation of the loss function for other pose parameter pairs were found to be similar .although a global minimum exists at the best pose of the real photograph , the nature of the loss function surface makes it more difficult to optimise when compared to artificial photos ( figure [ fig : f1surf ] ) .in particular , one can observe local minima in the periphery of the landscape , and the full 6d landscape is considerably more complex .as explained in section [ seclossfunction ] , the correct pose parameters will give the lowest loss value .the loss function landscape , as discussed in section [ seclosslandscape ] , shows that corresponds to the global minimum of the loss function .therefore , an optimisation was performed on the loss function to obtain based on the pose representation ( equation [ eqn : posex ] ) . the optimisation strategy and its reliability in different scenariosis discussed in this section .  to immunise the optimisation from pixel quantisation artefacts and noise in the images , direct search methods that do not calculate the derivative of the loss function were considered .the optimisation was performed using the well known_ downhill simplex method ( ds ) _ , owing to its efficiency and robustness .when optimising an -dimensional function with the ds method , a so called _ simplex _ consisting of points is used to traverse the -dimensional search space and find the optimum .the reliability of the optimisation is adversely affected by the existence of local minima .fortunately , the _ downhill simplex _ method has a useful property . in most cases ,if the simplex is reinitialised at the pose parameters of the local minimum and the optimisation is performed again , the solution converges to the global minimum .proper parameterisation is important for the optimiser to give good results .we have used a normalised pose parameterisation as follows .normalisation gives each pose parameter a comparable range during optimisation .the normalised pose was obtained by normalising and w.r.t .the dimensions of the photograph ._ n = ( , , , , _ x , _ y ) are the width and height of the photograph ( 2d image ) . is a unit vector and does not require normalisation .  the downhill simplex method , like all optimisation techniques , requires a reasonable starting position .there are many methods for selecting a starting point , from repeated random initialisation to structured partitioning of the optimisation volume .a disadvantage of these methods is that they require a number of optimisation runs to locate the optimal point , which can take significant time .depending on the application , it may be possible to develop a coarse location method which provides an estimate of the optimal pose .  the wheel match method described by hutter and brewer is one such method , providing an initial match for a vehicle pose if the vehicle s wheels are visible .estimations using this method generally locate the wheel centres with a high degree of accuracy , but perform less effectively when determining the axle direction .this indicates that it may be possible to perform staged optimisation , attempting to fix some parameters before others . in general , parameter estimationusing this method is within 5 - 15% of the true value .this initial pose selection is sufficient for our purposes .tests were carried out to asses the reliability of the pose estimate .we first generated synthetic `` photographs '' by rendering a 2d projection of a 3d car model at a known pose .the optimisation was performed from initial poses that had a known deviation from .test poses were selected at 1% , 2% , 4% , 8% and 16% deviation from the known initial parameters so as to investigate a large hyper - volume in 6d space .the parameter values for 50 such random starting poses , 10 for each range , are shown in figure [ fig : startingposedeviations50 ] .the reliability for each percentage range was defined as the proportion of correct matches in that deviation range .exact pose recovery for synthetic images and better than careful manual tuning for real photos were regarded as _correct_. with this definition , a reliability of 1 indicates that all test cases in the range converged .a reliability of 0 indicates that none of the test cases in the range converged .  to ensure that the selected optimisation method is appropriate, we first investigate a simple case in which an artificial image with known parameters is constructed and used to validate the optimisation method .the reliability of the optimisation ( with simplex re - initialisations ) was found to be 100% ( figure [ fig : combinedresultsgraph ] ) for initial poses with up to a 16% deviation from the matching pose .next we rendered artificial car models on a real background photo , and performed the same reliability tests .allowing simplex re - initialisations preserved a 100% convergence up to the 8% deviation range ( figure [ fig : combinedresultsgraph ] ) , although the simplex did not converge for certain starting poses at a 16% deviation .this shows that the effects of a real background can deteriorate the reliability of the pose estimate for higher deviations . in order to address this issue ,a further refinement of the algorithm was made by clipping the background in the photograph when calculating the loss .[ clipbg ] the methodology used to lower the effects of the image background is as follows : pixels in the projected image that do not correspond to points of the 3d model were treated as background .these pixels do not have surface normal components as they do not belong to the 3d model .therefore , they can easily be filtered out by identifying pixels in the projected image that have null values for all three components of the surface normal .only the remaining pixels were considered for the loss calculation ( figure [ fig : combinedresultsgraph ] ) .the reliability of the pose estimates on a real car photo are shown in figure [ fig : combinedresultsgraph ] .correct pose estimates with a high reliability were obtained for starting poses up to an 8% deviation .  the distance from the camera to the projection plane in the opengl perspective projection model was used as a seventh parameter when optimising using perspective projection .the extra pose parameter makes the optimisation harder at higher deviations as seen in the reliability graph in figure [ fig : combinedresultsgraph ] .the reliability of the pose estimate may be further improved by using more sophisticated optimisation methods .an example of a correctly estimated pose for a starting pose within a 16% deviation from the manually matched pose is shown in figures [ fig : realphotoinitialpose ] and [ fig : realphotoestimatedpose ] .given that we lack an absolute ground truth estimate , pose estimates were labelled as correct or incorrect based on their visual similarity to the input image .  .results for an artificial photograph ( a projection of the 3d model ) , an artificial photograph with a real background superimposed and a real photograph are included . ]in this section we describe some of the technical aspects of the proposed work .the initial code was implemented in matlab , however , components were gradually ported to c in order to improve performance .  in order to calculate the loss values described in section [ seclossfunction ] , it was required to render the surface normals and brightness of a 3d model at a given pose .initially , the rendering was done using _ model3d _ , a bsd licensed matlab class .as this rendering was not fast enough for our application , a separate module was written in c to render the model off - screen using opengl pbuffer extension and glx .this c module was used with the matlab code using the _ mex _ gateway .initially , only the rendering was done in c. the rendered 2d intensity and surface normal matrices were returned back to matlab using the mex gateway .this seemed to exhaust memory during the reliability tests described in section [ secoptim ] .therefore , the rendering and the loss calculation were also implemented in c , with only the loss value returned to matlab for use in optimisation ..rendering and loss calculation times .[ cols="<,^ , > " , ]   this second approach improved performance in terms of speed and memory usage .a summary of the time taken to render the image and to calculate the loss using these approaches are presented in table [ tblrenderlosscalctime ] .triangulated 3d car models of significant complexity and detail in the autodesk 3ds file format were used for the work in this paper .these models were purchased from online 3d model vendors and had in the order of 30,000 nodes and in the order of 50,000 triangles .  a typical downhill simplex minimisation required in the order of 100200 loss function evaluations .using the c based loss calculation and opengl rendering , pose estimation in artificial images took around 1 minute for models with more than 30,000 nodes .recent work done in on pose estimation using point correspondences , takes more than 3 minutes ( 200 seconds ) for an artificial image of a model with only 80 points .hence , despite being a pixel based method , the performance of our approach is very encouraging .the opengl context needs to be initialised each time the loss is calculated , when the c module ( mex ) to calculate the loss is invoked from matlab . a further speed - upcould be obtained by implementing the entire code ( rendering and optimisation ) in c , whereby the time spent on initialising the opengl context could be saved , as this needs to be done only once .it was also noted that although hardware accelerated opengl performs fast rendering , reading the rendered pixels back to main memory causes a performance bottleneck .the loss calculation may also be done in the graphics hardware itself , using glsl or gpu computing , in order to avoid this bottleneck .a method to register a known 3d model on a given 2d image is presented in this paper .the correlation between attributes in the 2d image and projected 3d model are analysed in order to arrive at a correct pose estimate .the method differs from existing 2d-3d registration methods found in the literature .the proposed method requires only a single view of the object . it does not require a motion sequence and works on a static image from a given viewalso , the method does not require the camera parameters to be known a priori .explicit point correspondences or matched features ( which are hard to obtain when comparing 3d models and image modalities ) need not be known beforehand .the method can recover the full 3d pose of an object .it does not require prior training or learning .as the method can handle 3d models of high complexity and detail , it could be used for applications that require detailed analysis of 2d images .it is particularly useful in situations where a known 3d model is used as a ground truth for analysing a 2d photograph .the method has been currently tested on real and artificial photographs of cars with promising results .a planned application of the method is to analyse images of damaged cars .a known 3d model of the damaged car will be registered on the image to be analysed , using the proposed registration method .this will be used as a ground truth .the method could be extended further to simultaneously identify the type of the car while estimating its pose , by optimising the loss function for a number of 3d models and selecting the model with the lowest loss value .more sophisticated optimisation methods may be used to improve results further .we conclude from our results that the linearly invariant loss function derived in section [ seclossfunction ] can be used to estimate the pose of cars from real photographs .we also demonstrate that the _ downhill simplex _ method can be effectively used to optimise the loss function in order to obtain the correct pose .allowing simplex re - initialisations makes the method more robust against local minima .the possibility of needing such re - initialisations can be significantly reduced by clipping the background of the image when calculating the loss . despite being a direct pixelbased method ( as opposed to a feature / point based method ) , the performance of our method is very encouraging in comparison with other recent approaches , as discussed in section [ sectech ] .g. j. gleason and g. j. agin . a modular system for sensor - controlled manipulation and inspection . in_ proceedings of the 9th international symposium on industrial robots _ , pages 5770 , washington , dc , usa , march 1979 .society of manufacturing engineers .

Abstract: the problem of identifying the 3d pose of a known object from a given 2d image has important applications in computer vision ranging from robotic vision to image analysis . our proposed method of registering a 3d model of a known object on a given 2d photo of the object has numerous advantages over existing methods : it does neither require prior training nor learning , nor knowledge of the camera parameters , nor explicit point correspondences or matching features between image and model . unlike techniques that estimate a partial 3d pose ( as in an overhead view of traffic or machine parts on a conveyor belt ) , our method estimates the complete 3d pose of the object , and works on a single static image from a given view , and under varying and unknown lighting conditions . for this purpose we derive a novel illumination - invariant distance measure between 2d photo and projected 3d model , which is then minimised to find the best pose parameters . results for vehicle pose detection are presented .   illumination - invariant loss ; 2d-3d pose estimation ; pixel - based ; featureless ; optimisation .