You are an expert at summarizing long articles. Proceed to summarize the following text: 
one goal of the inverse scattering problem is to find the shape of bulk or flaw in a medium using the scattered field data measured at the boundary . this is an old but it has several of applications such as detection of cracks in material engineering , ultrasound imaging in medical sciences , and scanning anti - personnel mines hidden in the ground in military services . related researches may be found in @xcite and the references therein . in many studies , various shape reconstruction algorithms have been developed , most of which are focused on the minimization of the least - square functional by using a newton - type iteration strategy , e.g. , level - set method @xcite . the merits of the iteration strategy is that it does not require a very large number of boundary measurements for complete shape reconstruction ; instead , in order to guarantee successful reconstruction , it demands very high computational expenditures , optimal regularization terms related to the problems at hand , calculation of a complex frchet derivative at each iteration , and _ a priori _ information of the unknown target to be reconstructed . nevertheless , even if these conditions are satisfied , the iteration strategy must begin with an initial guess that is close to the true one in order to avoid undesirable situations such as non - convergence or falling into the local minima . therefore , development of a mathematical theory and an algorithm for generating a good initial guess is an important research topic . conversely , certain non - iterative shape reconstruction algorithms have been proposed ; for example , multiple signal classification ( music ) algorithm @xcite , linear sampling method @xcite , kirchhoff migration @xcite , and fourier inversion based one @xcite . in contrast with the iterative strategy , these algorithms require a large amount of incident field data and a significant number of boundary measurements . however , if these conditions are satisfied , non - iterative shape reconstruction algorithms can prove to be effective , fast , and easy to extend to multiple targets . topological derivative concept is a non - iterative strategy . this concept was originally developed for the shape optimization problem , but its application to rapid shape reconstruction has only recently been proven . related works can be found in @xcite and references therein . one of the advantages of topological derivative concept is that it does not require a large amount of many incident field data ; however , a reduction in the amount of this data has been reported to result in poor resolution of the reconstructed shape . the purpose of this literature is to establish an effective and fast reconstruction algorithm for detecting both the location and shape of perfectly conducting crack(s ) from boundary - measured scattered field data by using only a small amount of incident field data . based on the structure of topological derivative imaging functional operated at a fixed - frequency @xcite and multi - frequency imaging schemes @xcite , we introduce a multi - frequency topological derivative concept . note that recent work ( * ? ? ? * formula ( 1.33 ) ) has briefly introduced a multi - frequency topological derivative but through a more detailed analysis , we investigate the structure of the proposed multi - frequency topological derivative and its benefits over the traditional topological derivative . the remainder of this paper is organized as follows . section [ sec2 ] briefly introduces the traditional topological derivative by creating a linear , narrow crack . section [ sec3 ] describes the design of a multi - frequency topological derivative imaging function and analyzes it to investigate its properties . section [ sec4 ] presents various numerical experiments to assess the performance of the proposed imaging function under various situations . section [ sec5 ] contains a short conclusion and some remarks on future work . assume that a perfectly conducting crack @xmath0 is completely hidden in a homogeneous domain @xmath1 with a smooth boundary @xmath2 . we denote @xmath3 as a two - dimensional vector that lies on crack @xmath0 .    throughout this paper , we consider the so - called transverse magnetic ( tm ) polarization case , letting @xmath4 be the ( single - component ) electric field that satisfies the following boundary value problem : @xmath5 where @xmath6 represents the unit outward normal to @xmath7 and @xmath8 , @xmath9 , denotes the two - dimensional vector on the unit circle @xmath10 . throughout this paper , we let @xmath11 denote a strictly positive wavenumber and presume @xmath12 to not be an eigenvalue of ( [ helmholtz ] ) . similarly , let @xmath13 be the solution of equation ( [ helmholtz ] ) without @xmath0 . then , the problem we consider here is the computation of the topological derivative of the residual function depending on the solution @xmath4 : @xmath14 where @xmath15 .    in order to compute the topological derivative , let us create a small linear crack @xmath16 of length @xmath17 centered at @xmath18 such that @xmath19 and denote @xmath20 as that domain . then , because of the change in the topology of @xmath21 , we can consider the topological derivative @xmath22 based on the residual function @xmath23 with respect to point @xmath24 as follows : @xmath25 where @xmath26 as @xmath27 . from relationship ( [ topderivative ] ) , we have the following asymptotic expansion : @xmath28    then , the topological derivative @xmath22 with @xmath29 different incident waves at a given wavenumber @xmath11 is as follows ( see @xcite for its derivation ) . [ thtop ] let @xmath30 denote the real part of @xmath31 . then , the topological derivative corresponding to ( [ energy ] ) is given by @xmath32,\ ] ] where @xmath33 satisfies the following adjoint problem : @xmath34 in this section , we introduce a multi - frequency topological derivative imaging function and explore its properties . based on recent research @xcite , a sufficiently large number of incident directions are required to obtain a good result . this means that if @xmath29 is small , an image with poor resolution will be reconstructed . moreover , when boundary measurement data are affected by a large amount of random noise , ( [ topderivative2 ] ) will fail to reconstruct a good image . inspired by multi - frequency imaging algorithms @xcite , we consider the following multi - frequency topological derivative for several wavenumbers @xmath35 : @xmath36 where @xmath37 is given by ( [ topderivative2 ] ) for @xmath38 , @xmath39 .    in this section , we investigate the structure of ( [ multifrequencytopologicalderivative ] ) . for this purpose , we recall a useful result ( see @xcite ) as follows . [ lemmkp ] let @xmath40 imply that there exists a constant @xmath41 such that @xmath42 . then , for @xmath43 and @xmath44 , ( [ topderivative2 ] ) satisfies @xmath45.\ ] ] here @xmath46 denotes the two - dimensional time - harmonic fundamental solution ( or green s function ) of helmholtz equation @xmath47 where @xmath48 , @xmath49 , and @xmath50 denote the hankel function of the first kind , bessel function , and neumann function of order zero , respectively . consequently , ( [ multifrequencytopologicalderivative ] ) is analyzed for the following three cases of interest . [ theoremsymmetric ] under the configuration of a small number of symmetric incident directions , the structure of ( [ multifrequencytopologicalderivative ] ) is @xmath51 for a sufficiently large @xmath52 and @xmath53 . assume that the symmetric incident directions are of the form : @xmath54 where @xmath29 is an even number , say , @xmath55 . based on this form , @xmath56 satisfies @xmath57 for @xmath58 . in contrast with lemma [ lemmkp ] , we assume that @xmath29 is not sufficiently large . let @xmath59 denote the imaginary part of @xmath31 then since @xmath60&=\re\bigg[\exp(ik\vec{\theta}_{l+n}\cdot(\vec{x}_c-\vec{x}_s))\bigg]\\    \im\bigg[\exp(ik\vec{\theta}_l\cdot(\vec{x}_c-\vec{x}_s))\bigg]&=-\im\bigg[\exp(ik\vec{\theta}_{l+n}\cdot(\vec{x}_c-\vec{x}_s))\bigg],\end{aligned}\ ] ] ( [ topderivative3 ] ) becomes @xmath61\\    & = \re\left[\frac{1}{2k_f}\sum_{m=1}^{n}\cos(k_f\vec{\theta}_m\cdot(\vec{x}_c-\vec{x}_s))\{j_0(k_f|\vec{x}_c-\vec{x}_s|)+iy_0(k_f|\vec{x}_c-\vec{x}_s|)\}\right]\\    & = \frac{1}{2k_f}\sum_{m=1}^{n}\cos(k_f\vec{\theta}_m\cdot(\vec{x}_c-\vec{x}_s))j_0(k_f|\vec{x}_c-\vec{x}_s|).\end{aligned}\ ] ] therefore , @xmath62    let us recall that for @xmath63 ( see ( * ? ? ? * formula 11.4.37 ( p. 487 ) ) ) @xmath64 let @xmath65 , @xmath66 , and @xmath67 . then since @xmath68\geq0,\ ] ] by letting @xmath53 , ( [ multifrequencytopologicalderivative ] ) becomes @xmath69    [ theoremnonsymmetric ] under the configuration of a small number of non - symmetric incident directions , the structure of ( [ multifrequencytopologicalderivative ] ) becomes @xmath70 for a sufficiently large @xmath52 and @xmath53 .    in constrast with theorem [ theoremsymmetric ] , we assume that the incident directions are non - symmetric . then , ( [ topderivative3 ] ) can be written as @xmath71\\    = & \re\left[\frac{1}{4k_f}\sum_{m=1}^{m}\exp(ik_f\vec{\theta}_m\cdot(\vec{x}_c-\vec{x}_s ) ) \{j_0(k_f|\vec{x}_c-\vec{x}_s|)+iy_0(k_f|\vec{x}_c-\vec{x}_s|)\}\right]\\    = & \frac{1}{4k_f}\sum_{m=1}^{m}\bigg[\cos(k_f\vec{\theta}_m\cdot(\vec{x}_c-\vec{x}_s))j_0(k_f|\vec{x}_c-\vec{x}_s|)\\ & -\sin(k_f\vec{\theta}_m\cdot(\vec{x}_c-\vec{x}_s))y_0(k_f|\vec{x}_c-\vec{x}_s|)\bigg],\end{aligned}\ ] ] and therefore @xmath72.\end{aligned}\ ] ] let us recall an integral formula in ( * ? ? ? * formula 11.4.40 ( p. 487 ) ) , @xmath73    then , if @xmath53 , we can evaluate @xmath74    hence , by combining ( [ inequalityab ] ) , ( [ structuremultitopderivative1 ] ) , and ( [ approximationsinexp ] ) , we can obtain @xmath75    [ theoremmanydirection ] under the configuration of a sufficiently large number of incident directions , the structure of ( [ multifrequencytopologicalderivative ] ) becomes @xmath76 for an adequately large @xmath52 . here , @xmath77 is defined as @xmath78 and @xmath79 denotes the dirac delta function . suppose that @xmath29 is large enough , i.e. , @xmath80 , then by ( * ? ? ? * lemma 4.1 ) , ( [ topderivative3 ] ) can be approximated as @xmath81\\          & \approx\re\left[\frac{1}{4k}\bigg(j_0(k|\vec{x}_c-\vec{x}_s|)+iy_0(k|\vec{x}_c-\vec{x}_s|)\bigg ) \int_{\mathcal{s}^1}\exp(ik\vec{\theta}\cdot(\vec{x}_c-\vec{x}_s))d\vec{\theta}\right]\\      & = \frac{\pi}{2k}j_0(k|\vec{x}_c-\vec{x}_s|)^2 . \end{aligned}\ ] ]    assume that @xmath52 is large enough and that @xmath82 . then applying an integral formula of the bessel function @xcite @xmath83 and a change of variable @xmath84 yields @xmath85 where @xmath86 is given in ( [ functionlambda ] ) . note that the term @xmath87 can be negligible since @xmath88 and @xmath89 , refer to @xcite .    finally , by applying @xmath53 in ( [ structuremultitopderivative3 ] ) , we can immediately obtain the following result : @xmath90    [ remark1]based on structures ( [ structuremultitopderivative1 ] ) , ( [ structuremultitopderivative2 ] ) , and ( [ structuremultitopderivative3 ] ) , we can summarize some properties of ( [ multifrequencytopologicalderivative ] ) as follows :    1 .   [ p1remark ] formulas ( [ structuremultitopderivative1 ] ) and ( [ structuremultitopderivative2 ] ) reach their maximum value at @xmath43 that satisfies @xmath91 therefore , at @xmath92 , the map of ( [ multifrequencytopologicalderivative ] ) will yield @xmath93 that the location of @xmath0 can be accurately determined . however , at point @xmath43 such that @xmath94 the map of ( [ multifrequencytopologicalderivative ] ) will show a large magnitude such that unexpected replicas @xmath95 will appear ( see figure [ remarkvector ] ) . [ p2remark ] if @xmath43 satisfies @xmath96 then both the numerator and denominator of ( [ structuremultitopderivative2 ] ) become @xmath97 . hence , at this point , ( [ structuremultitopderivative2 ] ) will yield a large value at not only @xmath3 but also other locations in @xmath21 . therefore , the configuration of symmetric incident directions will yield a better imaging result ( see figure [ gammanonsymmetric1 ] ) . [ p3remark ] if the number of incident directions is increased to as many as possible , i.e. , @xmath80 , it is clear that a more accurate location of @xmath44 can be obtained . in this case , an odd number of @xmath29 will not affect the imaging performance significantly . [ p4remark ] application of multiple frequencies enhances the imaging performance . illustration of remark [ remark1 ] . at the location @xmath24 such that @xmath98 ( black colored vector ) and @xmath99 ( green colored vector ) , true shape of @xmath0 and ghost replicas @xmath95 will appear , respectively.,scaledwidth=55.0% ] in this section , we exhibit some numerical examples to demonstrate the effectiveness of the proposed algorithm . throughout this section , the homogeneous domain @xmath21 is chosen as the interior region of the two - dimensional unit disk centered at the origin , i.e. , @xmath100\right\}}.\ ] ] the adopted wavenumber has the form @xmath101 for @xmath39 ; here , @xmath102 is the given wavelength that is equi - distributed between @xmath103 and @xmath104 . based on theorem [ theoremsymmetric ] , the incident direction @xmath8 is selected as @xmath54 for an even number @xmath29 .    motivated by @xcite , we choose two single cracks @xmath105 and @xmath106 as @xmath107\right\}}\\    \mathcal{c}_2&={\left\{\left(1.5\sin\frac{(3t+4)\pi}{8}-1 , 0.8\sin\frac{(3t+4)\pi}{4}\right):t\in[-1,1]\right\}},\end{aligned}\ ] ] and inspired by @xcite , we choose multiple cracks @xmath108 as @xmath109\}\cup\{(t+0.2,t^3+t^2 - 0.5):s\in[-0.5,0.5]\}.\ ] ]    note that in order to show the robustness of the proposed algorithm , a white gaussian noise with @xmath110 db signal - to - noise ratio ( snr ) is added to the unperturbed data . first , let us consider the imaging result of @xmath105 . the left - hand side of figure [ gamma1 ] shows the map of @xmath111 for @xmath112 different directions . in this result , because of its small magnitude at @xmath113 , @xmath105 appears divided . moreover , owing to the large magnitude at certain points ( for example , @xmath114 ) , we can not recognize its shape at this stage . the right - hand side of figure [ gamma1 ] is the map of @xmath115 . the figure shows that there is undivided or whole @xmath105 ; this means that we obtained a more accurate result than the previous one . however , because certain points have a large magnitude , we can not determine the true shape of @xmath105 . map of @xmath116 with @xmath117 ( left ) and @xmath118 ( right ) for @xmath105.,title="fig:",scaledwidth=49.0% ] map of @xmath116 with @xmath117 ( left ) and @xmath118 ( right ) for @xmath105.,title="fig:",scaledwidth=49.0% ]    figure [ gamma2 ] shows the map of @xmath116 for crack @xmath106 . in contrast with the result of @xmath105 , it is difficult to identify the shape of the crack because of the great number of points with large magnitudes . fortunately , similar to the results in figure [ gamma1 ] , map of @xmath115 yields a better image than that obtained by the map of @xmath111 . map of @xmath116 with @xmath117 ( left ) and @xmath118 ( right ) for @xmath106.,title="fig:",scaledwidth=49.0% ] map of @xmath116 with @xmath117 ( left ) and @xmath118 ( right ) for @xmath106.,title="fig:",scaledwidth=49.0% ]    the application of multiple cracks @xmath108 is shown in figure [ gammam ] . this figure shows the maps of @xmath119 and @xmath116 . throughout these results , we can observe that the map of @xmath22 yields a poor result even if the number of incident directions @xmath29 is large ; conversely , the multi - frequency imaging function @xmath116 yields an accurate result even if @xmath29 is small . maps of @xmath119 with @xmath120 ( top , left ) and @xmath121 ( top , right ) , and maps of @xmath116 with @xmath117 ( bottom , left ) and @xmath118 ( bottom , right ) for @xmath108.,title="fig:",scaledwidth=49.0% ] maps of @xmath119 with @xmath120 ( top , left ) and @xmath121 ( top , right ) , and maps of @xmath116 with @xmath117 ( bottom , left ) and @xmath118 ( bottom , right ) for @xmath108.,title="fig:",scaledwidth=49.0% ] + maps of @xmath119 with @xmath120 ( top , left ) and @xmath121 ( top , right ) , and maps of @xmath116 with @xmath117 ( bottom , left ) and @xmath118 ( bottom , right ) for @xmath108.,title="fig:",scaledwidth=49.0% ] maps of @xmath119 with @xmath120 ( top , left ) and @xmath121 ( top , right ) , and maps of @xmath116 with @xmath117 ( bottom , left ) and @xmath118 ( bottom , right ) for @xmath108.,title="fig:",scaledwidth=49.0% ]    in figure [ gammanonsymmetric1 ] , maps of @xmath115 with @xmath122 number of incident directions are displayed for cracks @xmath105 and @xmath108 in order to verify the [ p2remark ] of remark [ remark1 ] . by comparing figures [ gamma1 ] and [ gammam ] , it is difficult to discern the shape of true crack(s ) because of the appearance of unforseen replicas with large magnitudes . note that if @xmath29 is an odd number but is sufficiently large enough , the map of @xmath116 yield a good result , refer to figure [ gammanonsymmetric2 ] . map of @xmath116 with @xmath123 for @xmath105 ( left ) and @xmath108 ( right).,title="fig:",scaledwidth=49.0% ] map of @xmath116 with @xmath123 for @xmath105 ( left ) and @xmath108 ( right).,title="fig:",scaledwidth=49.0% ]    map of @xmath116 with @xmath124 for @xmath105 ( left ) and @xmath108 ( right).,title="fig:",scaledwidth=49.0% ] map of @xmath116 with @xmath124 for @xmath105 ( left ) and @xmath108 ( right).,title="fig:",scaledwidth=49.0% ]    at this moment , we increase the number of applied frequencies @xmath52 and incident directions @xmath29 . figures [ gamma12 - 6416 ] , [ gamma12 - 6464 ] , and [ gammam-64 ] depict the map of @xmath116 when @xmath125 and @xmath112 or @xmath121 . as we expected in [ p3remark ] of remark [ remark1 ] , the results yielded by @xmath116 improve with increasing number of @xmath52 and/or @xmath29 . map of @xmath116 with @xmath126 for @xmath105 ( left ) and @xmath106 ( right).,title="fig:",scaledwidth=49.0% ] map of @xmath116 with @xmath126 for @xmath105 ( left ) and @xmath106 ( right).,title="fig:",scaledwidth=49.0% ]    map of @xmath116 with @xmath127 for @xmath105 ( left ) and @xmath106 ( right).,title="fig:",scaledwidth=49.0% ] map of @xmath116 with @xmath127 for @xmath105 ( left ) and @xmath106 ( right).,title="fig:",scaledwidth=49.0% ]    map of @xmath116 with @xmath126 ( left ) and @xmath127 ( right ) for @xmath108.,title="fig:",scaledwidth=49.0% ] map of @xmath116 with @xmath126 ( left ) and @xmath127 ( right ) for @xmath108.,title="fig:",scaledwidth=49.0% ]    although multi - frequency topological derivative imaging yields very good imaging results for not only single but also multiple cracks , it still has the following limitations .    1 . let us consider the imaging of a crack of a large curvature near one of the crack tips . to illustrate this , we choose a crack from @xcite : @xmath128 + from the results in figure [ gammaa ] , the imaging results yielded by @xmath116 are satisfactory . however , at the point of the large curvature , the map of @xmath116 reconstructs a slightly different shape of the crack . we apply the proposed algorithm for imaging an oscillation crack represented as follows : @xmath129 + figure [ gammab ] shows the map of @xmath116 . the obtained result by the proposed algorithm does not improve even when the number of @xmath52 and @xmath29 are increased . the proposed algorithm requires further optiization . map of @xmath116 for @xmath130 with @xmath118 ( top , left ) , @xmath126 ( top , right ) , @xmath127 ( bottom , left ) and true shape ( bottom , right).,title="fig:",scaledwidth=49.0% ] map of @xmath116 for @xmath130 with @xmath118 ( top , left ) , @xmath126 ( top , right ) , @xmath127 ( bottom , left ) and true shape ( bottom , right).,title="fig:",scaledwidth=49.0% ] + map of @xmath116 for @xmath130 with @xmath118 ( top , left ) , @xmath126 ( top , right ) , @xmath127 ( bottom , left ) and true shape ( bottom , right).,title="fig:",scaledwidth=49.0% ] map of @xmath116 for @xmath130 with @xmath118 ( top , left ) , @xmath126 ( top , right ) , @xmath127 ( bottom , left ) and true shape ( bottom , right).,title="fig:",scaledwidth=49.0% ]    same as figure [ gammaa ] except the crack is @xmath131.,title="fig:",scaledwidth=49.0% ] same as figure [ gammaa ] except the crack is @xmath131.,title="fig:",scaledwidth=49.0% ] + same as figure [ gammaa ] except the crack is @xmath131.,title="fig:",scaledwidth=49.0% ] same as figure [ gammaa ] except the crack is @xmath131.,title="fig:",scaledwidth=49.0% ] we improved the traditional topological derivative concept to determine location and reconstruct the shape of arbitrary - shaped curve - like perfectly conducting cracks . for this purpose , we designed a reconstruction algorithm based on the multi - frequency topological derivative , analyzed its structure , and investigated its properties . we confirmed that the multi - frequency topological derivative contains singularity on the cracks , and therefore it reaches its maximum value at the location of the cracks . moreover , we demonstrated that why ghost replicas appeared in the neighborhood of cracks and why the configuration of symmetric incident directions yields better results than the non - symmetric configuration . numerical simulations under various situations depicted both the benefits and limitations of the proposed algorithm . because the algorithm requires only a one - step iteration procedure , it is very fast in terms of shape identification but does not yield the complete shape of cracks . however , by adopting the obtained result as a starting point of a newton - type based algorithm @xcite , it is expected that complete shape reconstruction can be successfully accomplished . this paper deals with the reconstruction of perfectly conducting cracks with dirichlet boundary condition . accordingly , the extension of this research to the crack with neumann boundary condition will be an interesting work . we believe that applying higher - 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this paper concerns a fast , one - step iterative technique of imaging extended perfectly conducting cracks with dirichlet boundary condition . in order to reconstruct the shape of cracks from scattered field data measured at the boundary 
 , we introduce a topological derivative - based electromagnetic imaging function operated at several nonzero frequencies . 
 the properties of the imaging function are carefully analyzed for the configurations of both symmetric and non - symmetric incident field directions . 
 this analysis explains why the application of incident fields with symmetric direction operated at multiple frequencies guarantees a successful reconstruction . 
 various numerical simulations with noise - corrupted data are conducted to assess the performance , effectiveness , robustness , and limitations of the proposed technique .    perfectly conducting cracks , topological derivative , multiple frequencies , numerical experiments