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645aa8085510ba0a5af47569d0a894c09dccca8a | subsection | 4 | 40 | Preliminaries | In this section, we provide a setting for later calculations. We start with a description of the bi-Killing structure of the bivector \Theta in the open-closed string map (). We recall that we are considering generic spacetime metrics G_{\mu \nu } with an isometry group. From the Killing vectors K_i, one can construct ... | {
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576dd8e733ee6796b50cb650c70b5a79e52ce0ca | subsection | 5 | 40 | Preliminaries | First we consider\Theta ^{\alpha \rho } \nabla _{\rho } \Theta ^{\beta \gamma } = r^{ij} r^{kl} \left( K_{i}^{\alpha } K_{l}^{\gamma } K^{\rho }_{j} \nabla _{\rho } K_k^{\beta } + K_{i}^{\alpha } K_k^{\beta } K^{\rho }_j \nabla _{\rho } K_{l}^{\gamma } \right),2.6before antisymmetrising,2.7[ ]
= Ki Kj Kk ( cl1 l2 i r... | {
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"raw": "J. i. Sakamoto, Y. Sakatani and K. Yoshida, “Homogeneous Yang-Baxter deformations as generalized diffeomorphisms,” J. Phys. A 50, no. 41, 415401 (2017) [arXiv:1705.07116 [hep-th]].",
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75ebeaa3564936d62b1fceafbe19bb8edee0b6bf | subsection | 6 | 40 | Preliminaries | Of course, it is more careful to state that the CYBE implies the Jacobi identity since there may be solutions to the Jacobi identity that are not bi-Killing.Moving on, we will now address the relation between the Killing vector I of generalized supergravity and the bivector \Theta (). In it was checked that this relati... | {
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a42e4e973dc07fbd7a8d71a2e2be4d6ffe02820b | subsection | 7 | 40 | Perturbative analysis | In this section, we will extract the CYBE from the EOMs of generalized supergravity. As stated earlier, we restrict our attention to the NS sector on the basis that repeating the calculations for the RR sector will not offer new insights. Indeed, since we are working perturbatively, yet ultimately interested in exact s... | {
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9978b80fcbd323da7ef826f71ce293183108de4d | subsection | 8 | 40 | Review of generalized supergravity | Let us begin by recalling the EOMs of generalized supergravity ,3.112 H = X H + X - X,112H2 = 2 X X- X,R = 14 H H - X - X,
where \hat{\nabla } and \hat{R}_{\mu \nu } denote the covariant derivative and curvature of the deformed solution g_{\mu \nu }, we have used the trace of the Einstein equation to eliminate \hat{R}... | {
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813987fa39d23d06403c866a0f6fd96f0cd4fbbc | subsection | 9 | 40 | Check of consistency of the EOMs. | Regardless of their \sigma -model roots, one can ask if the generalized supergravity EOMs provide a consistent set of differential equations. For the set of equations (REF ), (REF ), (REF ), this amounts to checking if the Bianchi identity \hat{\nabla }^\mu (\hat{R}_{\mu \nu }-\frac{1}{2} \hat{R} g_{\mu \nu })=0 holds ... | {
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ab04c7002da93f6466bf8c4eed7a6ec2e2980975 | subsection | 10 | 40 | Body | What we will do in this section is solve the EOMs by a perturbative expansion in powers of \Theta around a given solution at \Theta =0. This latter is given by background metric G_{\mu \nu } and dilaton \Phi . We start by expanding ()3.3g = G + +O(4),B = - - +O(5),= + 14 + O(4),where all indices are raised and lowered ... | {
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b3178e8a6963e8861cccc56bb24c01515b5f2931 | subsection | 11 | 40 | Body | Further, using the following identities:\partial ^\rho \phi \nabla _\rho \Theta _{\alpha \beta }=\Theta _{\alpha \rho }\nabla _{\beta }\nabla _{\rho }\phi +\Theta _{\rho \beta }\nabla _{\alpha }\nabla _{\rho }\phi ,\qquad \partial ^\rho \phi \nabla _\mu \Theta _{\alpha \rho }=-\Theta _{\alpha \rho }\nabla ^\mu \nabla _... | {
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1a829996f91262e14a621ecb6658f380a9c83512 | subsection | 12 | 40 | Zeroth order: | At this order the B-field equation is trivial and the other two equations read asR_{\mu \nu }+2\nabla _\mu \nabla _\nu \Phi =0,\qquad \nabla ^2\Phi -2(\nabla \Phi )^2=0,3.4where the curvature is computed using background metric G_{\mu \nu }. | {
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a89998fefb6846090a620936a5599e84d2b2aff6 | subsection | 13 | 40 | First order: | At first order the dilaton and Einstein equations (REF ) and (REF ), respectively yieldI^\mu \nabla _\mu \Phi =0,\qquad \nabla _\mu I_\nu +\nabla _\nu I_\mu =03.5which just confirm I as a Killing vector for the background solution, specified by G_{\mu \nu }, \Phi .The NSNS two-form EOM (REF ) using the bi-Killing struc... | {
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8485cf6432d27cd9e87042a24bfc05c560613aa1 | subsection | 14 | 40 | Second order: | The NSNS two-form EOM, once we use the first order results, takes a very simple form:\mathcal {L}_{I} \Theta = \textrm {d}i_{I} \Theta + i_{I} \textrm {d}\Theta = 0,3.7which essentially tells us that I is not only a Killing vector of the original geometry but also remains Killing in the deformed geometry.We next consid... | {
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fedfea55853bf408c25e602b5fab97be1986889c | subsection | 15 | 40 | Third order: | To work out equations at the third order, we recall (REF ) and that g_{\mu \nu } and \phi have even powers of \Theta while the NSNS two-form has odd powers and hence X has all powers from zero to three. Therefore, only the X-terms in the dilaton and Einstein equations contribute to third order. One may show that these ... | {
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2ece30051ecb709049047390986ca4939f6f61d6 | subsection | 16 | 40 | Higher orders: | From (REF ) one can readily see the following structure: For even powers of \Theta the NSNS two-form EOM is satisfied trivially if I is a Killing vector, while the dilaton and Einstein equations are non-trivial. Conversely, for odd powers of \Theta , the dilaton and Einstein equations are readily satisfied once we assu... | {
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185db03414cc6f5b7cdbe73009ea1047a1b73d65 | subsection | 17 | 40 | Examples | In this section, we provide examples of generalised Yang-Baxter deformations in a bid to get the reader better acquainted with the solution generating technique outlined in . We focus on two examples that fall outside the usual examples studied via the Yang-Baxter \sigma -model, before presenting a more familiar exampl... | {
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10283b20b6e5395d51188aff2e5089daf8504ac6 | subsection | 18 | 40 | Flat spacetime | Let us consider flat spacetime in three dimensions 3D. One may imagine that this is trivial compared to deformations of AdS spacetimes, but it turns out that generalising the Yang-Baxter \sigma -model to flat spacetime is complicated by the fact that the bilinear of the coset Poincaré group is degenerate . As a result,... | {
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5409967b51376dde66c042480a5fe4b3be8dd94f | subsection | 19 | 40 | Flat spacetime | One can determine the corresponding Killing vector from (),I = \alpha ( x \partial _y - y \partial _x ) - \beta (t \partial _x + x \partial _t ) - \gamma ( y \partial _t + t \partial _y ).4.8At this stage, one generates a deformed supergravity solution from () and (),4.9g dx dx = 1[1 + 2(t2 - x2 - y2)] [ - dt2 + dx2 + ... | {
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9011bb4cfaab6d7cbc08fb254736e59239abbfcd | subsection | 20 | 40 | Bianchi III | The previous example involved a deformation of flat spacetime. In a bid to consider spacetimes that are not Ricci-flat, let us consider the following Bianchi III spacetime,4.11ds2 = - a12 a22 a32 e-4 dt2 + a12 12 + a22 22 + a32 32,= t,
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dba4a71d68ac3a3e8ef86388cd90c27403b3adae | subsection | 21 | 40 | Bianchi III | It should be noted that both of these Killing vectors commute and the r-matrix is Abelian, so it is a TsT transformation .On the other hand, setting \alpha = 0, we encounter the geometry4.21g dx dx = - a12 a22 a32 e-4 t dt2 + a22 dy2 + 1[ 1+ e2 x 2 a12 a32] [ a12 dx2 + a32 e2 x dz2 ],B = e2 x a12 a32[ 1+ e2 x 2 a12 a32... | {
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33a107d124c80bd21dee7ec465cd1467f5fca1c2 | subsection | 22 | 40 | Lunin-Maldacena-Frolov | As promised we give one example of a geometry with an RR sector simply to illustrate the utility of the methods outlined in . While it is easy to consider a new example, and we invite readers to do so, this risks distracting the reader from our main message. For this reason, we find it instructive to study an example f... | {
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ef0f3154ec38c5618d3e20a85c367865aa1b7bad | subsection | 23 | 40 | Lunin-Maldacena-Frolov | Inverting this matrix, while redefining \hat{\gamma }_i = R^2 \gamma _i, we get the following metric and NSNS two-form:4.25ds2 = R2 [ ds2 (AdS5) + i=13 ( dri2 + G ri2 di2) + G r12 r22 r32 ( i=13 i di )2 ],B = - R2 G ( 3 r12 r22 d1 d2 + 1 r22 r32 d2 d3 + 2 r32 r12 d3 d1) ,
where we have defined4.26G-1 = 1 + 32 r12 r22 +... | {
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ab521704a0d4d2ab43a0ba49ada099921ac37d5b | subsection | 24 | 40 | Lunin-Maldacena-Frolov | As a result, we have\tilde{F}_{3 \, \rho _1 \rho _2 \rho _3} = Q_{3 \, \rho _1 \rho _2 \rho _3} = \frac{1}{2!} \Theta ^{\mu \nu } Q_{5 \, \mu \nu \rho _1 \rho _2 \rho _3} .4.30Following this procedure, we get\tilde{F}_3 = 4 R^2 \sin ^3 \alpha \cos \alpha \sin \theta \cos \theta \textrm {d}\alpha \wedge \textrm {d}\thet... | {
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9e3ee0e74e16626331e8da4c4f84786b2fb7b2c5 | subsection | 25 | 40 | Discussion | In this work, we focused on the generalized supergravity EOMs and analysed what they imply on solutions obtained from deformations generated through the open-closed string map, and in this way, substantiated the claims of our earlier letter . Assuming the bivector \Theta to be a generic linear combination of anti-symme... | {
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09833610918c1962e4d3a1f0830c375fc4a34d93 | subsection | 26 | 40 | Discussion | Nevertheless, one can consider the more general framework of O(d,d) and \beta -transformations (also ), which include both non-Abelian T-duality and Yang-Baxter deformations as special cases We thank Y. Sakatani and J. Sakamoto for correspondence on this issue.. | {
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... | 10.1007/JHEP06(2018)161 | 1803.07498 | Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT) | [
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f32c44ee323cece462d6feb92863e6da42e035e5 | subsection | 27 | 40 | Consistency of the generalized supergravity field equations | To check the consistency of equations (REF -REF ), we first rewrite the Einstein equation (REF ) as follows\hat{R}_{\mu \nu }-\frac{1}{2}g_{\mu \nu }\hat{R}=\frac{1}{4} (H_{\mu \alpha \beta }H_\nu ^{\;\;\alpha \beta }-\frac{1}{2} H^2 g_{\mu \nu })-(\hat{\nabla }_\mu X_\nu +\hat{\nabla }_\nu X_\mu -g_{\mu \nu } \hat{\na... | {
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} | 10.1007/JHEP06(2018)161 | 1803.07498 | Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT) | [
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321c3e03486b9e51406da59bb3a364cb82d9a2f5 | subsection | 28 | 40 | Consistency of the generalized supergravity field equations | From () we can argue that f_{\mu \nu } should contain some term proportional to H. Therefore we write \partial _\mu \lambda _\nu -\partial _\nu \lambda _\mu =z^\alpha H_{\alpha \mu \nu }, where z is an H-independent vector field. Replacing this ansatz in () , we find6.142 I-I- H z- zH -12H I+12H I+ 12 zH H -2XI+2XI+ 2X... | {
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} | 10.1007/JHEP06(2018)161 | 1803.07498 | Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT) | [
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02cfafe540720518743dc97ccd3fcc8e33c233ed | subsection | 29 | 40 | Details of the perturbative analysis | In this section, we provide some details of the results quoted in the text. | {
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1c620c1e093d4ec169401631723406010b33a1f8 | subsection | 30 | 40 | Some useful Killing identities. | In our perturbative analysis we have heavily used Killing vectors and their properties. So we start with some useful identities. Given a set of Killing vectors K_i^\mu ,\nabla _{\mu }K_{i \, \nu }+\nabla _{\nu }K_{i \, \mu }=0,7.1there is a well-known identity,\nabla _{\alpha } \nabla _{\beta } K^{\gamma } = R^{\gamma ... | {
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e017470013d477678ffa89e87b5d153f362e84cd | subsection | 31 | 40 | Perturbative expansion. | Expanding the metric g_{\mu \nu }, B-field and dilaton for small \Theta , we get7.4g=G++O(4),B=-- +O(5),=+14 +O(4).
Plugging the above expressions directly into the EOMs, we can also expand them for small \Theta . We now detail the information extracted at each order from the EOMs. | {
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} | 10.1007/JHEP06(2018)161 | 1803.07498 | Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT) | [
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a23b59246e5d3b388bd428552127f53db22b171d | subsection | 32 | 40 | Zeroth Order | At zeroth order in \Theta , equations (REF -REF ) become7.5X-X=0 - =0R +X+X=0 R+2=0X -2 X2=0 2-2=0
where we remark that the first equation is trivial, whereas the second and third are simply the EOMs satisfied by the original undeformed solution, in line with expectations. | {
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4001ac8645dfd933347572816467932ad9d93b0f | subsection | 33 | 40 | First Order | At first order the linear terms in \Theta give the following contribution to the EOMs,7.612H-() H-I+I=0,I+I=0,I-4 I=0 I=0.
Now we assume that \Theta is bi-Killing () and use the identities ()\nabla \cdot K=0,\qquad K\cdot \nabla \Psi =0,7.7where the latter is valid for any field \Psi . This allows us to write H= ... | {
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8357447ae6387e91f7dfe0ad110da8ef77f5b461 | subsection | 34 | 40 | Second Order | Before trying to expand and solve the second order equations, it would be useful to simplify the EOMs using what we have found from the zeroth and the first order equations. Using the fact that I^\mu is a Killing vector we find{\mathcal {L}}_I\phi =I^\mu \partial _\mu \Phi =0,7.12{\mathcal {L}}_Ig=\hat{\nabla }_\mu I_\... | {
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3522d97db59add360a53f691df284db9243d819e | subsection | 35 | 40 | The Einstein equations | The quadratic term in the Einstein equation can be organised as follows,D_{\mu \nu }-\frac{1}{4}H_{\mu \alpha \beta }H_{\nu }^{\;\;\alpha \beta }+\nabla _\mu (\Theta _{\nu \rho }I^\rho )+\nabla _\nu (\Theta _{\mu \rho }I^\rho ) +2 C^\alpha _{\;\;\mu \nu } \partial _\alpha \Phi =0,7.20where D denotes the second order te... | {
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f21c6e228bde9394db40db0a6fc4713049a01091 | subsection | 36 | 40 | Dilaton equation | Having discussed the more involved Einstein equation, we focus on the dilaton equation at second order. It takes the form,\frac{1}{12}H^2+\nabla ^\mu (\Theta _{\mu \nu } I^\nu )+C^\mu _{\;\;\;\mu \rho }\partial ^\rho \Phi -4\partial ^\mu \Phi \;\Theta _{\mu \nu }I^\nu -2I_\mu I^\mu =0.7.28The second and the last term c... | {
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15bde93fff5188320e4efd0a68057b8c9d167d4a | subsection | 37 | 40 | Third order equations | To test that nothing funny happens at the higher order, we study the EOMs to third order in \Theta . We will see in this order that the dilaton and Einstein equation are somewhat trivial, while the B-field EOM encapsulates information of the CYBE. | {
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933fa3f3a73300d054c604b619ea549093d5eef6 | subsection | 38 | 40 | Dilaton equations at third order | In order to expand the EOMs, it is useful to note thatX_\mu =\partial _\mu \phi + g_{\mu \nu }I^\nu -B_{\mu \nu }I^\nu = \partial _\mu \phi +\nabla ^\alpha \Theta _{\alpha \mu } +\Theta _{\mu \nu }\nabla _\alpha \Theta ^{\alpha \nu } +\Theta ^2_{\mu \nu } \nabla _\alpha \Theta ^{\alpha \nu }+ {\mathcal {O}}(\Theta ^4).... | {
"cite_spans": []
} | 10.1007/JHEP06(2018)161 | 1803.07498 | Yang-Baxter Deformations Beyond Coset Spaces (a slick way to do TsT) | [
"I. Bakhmatov",
"E. Ó Colgáin",
"M. M. Sheikh-Jabbari",
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3d03d369c42b568b937358348175f3730bdd899a | subsection | 39 | 40 | Einstein equation at third order | At cubic order, the Einstein equation turns out to be the following:\nabla _\mu (\Theta _{\nu \beta }^2\nabla _\alpha \Theta ^{\alpha \beta })-C^\alpha _{\mu \nu }\nabla ^\beta \Theta _{\beta \alpha } + (\mu \leftrightarrow \nu ) =0.7.42Using () and noting again that I is Killing, we find that above equation is also tr... | {
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e095d5db7c2449ae29a979c846d608871d436ecb | abstract | 0 | 98 | Abstract | Nowadays, the field computed tomography (CT) encompasses a large variety of
settings, ranging from nanoscale to meter-sized objects imaged by different
kinds of radiation in various acquisition modes. This experimental diversity
challenges the flexibility of tomographic reconstruction methods. Kaczmarz-type
methods, wh... | {
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} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
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9c9f00b49aa5be9d81c842c77488909c623da1a5 | subsection | 1 | 98 | Introduction | Since the pioneering works of Cormack , and Hounsfield , the field of computed tomography (CT) has broadened considerably. While classical CT based on the partial attenuation of X-rays by matter continues to be a principal workhorse of medical diagnosis, several other applications have emerged over the past decades: fo... | {
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6abcaa54b5095a5827db1d7b7cc99bbdfbb9eb73 | subsection | 2 | 98 | Introduction | Moreover, Kaczmarz-methods are often observed to exhibit fast semi-convergence , , arriving at accurate reconstructions already after \mathcal {O}(1) fitting-cycles over the tomographic data set.
If the iterations are sufficiently cheap to compute, this enables image-recovery at overall computational costs comparable t... | {
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99c1aeab7f9c65f3f3b2c1a921b75d1ba6dcf37c | subsection | 3 | 98 | Tomographic imaging model | We consider general tomographic inverse problems, for which the dependence of the data g_{{\textup {tot}}} from the sought object f can be modeled asg_{{\textup {tot}}} = \begin{pmatrix}
g_{1} \\
\vdots \\
g_{{N_{\textup {proj}}}}
\end{pmatrix}
= \begin{pmatrix}
F_1(P_1 (f)) \\
\vdots \\
F_{{N_{\textup {proj}}}}(P_{{N_... | {
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} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
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ebf94f1c2fc811fe832d61cb233e68175d28c50a | subsection | 4 | 98 | Tomographic imaging model | Bottom: cone-beam setup with absorption-contrast as in conventional CT.]In a parallel-beam tomography setup, each P_j = {P}_{{\theta }_{(j)}} maps f onto its line integrals along a certain incident direction {\theta }\in \mathbb {S}^2 := \lbrace {x}\in \mathbb {R}^3: |{x}| = 1\rbrace :{P}_{{\theta }} (f)( {x}_\perp ) :... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
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dae5cf807056adb65a6790795e1c83ae9ff897a1 | subsection | 5 | 98 | Inverse problem and a priori constraints | This work is concerned with
reconstructing the an unknown object f from tomographic data g_{{\textup {tot}}} modeled by (REF ), i.e. in solving the following inverse problem:[Tomographic reconstruction]
For a given domain \Omega \subset \mathbb {R}^3, parallel- or cone-beam projectors P_{1}, \ldots , P_{{N_{\textup {pr... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
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49bbb65d80c9edff12f72997f3424b69c22db541 | subsection | 6 | 98 | Variational methods: | The widely used filtered back-projection- (FBP) and Feldkamp-Davis-Kress (FDK) reconstruction algorithms often lack flexibility to accurately account for specific tomographic settings and available a priori knowledge.
As a remedy, variational methods have been proposed.
The idea is to minimize a generalized Tikhonov fu... | {
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319bcd0c71f010ede55a551ad6d427e2b05e895f | subsection | 7 | 98 | Kaczmarz-type methods: | One approach to decrease the number of expensive evaluations of P_{{\textup {tot}}} and P^\ast _{{\textup {tot}}} compared to (bulk) variational methods is to exploit the block-structure of Inverse Problem REF by performing cyclic iterations on the sub-problems g_{j}^{\textup {obs}}= F_j(P_j(f)) + {\epsilon }_j. We con... | {
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68ee6594f2f0472ed3c97a772a0545865c257620 | subsection | 8 | 98 | Convergence of Kaczmarz-iterations and relation to Tikhonov regularization | Classical Kaczmarz-iterations of the form (REF ) are known to exhibit fast semi-convergence, typically yielding a regularized reconstruction within \mathcal {O}(1) cycles while increasingly amplifying data-noise if more iterations are preformed, see and references therein. Recently, Kindermann and Leitão obtained a muc... | {
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... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
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799e253acb57954bab3b543d89bbfe3592425006 | subsection | 9 | 98 | Convergence of Kaczmarz-iterations and relation to Tikhonov regularization | Then, after a symmetric Kaczmarz-cycle,f_{k+1} = {\left\lbrace \begin{array}{ll}
\operatornamewithlimits{argmin}_{f \in X} \Vert A_{k+1} f - g^{\textup {obs}}_{k+1} \Vert _{Y_j}^2 + \alpha \Vert f - f_{k} \Vert _X^2 &\text{for } k < N \\
\operatornamewithlimits{argmin}_{f \in X} \Vert A_{2N-k} f - g^{\textup {obs}}_{2N... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
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8ae28ac56d0a06e34515fbd25706a85c18fa68b6 | subsection | 10 | 98 | Convergence of Kaczmarz-iterations and relation to Tikhonov regularization | On the contrary,
for sufficiently large \alpha ,
thm:KindermannLeitao shows that regularized Kaczmarz-iterations may be used to approximate Tikhonov-minimizers, i.e. to emulate bulk variational methods – at least in the considered quadratic setting.
Other convergence results pointing in a similar direction are given in... | {
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"Simon Maretzke"
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40bfe282d6693360788e8f976e8d19b859274c6b | subsection | 11 | 98 | Contribution | In order for Kaczmarz-methods to provide a truly efficient alternative to bulk variational approaches, essentially two conditions have to be satisfied:The iterates f_k arrive at a reasonable reconstruction after few cycles (ideally one).
The individual Kaczmarz-iterations may be evaluated at low computational costs.T... | {
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} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
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31ce9aeccd70d64f26dc93550461728bd4cad729 | subsection | 12 | 98 | Preparations: notation and analysis of the projectors | In order to analyze the Kaczmarz-iterations (REF ), we need some properties of the projection operators P\in \lbrace {P}_{{\theta }}, {D}_{{s}}\rbrace for a single incident direction {\theta } or source position {s}, see §REF . First of all, we note that different values of {\theta } and {s} merely correspond to a rota... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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0bb663d74d6e0a7bd1bab15c5e035d3afebcfa07 | subsection | 13 | 98 | Preparations: notation and analysis of the projectors | To enable a unified treatment of parallel- and cone-beam settings, we set \mathbb {D}:= \mathbb {R}^2 if P= {P} and \mathbb {D}:= \mathbb {S}^2 if P= {D}.We define the ray-density functions w_{P} and (weighted) unit-projections u_P, {\tilde{u}_P}:w_{P}: \Omega \rightarrow \mathbb {R}_{>0}; \; {x}\mapsto {\left\lbrace \... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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579716066bc5d149eb4c8522a329b0b24365a963 | subsection | 14 | 98 | Back-projections: | We introduce the back-projections corresponding to {P} and {D}:{P}^{\textup {B}}(p)({x}_\perp , z) &:= {\left\lbrace \begin{array}{ll}
p({x}_\perp ) &\textup {if } ({x}_\perp , z) \in \Omega \\
0 &\textup {else}
\end{array}\right.}, \qquad {D}^{\textup {B}}(p)({x}) := {\left\lbrace \begin{array}{ll}
p({x}/|{x}|) &\text... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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bcbf12ad9613c2ef07e70d7156cebe4bfd226e94 | subsection | 15 | 98 | Spaces: | We study the maps {P}, {D} as operators between spaces of square-integrable functions L^2(\Omega ) := \lbrace f \in L^2(\mathbb {R}^3): \operatornamewithlimits{supp}(f) \subset \Omega \rbrace and {L^2({\mathbb {D}\!_{P}})}:= \lbrace p \in L^2(\mathbb {D}): \operatornamewithlimits{supp}(p) \subset {\mathbb {D}\!_{P}}\rb... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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41675d315a17efd70ac0af2591b8a06cd2d1042a | subsection | 16 | 98 | Adjoints: | For P\in \lbrace {P}, {D}\rbrace , it can be shown that P: L^2(\Omega ) \rightarrow {L^2({\mathbb {D}\!_{P}})} is a bounded linear operator. The adjoints
are given by (weighted) back-projections (see e.g. , ):P^\ast : {L^2({\mathbb {D}\!_{P}})}\rightarrow L^2(\Omega ); \; P^\ast (p) = w_{P} \cdot P^{\textup {B}}(p) \;\... | {
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3586b6b002a8b29a61908320439dd5fdd9bd027e | subsection | 17 | 98 | Geometric characterization: | In the analysis, we will need to consider weighted projectors:P_{\textup {iso}}: f \mapsto {\tilde{u}_P}^{-1/2} \cdot P(f) \;\;\;\;\;\text{for}\;\;\;\;\; P\in \lbrace {P}, {D}\rbrace .Note that the expression is well-defined by conv:ProjOps since {\tilde{u}_P}(x) > 0 for all x \in {\mathbb {D}\!_{P}}.
At the first glan... | {
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"Simon Maretzke"
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e3e721d9847b26409031d6f82d0a36a294e5d028 | subsection | 18 | 98 | Body | As a motivation for the general result presented in the subsequent section, we consider an example of Kaczmarz-iterations, that turn out to be computable via a simple analytical formula. Let the iterates be defined byf_{k+1} \in \operatornamewithlimits{argmin}_{f \in L^2(\Omega )} \Vert P_{j_k} ( f) - g^{\textup {obs}}... | {
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"Simon Maretzke"
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b53aaa92e5f7c368708f5219fcbea5595453918b | subsection | 19 | 98 | Body | An extension to the cone-beam case may be possible.[Admissibility of L^q-penalties]
Let {\tilde{P}}= {P}_{\textup {iso}}: L^2(\Omega ) \rightarrow {L^2({\mathbb {D}\!_{P}})} and let \mathcal {R}: L^2(\Omega ) \rightarrow \mathbb {R}\cup \lbrace \infty \rbrace be defined by (REF ). Then A1 is satisfied and\mathcal {R}(... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
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89f7002e3071958d2f636d84e8cf68f9d9c0be2c | subsection | 20 | 98 | Relation to SART: | By exchanging the continuous object density f_k, projection data g^{\textup {obs}}_{j_k}, projector P_{j_k} and unit-projection {\tilde{u}_{j_k}} in (REF ) with suitable discretizations {P}_{j_k} \in \mathbb {R}^{m\times n}, {f}_k \in \mathbb {R}^n, {\tilde{u}}_{j_k}, {g}_{j_k}^{\textup {obs}}\in \mathbb {R}^m, a numer... | {
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"raw": "A. H. Andersen and A. C. Kak, Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm, Ult... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
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4eb5e37b770a591af6d8422565edee9e1eaee277 | subsection | 21 | 98 | Relation to SART: | Since (REF ) has been derived as a discretization of (REF ), SART (REF ) may thus be interpreted as a formula to compute the classical Kaczmarz-iterations in (REF ).
Conversely, (REF ) can be seen as an L^2-regularized SART-variant. | {
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} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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587fcb17b1c8ab44da3c21641aefb0641bb08303 | subsection | 22 | 98 | The SART-Scheme: | Analogously to classical SART, the update-formula (REF ) computes the Kaczmarz-iterate in (REF ) via a highly efficient non-iterative scheme, that requires only a single evaluation of {P}_{j_k} and {P}_{j_k}^\ast each:[L^2-regularized SART]{(1)}
Forward-project the current iterate: {p}_k = {P}_{j_k} ({f}_k )
{(2)}
Com... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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b817fbf7485155ce1dea34bb3f1e52d9195e366a | subsection | 23 | 98 | Generalized SART framework | In §REF , it has been shown that L^2-Kaczmarz-iterations (REF ) can be evaluated in a simple and efficient manner although they involve a seemingly complicated optimization problem. Motivated by this result, we explore in how far more general Kaczmarz-iterations permit an efficient computation analogous to the SART-lik... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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739b044cf3e1c23c241365a13b4672b6179f767e | subsection | 24 | 98 | Generalized SART framework | Assume that there exists an f_{\textup {ref}} \in X such that\mathcal {R}(f_{\textup {ref}} + {\tilde{P}}^\ast (p) + f_0) \ge \mathcal {R}(f_{\textup {ref}} + {\tilde{P}}^\ast (p) ) \;\;\;\;\;\text{for all}\;\;\;\;\; p \in Y, \, f_0 \in \operatornamewithlimits{kern}({\tilde{P}}).
\qquad \mathrm {(A)}A1 ensures that th... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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eeb03112e430d45e5ff187bfd913e22bd1ca01b4 | subsection | 25 | 98 | Generalized SART framework | Now define \tilde{f}_{\textup {new}} := f_{\textup {ref}} + {\tilde{P}}^\ast (\Delta p). Then (REF ) implies that\mathcal {R}(\tilde{f}_{\textup {new}}) = \mathcal {R}(f_{\textup {ref}} + {\tilde{P}}^\ast (p) ) \le \mathcal {R}(f_{\textup {ref}} + {\tilde{P}}^\ast (p) + f_0) = \mathcal {R}( f_{\textup {new}}).Moreover,... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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ea9e4725e1363edd525ade3f671eb0ead3804fdd | subsection | 26 | 98 | Generalized SART framework | Then \tilde{f}_{\textup {new}} minimizes the cost-functional \mathcal {C}(f):= \tilde{\mathcal {S}}( {\tilde{P}}(f) ) + \mathcal {R}( f ) over all f \in A := f_{\textup {ref}} + \operatornamewithlimits{range}({\tilde{P}}^\ast ). Since A1 implies that
\mathcal {C} (\tilde{f}_{\textup {new}} + f_0) \ge \mathcal {C}(\tild... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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8b729351ccf37a07df652b0473d245a9bfc4a0c4 | subsection | 27 | 98 | Admissible penalty functionals | The aim of this section is to identify penalty functionals \mathcal {R}= \mathcal {R}_k that satisfy A1, in which case the Kaczmarz-iterations in (REF ) can be computed via GenSART-schemes by virtue of thm:genSART. | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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feae179bf94c0c64e21eea336fb1163381548077 | subsection | 28 | 98 | Preliminary insights | In order to gain an intuition for the admissible penalties, let us first discuss the meaning of the condition (REF ). It asserts that – relative to a certain reference object f_{\textup {ref}} – any deviation by an element from the null-space \operatornamewithlimits{kern}({\tilde{P}}) is penalized or at least does not ... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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c1bb1df024478c02d3080d40907ba8e2e513856e | subsection | 29 | 98 | Preliminary insights | Then \mathcal {R}:= \alpha _1 \mathcal {R}_1 + \alpha _2 \mathcal {R}_2 satisfies (REF ).This follows by summing scaled versions of (REF ) for \mathcal {R}_1 and \mathcal {R}_2. | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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638eccd990f61f46123dbaaa5ed4284639bffc8f | subsection | 30 | 98 | (Weighted) | As a simple candidate for admissible penalty functionals \mathcal {R} within the framework of thm:genSART, we consider quadratic penalties of the form\mathcal {R}(f):= \Vert f - f_{\textup {ref}} \Vert _{X}^2,where \Vert \cdot \Vert _X denotes the norm of the Hilbert space X. Owing to the geometric nature of the condit... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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9633dc8745cb89774a4a118da64e1a6a4e409670 | subsection | 31 | 98 | (Weighted) | The equality in the third line simply follows by the defining property of the adjoint.The abstract result from lem:admissibleQuadratic can be applied to establish GenSART-schemes for general Kaczmarz-iterations with L^2-penalty:[Generalized SART with L^2-penalty]
Let P\in \lbrace {P}, {D}\rbrace : L^2(\Omega ) \righta... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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8c86a04b7017d0648c6ec547498674c946a6f24d | subsection | 32 | 98 | (Weighted) | Moreover, let \lambda : \Omega \rightarrow \mathbb {R} with \lambda _{\min } \le |\lambda ({x})| \le \lambda _{\max } for almost all {x}\in \Omega and some constants 0 < \lambda _{\min } \le \lambda _{\max } < \infty . Then the minimizers off_{\textup {new}} \in \operatornamewithlimits{argmin}_{ f \in L^2(\Omega ) } \m... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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6f249ea0e3f582d3a3ef01b2b9a2b0a8da57b26c | subsection | 33 | 98 | (Weighted) | By setting \tilde{\mathcal {S}}(p) := \mathcal {S}({\tilde{u}_P}^{1/2} \cdot p) and \mathcal {R}(f) := \alpha \Vert f - f_{\textup {ref}} \Vert _{L^2}^2, (REF ) is cast to the form (REF ).According to lem:admissibleQuadratic,lem:ConvCombPenalties, A1 is satisfied in the strict-inequality-version so that the GenSART-thm... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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9a2b1a62ee8783b71a8251f54cbff0c30a78d126 | subsection | 34 | 98 | (Weighted) | Substituting the expressions for \tilde{\mathcal {S}}, {\tilde{P}}, \mathcal {R} and exploiting that \mathcal {R}(f_{\textup {ref}} + {\tilde{P}}^\ast (p)) = \langle p, {\tilde{P}}{\tilde{P}}^\ast (p)\rangle _Y according to (REF ) yields\tilde{p}_{\textup {ref}} &= {\tilde{P}}(f_{\textup {ref}} ) = {\tilde{u}_P}^{-1/2}... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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9898823134def53eeeaf80834f22364c7d09aa65 | subsection | 35 | 98 | (Weighted) | Moreover, the theorem enables GenSART for weighted L^2-penalties:[Generalized SART with weighted L^2-penalties]
Let P\in \lbrace {P}, {D}\rbrace : L^2(\Omega ) \rightarrow {L^2({\mathbb {D}\!_{P}})}, f_{\textup {ref}} \in L^2(\Omega ), \alpha > 0 and let \mathcal {S}: {L^2({\mathbb {D}\!_{P}})}\rightarrow \mathbb {R}\... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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5effafd6dd23a2fe24c63053f14cbb4e52cdc67c | subsection | 36 | 98 | (Weighted) | Moreover, note that the assumption that \lambda or w are bounded from below can be dropped at the cost of a more technical proof. Indeed, the formulas (REF ), (REF ) still make sense if \lambda or w vanishes in parts of \Omega .For completeness, we mention the case of (weighted) L^2-data fidelities \mathcal {S}\big ( g... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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598ea84a30b7ff1ed7eefb57cb5181116204221d | subsection | 37 | 98 | Gradient-penalties | In order to enforce a certain smoothness of the reconstructed object, variational methods often use penalties that involve derivatives. Similarly, we seek to extend the above results to Kaczmarz-iterations (REF ) with quadratic gradient penalties:\mathcal {R}: L^2(\Omega ) \rightarrow \mathbb {R}\cup \lbrace \infty \rb... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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2757922a01ec3bc33e9f9f313ba99c2fac5dd79c | subsection | 38 | 98 | Gradient-penalties | Then we have that\mathcal {R}(f_{\textup {ref}} + {\tilde{P}}^\ast (p) + f_0 ) &= \int _{\Omega } | \nabla ( {P}_{\textup {iso}}^\ast (p) + f_0 ) |^2 \; \\&= \Vert \nabla {P}_{\textup {iso}}^\ast (p) \Vert _{L^2}^2 + \Vert \nabla f_0 \Vert _{L^2}^2 + 2 \int _{\Omega } \nabla {P}_{\textup {iso}}^\ast (p) \cdot { \nabla ... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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da06548aacbd5ef8b24bab920a6a853cbf8b83d8 | subsection | 39 | 98 | Gradient-penalties | Since a and b are continuously differentiable within the open set U := \lbrace {x}_\perp \in \mathbb {R}^2 : a({x}_\perp ) < b({x}_\perp )\rbrace by the assumptions on \Omega , we can apply Leibniz' rule to the inner integrals in (REF ):\int _{a({x}_\perp )}^{b({x}_\perp )} &\nabla _\perp f_0({x}_\perp , z) \; z̥ = \na... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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3c3eefe30d31d52be4988f6cbafecc30fdc2168a | subsection | 40 | 98 | Gradient-penalties | Denote by \nabla _{\mathbb {D}} the gradient on the detection domain \mathbb {D}\in \lbrace \mathbb {R}^2 , \mathbb {S}^2\rbrace and by \nabla \!_P the component of gradient in \mathbb {R}^3 perpendicular to the local ray-direction of the projector P. Then it holds thatP(w_{P}^{-1/2} \cdot \nabla \!_Pf) &= \nabla _{\ma... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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e88a474338228c51965dafd4df596fb640ee9167 | subsection | 41 | 98 | Gradient-penalties | Let X := ( L^2(\Omega ), \langle \cdot , \cdot \rangle _P) be equipped with the inner product \langle f_1, f_2\rangle _P:= \langle w_P\cdot f_1, f_2\rangle _{L^2}.
and define {\tilde{P}}: X \rightarrow {L^2({\mathbb {D}\!_{P}})}; \, f \mapsto u_P^{- 1 /2 } \cdot P\left(f\right). Then A1 is satisfied if we restrict to e... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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ca79069d936d2ff2f669c736f6f43281859c78b8 | subsection | 42 | 98 | Gradient-penalties | Hence, we have for all p \in {L^2({\mathbb {D}\!_{P}})}{\tilde{P}}^\ast ( p ) = \iota _X^\ast \big ( P^\ast \big ( u_P^{-1/2} \cdot p \big ) \big ) = w_{P}^{-1} \cdot P^\ast \big ( u_P^{-1/2} \cdot p \big ) \stackrel{(\ref {eq:ProjAdj})}{=} P^{\textup {B}}\big ( u_P^{-1/2} \cdot p \big ).Now let p\in {L^2({\mathbb {D}\... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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d510678263ba64641b12bbe3916f7e4fc48c7734 | subsection | 43 | 98 | Gradient-penalties | However, this is a simple consequence of lem:ProjGradient: since {\tilde{P}}^\ast (p) \in W^{1,2}(\Omega ) and{\tilde{P}}^\ast (p) = \iota ^\ast _X \circ P_{\textup {iso}}^\ast \circ M^\ast (p) = w_{P}^{-1} \cdot w_{P} \cdot P^{\textup {B}}\big ( {\tilde{u}_P}^{-1/2} \cdot ({\tilde{u}_P}/u_P)^{ 1/2 } \cdot p \big ) = P... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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2e843208b1935abf3d182640b6fbcb75b19397a3 | subsection | 44 | 98 | Gradient-penalties | Finally, combining (REF ) and (REF ) yields\mathcal {R}(f_{\textup {ref}} + P^\ast (p)) = \big \Vert \nabla \big ( P^{\textup {B}}(u_P^{-1/2} \cdot p) \big ) \big \Vert _{L^2}^2 = \big \Vert u_P^{1/2} \cdot \nabla _{\mathbb {D}} \big ( u_P^{-1/2} \cdot p \big ) \big \Vert _{L^2}^2,which, by lem:ProjGradient, remains va... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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466f4490426069da6b57c3bb348b40429594b001 | subsection | 45 | 98 | Gradient-penalties | The general result reads as follows:[Generalized SART with W^{1,2}-penalties]
Let P\in \lbrace {P}, {D}\rbrace : L^2(\Omega ) \rightarrow {L^2({\mathbb {D}\!_{P}})}, f_{\textup {ref}} \in L^2(\Omega ), \alpha > 0 and let \mathcal {S}: {L^2({\mathbb {D}\!_{P}})}\rightarrow \mathbb {R}\cup \lbrace \infty \rbrace be any ... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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4b5fb2d1289a988215d6e04392f18be581601416 | subsection | 46 | 98 | Gradient-penalties | Then the optimization problem (REF ) can be written in the formf_{\textup {new}} \in \operatornamewithlimits{argmin}_{ f \in X } \tilde{\mathcal {S}}\left( P(f) \right) + \alpha ( 1- \gamma ) \mathcal {R}_1 (f) + \alpha \gamma \mathcal {R}_2 (f) .where the functionals \mathcal {R}_1, \mathcal {R}_2: X \rightarrow \math... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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85d85d973b8d6b4e399528f7f43be74a20cc0bb5 | subsection | 47 | 98 | Gradient-penalties | Hence, the GenSART-thm:genSART is applicable to (REF )
so that a minimizers f_{\textup {new}} can be found via the scheme (REF ):\tilde{p}_{\textup {ref}} &= {\tilde{P}}(f_{\textup {ref}} ) = u_P^{-1/2} \cdot P( f_{\textup {ref}} ) \\
\Delta p &\in \operatornamewithlimits{argmin}_{p \in {L^2({\mathbb {D}\!_{P}})}} \til... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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61a9e37e33944ec3525584705df1c194857f3896 | subsection | 48 | 98 | Gradient-penalties | Accordingly, the ray-density-weighting of the back-projection in the cone-beam case is omitted. This is quite intuitive since back-projecting uniformly along the rays results in smaller values of the gradient-penalty functional (REF ) and thus a non-weighted back-projection can be regarded as the natural one in the con... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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23c6bcb5eb901427afd10b853bfcf66ab2e0aab4 | subsection | 49 | 98 | Applications | In the preceding sections, it has been analyzed in which abstract situations Kaczmarz-iterations of the form (REF ) can be computed via a generalized SART-scheme. In the following, the principal theory is applied to design tailored methods for various settings of tomographic imaging. Specifically, the aim is to exploit... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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ec83dcdac1133ce3f28a55006d15b7886a889359 | subsection | 50 | 98 | Noise-model-adapted GenSART | As outlined in §REF , variational- and Kaczmarz-type reconstruction methods may account for the expected statistics of the data errors {\epsilon } in Inverse Problem REF by suitably choosing the data-fidelity functionals \mathcal {S}_k in (REF ).
We illustrate this for Kaczmarz-iterations with a simple L^2-penalty and ... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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75cbb18caee41aeac32fda49f978b6cafbba4d75 | subsection | 51 | 98 | Efficient closed-form optimization in projection-space: | For general noise-models and image-formation operators F_j, the optimization problem in () could still be hard to solve, in spite of being cast to the low-dimensional projection-space via the GenSART-approach. In the following, we therefore outline practically relevant settings where the optimization-step in the GenSAR... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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96a160556fedc0eb4c0bfa37c3004c4f6a225b23 | subsection | 52 | 98 | Efficient closed-form optimization in projection-space: | As a consequence, the optimization in () is equivalent to a family of scalar problems:\Delta p _k &\in \operatornamewithlimits{argmin}_{p \in {L^2({\mathbb {D}\!_{{j_k}}})}} \mathcal {S}\big ( g_{j_k}^{\textup {obs}}; \, F_{j_k} \big ( p_k + {\tilde{u}_{j_k}}^{1/2}\cdot p \big ) \big ) + \alpha \Vert p \Vert _{L^2}^2 \... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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38b01d707678d36185f1d22670d95b58603bfec2 | subsection | 53 | 98 | Robust GenSART | As a first non-standard application, we consider the problem of robust tomographic reconstruction: systematic errors in the acquisition geometry or modeling-inaccuracies due to nonlinear effects, as arising from metal-inclusions in soft tissue for example , tend to produce large outliers in the data, i.e. errors with h... | {
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"raw": "J. F. Barrett and N. Keat, Artifacts in CT: recognition and avoidance, Radiographics, 24 (2004), pp. 1679–1691.",
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"start": 0
... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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14865bc5365cc1682f97fe0d40b1410a2e3385dd | subsection | 54 | 98 | Robust GenSART | The proximal maps of s_{L^1_{\textup {H}}, \nu }, s_{\textup {s-t}, \nu } are given by\operatornamewithlimits{prox}( s_{L^1_{\textup {H}}, \nu })(y, \tau )&= y - \frac{2\nu \tau y}{\max \lbrace |y|, 2\nu \tau + 1\rbrace } \\
\operatornamewithlimits{prox}(s_{\textup {s-t}, \nu })(y, \tau )&= \operatornamewithlimits{argm... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s11075-013-9778-8",
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"raw": "M. S. Andersen and P. C. Hansen, Generalized row-action methods for tomographic imaging, Numerical Algorithms, 67 (2014), pp. 121–144.",
... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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e9ef4222554538a3d878c0c68695d38e029dc1bd | subsection | 55 | 98 | Poisson-noise-adapted GenSART | In many practical applications of X-ray- or electron tomography, the data errors are primarily due to the Poisson-statistics of the detection process: detector pixels actually count a discrete number of incident photons or electrons over some exposure time t>0, where the counts follow a Poisson-distribution. Disregardi... | {
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{
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"raw": "I. Mori, Y. Machida, M. Osanai, and K. Iinuma, Photon starvation artifacts of X-ray CT: their true cause and a solution, Radiological physics... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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d39ccd6ea83ed82ff0f17c85a41da373dbc659cb | subsection | 56 | 98 | Poisson-noise-adapted GenSART | In this setting, the log-likelihood in (REF ) leads to the discrete Kullback-Leibler-divergence, see for details:\mathcal {S}^{\textup {Poi}} \left( g^{\textup {obs}}_j; g_j \right) &:= \sum _{ i = 1}^{ {m_{\textup {proj}}}} \textup {KL}(g^{{\textup {obs}}}_{ji}; t \mathcal {M}_i \left( g_j ) \right) ), \quad \textup {... | {
"cite_spans": [
{
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"raw": "T. Hohage and F. Werner, Inverse problems with poisson data: statistical regularization theory, applications and algorithms, Inverse Prob... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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... | |
5c5e24fb147721225544725a3519dff326f793e0 | subsection | 57 | 98 | Poisson-noise-adapted GenSART | This is the model for classical (monochromatic) X-ray computed tomography.Inserting these models into (REF ), it can be seen that the resulting data-term \mathcal {S}( g^{{\textup {obs}}}_{j }; \, F_j(p) ) is of the integral-form (REF ) withs_j(x,y) =
{\left\lbrace \begin{array}{ll}
\textup {KL}\left(g^{{\textup {obs}}... | {
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"doi": "10.1088/0266-5611/32/9/093001",
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"raw": "T. Hohage and F. Werner, Inverse problems with poisson data: statistical regularization theory, applications and algorithms, Inverse Prob... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
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d2f6e913595879ca037db600ac3da7bb3af11e47 | subsection | 58 | 98 | Regularized Newton-Kaczmarz-GenSART | Regularized Newton-Kaczmarz methods have been proposed in for the solution of general block-structured inverse problems G(f) = (G_1(f), \ldots , G_N(f)) = (g_1^{\textup {obs}}, \ldots , g_N^{\textup {obs}}) with nonlinear forward operators G_j: X \rightarrow Y_j between Hilbert spaces X, Y_1, \ldots , Y_N. In its simpl... | {
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"raw": "M. Burger and B. Kaltenbacher, Regularizing Newton–Kaczmarz methods for nonlinear ill-posed problems, SIAM Journal on Numerical Analysis, 44 (2006), pp. 153–182.",
"source_ref_id": "eb948c5798... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
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65b157677bc8d0782e2e0d9ecdcdde3446c8f431 | subsection | 59 | 98 | Propagation-based X-ray phase contrast tomography | We consider the setting of (propagation-based) X-ray phase contrast tomography (XPCT), see e.g. , , , , .
In this experimental setup, the recorded data is given by near-field diffraction patterns, that relate to tomographic projections of the object density via a highly non-trivial image-formation operator F_j = F : un... | {
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"raw": "P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. Guigay, and M. Schlenker, Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiati... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
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553e376679702867d933657d5d5a526b654c8f39 | subsection | 60 | 98 | Propagation-based X-ray phase contrast tomography | Newton-Kaczmarz iterations for this problem with L^2-data-fidelity and Sobolev-W^{1,2}-penalty, as first proposed in \cite {MaretzkeEtAl2016OptExpr}, are of the form \begin{align}
f_{k+1} = \operatornamewithlimits{argmin}_{f \in L^2(\Omega ) } \big \Vert F \left( {P}_{j_k} (f_k) \right) + F^{\prime }[{P}_{j_k} (f_k) ] ... | {
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6a903db6e9073ab3aac25d4ff31d576db8f7a842 | subsection | 61 | 98 | Polychromatic CT | If the polychromatic nature of the X-rays in conventional CT-scanners is neglected, so called beam-hardening artifacts may arise .
In , , a simplified model for polychromatic CT has been proposed, which partially accounts for the arising nonlinear effects. Within this model, the detected intensity data g_j for the jth ... | {
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ca2588c6cc5659d216e5297c00229628a87df58f | subsection | 62 | 98 | Polychromatic CT | However, we may still compute the Fréchet-derivative:G_j^{\prime }[f]h_f &= - G_j^\Phi (f) \cdot P_j\left( \phi ^{\prime } (f) \cdot h_f \right) - G_j^\Theta (f) \cdot P_j\left( \theta ^{\prime } (f) \cdot h_f \right) \\
G_j^\Phi (f) &:= \int \Phi (\varepsilon ) G_{j, \varepsilon }(f) \, , \quad G_j^\Theta (f) := \int ... | {
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} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
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74db661b3461901805b9f6ea3ed2e23209cd9fe0 | subsection | 63 | 98 | Polychromatic CT | Including the necessary computations of G_{j_k} (f_k) and \lambda _{j_k}(f_k), evaluating (REF ) requires three evaluations of the forward- and back-projectors P_{j_k} and P_{j_k}^\ast , plus computationally inexpensive pointwise operations.Similarly efficient formulas may be obtained if the L^2-data-fidelity in (REF )... | {
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