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02974dc5c54fcae223acdabb5e545109e1364d6c | subsection | 94 | 95 | The case of derivators | In fact, Street's one is presumably only one among a complicated web of Yoneda structures: notably, we can record/conjecture the existence ofThe Muro-Raptis Yoneda structure, built taking the represented derivator on \text{\fontseries {b}\selectfont {\upshape sSet}}; this is again representable, and the canonical map \... | {
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db8a1dd87856f958aaa78c4c74328f5140773813 | abstract | 0 | 2 | Abstract | Recently it was shown that the transitive closure of a directed graph can be
updated using first-order formulas after insertions and deletions of single
edges in the dynamic descriptive complexity framework by Dong, Su, and Topor,
and Patnaik and Immerman. In other words, Reachability is in DynFO.
In this article we e... | {
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9f037be8655d00703f40061d7c5b7a7005a8ed8d | subsection | 1 | 2 | Body | Reachability and Distances under Multiple ChangesSamir DattaChennai Mathematical Institute & UMI ReLaX, Chennai, Indiasdatta@cmi.ac.in
Anish MukherjeeChennai Mathematical Institute, Chennai, Indiaanish@cmi.ac.in
Nils VortmeierTU Dortmund University, Dortmund, Germanynils.vortmeier@tu-dortmund.de
Thomas ZeumeTU Dortmund... | {
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fd74950fb00c67ac37d768d66ab9f83227dce0f1 | abstract | 0 | 19 | Abstract | We study the fundamental problem of polytope membership aiming at large
convex polytopes, i.e. in high dimension and with many facets, given as an
intersection of halfspaces. Standard data-structures as well as brute force
methods cannot scale, due to the curse of dimen- sionality. We design an
efficient algorithm, by ... | {
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60a0c92c4ac4dc0ada22159a538c893a20203805 | subsection | 1 | 19 | Introduction | In geometric optimization, convex polytopes are very important objects appearing also as feasible regions in linear programming.
Let us consider a convex polytope P in H-representation, that is as the intersection of a finite set of linear inequalities:
P = \lbrace x\in {\mathbb {R}}^d \mid Ax\le b,\, A\in {\mathbb {R}... | {
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00902562a91eea6c7aa5a676450a7999ba1d0c5a | subsection | 2 | 19 | Previous Work. | There are two classical results for the approximate membership problem, both based on creating \epsilon -approximating polytopes and answering membership on them.
Any convex body is \epsilon -approximated by a polytope with O(1/\epsilon ^{(d-1)/2}) facets, which is asymptotically tight in the worst case .
This leads to... | {
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63c34a4b1397175ee6341bafd9d9db95535ac5d5 | subsection | 3 | 19 | Our contribution. | We describe a simple constructive reduction from the polytope membership
problem to ANN, then show under which conditions this
reduction holds for the respective approximate versions of the problems. This gives
us the flexibility to exploit advances in the research of ANN in order to offer, the first (as far as the aut... | {
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550ed2efa88f1cfbad843737b0d611d4bde566a3 | subsection | 4 | 19 | Approximate Polytope Membership | We assume that the given H-polytope P is full dimensional and that its representation is minimal, i.e. that it does not contain redundant inequalities.We denote the i-th (in)equality of P as a_i x \le b_i, 1 \le i \le n. We associate each facet of the polytope with a corresponding (in)equality and denote it as F_i. For... | {
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"Evangelos Anagnostopoulos",
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4be731e6a3c98ce8c2e736403f666ab0503a955b | subsection | 5 | 19 | Exact Polytope Membership Oracle | A reduction from the exact polytope membership problem to the exact nearest neighbor problem was established in , where it was shown that there is a connection between the boundaries of polytopes in {\mathbb {R}}^d and power diagrams in {\mathbb {R}}^{d-1}. Power diagrams define a partition of the Euclidean space into ... | {
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8add883a22ccff161cca67c1c5723fc7883aec62 | subsection | 6 | 19 | Exact Polytope Membership Oracle | By the Voronoi property, the following holds:
q \in P \Leftrightarrow ||p^*- q||_2 \le ||q - s||_2, \; \forall s \in S.
Polytope membership returns “YES" iff the nearest neighbor of q is p^*.
[Figure: A conceptual presentation of the constructive proof in the case of d=2. Each p_i corresponds to the symmetric point o... | {
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} | 1804.11295 | Algorithms for Deciding Membership in Polytopes of General Dimension | [
"Evangelos Anagnostopoulos",
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bb7de9732862763719a61738c1f315c61847d6b5 | subsection | 7 | 19 | Approximate Polytope Membership Oracle | Let us consider the following relaxation.Definition (Approximate Polytope Membership Problem)
Given a convex polytope P\subset {\mathbb {R}}^d and an approximation parameter \epsilon \in (0,1), an \epsilon -approximate polytope membership query decides whether a query point q\in {\mathbb {R}}^d lies inside or outside ... | {
"cite_spans": []
} | 1804.11295 | Algorithms for Deciding Membership in Polytopes of General Dimension | [
"Evangelos Anagnostopoulos",
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1b0981277ca1a7cbbd55bac4031fb454fe1771e3 | subsection | 8 | 19 | Approximate Polytope Membership Oracle | We distinguish two cases when q \in P^{-\epsilon } and q \in \lbrace {\mathbb {R}}^d \mid q \notin P \ \ \wedge \ \ d(q, \partial P) > \epsilon \cdot diam(P)\rbrace .– Let q \in P^{-\epsilon }, we wish to select an \epsilon ^{\prime } for the ANN problem such that:(1+\epsilon ^{\prime }) < ||p_i - q||_2/||p^* - q||_2Es... | {
"cite_spans": []
} | 1804.11295 | Algorithms for Deciding Membership in Polytopes of General Dimension | [
"Evangelos Anagnostopoulos",
"Ioannis Z. Emiris",
"Vissarion Fisikopoulos"
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163380ecb01c8ab1491ce783bb0d36ee470cc9a1 | subsection | 9 | 19 | Approximate Polytope Membership Oracle | It follows that,\frac{||p_i - q||_2^2}{||p^* - q||_2^2} = \frac{(r_i + a_i)^2 + k_i^2}{(r_i - a_i)^2 + k_i^2} = 1 + \frac{4r_i a_i}{(r_i-a_i)^2+k_i^2} \ge \\
\ge 1 + \frac{4\epsilon ^2 (diam(P))^2}{(r_i-a_i)^2+k_i^2} \ge 1 + \frac{4\epsilon ^2 (diam(P))^2}{2 (diam(P))^2} \ge 1 + 2\epsilon ^2Substituting in (REF ), yiel... | {
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} | 1804.11295 | Algorithms for Deciding Membership in Polytopes of General Dimension | [
"Evangelos Anagnostopoulos",
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2676bfed04604c74c349bff42973dbda23e1c93e | subsection | 10 | 19 | Approximate Polytope Membership Oracle | Now, like before:\frac{||p^*-q||^2_2}{||p_i-q||^2_2} = \frac{(r_i + a_i)^2 + k_i^2}{(r_i - a_i)^2 + k_i^2} = 1 + \frac{4r_i a_i}{(r_i-a_i)^2+k_i^2} \ge \\
\ge 1 + \frac{4(\epsilon \cdot diam(P))^2}{(r_i-a_i)^2+k_i^2} \ge 1 + \frac{4(\epsilon \cdot diam(P))^2}{2 \left(\frac{2 \Delta (P)}{2\epsilon }\right)^2} \ge \\ \ge... | {
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"Evangelos Anagnostopoulos",
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d78ca818e3aa5b69385ec4fd0674331c0c6148a3 | subsection | 11 | 19 | Approximate Polytope Membership Oracle | Below, \tilde{O} omits logarithmic factors.[AMO in High Dimension]
For an H-polytope P \subset {\mathbb {R}}^d and an approximation parameter \epsilon , such that P^{-\epsilon }\ne \emptyset , we can solve the Approximate Polytope membership problem on P by building a data structure on P answering queries in \tilde{O}(... | {
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be7a088d9157157a787982f0715ab25e662d9007 | subsection | 12 | 19 | Application to Polytope Boundary Problem | The polytope boundary problem consists of creating a data structure for an H-polytope P such that, given a query ray emanating from inside the polytope, we can efficiently compute the point p=r \cap \partial P. It is possible to achieve query time in O(\log n) by using space in O(n^d/\log ^{\lfloor d/2 \rfloor }n) . Th... | {
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9f59aac384fd8ee8b98aed7146f7c3de362f2a95 | subsection | 13 | 19 | Exact Polytope Boundary Oracle. | We now describe an iterative procedure for P based on an exact nearest neighbor data structure E_MEM defined on the pointset S of Corollary REF that we described in section REF . This exact nearest neighbor data structure will act as the exact membership oracle for the polytope P. We call this algorithm BoundaryOracle.... | {
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} | 1804.11295 | Algorithms for Deciding Membership in Polytopes of General Dimension | [
"Evangelos Anagnostopoulos",
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ed8487cd07f77bf326fd32af49abcf587e8d3ecb | subsection | 14 | 19 | Approximate Polytope Boundary Oracle. | Now, we define an approximate version of the polytope boundary problem.Definition (Approximate Polytope Boundary Problem)
Given a convex H-polytope P \subset \mathbb {R}^d and an approximation
parameter \epsilon \in (0,1), preprocess P into a data structure such that, given
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60e525c3b1c07f2c4b8a588ca52dc927b342c176 | subsection | 15 | 19 | Implementation and Experiments | Implementation. All of our codehttps://github.com/van51/volume_approximation is linked to the software of . It is written in C++11 based on using the CGALhttp://www.cgal.org/ library for the readily available data structures of d-dimensional objects, Eigen3 for some linear algebra computations and FALCONN for the appro... | {
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9379be62766c70035659f17af66e1cd4699634bd | subsection | 16 | 19 | Implementation and Experiments | For the membership oracle we sample 1000 query points inside the polytope via the popular hit-and-run paradigm and then move these points sufficiently far from the origin so that they lie outside the polytope. This generates another 1000 points to form a total of 2000 points. Similarly for the boundary oracle we use 10... | {
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ffa2378c243284a56c05b85a7d8153233b703819 | subsection | 17 | 19 | Proofs of section | Proof of lemma REF .Let t_1, t_2, \ldots denote the sequence of successive points computed on the ray r by the above algorithm. Let x_1, x_2, \ldots be a sequence of points in S, each representing the nearest neighbor of the point t_i. This means that the nearest neighbor of t_i was x_i at the i-th step. We assume with... | {
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bf6d491ebef233a0b0c686c184268c81d36f8328 | subsection | 18 | 19 | Proofs of section | In that case, s is identified as lying at distance at most \epsilon diam(P) + \epsilon from \partial P and in which case we return point s + \frac{v}{||v||_2}\epsilon which lies in r at distance < \epsilon \cdot diam(P) from \partial P.Eventually, the algorithm returns point t: t \in P and d(t, \partial P) \ge \epsilon... | {
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da9e1f96601c9f12057775756fe3aa0e4575bea2 | abstract | 0 | 58 | Abstract | We consider the dynamical super-resolution problem consisting in the recovery
of positions and velocities of moving particles from low-frequency static
measurements taken over multiple time steps. The standard approach to this
issue is a two-step process: first, at each time step some static
reconstruction method is ap... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
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0e46be65b6669ce3cc70d9015ab6d94f7af9b7db | subsection | 1 | 58 | Introduction | It is well-known that the resolution of any wave imaging method is limited by the diffraction limit . Super-resolution is understood as any technique whose resolution surpasses this fundamental limit, which is of order of half the operating wavelength. More precisely, the super-resolution problem can be stated as follo... | {
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Imaging | [
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ce2b8ac6839488b75bddd45a261a06b9c1a1f9e8 | subsection | 2 | 58 | Introduction | This approach suffers from three main drawbacks: first, a lot of data are discarded whenever static reconstruction cannot be performed because of particles being too close, second, the information from neighboring frames is ignored in the first step of the reconstruction and, third, tracking algorithms are computationa... | {
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Imaging | [
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e1b52a12bf0e49a4cdf45f927e1dbc459e3895b2 | subsection | 3 | 58 | The space-velocity model | Let us now introduce our model for super-resolution of dynamic spikes. Instead of considering a single measure \mu , we consider a time-varying measure \mu _t, where t\in [-\delta , \delta ], and \delta >0 defines our observation window. Since \delta is expected to be small, we can approximate the dynamics of each poin... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
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cadb486804690a935978e5668cdbcda77830059b | subsection | 4 | 58 | The space-velocity model | Applying \mathcal {F} to the measures \mu _{k}:=\mu _{t_k} for every time step k\in \left\lbrace -K,\dots ,K\right\rbrace gives\mathcal {F}\mu _{k} = \left( \left< \mu _{k}, \varphi _l \right> \right)_{l=1}^{n} = \left( \sum _{i=1}^{N} w_i \varphi _l (x_i + k \tau v_i ) \right)_{l=1}^{n}.By construction, these measurem... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
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770f380a6dca73c79de1dabd94e1b239a3cae785 | subsection | 5 | 58 | The perfect low-pass case | Instead of studying the general framework outlined so far, in order to highlight the main features of this approach we prefer to focus on the particular case of low-frequency Fourier measurements, which represents a simplified model for many different applications. Thus, the theoretical analysis discussed below refers ... | {
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Imaging | [
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a0a9941062cf13d4202b54c7f8ead834a66cafe5 | subsection | 6 | 58 | Dual certificates | As it is standard in convex optimization, it is useful to consider the dual problem to study the exact recovery for (REF ).
In order to do this, we need to introduce the concept of dual certificate. We use the notation \operatornamewithlimits{sgn}{K}^N = \lbrace \eta \in {K}^N:|\eta _i|=1\text{ for every $i=1,\dots ,N$... | {
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Imaging | [
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e3995dd7802cbd1e986981a6a59c8dda48eae033 | subsection | 7 | 58 | Dual certificates | As mentioned above, the static recovery problem was treated in , but their methodology cannot be transferred directly to our case since the static dual certificates are constructed with all low frequency coefficients, whereas in our case we have access only to the frequencies given by the set (REF ).The particular stru... | {
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Imaging | [
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c885ff708bb4d902bc9f435bf5c93134678d1a0c | subsection | 8 | 58 | Dual certificates | The values of q are fixed by \gamma _{i,k} on each line L_{i,k}, and in particular in their points of intersection. As we shall see below, the problematic points are those where several lines (or d-dimensional affine subspaces) intersect, as in the two circled dots in the figure.
[Figure: The geometries of the problem ... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
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2ee22cd1e35588ea12f3eed20c26ea4373bfd5f5 | subsection | 9 | 58 | Dual certificates | Take as example the configuration of points given in {fig:ghostb} with 1= 2 = 3 = 1 and K={-1,0,1}. The static average certificate will value 1 in each of the particles and by construction i,k = 1. As a consequence, q will have value 1 also in the circled points, in which |K|=3 lines Li,k intersect, hence breaking cond... | {
"cite_spans": []
} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
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b02df7177f260e5e74fd4cff910b66509a7f17e6 | subsection | 10 | 58 | Main result | We are now ready to state the main result of this section. Several comments on the assumptions are given after the proof.Theorem 5 Let T=\lbrace (x_i,v_i)\rbrace _{i=1,\dots ,N} be a configuration of N particles, w \in {K}^N and \mathcal {K}\subseteq \lbrace -K,\dots ,K\rbrace be such that |\mathcal {K}|\ge 3. Let\omeg... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
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5675c36992b725f4e605587082c5959d0cd850ce | subsection | 11 | 58 | Main result | In the case none of the indexes i_k repeats, then by definition (x,v) is a ghost particle. | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
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"Timothée Wintz"
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138abe7642fc28dcc737153f128220b8306d62d5 | subsection | 12 | 58 | Comments on the hypotheses of thm:main | Let us now comment on assumptions (1) and (2), and show why these are easily satisfied. Let us start from assumption (1), namely the existence of static dual certificates.Remark 6
Take k\in \lbrace -K,\ldots ,K\rbrace . There exists a static dual certificate \tilde{q}_k(x) satisfying (REF ) in any of the following sit... | {
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Imaging | [
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5cf24b232ef44265975ba7344a8937163d39bc4d | subsection | 13 | 58 | Comments on the hypotheses of thm:main | We say that (g,w)\in \tilde{G}(T_{n-1}) if there exists a particle P_{n}\in \Omega such that (g,w) \in G(T_{n-1} \cup \lbrace P_{n}\rbrace ), namely\tilde{G}(T_{n-1}):= \bigcup _{P_n\in \Omega } G(T_{n-1} \cup \lbrace P_{n}\rbrace ).By definition of ghost points, we have that if
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
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b34defb032e54f26059462b327a26ec99123a4bf | subsection | 14 | 58 | Comments on the hypotheses of thm:main | More precisely, if G(T_{n-1}) = \emptyset then\mu _n \left( \left\lbrace P_n \in \Omega : G(T_{n-1} \cup \lbrace P_n\rbrace ) \ne \emptyset \right\rbrace \right) = 0.In order to prove this, notice that if G(T_{n-1}) = \emptyset we haveG(T_{n-1} \cup \lbrace P_n\rbrace ) \subseteq \lbrace (g,w) \in \tilde{G}(T_{n-1}): \... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
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"Timothée Wintz"
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5a97a4774614f90a96995797c0dc63944708cf8d | subsection | 15 | 58 | Comments on the hypotheses of thm:main | Using this property N-m+1 times for n=N,N-1,\dots ,m and setting \mu =\otimes _{i=1}^N \mu _{i}, we obtain\mu (\left\lbrace (P_1,\ldots ,P_N) : G(T_N) \ne \emptyset \right\rbrace ) &= \int _\Omega \dots \int _\Omega {1}_{G(T_N) \ne \emptyset } \prod _{i=1}^{N} f_{i}(P_i) dP_1\dots dP_N \\ & = \int _\Omega \dots \int _\... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
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3e0e69346ec0cc1fb1773fc7d41fb78511b21edb | subsection | 16 | 58 | Comments on the hypotheses of thm:main | On the right hand side we can observe the relative position of each particle at each time step.]Proposition 8
Take w\in {R}_+^{m}, with m = \vert \mathcal {K}\vert \ge 3, and let \lbrace (x_i,v_i)\rbrace _{i=1}^m \subseteq \Omega be a configuration of m distinct particles admitting m distinct ghost particles \lbrace (... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
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078ea58544d22266f003d0399cab4ad421e375b6 | subsection | 17 | 58 | Comments on the hypotheses of thm:main | Therefore, each term of the first sum cancels out with one term of the second sum, as desired.For \beta \in [0, \min _i w_i], consider the measure\omega _\beta = \omega - \beta h = \sum _{i = 1}^{m} ( w_i - \beta ) \delta _{(x_i,v_i)} +\beta \sum _{j = 1}^{m} \delta _{(g_j, w_j)}.Since \mathcal {G}\omega _\beta = \math... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
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8f2f03321993ad5b5f4172fc6d163ac598a1d790 | subsection | 18 | 58 | Other constructions of dynamical dual certificates | In this section we show that the construction of dynamical dual certificates as static average certificates, although natural and efficient, is not the end of the story. In other words, exact recovery may be guaranteed even if assumptions (1) and (2) of thm:main are not satisfied.
More precisely, we provide alternative... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
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e91e3552cd80e32725e974d8926704f5d2d30722 | subsection | 19 | 58 | Dual certificates with no static separation condition | The following example of dual certificate is purely numerical, but shows the possibility of constructing a dual certificate in cases in which static reconstructions are expected to fail.
[Figure: An example of a dynamical dual certificate whithout a static minimum separation condition.]The chosen configuration is prese... | {
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2b1ab69f21ab8d71e190d1e9852c492cc50dbdcb | subsection | 20 | 58 | Dual certificates in presence of ghost particles | As we saw in fig:ghostb, in the presence of ghost points the static average dual certificate constructed in (REF ) is not a valid dynamical dual certificate if the values \eta _i have a constant sign. Indeed, the static average dual certificate will have absolute value equal to 1 in the ghost particles, since the value... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
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baafff82e661fd1ec3acbc4904a81edab5fae620 | subsection | 21 | 58 | Dual certificates in presence of ghost particles | In view of the geometrical configuration (see fig:ghostb), these conditions are:\gamma _{i,{\text{\scalebox {0.4}[1.0]{$ - $}}}1} + \gamma _{i,0} + \gamma _{i,1} & = 3, \qquad i \in \lbrace 1,2,3\rbrace , \\
|\gamma _{1,1} + \gamma _{2,0} + \gamma _{3,{\text{\scalebox {0.4}[1.0]{$ - $}}}1}| & < 3, \\
| \gamma _{1,{\tex... | {
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35cd5d501cca1f8bf25e724b28730dab720e2b21 | subsection | 22 | 58 | Dual certificates in presence of ghost particles | Finally, by continuity of q(x,v) with respect to \varepsilon , there exists an \varepsilon sufficiently small such that |q(x,v)| < 1 for every element in \Omega that is not a particle.The methodology presented in the proof of prop:removeGhost can be applied to more general examples and it can be iterated to deal with t... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
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52fe155d8eddfe2d8ab0e549b513066a6eb2ca57 | subsection | 23 | 58 | Dual certificates in presence of ghost particles | In view of the geometrical configuration (see fig:ghostb), these conditions are:\gamma _{i,{\text{\scalebox {0.4}[1.0]{$ - $}}}1} + \gamma _{i,0} + \gamma _{i,1} & = 3, \qquad i \in \lbrace 1,2,3\rbrace , \\
|\gamma _{1,1} + \gamma _{2,0} + \gamma _{3,{\text{\scalebox {0.4}[1.0]{$ - $}}}1}| & < 3, \\
| \gamma _{1,{\tex... | {
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Imaging | [
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1b1975bc38fc1919c54fffa41f5ede7b4fed92e2 | subsection | 24 | 58 | Dual certificates in presence of ghost particles | Finally, by continuity of q(x,v) with respect to \varepsilon , there exists an \varepsilon sufficiently small such that |q(x,v)| < 1 for every element in \Omega that is not a particle.The methodology presented in the proof of prop:removeGhost can be applied to more general examples and it can be iterated to deal with t... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
] | [
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bf02292ed0d7a3c8b5a23e488e96c0a96820f83d | subsection | 25 | 58 | Stable reconstruction with noise | Following the analysis done in for the static case, we present a stability result for the dynamical problem. We will review their setting and adapt it to our case under the same hypotheses: one dimensional case (d=1), discrete setting and a specific type of noise model with bounded total variation norm. Even though mor... | {
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Imaging | [
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dc64192942cf7d523fcf3c4abf3f8ce1e40aca8e | subsection | 26 | 58 | Stable reconstruction with noise | We consider a relaxed version of the noiseless problem (REF )\min _{\lambda \in \mathcal {M}\left( \Omega _\#\right)}\left\Vert \lambda \right\Vert _{1} \quad \text{subject to } \;\Vert \mathcal {G}\lambda - y \Vert _1 \le \frac{ \delta }{\Delta _x \Vert \mathcal {G}^* \Vert _{\ell ^1 \rightarrow \mathcal {M}}} ,where ... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
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c894902102e8145356c4e55020d0d6147b7e9f17 | subsection | 27 | 58 | Stable reconstruction with noise | Let q(x,v) be the static average dual certificate given in (REF ), where the each static dual certificate \tilde{q}_k is constructed as in for every k\in \mathcal {K}, thanks to assumption (i).
With an abuse of notation, set q = q_{i,j} = q\left(i \Delta _x , j \Delta _v \right). Let P_T denote the projection onto the ... | {
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Imaging | [
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86048bf49ba844345e3204e3003d7357a4b06119 | subsection | 28 | 58 | Stable reconstruction with noise | Therefore
|q(x_\#, v_\#) | & = \frac{1}{2K+1}\left|\sum _{k=-K}^K \tilde{q}_k\big ( x_i + k \tau v_i + (n_x\Delta _x + k \tau n_v \Delta _v) \big )\right|\\
& \le \frac{1}{2K+1} \sum _{k=-K}^K \left( 1 - C_1 f_c^2(n_x \Delta _x + k \tau n_v \Delta _v)^2 \right) \\
& = \frac{1}{2K+1} \sum _{k=-K}^K \left( 1 - C_1 f_c^2... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
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"Timothée Wintz"
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541c1a5ebbb83e6b01f57d24d1536e625a73129b | subsection | 29 | 58 | Stable reconstruction with noise | Hence, by () we have
|q(x_\#&, v_\#)| = \frac{1}{2K+1} \left| \sum _{k = -K}^K \tilde{q}_k( x_{i_k} + k \tau v_{i_k} - \mathcal {I}_k)\right| \\
& \le 1 - \frac{C_1 f_c^2}{2K+1} \sum _{k = -K}^K \left(\mathcal {I}_k^2\, {1}_{(-\infty ,0]}\Bigl (|\mathcal {I}_k|- \frac{C_2}{f_c}\Bigr ) + \frac{C_2^2}{f_c^2}\,{1}_{(0,+\... | {
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Imaging | [
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2de00f3a5565d7e51c0d48d14026f30a46f0b3f3 | subsection | 30 | 58 | Stable reconstruction with noise | Since (REF ) admits a unique minimizer, every \tilde{\mu } \ne \mu such that \mathcal {F} \tilde{\mu } = Y verifies \Vert {\tilde{\mu }} \Vert _{TV} > M. Therefore, \tilde{\mu } is the unique minimizer of (REF ).
The codes of the simulations of this paper are available at https://github.com/panchoop/dynamic_spike_super... | {
"cite_spans": []
} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
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7b3ff8c6103c0e35d8b1d6b9fd90ed98a0ef6d74 | subsection | 31 | 58 | Stable reconstruction with noise | Let us introduce the following measure of separation of a configuration of particles T=\lbrace (x_i,v_i)\rbrace _i\subseteq \Omega :
\Delta _{dyn}(T) = \max ^3_{k \in \lbrace -K,\dots ,K\rbrace } \min _{i\ne j} \vert x_i - x_j + \tau k ( v_i - v_j) \vert ,
where \displaystyle \max ^3 is the third highest element of ... | {
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6f330785c5e17cffd455e41e0e38603ef0d75aad | subsection | 32 | 58 | Stable reconstruction with noise | More precisely, for a frequency l and time sample k, our measurements are
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09c54d94ab912eaffa99ce50716c940bf5fe5b67 | subsection | 33 | 58 | Stable reconstruction with noise | We assume that we can filter out clutter signal coming from other sources than these reflectors. The recorded images are then convolutions of these point sources by the point spread function (PSF). The PSF of
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08b6295714adc5442aa5753042a9619bf9316489 | subsection | 34 | 58 | Stable reconstruction with noise | As an illustration, in fig:threeframes we show three frames of the simulated measurements. In fig:bmode we present the B-mode image, i.e. the average signal intensity over all the frames, which shows that the resolution does not allow for separation of the vessels. fig:superres presents the reconstructed position and v... | {
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bd0ebc6a601e4301adc27d65ff6f965d1aca7a78 | subsection | 35 | 58 | Stable reconstruction with noise | We also recall from the super-resolution factor in space:SRF_x = \frac{1}{\Delta _x f_c},which can be understood as the ratio between the desired space resolution and the permitted resolution given by the diffraction limit.We consider, instead of (REF ), the following input noise model:y = \mathcal {G} (\omega + z), \q... | {
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7b5d184a4404f780eddc4bf232fb3d5cba4b3807 | subsection | 36 | 58 | Stable reconstruction with noise | Let
\omega = \sum _{i=1}^N w_i\delta _{(x_i,v_i)}\in \mathcal {M}(\Omega )
be the unknown measure to be recovered.Suppose thatfor every k\in \mathcal {K}, the minimum separation condition (REF ) holds;
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333cdff06a28fcff3bdc445ce421d1f083db179b | subsection | 37 | 58 | Stable reconstruction with noise | In order to do so, we shall use the following estimates from \cite [Lemma~2.5]{candes2014towards}:
\begin{equation}
|\tilde{q}_k(t)| \le 1 - C_1 f_c^2(t-(x_i + k \tau v_i))^2, \quad \textnormal { when } \quad |t - (x_i+k \tau v_i)| \le C_2/f_c,
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\begin{equation}
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
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e4515fca5ffe84529e06246718831f77cf4f7f3f | subsection | 38 | 58 | Stable reconstruction with noise | Therefore
|q(x_\#, v_\#) | & = \frac{1}{2K+1}\left|\sum _{k=-K}^K \tilde{q}_k\big ( x_i + k \tau v_i + (n_x\Delta _x + k \tau n_v \Delta _v) \big )\right|\\
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& = \frac{1}{2K+1} \sum _{k=-K}^K \left( 1 - C_1 f_c^2... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
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b73cfef884abc4372498d068bf57b998a498b2f6 | subsection | 39 | 58 | Stable reconstruction with noise | Hence, by () we have
|q(x_\#&, v_\#)| = \frac{1}{2K+1} \left| \sum _{k = -K}^K \tilde{q}_k( x_{i_k} + k \tau v_{i_k} - \mathcal {I}_k)\right| \\
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Imaging | [
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8b34193cc42861aa4b0d96e35f4b713803f747bd | subsection | 40 | 58 | Stable reconstruction with noise | Since (REF ) admits a unique minimizer, every \tilde{\mu } \ne \mu such that \mathcal {F} \tilde{\mu } = Y verifies \Vert {\tilde{\mu }} \Vert _{TV} > M. Therefore, \tilde{\mu } is the unique minimizer of (REF ).
The codes of the simulations of this paper are available at https://github.com/panchoop/dynamic_spike_super... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
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7a403746081f3701bcc640ef2b716ef658571c56 | subsection | 41 | 58 | Stable reconstruction with noise | Let us introduce the following measure of separation of a configuration of particles T=\lbrace (x_i,v_i)\rbrace _i\subseteq \Omega :
\Delta _{dyn}(T) = \max ^3_{k \in \lbrace -K,\dots ,K\rbrace } \min _{i\ne j} \vert x_i - x_j + \tau k ( v_i - v_j) \vert ,
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e57b61492fc08a5b0b32fce616cea8c134038570 | subsection | 42 | 58 | Stable reconstruction with noise | More precisely, for a frequency l and time sample k, our measurements are
\left( \mathcal {G}\omega \right)_{l,k} = \sum _{i=1}^N w_i e^{-i2\pi l (x_i + k \tau v_i)} + \alpha \left( \mathcal {N}_{l,k,1} + i \mathcal {N}_{l,k,2} \right),
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
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8590237d86e1ee7f1a9109c18333030104485466 | subsection | 43 | 58 | Stable reconstruction with noise | We assume that we can filter out clutter signal coming from other sources than these reflectors. The recorded images are then convolutions of these point sources by the point spread function (PSF). The PSF of
ultrafast ultrasound imaging was derived in , but filtering out clutter signal changes the shape of the PSF, wh... | {
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Imaging | [
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5577a32b2594093f999e4ad4d998b887e0d8d2db | subsection | 44 | 58 | Stable reconstruction with noise | As an illustration, in fig:threeframes we show three frames of the simulated measurements. In fig:bmode we present the B-mode image, i.e. the average signal intensity over all the frames, which shows that the resolution does not allow for separation of the vessels. fig:superres presents the reconstructed position and v... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
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a0a0f03869633195cc24f590f2eac88c8db3dc34 | subsection | 45 | 58 | Methods | Solving minimization problem (REF ) in all its generality is not an easy task, since it is nonlinear and infinite dimensional. It is possible to use an analogue discrete problem, where the locations and velocities are fixed on a grid whose size determines the resolution we want to obtain. However, this methods becomes ... | {
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0cf3713312872019a7788ff2397a56d26c5a429b | subsection | 46 | 58 | The measurements | We consider the perfect low-pass filter described in subsec:lowFreq with d=1 as forward measurement operator, where the measured Fourier frequencies are \lbrace -f_c, \dots , f_c \rbrace for some f_c \in {N} and the number of time samples are 2K+1 with sampling rate \tau . The considered parameters for the simulations ... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
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59fb40c60ae04fbc30e97c95838eaf5cf3f50135 | subsection | 47 | 58 | The measurements | The quantity \Delta _{dyn} represents condition (1) of thm:main, for any subset of measurements \mathcal {K}\subseteq \lbrace -K,\ldots ,K\rbrace with \vert \mathcal {K}\vert = 3. We will evaluate the simulated reconstructions against this measure of separation, which will be scaled by \frac{1}{f_c}, as the theoretical... | {
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1742a35efda7306e98d6b9f5d0caec17888e9ea7 | subsection | 48 | 58 | Comparison of dynamical and static reconstructions | For a given configuration of particles, we consider three reconstruction procedures.The dynamical reconstruction: we take the whole data and recover both positions and velocities of all particles.
The static reconstruction: we perform static reconstruction of the positions at each time step independently, and we call ... | {
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Imaging | [
"Giovanni S. Alberti",
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747ca035783af5acc3ce011d3009c6b902e2f065 | subsection | 49 | 58 | Robustness to noise | We now study the stability of the dynamical reconstruction method and compare it to the stability for the static approaches. We consider a measurement noise model, given by a normally distributed noise scaled by a factor \alpha \ge 0 . More precisely, for a frequency l and time sample k, our measurements are\left( \mat... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
] | [
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2c3675d0f85a0d2dadc57d18ddba84fbfc77b795 | subsection | 50 | 58 | Robustness to curvature of trajectory | We study how the reconstruction algorithm fares when instead of a constant velocity, we consider a curved trajectory for the imaged particles. For this purpose we consider the following dynamics for a particle \delta _{x_i,v_i}:x(t) = x_i + v_i t + \frac{a}{2} t^2 = x_i + v_i t (1 _+ \frac{a}{2 v_i}t).We consider \beta... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
] | [
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8c825985e2735c28bb432700d0b190837b106837 | subsection | 51 | 58 | Applications to ultrafast ultrasonography | In this section, we describe a protocol to apply our method to the problem of super-resoluted imaging of blood vessels arising from ultrafast ultrasonography, as mentioned in the Introduction. The setting is the following: we have a sequence of images of a medium containing blood vessels, in which point reflectors are ... | {
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Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
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292ef79d8b00c137a1fd61613fbb7a09ce7644fd | subsection | 52 | 58 | Fully automated imaging protocol | We must make sure that the reconstruction algorithm works in this sequence, and that reflectors do not appear or disappear in a chosen sequence. Given all these remarks, we propose the following imaging procedure, which fully automatically produces a super-resoluted image of blood vessels with velocities. It is compose... | {
"cite_spans": []
} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
] | [
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73e5023ee923de6df6f670905e899e6a893c896c | subsection | 53 | 58 | Choosing reconstruction intervals. | In order to apply our dynamical reconstruction algorithm to this problem, we have to select consecutive frames during which the particles do not appear or vanish from one frame to another. One way to ensure this condition is to select time intervals during which the \ell ^2 norm of the observations is constant. Particl... | {
"cite_spans": []
} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
] | [
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50e850c3d50a7c845fe2bb46dc1ae4bd39b98422 | subsection | 54 | 58 | Reconstructing position and velocities. | We then propose to reconstruct positions and velocities using the algorithm presented in the previous sections, using the PSF of ultrafast ultrasound, in each of the intervals chosen in the previous step. By aggregating all positions obtained using this algorithm, we obtain a super-resoluted image of the blood vessels. | {
"cite_spans": []
} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
] | [
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1100a2116ecec40362e326e7dba8ef77fa9d268d | subsection | 55 | 58 | Setting | In order to test this procedure, we generate images using a toy example in which we have two slightly separated curved blood vessels with two branches and opposite blood flows. In Figure REF there is an illustration of the considered geometry. The velocities of the particles have constant modulus and the particles' wei... | {
"cite_spans": []
} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
] | [
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95eeb4ea6e372f32736026a99a8d590069ff26cb | subsection | 56 | 58 | Results | In fig:norm we present the \ell ^2 norms of the measurements at each time step. As an illustration, in fig:threeframes we show three frames of the simulated measurements. In fig:bmode we present the B-mode image, i.e. the average signal intensity over all the frames, which shows that the resolution does not allow for s... | {
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} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
] | [
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77e0cfc6ea9119139dff351c7f3b9f75e6d1b484 | subsection | 57 | 58 | Conclusion | In this paper, we have introduced and studied a new framework for dynamical super-resolution imaging, which allows for super-resolved recovery of positions and velocities of particles from low-frequency measurements. The presented theoretical results are validated by extensive simulations, related to low-frequency one-... | {
"cite_spans": []
} | 1803.03251 | Dynamic Spike Super-resolution and Applications to Ultrafast Ultrasound
Imaging | [
"Giovanni S. Alberti",
"Habib Ammari",
"Francisco Romero",
"Timothée Wintz"
] | [
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e2d1b2fb2897ff7dcb565fdeb11fb3fa77e04f58 | abstract | 0 | 130 | Abstract | String and M-theory contain a family of branes forming U-duality multiplets.
In particular, standard branes with codimension higher than or equal to two,
can be explicitly found as supergravity solutions. However, whether domain-wall
branes and space-filling branes can be found as supergravity solutions is still
unclea... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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c308476c1d6d46dc531834036dcefdaa7a33be47 | subsection | 1 | 130 | Background | The ten-dimensional type II superstring theories contain a rich variety of extended objects such as the D-branes and the NS5-brane.
The tension of a Dp-brane is proportional to g_s^{-1} (g_s: string coupling constant) and that of the NS5-brane is proportional to g_s^{-2} .It is conjectured that string theories are rela... | {
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"source_ref_id": "088e9c46d5d8dad95de576902... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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dfd9dca6952de53a6afc4aaa628553ee555ebdfb | subsection | 2 | 130 | Background | Lately, the so-called wrapping rules were formulated , , , , .
This set of rules allows one to construct a set of mixed-symmetry potentials, depending on the type of T- and S-duality transformations that one performs.
This approach is in full agreement with the predictions given by the E_{11} decomposition method.Despi... | {
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"doi": "10.1016/j.physletb.2011.09.043",
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"source_... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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5a0ae902294c34f3ae7bcfabef9fbf8740b99c8e | subsection | 3 | 130 | Background | Such deformations can be studied systematically:
using the embedding tensor formalism , and constructing the tensor hierarchy of a theory, one can scan all the possible deformations of a particular lower-dimensional supergravity.
Then, a dictionary between the fluxes associated to these deformed supergravities and the ... | {
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"raw": "F. Cordaro, P. Fré, L. Gualtieri, P. Termonia and M. Trigiante, “N=8 gaugings revisited: an exhaustive classification,” Nucl. Phys. B 532, 245 (1998) [hep-th/9804056].",
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024eebb097359b5f132a3ca7cba9e2331a97e397 | subsection | 4 | 130 | Main results | In this subsection we summarize the results that we have obtained.In Section , by brute force application of the S- and T-duality transformations, we have generated the full web of supersymmetric branes for each p-brane multiplet at any dimension d\ge 3.
In this classification, we have distinguished the defect, domain-... | {
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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0683c7c1ebecff6927c6229f4b1788e1e50d551c | subsection | 5 | 130 | Main results | As mentioned earlier, these theories can enjoy one or more isometry directions, each of them characterized by a Killing vector.
For a given exotic brane, we have provided a prescription to identify the lower-dimensional deformed supergravity theory that realizes that background.
While standard and defect branes do not ... | {
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"raw": "P. Meessen and T. Ortín, “An Sl(2,Z) multiplet of nine-dimensional type II supergravity theories,” Nucl. Phys. B 541, 195 (1999) [hep-th/9806120].",
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
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7ebc8b6850866b17583e1134510f4b627e1dec25 | subsection | 6 | 130 | Future directions | Let us comment on several open questions that have not been addressed in this paper.In this work, we have concentrated on the branes which are connected through Weyl reflections (which is a part of U-duality transformations).
For the disconnected “missing states,” we have only translated their Dynkin labels to the mass... | {
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"raw": "P. P. Cook, “Exotic E_{11} branes as composite gravitational solutions,” Class. Quant. Grav. 26, 235023 (2009) [arXiv:0908.0485 [hep-th]].",
"source_ref_id": "e202510e1bffa791c2881be066c9a6a26... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
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7fb65df118c66555ac470ed01d2561dc8cbacaf8 | subsection | 7 | 130 | Future directions | As we have explicitly shown in this paper, these p-brane multiplets include all of the exotic branes and these (p+1)-form fields should be composed of the mixed-symmetry potentials.
The explicit parameterizations of these (p+1)-form fields will be important.
More importantly, there must be constraints for the derivativ... | {
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fa0c9d5c7b0355f45b713a42d5b7ae6fc7af769b | subsection | 8 | 130 | Plan of the paper | The paper is organized as follows.
In Section we briefly review the notation for exotic branes.
In Section , we review the relation between exotic branes and the weights of the U-duality group.
We then construct the full web of “elementary” exotic branes and give the duality transformations that relate them.
In Section... | {
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"raw": "D. S. Berman, E. T. Musaev and R. Otsuki, “Exotic Branes in Exceptional Field Theory: E_{7(7)} and Beyond,” arXiv:1806.00430 [hep-th].",
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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89dac3d48a77bcae41ed7e23daff4d73c52caa4f | subsection | 9 | 130 | A brief review of duality rules and exotic branes | In this section, we provide a brief review of exotic branes in type II string theories and M-theory toroidally compactified to d-dimensions. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
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] | 2,018 | en | Physics | [
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9b23467ec26df3fa467e9dbf9338ba3889b565f3 | subsection | 10 | 130 | Type II branes | In type II string theory, by denoting the radius of the torus along the x^i-direction as R_i, the mass of a fundamental string (denoted as F1) wrapped along the x^i-direction is given byM_{\text{F1}(i)} = \frac{1}{2\pi l_s^2}\times (2\pi R_i) = \frac{R_i}{l_s^2} \qquad \bigl (l_s\equiv \sqrt{\alpha ^{\prime }}\bigr )\,... | {
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"doi": "10.1016/s0370-1573(99)00004-6",
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"raw": "N. A. Obers and B. Pioline, “U-duality and M-theory,” Phys. Rept. 318, 113 (1999) [hep-th/9809039].",
"source_ref_id": "c550bf7360... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
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065740791d3b8875c434ba719e527f6cab59bdfe | subsection | 11 | 130 | Type II branes | If an object wrapped along x^{n_1},\cdots ,x^{n_b}-directions has a mass,M = \frac{1}{g_s^{n}\,l_s} \Bigl (\frac{R_{n_1}\cdots R_{n_b}}{l_s^b}\Bigr ) \Bigl (\frac{R_{m_1}\cdots R_{m_{c_2}}}{l_s^{c_2}}\Bigr )^2\cdots \Bigl (\frac{R_{p_1}\cdots R_{p_{c_s}}}{l_s^{c_s}}\Bigr )^{s}\,,we denote the brane asb^{(c_s,\cdots ,c_... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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de87d999904208def746fa645bdb2b0d5fa86af1 | subsection | 12 | 130 | M-theory branes | We can uplift the mass of type IIA branes to the mass of M-theory branes by using the usual relation connecting 11D and 10D,R_{\text{\tiny M}} = g_sl_s\,,\qquad l_p= g_s^{1/3}\,l_s\,,where R_{\text{\tiny M}} represents the radius of the M-theory circle.
After the uplift, M-theory branes generally have masses of the for... | {
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{
"arxiv_id": "",
"doi": "10.1016/0550-3213(96)00321-5",
"end": 1178,
"openalex_id": "https://openalex.org/W3102005076",
"raw": "O. Aharony, “String theory dualities from M-theory,” Nucl. Phys. B 476, 470 (1996) [hep-th/9604103].",
"source_ref_id": "b79a5195ce... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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d443c43aaacc7feb03b816664ccb6f4334d93c05 | subsection | 13 | 130 | Brane tension | The b^{(c_s,\cdots ,c_2)}_{n}-brane in type II/M-theory can extend along the external spacetime up to b number of spatial dimensions, although the indices in the second slots (i.e. j_1,\cdots ,j_{c_2}) or later should be internal ones.
Namely, we can consider an external p-brane,\begin{split}
\text{Type II}:\quad &b^{(... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
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5d344a252185163f5fef1a0d41d86e4efea0dc15 | subsection | 14 | 130 | Brane tension | In the next section, we provide a full list of branes obtained by the U-duality transformations. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
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aeaae3e629cb0429cd1b9e39f82f397fbdd96a19 | subsection | 15 | 130 | Full duality web for | In this section, we provide the full duality web for string/M-theory compactified to d-dimensions with d\ge 3. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
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2dbc3b10b83fcd64ad38ded03826e9ee41389bdc | subsection | 16 | 130 | Duality rotations as Weyl reflections | Before showing the duality web, here we explain that the chain of T- and S-duality can be regarded as Weyl reflections by closely following the discussion of , . | {
"cite_spans": [
{
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"raw": "S. Elitzur, A. Giveon, D. Kutasov and E. Rabinovici, “Algebraic aspects of matrix theory on T^d,” Nucl. Phys. B 509, 122 (1998) [hep-th/9707217].",
"source_ref_id": "088e9c46d5d8dad95de5769026... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
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c41d243e6c34f3f5986a1283ccf722d7c1ae0945 | subsection | 17 | 130 | Setup | Let us rewrite the brane tension (REF ) as\mathcal {T}_p &\equiv (2\pi )^p\,T_p \equiv l_p^{3 v^0} (R_{i_1}\cdots R_{i_{b-p}}) (R_{j_1}\cdots R_{j_{c_2}})^2\cdots (R_{k_1}\cdots R_{k_{c_s}})^s
\\
&\equiv \operatorname{e}^{v^0\,x_0+v^i\,x_i} \qquad \bigl (\operatorname{e}^{x_0} \equiv l_p^{3}\,,\quad \operatorname{e}^{x... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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d564f57563c0098e59f3550536b4e2bdc10ff355 | subsection | 18 | 130 | Setup | Therefore, the U-duality (REF ) contains all possible permutations R_i\leftrightarrow R_j in 11D. and P_{i} can be also realized as a reflection,v \ \rightarrow \ v - 2\, \frac{v\cdot \alpha _i}{\alpha _i\cdot \alpha _i} \,\alpha _i \,,\qquad \alpha _i \equiv e_{i} - e_{i+1} \qquad (i=1,\cdots ,n-1)\,.Since the inner p... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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0e2a5e0d323b51a4aa922909f3cac3fa046621f8 | subsection | 19 | 130 | Setup | This shows that, for an arbitrary vector v=v^\mu \,e_\mu , a quantity 3\,v^0 + v^1+\cdots +v^d\,(=p) is invariant under the Weyl reflections.Now, we introduce the fundamental weights \lambda ^i associated with the simple roots (satisfying \lambda ^i \cdot \alpha _j=\delta ^i_j) as follows:\begin{split}
&\lambda ^1\equ... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
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