Filtered backprojection inversion of the cone beam transform for a general class of curves

We extend a cone beam transform inversion formula, proposed earlier for helices by one of the authors, to a general class of curves. The inversion formula remains efficient, because filtering is shift-invariant and is performed along a one-parametric family of lines. The conditions that describe the class are very natural. Curves C are smooth, without self-intersections, have positive curvature and torsion, do not bend too much, and do not admit lines which are tangent to C at one point and intersect C at another point. The notions of PI lines and PI segments are generalized, and their properties are studied. The domain U is found, where PI lines are guaranteed to be unique. Results of numerical experiments demonstrate very good image quality

Expander Decomposition in Dynamic Streams

In this paper we initiate the study of expander decompositions of a graph $G=(V, E)$ in the streaming model of computation. The goal is to find a partitioning $\mathcal{C}$ of vertices $V$ such that the subgraphs of $G$ induced by the clusters $C \in \mathcal{C}$ are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of $V$) to within a $(\delta, \epsilon)$-multiplicative/additive error with high probability. The power cut sparsifier uses $\tilde{O}(n/\epsilon\delta)$ space and edges, which we show is asymptotically tight up to polylogarithmic factors in $n$ for constant $\delta$.Comment: 31 pages, 0 figures, to appear in ITCS 202

Expander Decomposition in Dynamic Streams

In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning ? of vertices V such that the subgraphs of G induced by the clusters C ? ? are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (?, ?)-multiplicative/additive error with high probability. The power cut sparsifier uses O?(n/??) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant ?

Toeplitz Low-Rank Approximation with Sublinear Query Complexity

We present a sublinear query algorithm for outputting a near-optimal low-rank approximation to any positive semidefinite Toeplitz matrix $T \in \mathbb{R}^{d \times d}$. In particular, for any integer rank $k \leq d$ and $\epsilon,\delta > 0$, our algorithm makes $\tilde{O} \left (k^2 \cdot \log(1/\delta) \cdot \text{poly}(1/\epsilon) \right )$ queries to the entries of $T$ and outputs a rank $\tilde{O} \left (k \cdot \log(1/\delta)/\epsilon\right )$ matrix $\tilde{T} \in \mathbb{R}^{d \times d}$ such that $\| T - \tilde{T}\|_F \leq (1+\epsilon) \cdot \|T-T_k\|_F + \delta \|T\|_F$. Here, $\|\cdot\|_F$ is the Frobenius norm and $T_k$ is the optimal rank-$k$ approximation to $T$, given by projection onto its top $k$ eigenvectors. $\tilde{O}(\cdot)$ hides $\text{polylog}(d)$ factors. Our algorithm is \emph{structure-preserving}, in that the approximation $\tilde{T}$ is also Toeplitz. A key technical contribution is a proof that any positive semidefinite Toeplitz matrix in fact has a near-optimal low-rank approximation which is itself Toeplitz. Surprisingly, this basic existence result was not previously known. Building on this result, along with the well-established off-grid Fourier structure of Toeplitz matrices [Cybenko'82], we show that Toeplitz $\tilde{T}$ with near optimal error can be recovered with a small number of random queries via a leverage-score-based off-grid sparse Fourier sampling scheme.Comment: Accepted in SODA 202

Motif Cut Sparsifiers

A motif is a frequently occurring subgraph of a given directed or undirected graph $G$. Motifs capture higher order organizational structure of $G$ beyond edge relationships, and, therefore, have found wide applications such as in graph clustering, community detection, and analysis of biological and physical networks to name a few. In these applications, the cut structure of motifs plays a crucial role as vertices are partitioned into clusters by cuts whose conductance is based on the number of instances of a particular motif, as opposed to just the number of edges, crossing the cuts. In this paper, we introduce the concept of a motif cut sparsifier. We show that one can compute in polynomial time a sparse weighted subgraph $G'$ with only $\widetilde{O}(n/\epsilon^2)$ edges such that for every cut, the weighted number of copies of $M$ crossing the cut in $G'$ is within a $1+\epsilon$ factor of the number of copies of $M$ crossing the cut in $G$, for every constant size motif $M$. Our work carefully combines the viewpoints of both graph sparsification and hypergraph sparsification. We sample edges which requires us to extend and strengthen the concept of cut sparsifiers introduced in the seminal work of to the motif setting. We adapt the importance sampling framework through the viewpoint of hypergraph sparsification by deriving the edge sampling probabilities from the strong connectivity values of a hypergraph whose hyperedges represent motif instances. Finally, an iterative sparsification primitive inspired by both viewpoints is used to reduce the number of edges in $G$ to nearly linear. In addition, we present a strong lower bound ruling out a similar result for sparsification with respect to induced occurrences of motifs.Comment: 48 pages, 3 figure

Traversing the FFT Computation Tree for Dimension-Independent Sparse Fourier Transforms

We consider the well-studied Sparse Fourier transform problem, where one aims to quickly recover an approximately Fourier $k$-sparse vector $\widehat{x} \in \mathbb{C}^{n^d}$ from observing its time domain representation $x$. In the exact $k$-sparse case the best known dimension-independent algorithm runs in near cubic time in $k$ and it is unclear whether a faster algorithm like in low dimensions is possible. Beyond that, all known approaches either suffer from an exponential dependence on the dimension $d$ or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is $O(N\log N)$ in any dimension $d$ for $N=n^d$. Our work aims to address the above issues. First, we provide a translation/reduction of the exactly $k$-sparse FT problem to a concrete tree exploration task which asks to recover $k$ leaves in a full binary tree under certain exploration rules. Subsequently, we provide (a) an almost quadratic in $k$ time algorithm for this task, and (b) evidence that a strongly subquadratic time for Sparse FT via this approach is likely impossible. We achieve the latter by proving a conditional quadratic time lower bound on sparse polynomial multipoint evaluation (the classical non-equispaced sparse FT) which is a core routine in the aforementioned translation. Thus, our results combined can be viewed as an almost complete understanding of this approach, which is the only known approach that yields sublinear time dimension-independent Sparse FT algorithms. Subsequently, we provide a robustification of our algorithm, yielding a robust cubic time algorithm under bounded $\ell_2$ noise. This requires proving new structural properties of the recently introduced adaptive aliasing filters combined with a variety of new techniques and ideas

Sensing of surface and bulk refractive index using magnetophotonic crystal with hybrid magneto-optical response

We propose an all-dielectric magneto-photonic crystal with a hybrid magneto-optical response that allows for the simultaneous measurements of the surface and bulk refractive index of the analyzed substance. The approach is based on two different spectral features of the magneto-optical response corresponding to the resonances in p-and s-polarizations of the incident light. Angular spectra of p-polarized light have a step-like behavior near the total internal reflection angle which position is sensitive to the bulk refractive index. S-polarized light excites the TE-polarized optical Tamm surface mode localized in a submicron region near the photonic crystal surface and is sensitive to the refractive index of the near-surface analyte. We propose to measure a hybrid magneto-optical intensity modulation of p-polarized light obtained by switching the magnetic field between the transverse and polar configurations. The transversal component of the external magnetic field is responsible for the magneto-optical resonance near total internal reflection conditions, and the polar component reveals the resonance of the Tamm surface mode. Therefore, both surface-and bulk-associated features are present in the magneto-optical spectra of the p-polarized light

Efficient Inversion Of The Cone Beam Transform For A General Class Of Curves

We extend an efficient cone beam transform inversion formula, proposed earlier for helices, to a general class of curves. The conditions that describe the class are very natural. Curves C are smooth, without self-intersections, have positive curvature and torsion, do not bend too much in a certain sense, and do not admit lines which are tangent to C at one point and intersect C at another point. A domain U is found where reconstruction is possible with a filtered backprojection type algorithm. Results of numerical experiments demonstrate very good image quality. The algorithm developed is useful for image reconstruction in computerized tomography

Sensing of surface and bulk refractive index using magnetophotonic crystal with hybrid magneto-optical response

We propose an all-dielectric magneto-photonic crystal with a hybrid magneto-optical response that allows for the simultaneous measurements of the surface and bulk refractive index of the analyzed substance. The approach is based on two different spectral features of the magneto-optical response corresponding to the resonances in p-and s-polarizations of the incident light. Angular spectra of p-polarized light have a step-like behavior near the total internal reflection angle which position is sensitive to the bulk refractive index. S-polarized light excites the TE-polarized optical Tamm surface mode localized in a submicron region near the photonic crystal surface and is sensitive to the refractive index of the near-surface analyte. We propose to measure a hybrid magneto-optical intensity modulation of p-polarized light obtained by switching the magnetic field between the transverse and polar configurations. The transversal component of the external magnetic field is responsible for the magneto-optical resonance near total internal reflection conditions, and the polar component reveals the resonance of the Tamm surface mode. Therefore, both surface-and bulk-associated features are present in the magneto-optical spectra of the p-polarized light

Garnet-based magnetoplasmonic heterostructures with 1D photonic crystals for highly effective chemo- and biosensing

Summary form only given. We present a novel type of magnetoplasmonic heterostructures containing one-dimensional photonic crystals. We performed the design of the magnetoplasmonic heterostructures and tuned the parameters of the structure in order to enhance the magneto-optical response via excitation of the ultralong-range propagating surface plasmon polariton mode. This structure is designed for gas sensing at the operating wavelength of 790 nm. One-dimensional photonic crystal (PC) is used to tune the impedance of the heterostructure so that the long-range propagating modes can be excited. Our sample contains 1D PC made of alternating SiO2 and Ta2O5 layers of thickness 164 nm and 119.4 nm correspondingly. PC structure is coated with a 125-nm thick ferromagnetic film of bismuth-substituted iron garnet and 8-nm thick gold film for the excitation of the SPPs. Some part of the structure is not coated with a gold film therefore providing a possibility to excite another type of the surface electromagnetic modes in fully dielectric magnetophotonic structure