Sampling and Reconstruction of Shapes with Algebraic Boundaries

We present a sampling theory for a class of binary images with finite rate of innovation (FRI). Every image in our model is the restriction of \mathds{1}_{\{p\leq0\}} to the image plane, where \mathds{1} denotes the indicator function and $p$ is some real bivariate polynomial. This particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial $p$. We show that the image parameters --i.e., the polynomial coefficients-- satisfy a set of linear annihilation equations with the coefficients being the image moments. The inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels. As a remedy to these problems, we replace conventional moments with more stable \emph{generalized moments} that are adjusted to the given sampling kernel. The benefits are threefold: (1) it relaxes the requirements on the sampling kernels, (2) produces annihilation equations that are robust at numerical precision, and (3) extends the results to images with unbounded boundaries. We further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points, and sequentially enforcing measurement consistency. We consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images, including low to moderate noise levels and a range of realistic sampling kernels.Comment: 12 pages, 14 figure

Shapes From Pixels

Continuous-domain visual signals are usually captured as discrete (digital) images. This operation is not invertible in general, in the sense that the continuous-domain signal cannot be exactly reconstructed based on the discrete image, unless it satisfies certain constraints (\emph{e.g.}, bandlimitedness). In this paper, we study the problem of recovering shape images with smooth boundaries from a set of samples. Thus, the reconstructed image is constrained to regenerate the same samples (consistency), as well as forming a shape (bilevel) image. We initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes. Next, we relax the non-convex shape constraint to transform the problem into minimizing the total variation over consistent non-negative-valued images. We also introduce a requirement (called reducibility) that guarantees equivalence between the two problems. We illustrate that the reducibility property effectively sets a requirement on the minimum sampling density. One can draw analogy between the reducibility property and the so-called restricted isometry property (RIP) in compressed sensing which establishes the equivalence of the $\ell_0$ minimization with the relaxed $\ell_1$ minimization. We also evaluate the performance of the relaxed alternative in various numerical experiments.Comment: 13 pages, 14 figure

Shapes from Pixels

In today's digital world, sampling is at the heart of any signal acquisition device. Imaging devices are ubiquitous examples that capture two-dimensional visual signals and store them as the pixels of discrete images. The main concern is whether and how the pixels provide an exact or at least a fair representation of the original visual signal in the continuous domain. This motivates the design of exact reconstruction or approximation techniques for a target class of images. Such techniques benefit different imaging tasks such as super-resolution, deblurring and compression. This thesis focuses on the reconstruction of visual signals representing a shape over a background, from their samples. Shape images have only two intensity values. However, the filtering effect caused by the sampling kernel of imaging devices smooths out the sharp transitions in the image and results in samples with varied intensity levels. To trace back the shape boundaries, we need strategies to reconstruct the original bilevel image. But, abrupt intensity changes along the shape boundaries as well as diverse shape geometries make reconstruction of this class of signals very challenging. Curvelets and contourlets have been proved as efficient multiresolution representations for the class of shape images. This motivates the approximation of shape images in the aforementioned domains. In the first part of this thesis, we study generalized sampling and infinite-dimensional compressed sensing to approximate a signal in a domain that is known to provide a sparse or efficient representation for the signal, given its samples in a different domain. We show that the generalized sampling, due to its linearity, is incapable of generating good approximation of shape images from a limited number of samples. The infinite-dimensional compressed sensing is a more promising approach. However, the concept of random sampling in this scheme does not apply to the shape reconstruction problem. Next, we propose a sampling scheme for shape images with finite rate of innovation (FRI). More specifically, we model the shape boundaries as a subset of an algebraic curve with an implicit bivariate polynomial. We show that the image parameters are solutions of a set of linear equations with the coefficients being the image moments. We then replace conventional moments with more stable generalized moments that are adjusted to the given sampling kernel. This leads to successful reconstruction of shapes with moderate complexities from samples generated with realistic sampling kernels and in the presence of moderate noise levels. Our next contribution is a scheme for recovering shapes with smooth boundaries from a set of samples. The reconstructed image is constrained to regenerate the same samples (consistency) as well as forming a bilevel image. We initially formulate the problem by minimizing the shape perimeter over the set of consistent shapes. Next, we relax the non-convex shape constraint to transform the problem into minimizing the total variation over consistent non-negative-valued images. We introduce a requirement -called reducibility- that guarantees equivalence between the two problems. We illustrate that the reducibility effectively sets a requirement on the minimum sampling density. Finally, we study a relevant problem in the Boolean algebra: the Boolean compressed sensing. The problem is about recovering a sparse Boolean vector from a few collective binary tests. We study a formulation of this problem as a binary linear program, which is NP hard. To overcome the computational burden, we can relax the binary constraint on the variables and apply a rounding to the solution. We replace the rounding procedure with a randomized algorithm. We show that the proposed algorithm considerably improves the success rate with only a slight increase in the computational cost

Randomized recovery for boolean compressed sensing

We consider the problem of boolean compressed sensing, which is alternatively known as group testing. The goal is to recover a small number of defective items in a large set from a few collective binary tests. This problem can be formulated as a binary linear program, which is NP hard in general. To overcome the computational burden, it was recently proposed to relax the binary constraint on the variables, and apply a rounding to the solution of the relaxed linear program. In this paper, we introduce a ran- domized algorithm to replace the rounding procedure. We show that the proposed algorithm considerably improves the success rate by slightly increasing the computational cost

A Multi-stage Secret Sharing Scheme Using All-or-Nothing Transform Approach

A multi-stage secret sharing (MSS) scheme is a method of sharing a number of secrets among a set of participants, such that any authorized subset of participants could recover one secret in every stage. The first MSS scheme was proposed by He and Dawson in 1994, based on Shamir’s well-known secret sharing scheme and one-way functions. Several other schemes based on different methods have been proposed since then. In this paper, the authors propose an MSS scheme using All-Or-Nothing Transform (AONT) approach. An AONT is an invertible map with the property that having “almost all” bits of its output, one could not obtain any information about the input. This characteristic is employed in the proposed MSS scheme in order to reduce the total size of secret shadows, assigned to each participant. The resulted MSS scheme is computationally secure. Furthermore, it does not impose any constraint on the order of secret reconstructions. A comparison between the proposed MSS scheme and that of He and Dawson indicates that the new scheme provides more security features, while preserving the order of public values and the computational complexity

Anonymous roaming in universal mobile telecommunication system mobile networks

A secure roaming protocol for mobile networks is proposed. Roaming has been analysed in some schemes from the security point of view; however, there are vulnerabilities in most of them and so the claimed security level is not achieved. The scheme offered by Wan et al. recently is based on hierarchical identity-based encryption, in which the roaming user and the foreign network mutually authenticate each other without the help of the home network. Although the idea behind this proposal is interesting, it contradicts technical considerations such as routing and billing. The proposed protocol makes use of similar functions used in Wan et al.'s scheme but contributes a distinguished structure that overcomes the previous shortcomings and achieves a higher possible level of security in mobile roaming as well as enhancing the security of the key issuing procedure

An Efficient Multistage Secret Sharing Scheme Using Linear One-way Functions and Bilinear Maps

In a Multistage Secret Sharing (MSSS) scheme, the authorized subsets of participants could reconstruct a number of secrets in consecutive stages. A One-Stage Multisecret Sharing (OSMSS) scheme is a special case of MSSS schemes that all secrets are recovered simultaneously. In these schemes, in addition to the individual shares, the dealer should provide the participants with a number of public values related to the secrets. The less the number of public values, the more efficient the scheme. It is desired that MSSS and OSMSS schemes provide the computational security; however, we show in this paper that OSMSS schemes do not fulfill the promise. Furthermore, by introducing a new multi-use MSSS scheme based on linear one-way functions, we show that the previous schemes can be improved in the number of public values. Compared to the previous MSSS schemes, the proposed scheme has less complexity in the process of share distribution. Finally, using bilinear maps, the participants are provided with the ability of verifying the released shares from other participants. To the best of our knowledge, this is the first verifiable MSSS scheme in which the number of public values linearly depends on the number of the participants and the secrets and which does not require secure communication channels

Optimal Sampling Rates in Infinite-Dimensional Compressed Sensing

The theory of compressed sensing studies the problem of recovering a high dimensional sparse vector from its projections onto lower dimensional subspaces. The recently introduced framework of infinite-dimensional compressed sensing [1], to some extent generalizes these results to infinite-dimensional scenarios. In particular, it is shown that the continuous-time signals that have sparse representations in a known domain can be recovered from random samples in a different domain. The range M and the minimum number m of samples for perfect recovery are limited by a balancing property of the two bases. In this paper, by considering Fourier and Haar wavelet bases, we experimentally show that M can be optimally tuned to minimize the number of samples m that guarantee perfect recovery. This study does not have any parallel in the finite-dimensional CS

Equivalence of synthesis and atomic formulations of sparse recovery

Les relations d’interdépendance entre la télévision et les salles de cinéma ont largement été analysées ces dernières décennies, et l’exposition du film cinématographique comme produit d’appel et de vente pour la filière audiovisuelle n’est plus à démontrer. Cependant, à la suite du profond bouleversement du paysage audiovisuel français engagé depuis les années 1984-1986, il est intéressant de se pencher sur le rôle que peuvent désormais jouer les films – et d’abord, quels films ? – dans le s..

Equivalence of synthesis and atomic formulations of sparse recovery

Recovery of sparse signals from linear, dimensionality reducing measurements broadly fall under two well-known formulations, named the synthesis and the analysis a ́ la Elad et al. Recently, Chandrasekaran et al. introduced a new algorithmic sparse recovery framework based on the convex geometry of linear inverse prob- lems, called the atomic norm formulation. In this paper, we prove that atomic norm formulation and synthesis formulation are equiva- lent for closed atomic sets. Hence, it is possible to use the synthesis formulation in order to obtain the so-called atomic decompositions of signals. In order to numerically observe this equivalence we derive exact linear matrix inequality representations, also known as the theta bodies, of the centrosymmertic polytopes formed from the columns of the simplex and their antipodes. We then illustrate that the atomic and synthesis recovery results agree on machine precision on randomly generated sparse recovery problems