Yi Ming Zou

Toeplitz block matrices in compressed sensing and their applications in imaging

Recent work in compressed sensing theory shows that sensing matrices whose entries are drawn independently from certain probability distributions guarantee exact recovery of a sparse signal from a small number of measurements with high probability. In most medical imaging systems, the encoding matrices cannot take that form. Instead, they are Toeplitz block matrix. Motivated by this fact, we consider Toeplitz block matrices as the sensing matrices. We show that the probability of perfect reconstruction from a smaller number of filter outputs is also high if the filter coefficients are independently and identically-distributed random variable. Their applications in medical imaging is discussed. Simulation results are also shown to validate the theorem.

Sparsesense: Application of compressed sensing in parallel MRI

Compressed sensing (CS) has recently drawn great attentions in the MRI research community. The most desirable property of CS in MRI application is that it allows sampling of k-space well below Nyquist sampling rate, while still being able to reconstruct the image if certain conditions are satisfied. Recent work has successfully applied CS to reduce scanning time in conventional Fourier imaging. In this paper, the application of CS to parallel imaging, a fast imaging technique, is investigated to achieve an even higher imaging speed. The sampling scheme for incoherence is discussed and reconstruction method using Begman iteration is proposed. Our experiments show that the combined method, named SparseSENSE, can achieve a reduction factor higher than the number of channels.

Structures of coincidence symmetry groups.

The structure of the coincidence symmetry group of an arbitrary n-dimensional lattice in the n-dimensional Euclidean space is considered by describing a set of generators. Particular attention is given to the coincidence isometry subgroup (the subgroup formed by those coincidence symmetries that are elements of the orthogonal group). Conditions under which the coincidence isometry group can be generated by reflections defined by vectors of the lattice are discussed and an algorithm to decompose an arbitrary element of the coincidence isometry group in terms of reflections defined by vectors of the lattice is given.

Linear dynamical systems over finite rings

The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem of determining whether the system is a fixed point system can be answered by computing and factoring the systems characteristic polynomial and minimal polynomial. It has become clear recently that the study of finite linear dynamical systems must be extended to embrace finite rings. The difficulty of dealing with an arbitrary finite commutative ring is that it lacks of unique factorization. In this paper, an efficient algorithm is provided for analyzing the cycle structure of a linear dynamical system over a finite commutative ring. In particular, for a given commutative ring R such that |R|=q, where q is a positive integer, the algorithm determines whether a given linear system over Rn is a fixed point system or not in time O(n3log(nlog(q))).

Indices of coincidence isometries of the hypercubic lattice Z n.

The problem of computing the index of a coincidence isometry of the hypercubic lattice Z{n} is considered. The normal form of a rational orthogonal matrix is analyzed in detail and explicit formulas for the indices of certain coincidence isometries of Z{n} are obtained. These formulas generalize the known results for n < or = 4.

Integrated network model provides new insights into castration-resistant prostate cancer

Castration-resistant prostate cancer (CRPC) is the main challenge for prostate cancer treatment. Recent studies have indicated that extending the treatments to simultaneously targeting different pathways could provide better approaches. To better understand the regulatory functions of different pathways, a system-wide study of CRPC regulation is necessary. For this purpose, we constructed a comprehensive CRPC regulatory network by integrating multiple pathways such as the MEK/ERK and the PI3K/AKT pathways. We studied the feedback loops of this network and found that AKT was involved in all detected negative feedback loops. We translated the network into a predictive Boolean model and analyzed the stable states and the control effects of genes using novel methods. We found that the stable states naturally divide into two obvious groups characterizing PC3 and DU145 cells respectively. Stable state analysis further revealed that several critical genes, such as PTEN, AKT, RAF, and CDKN2A, had distinct expression behaviors in different clusters. Our model predicted the control effects of many genes. We used several public datasets as well as FHL2 overexpression to verify our finding. The results of this study can help in identifying potential therapeutic targets, especially simultaneous targets of multiple pathways, for CRPC.

Spherical functions on homogeneous superspaces

Homogeneous superspaces arising from the general linear supergroup are studied within a Hopf algebraic framework. Spherical functions on homogeneous superspaces are introduced, and the structures of the superalgebras of the spherical functions on classes of homogeneous superspaces are described explicitly.

Linear transformations and Restricted Isometry Property

The Restricted Isometry Property (RIP) introduced by Candés and Tao is a fundamental property in compressed sensing theory. It says that if a sampling matrix satisfies the RIP of certain order proportional to the sparsity of the signal, then the original signal can be reconstructed even if the sampling matrix provides a sample vector which is much smaller in size than the original signal. This short note addresses the problem of how a linear transformation will affect the RIP. This problem arises from the consideration of extending the sensing matrix and the use of compressed sensing in different bases. As an application, the result is applied to the redundant dictionary setting in compressed sensing.

Gaussian binomials and the number of sublattices.

The purpose of this short communication is to make some observations on the connections between various existing formulas of counting the number of sublattices of a fixed index in an n-dimensional lattice and their connection with the Gaussian binomials.

FINITELY GENERATED NIL BUT NOT NILPOTENT EVOLUTION ALGEBRAS

To use evolution algebras to model population dynamics that both allow extinction and introduction of certain gametes in finite generations, nilpotency must be built into the algebraic structures of these algebras with the entire algebras not to be nilpotent if the populations are assumed to evolve for a long period of time. To adequately address this need, evolution algebras over rings with nilpotent elements must be considered instead of evolution algebras over fields. This paper develops some criteria, which are computational in nature, about the nilpotency of these algebras, and shows how to construct finitely generated evolution algebras which are nil but not nilpotent.