Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Industry experience indicates that the ability to incrementally expand data centers is essential. However, existing high-bandwidth network designs have rigid structure that interferes with incremental expansion. We present Jellyfish, a high-capacity network interconnect which, by adopting a random graph topology, yields itself naturally to incremental expansion. Somewhat surprisingly, Jellyfish is more cost-efficient than a fat-tree, supporting as many as 25% more servers at full capacity using the same equipment at the scale of a few thousand nodes, and this advantage improves with scale. Jellyfish also allows great flexibility in building networks with different degrees of oversubscription. However, Jellyfish's unstructured design brings new challenges in routing, physical layout, and wiring. We describe approaches to resolve these challenges, and our evaluation suggests that Jellyfish could be deployed in today's data centers.

/pdf/jellyfish-networking-data-centers-randomly-275suk1xr5.pdf

Jellyfish: networking data centers randomly

Data centers (DCs), owing to the exponential growth of Internet services, have emerged as an irreplaceable and crucial infrastructure to power this ever-growing trend. A DC typically houses a large number of computing and storage nodes, interconnected by a specially designed network, namely, DC network (DCN). The DCN serves as a communication backbone and plays a pivotal role in optimizing DC operations. However, compared to the traditional network, the unique requirements in the DCN, for example, large scale, vast application diversity, high power density, and high reliability, pose significant challenges to its infrastructure and operations. We have observed from the premium publication venues (e.g., journals and system conferences) that increasing research efforts are being devoted to optimize the design and operations of the DCN. In this paper, we aim to present a systematic taxonomy and survey of recent research efforts on the DCN. Specifically, we propose to classify these research efforts into two areas: 1) DCN infrastructure and 2) DCN operations. For the former aspect, we review and compare the list of transmission technologies and network topologies used or proposed in the DCN infrastructure. For the latter aspect, we summarize the existing traffic control techniques in the DCN operations, and survey optimization methods to achieve diverse operational objectives, including high network utilization, fair bandwidth sharing, low service latency, low energy consumption, high resiliency, and etc., for efficient DC operations. We finally conclude this survey by envisioning a few open research opportunities in DCN infrastructure and operations.

A Survey on Data Center Networking (DCN): Infrastructure and Operations

The present communication relates to the theory of the solution, in positive and negative integral numbers, of systems of linear indeterminate equations, having integral coefficients. In connexion with this theory, a solution is also given of certain problems relating to rectangular matrices, composed of integral numbers, which are of frequent use in the higher arithmetic. Of this kind are the two following:—1. 1. \"Given (in integral numbers) the values of the determinants of any rectangular matrix of given dimensions, to find all the matrices, the constituents of which are integers, and the determinants of which have those given values.

http://rspl.royalsocietypublishing.org/content/11/86.full.pdf

I. On systems of linear indeterminate equations and congruences

Numerical simulation of electromagnetic, thermal, and mechanical responses of the human body to different stimuli in magnetic resonance imaging safety, antenna research, electromagnetic tomography, and electromagnetic stimulation is currently limited by the availability of anatomically adequate and numerically efficient cross-platform computational models or “virtual humans.” The objective of this study is to provide a comprehensive review of modern human models and body region models available in the field and their important features.

Virtual Human Models for Electromagnetic Studies and Their Applications

Despite the many proposals for data center network (DCN) architectures, designing a DCN remains challenging. DCN design is especially difficult when expanding an existing network, because traditional DCN design places strict constraints on the topology (e.g., a fat-tree). Recent advances in routing protocols allow data center servers to fully utilize arbitrary networks, so there is no need to require restricted, regular topologies in the data center. Therefore, we propose a data center network design framework, that we call REWIRE, to design networks using an optimization algorithm. Our algorithm finds a network with maximal bisection bandwidth and minimal end-to-end latency while meeting user-defined constraints and accurately modeling the predicted cost of the network. We evaluate REWIRE on a wide range of inputs and find that it significantly outperforms previous solutions—its network designs have up to 100–500% more bisection bandwidth and less end-to-end network latency than equivalent-cost DCNs built with best practices.

REWIRE: An optimization-based framework for unstructured data center network design

We introduce a reliable intersection algorithm for manifold surface meshes. The proposed algorithm builds conforming surface meshes from a set of intersecting triangulated surfaces. This algorithm effectively handles all degenerate triangle---triangle intersection cases. The key idea of the algorithm is based on an extensive set of triangle---edge intersection cases, combined with an intersection curve tracking method. The intersection operations do not rely on global spatial search operations and no remeshing steps are needed. The intersection curves are introduced into each surface mesh using a unique curve imprinting algorithm. The imprinting algorithm naturally handles degenerate intersection cases of many surfaces at an edge or at a point. The algorithm produces a consistent mesh data structure for subsequent mesh optimization operations. The mesh intersection algorithm is used within a general framework for modelling and meshing of geological formations, which are essential for reliable mathematical modelling of oil reservoirs.

A reliable triangular mesh intersection algorithm and its application in geological modelling

We present the design and implementation of a generative geometric kernel. The kernel generator is generic, type-safe, parametrized by many design-level choices and extensible. The resulting code has minimal traces of the design abstractions. We achieve genericity through a layered design deriving concepts from affine geometry, linear algebra and abstract algebra. We achieve parametrization and type-safety by using OCaml's module system, including higher order modules. The cost of abstraction is removed by using MetaOCaml's support for code generation coupled with some annotations atop the code type.

/pdf/a-generative-geometric-kernel-5d42z2ocf4.pdf

A generative geometric kernel

We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. 
The algorithms use the \emph{black-box} model which is suitable for sparse and structured matrices. The algorithms depend on a number of tools, such as matrix rank computation over finite fields, for which the best-known time- and memory-efficient algorithms are probabilistic. 
For an $\nxn$ matrix $A$ over the ring $\Fzfe$, where $f^e$ is a power of an irreducible polynomial $f \in \Fz$ of degree $d$, our algorithm requires $\bigO(\eta de^2n)$ operations in $\F$, where our black-box is assumed to require $\bigO(\eta)$ operations in $\F$ to compute a matrix-vector product by a vector over $\Fzfe$ (and $\eta$ is assumed greater than $\Pden$). The algorithm only requires additional storage for $\bigO(\Pden)$ elements of $\F$. In particular, if $\eta=\softO(\Pden)$, then our algorithm requires only $\softO(n^2d^2e^3)$ operations in $\F$, which is an improvement on known dense methods for small $d$ and $e$. 
For the ring $\ZZ/p^e\ZZ$, where $p$ is a prime, we give an algorithm which is time- and memory-efficient when the number of nontrivial invariant factors is small. We describe a method for dimension reduction while preserving the invariant factors. The time complexity is essentially linear in $\mu n r e \log p,$ where $\mu$ is the number of operations in $\ZZ/p\ZZ$ to evaluate the black-box (assumed greater than $n$) and $r$ is the total number of non-zero invariant factors. 
To avoid the practical cost of conditioning, we give a Monte Carlo certificate, which at low cost, provides either a high probability of success or a proof of failure. The quest for a time- and memory-efficient solution without restrictions on the number of nontrivial invariant factors remains open. We offer a conjecture which may contribute toward that end.

Fast Computation of Smith Forms of Sparse Matrices Over Local Rings

We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the black-box model which is suitable for sparse and structured matrices. The algorithms depend on a number of tools, such as matrix rank computation over finite fields, for which the best-known time- and memory-efficient algorithms are probabilistic.For an n x n matrix A over the ring F[z]/(fe), where fe is a power of an irreducible polynomial f ∈ F[z] of degree d, our algorithm requires O(ηde2n) operations in F, where our black-box is assumed to require O(η) operations in F to compute a matrix-vector product by a vector over F[z]/(fe) (and η is assumed greater than nde). The algorithm only requires additional storage for O(nde) elements of F. In particular, if η = O(nde), then our algorithm requires only O(n2d2e3) operations in F, which is an improvement on known dense methods for small d and e.For the ring Z/peZ, where p is a prime, we give an algorithm which is time- and memory-efficient when the number of nontrivial invariant factors is small. We describe a method for dimension reduction while preserving the invariant factors. The time complexity is essentially linear in μnre log p, where μ is the number of operations in Z/pZ to evaluate the black-box (assumed greater than n) and r is the total number of non-zero invariant factors. To avoid the practical cost of conditioning, we give a Monte Carlo certificate, which at low cost, provides either a high probability of success or a proof of failure. The quest for a time- and memory-efficient solution without restrictions on the number of nontrivial invariant factors remains open. We offer a conjecture which may contribute toward that end.

Mustafa Elsheikh

Papers

REWIRE: An optimization-based framework for unstructured data center network design

A reliable triangular mesh intersection algorithm and its application in geological modelling

A generative geometric kernel

Fast Computation of Smith Forms of Sparse Matrices Over Local Rings

Fast computation of Smith forms of sparse matrices over local rings