What's in a set of points? [straight line fitting]

The problem of fitting a straight line to a planar set of points is reconsidered. A parameter space computational approach capable of fitting one or more lines to a set of points is presented. The suggested algorithm handles errors in both coordinates of the data points, even when the error variances vary between coordinates and among points and can be readily made robust to outliers. The algorithm is quite general and allows line fitting according to several useful optimality criteria to be performed within a single computational framework. It is observed that certain extensions of the Hough transform can be turned to be equivalent to well-known M estimators, thus allowing computationally efficient approximate M estimation     

On reliable computation with formal neurons

The authors investigate the computing capabilities of formal McCulloch-Pitts neurons when errors are permitted in decisions. They assume that m decisions are to be made on a randomly specified m set of points in n space and that an error tolerance of ϵm decision errors is allowed, with 0⩽ϵ<1/2. The authors are interested in how large an m can be selected such that the neuron makes reliable decisions within the prescribed error tolerance. Formal results for two protocols for error-tolerance-a random error protocol and an exhaustive error protocol-are obtained. The results demonstrate that a formal neuron has a computational capacity that is linear in n and that this rate of capacity growth persists even when errors are tolerated in the decisions     

True 3D animation for displaying VLSI modeling data

The authors report on a flexible system that provides visual feedback to VLSI designers with a novel display method. The basic problem is to display simultaneously two or more functions f ₁, f₂, . . ., each of which depends on the two spatial variables x and y. The method is based on results from visual perception experiments, which indicated that the human visual system can view simultaneously two or more images that are separated in depth, even if they are combined (superimposed) over a common spatial domain. Two display modes have been implemented: a gray-level mode and a contour mode in which only the edges separating adjacent regions are displayed in an effort to make things easier for the user. This depth-separation technique allows the viewer to register spatially the x,y distribution of multiple variables in an animation sequence. The technique can be applied to many other situations in which a single-plane episode involves several variables     

Output stabilizability of discrete-event dynamic systems

The authors investigate the problem of designing stabilizing feedback compensators for discrete-event dynamic systems (DEDS) modeled as finite-state automata in which some transition events are controllable and some events are observed. The problem of output stabilization is defined as the construction of a compensator such that all state trajectories in the closed-loop system go through a given set E infinitely often. The authors also define a stronger notion of output stabilizability which requires that the state not only pass through E infinitely often but that the set of instants when the state is in E and one knows it is in E is also infinite. Necessary and sufficient conditions are presented for both notions. The authors also introduce and characterize a notion of resiliency that corresponds to the system being able to recover from observation errors. In addition, they provide some general bounds for the algorithms considered and discuss several conditions under which far smaller bounds can be achieved     

Out-of-roundness problem revisited

The properties and computation of the minimum radial separation (MRS) standard for out-of-roundness are discussed. Another standard out-of-roundness measurement called the minimum area difference (MAD) center is introduced. Although the two centers have different characteristics, the approach to finding both centers shares many commonalities. An O(n log n+k) time algorithm which is used to compute the MRS center is presented. It also computes the MAD center of a simple polygon G, where n is the number of vertices of G, and k is the number of intersection points of the medial axis and the farthest-neighbor Voronoi diagram of G. The relationship between MRS and MAD is discussed     

Constant time algorithms for the transitive closure and somerelated graph problems on processor arrays with reconfigurable bussystems

The transitive closure problem in O(1) time is solved by a new method that is far different from the conventional solution method. On processor arrays with reconfigurable bus systems, two O (1) time algorithms are proposed for computing the transitive closure of an undirected graph. One is designed on a three-dimensional n×n×n processor array with a reconfigurable bus system, and the other is designed on a two-dimensional n²×n² processor array with a reconfigurable bus system, where n is the number of vertices in the graph. Using the O(1) time transitive closure algorithms, many other graph problems are solved in O(1) time. These problems include recognizing bipartite graphs and finding connected components, articulation points, biconnected components, bridges, and minimum spanning trees in undirected graphs     

Distributed detection with consulting sensors and communicationcost

The problem of distributed detection with consulting sensors in the presence of communication cost associated with any exchange of information (consultation) between sensors is considered. The system considered has two sensors, S1 and S2; S1 is the primary sensor responsible for the final decision u₀ , and S2 is a consulting sensor capable of relaying its decision u₂ to S1 when requested by S 1. The final decision u₀ is either based on the raw data available to S1 only, or, under certain request conditions, also takes into account the decision u₂ of sensor S2. Random and nonrandom request schemes are analyzed and numerical results are presented and compared for Gaussian and slow-fading Rayleigh channels. For each decision-making scheme, an associated optimization problem is formulated whose solution is shown to satisfy certain set design criteria that the authors consider essential for sensor fusion     

Efficient routing schemes for multiple broadcasts in hypercubes

The authors analyze the problem in which each node of the binary hypercube independently generates packets according to a Poisson process with rate λ; each of the packets is to be broadcast to all other nodes. Assuming unit packet length and no other communications taking place, it is observed that the system can be stable in steady-state only if the load factor ρ≡λ (2^d-1)/d satisfies ρ<1 where d is the dimensionality (diameter) of the hypercube. Moreover, the authors establish some lower bounds for the steady-state average delay D per packet and devise and analyze two distributed routing schemes that are efficient in the sense that stability is maintained for all ρ<ρ* where ρ* does not depend on the dimensionality d of the network, while the average delay D per packet satisfies D⩽Kd(1+ρ) for small values of ρ (with constant K). The performance evaluation is rigorous for one scheme, while for the other the authors resort to approximations and simulations     

On quantization errors in computer vision

The author considers the error resulting in the computation of multivariable functions h(X₁, X, . . ., X_n), where all the X_is are only available in the quantized form. In image processing and computer vision problems, the variables are typically a mixture of the spatial coordinates and the intensity levels of objects in an image. A method is introduced using a first-order Taylor series expansion together with a periodic extension of the resulting expression and its Fourier series representation so that the moments and the probability distribution function of the error can be computed in closed form. This method only requires that the joint probability density function of X_i s be known and makes no assumption on the behavior on the quantization errors of the variables. Examples are also given where these results are applied     

A structured approximation problem with applications to frequencyweighted model reduction

The authors consider the problem of finding necessary and sufficient conditions for the existence of a Q with no more than k poles in the closed left-half plane such that the norm of a certain matrix ⩽1. If solutions exist, a formula for all solutions is given. Special attention is given to the characterization of all optimal solutions. As an application, the frequency weighted optimal Hankel norm model reduction problem is recast as a problem of characterizing every Q which satisfies a certain condition. After reformulation, the model reduction problem is easily solved using the techniques presented in this work     

On the number of digital straight line segments

A closed-form expression has been reported in the literature for L_N, the number of digital line segments of length N that correspond to lines of the form y=ax+β, O⩽α, β<1. The authors prove an asymptotic estimate for L_N that might prove useful for many applications, namely, L_N=N ³/π²+O(N² log N). An application to an image registration problem is given     

Efficient algorithms for list ranking and for solving graphproblems on the hypercube

A hypercube algorithm to solve the list ranking problem is presented. Let n be the length of the list, and let p be the number of processors of the hypercube. The algorithm described runs in time O(n/p) when n=Ω(p ^1+ε) for any constant ε>0, and in time O(n log n/p+log³ p) otherwise. This clearly attains a linear speedup when n=Ω(p ^1+ε). Efficient balancing and routing schemes had to be used to achieve the linear speedup. The authors use these techniques to obtain efficient hypercube algorithms for many basic graph problems such as tree expression evaluation, connected and biconnected components, ear decomposition, and st-numbering. These problems are also addressed in the restricted model of one-port communication     

Some properties of the E matrix in two-view motionestimation

In the eight-point linear algorithm for determining 3D motion/structure from two perspective views using point correspondences, the E matrix plays a central role. The E matrix is defined as a skew-symmetrical matrix (containing the translation components) postmultiplied by a rotation matrix. The authors show that a necessary and sufficient condition for a 3×3 matrix to be so decomposable is that one of its singular values is zero and the other two are equal. Several other forms of this property are presented. Some applications are briefly described     

Handling timing errors in distributed programs

The authors describe a tool called TAP, which is defined to aid the programmer in discovering the causes of timing errors in running programs. TAP is similar to a postmortem debugger, using the history of interprocess communication to construct a timing graph, a directed graph where an edge joins node x to node y if event x directly precedes event y in time. The programmer can then use TAP to look at the graph to find the events that occurred in an unacceptable order. Because of the nondeterministic nature of distributed programs, the authors feel a history-keeping mechanism but always be active so that bugs can be dealt with as they occur. The goal is to collect enough information at run time to construct the timing graph if needed. Since it is always active, this mechanism must be efficient. The authors also describe experiments run using TAP and report the impact that TAP's history-keeping mechanism has on the running time of various distributed programs     

Decomposition of 2-D linear systems into 1-D systems in thefrequency domain

A method is presented for the decomposition of the frequency domain of 2-D linear systems into two equivalent 1-D systems having dynamics in different directions and connected by a feedback system. It is shown that under some assumptions the decomposition problem can be reduced to finding a realizable solution to the matrix polynomial equation X(z₁)P(z₂ )+Q(z₁)Y(z₂ )=D(z₁, z₂). A procedure for finding a realizable solution X(z₁ ), Y(z₂) to the equation is given     

Output feedback eigenstructure assignment using two Sylvesterequations

The eigenstructure assignment problem with output feedback is studied for systems satisfying the condition p+m> n. The main tool used is the concept of (C, A, B)-invariance and two coupled Sylvester equations, the solution of which leads to the computation of an output stabilizing feedback. A computationally efficient algorithm for the solution of these two coupled equations, which leads to the computation of a desired output feedback, is presented     

A note on the survival time of a dynamic system in an interval

The one-dimensional system dx(t=bu(t)dt+(ct ²)^1/2dW(t), where b (≠0) and c (⩾0) are real constants and W(t ) is a standard Brownian motion, is considered. The aim is to obtain the control u* that minimizes the expected value of a cost function with terminal cost equal to 0 or +∞ depending on whether the survival time in a given region is at least equal to or less than a fixed time     

Breaking substitution cyphers using stochastic automata

Let Λ be a finite plaintext alphabet and V be a cypher alphabet with the same cardinality as Λ. In all one-to-one substitution cyphers, there exists the property that each element in V maps onto exactly one element in Λ and vice versa. This mapping of V onto Λ is represented by a function T*, which maps any v∈V onto some λ∈Λ (i.e., T*(v)=λ). The problem of learning the mapping of T* (or its inverse (T *)^-1) by processing a sequence of cypher text is discussed. The fastest reported method to achieve this is a relaxation scheme that utilizes the statistical information contained in the unigrams and trigrams of the plaintext language. A new learning automaton solution to the problem called the cypher learning automaton (CLA) is given. The proposed scheme is fast, and the advantages of the scheme in terms of time and space requirements over the relaxation method have been listed. Simulation results comparing both cypher-breaking techniques are presented     

Computing the width of a set

For a set of points P in three-dimensional space, the width of P, W (P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in O(n log n +I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and n is the number of vertices; in the worst case, I=O( n²). For a convex polyhedra the time complexity becomes O(n+I). If P is a set of points in the plane, the complexity can be reduced to O(nlog n). For simple polygons, linear time suffices     

Mapping nested loop algorithms into multidimensional systolicarrays

Consideration is given to transforming depth p-nested for loop algorithms into q-dimensional systolic VLSI arrays where 1⩽q⩽p-1. Previously, there existed complete characterizations of correct transformation only for the cases where q=p-1 or q=1. This gap is filled by giving formal necessary and sufficient conditions for correct transformation of a p-nested loop algorithm into a q-dimensional systolic array for any q, 1⩽q⩽p-1. Practical methods are presented. The techniques developed are applied to the automatic design of special purpose and programmable systolic arrays. The results also contribute toward automatic compilation onto more general purpose programmable arrays. Synthesis of linear and planar systolic array implementations for a three-dimensional cube-graph algorithm and a reindexed Warshall-Floyd path-finding algorithm are used to illustrate the method