Computing Moments by Prefix Sums

Moments of images are widely used in pattern recognition, because in suitable form they can be made invariant to variations in translation, rotation and size. However the computation of discrete moments by their definition requires many multiplications which limits the speed of computation. In this paper we express the moments as a linear combination of higher order prefix sums, obtained by iterating the prefix sum computation on previous prefix sums, starting with the original function values. Thus the pth moment can be computed by O (N · p) additions followed by p multiply-adds. The prefix summations can be realized in time O(N) using p + 1 simple adders, and in time O(p log N) using parallel prefix computation and O(N) adders. The prefix sums can also be used in the computation of two-dimensional moments for any intensity function f(x,y). Using a simple bit-serial addition architecture, it is sufficient with 13 full adders and some shift registers to realize the 10 order 3 image moment computations for a 512 × 512 size image at the TV rate. In 1986 Hatamian published a computationally equivalent algorithm, based on a cascade of filters performing the summations. Our recursive derivation allows for explicit expressions and recursive equations for the coefficients used in the final moment calculation. Thus a number of alternative forms for the moment computation can be derived, based on different sets of prefix sums. It is also shown that similar expressions can be obtained for the moments introduced by Liao and Pawlak in 1996, forming better approximations to the exact geometric moments, at no extra computational cost.     

2D and 3D image localization,compression and reconstruction using new hybrid moments

In this article, we will present a new set of hybrid polynomials and their corresponding moments, with a view to using them for the localization, compression and reconstruction of 2D and 3D images. These polynomials are formed from the Hahn and Krawtchouk polynomials. The process of calculating these is successfully stabilized using the modified recurrence relations with respect to the n order, the variable x and the symmetry property. The hybrid polynomial generation process is carried out in two forms: the first form contains the separable discrete orthogonal polynomials of Krawtchouk–Hahn (DKHP) and Hahn–Krawtchouk (DHKP). The latter are generated as the product of the discrete orthogonal Hahn and Krawtchouk polynomials, while the second form is the square equivalent of the first form, it consists of discrete squared Krawtchouk–Hahn polynomials (SKHP) and discrete polynomials of Hahn–Krawtchouk squared (SHKP). The experimental results clearly show the efficiency of hybrid moments based on hybrid polynomials in terms of localization property and computation time of 2D and 3D images compared to other types of moments; on the other hand, encouraging results have also been shown in terms of reconstruction quality and compression despite the superiority of classical polynomials.

     

Immittance-type tabular stability test for 2-D LSI systems based on a zero location test for 1-D complex polynomials

A new algebraic test is developed to determine whether or not a two-variable (2-D) characteristic polynomial of a recursive linear shift invariant (LSI, discrete-time) system is stable (i.e., it does not vanish in the closed exterior of the unit bi-circle). The method is based on the original form of a unit-circle zero location test for one variable (1-D) polynomials with complex coefficients proposed by the author. The test requires the construction of a table, in the form of a sequence of centrosymmetric matrices or 2-D polynomials, that is obtained using a certain three-term recursion, and examination of the zero location with respect to the unit circle of a few associated 1-D polynomials. The minimal set necessary and sufficient conditions for 2-D stability involves one 1-D polynomial whose zeros must reside inside the unit circle (which may be examined before the table is constructed), and one symmetric 1-D polynomial (which becomes available after completing the table) that is required not to have zeros on the unit circle. A larger set of intermediate necessary conditions for stability (which may be examined during the table's construction) are also given. The test compares favorably with Jury's recently improved 2-D stability test in terms of complexity and munerical stability.     

Spectral factorization of two-variable para-Hermitian polynomial matrices

This paper presents an algorithm for the so-called spectral factorization of two-variable para-Hermitian polynomial matrices which are nonnegative definite on thej axis, arising in the synthesis of two-dimensional (2-D)passive multiports, Wiener filtering of 2-D vector signals, and 2-D control systems design. First, this problem is considered in the scalar case, that is, the spectral factorization of polynomials is treated, where the decomposition of a two-variable nonnegative definite real polynomial in a sum of squares of polynomials in one of the two variables having rational coefficients in the other variable plays an important role (cf. Section 4). Second, by using these results, the matrix case can be accomplished, where in a first step the problem is reduced to the factorization of anunimodular para-Hermitian polynomial matrix which is nonnegative definite forp=j , and in a second step this simplified problem is solved by using so-called elementary row and column operations which are based on the Euclidian division algorithm. The matrices considered may be regular or singular and no restrictions are made concerning the coefficients of their polynomial entries; they may be either real or complex.     

An algorithm for stability determination of two-dimensional delta-operator formulated discrete-time systems

The recent interest in delta-operator (or, -operator) formulated discrete-time systems (or, -systems) is due mainly to (a) their superior finite wordlength characteristics as compared to their more conventional shift-operator (or,q-operator) counterparts (or,q-systems), and (b) the possibility of a more unified treatment of both continuous- and discrete-time systems. With such advantages, design, analysis, and implementation of two-dimensional (2-D) discrete-time systems using the -operator is indeed warranted. Towards this end, the work in this paper addresses the development of an easily implementabledirect algorithm for stability checking of 2-D -system transfer function models.Indirect methods that utilize transformation techniques are not pursued since they can be numerically unreliable. In developing such an algorithm, a tabular form for stability checking of -system characteristic polynomials with complex-valued coefficients and certain quantities that may be regarded as their corresponding Schur-Cohn minors are also proposed.     

Image analysis by Krawtchouk moments

A new set of orthogonal moments based on the discrete classical Krawtchouk polynomials is introduced. The Krawtchouk polynomials are scaled to ensure numerical stability, thus creating a set of weighted Krawtchouk polynomials. The set of proposed Krawtchouk moments is then derived from the weighted Krawtchouk polynomials. The orthogonality of the proposed moments ensures minimal information redundancy. No numerical approximation is involved in deriving the moments, since the weighted Krawtchouk polynomials are discrete. These properties make the Krawtchouk moments well suited as pattern features in the analysis of two-dimensional images. It is shown that the Krawtchouk moments can be employed to extract local features of an image, unlike other orthogonal moments, which generally capture the global features. The computational aspects of the moments using the recursive and symmetry properties are discussed. The theoretical framework is validated by an experiment on image reconstruction using Krawtchouk moments and the results are compared to that of Zernike, pseudo-Zernike, Legendre, and Tchebyscheff moments. Krawtchouk moment invariants are constructed using a linear combination of geometric moment invariants; an object recognition experiment shows Krawtchouk moment invariants perform significantly better than Hu's moment invariants in both noise-free and noisy conditions.     

Computation of simple and group factors of multivariate polynomials

This paper generalizes a recent result onsimple factorization of 2-variable (2-v) polynomials to simple andgroup factorization ofn-variate (n-v), (n3) polynomials. The emphasis is on developing a reliablenumerical technique for factorization. It is shown that simple as well as group factorization can be achieved by performing singular value decomposition (SVD) on certain matrices obtained from the coefficients of the givenn-v polynomial expressed in a Kronecker product form. For the polynomials that do not have exact simple and/or group factors, the concepts of approximate simple and group factorization are developed. The use of SVD leads to an elegant solution of an approximaten factorization problem. Several nontrivial examples are included to illustrate the results presented in this paper.Research supported by WRDC grant F33615-88-C-3605, NSF grant ECS-9110636, and NSERC of Canada grant A1345.     

Built-in Test with Modified-Booth High-Speed Pipelined Multipliers and Dividers

An embedded test pattern generator scheme for large-operand (unlimited bit length) multiplier and divider is presented by employing a simple digital circuit. This scheme is based on the generation of cyclic code polynomials from a characterized polynomials generator G(X) and incorporated with Modified-Booth algorithm. Due to the advantages of the former, the hardware complexity is simple, and moreover, the multiplier and divider can share the same hardware with a small change of control lines. Due to the advantages of latter's schemes, the numbers of sub/add operations are reduced to one half of the multiplicand for the result of final product. Therefore, the proposed pipelined multipliers permit very high throughput for arbitrary value of digit size. Only full adders/subtractors and shift registers are used in the proposed multiplier and divider hardware. The input data of the multiplier/divider can be processed in parallel or in pipelined without considering carry/borrow delays during the operations. The speed of computation has therefore been greatly improved by approximately a factor of 2. Since most parts of the components can be used for both the multiplier and divider, with full adders replaced by subtractors for switching from a multiplier to a divider, the structure is therefore tremendously reduced. In addition, these function units are involved with cyclic code generators, so that they can be used as a built-in self-test (BIST).     

Real-Polynomial Based Immittance-Type Tabular Stability Test for Two-Dimensional Discrete Systems

The paper proposes a new stability test for two-dimensional (2D) discrete systems. It is a tabular test; that is, it builds for the tested bivariate polynomial of degree (n₁, n₂) a sequence of n₂ (or n₁) bivariate polynomials or matrices (the 2D table) of increasing row and decreasing column sizes. It is an immittance-type test, which means that it uses a three-term recurrence relation to obtain a sequence of matrices with certain symmetry. It differs from some recent immittance tabular tests in that it is derived from the authors stability test for real polynomials instead of complex-coefficient polynomials. In comparison with related 2D stability tests developed before by Karan and Sarisvastava and by Premaratne, it simplifies the number of stability conditions and reduces the overall cost of computation from an exponential to a polynomial order of complexity.     

Full-wave analysis of radiation effect of microstrip transmission lines

A full-wave analysis of radiation effects produced by discontinuities in microstrip and buried microstrip transmission lines is presented. Beginning with the dyadic Green's function for a dielectric slab, with an embedded source, an integral equation is formulated. This equation is then solved by the method of moments to obtain the current distributions along the transmission line, in particular, near the discontinuities. Employing these results, the near- and far-zone fields, as well as radiation patterns are computed. The results from our method showed good agreement with those of previous publications in complex reflection and transmission coefficients, and equivalent capacitance values. It is found that under resonance conditions the radiation efficiency of a simple structure can exceed 41%, which may cause a potential problem in electromagnetic compatibility. Our analytic result also shows that the maximum radiation occurs when the source is located at the height of from the bottom ground plane, which should be prevented.     

Reflections on Schur-Cohn matrices and Jury-Marden tables and classification of related unit circle zero location criteria

We use the so-calledreflection coefficients (RCs) to examine, review, and classify the Schur-Cohn and Marden-Jury (SCMJ) class of tests for determining the zero location of a discrete-time system polynomial with respect to the unit circle. These parameters are taken as a platform to propose a partition of the SCMJ class into four useful types of schemes. The four types differ in the sequence of polynomials (the table) they associate with the tested polynomials by scaling factors: (A) a sequence of monic polynomials, (B) a sequence of least arithmetic operations, (C) a sequence that produces the principal minors of the Schur-Cohn matrix, and (D) a sequence that avoids division arithmetic. A direct derivation of a zero location rule in terms of the RCs is first provided and then used to track a proper zero location rule in terms of the leading coefficients of the polynomials of the B, C, and D scheme prototypes. We review many of the published stability tests in the SCMJ class and show that each can be sorted into one of these four types. This process is instrumental in extending some of the tests from stability conditions to zero location, from real to complex polynomial, in providing a proof of tests stated without a proof, or in correcting some inaccuracies. Another interesting outcome of the current approach is that a byproduct of developing a zero location rule for the Type C test is one more proof for the relation between the zero location of a polynomial and the inertia of its Schur-Cohn matrix.This research was supported by Grant No. 88-425 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.     

A Signal Conditioning Circuit for Piezoresistive Pressure Sensors With Variable Pulse-Rate Output

The equivalent impedance of the conventional ideal inductance implemented from two second-generation current conveyors is firstly calculated taking all the parasitic elements into account. Its equivalent electrical schema, which comprises six components is characterized. It is demonstrated that the most important deviation at high frequencies comes from the phase shifts of the transfers of the conveyors. Compensation of these effects are obtained from the first-order compensation method using a single additive resistor. SPICE simulations using an industrial BiCMOS process are used to demonstrate the validity of this approach. As an example, the current conveyors being DC biased with I₀ = 100 A and supplied under ±2.5 V, an inductance of 0.67 H was found directly usable without compensation up to about 15 MHz. This frequency range is then extended up to about 100 MHz when the circuit has been compensated from a single resistor of 75 .     

A numerical algorithm for the diffusion equation using 3D FEM and the Arnoldi method

In this paper we introduce a new computational method for solving the diffusion equation. In particular, we construct a generalized state-space system and compute the impulse response of an equivalent truncated state-space system. In this effort, we use a 3D finite element method (FEM) to obtain the state-space system. We then use the Arnoldi iteration to approximate the state impulse response by projecting on the dominant controllable subspace. The idea exploited here is the approximation of the impulse response of the linear system. We study the homogeneous and heterogeneous cases and discuss the approximation error. Finally, we compare our computational results to our experimental setup.This research was supported by the Central Research Laboratory, Hamamatsu Photonics K.K., 5000 Hirakuchi, Hamakita 434, Japan.     

Diagnostic simulation of stuck-at faults in combinational circuits

Two faults are said to be equivalent, with respect to a test set , iff they cannot be distinguished by any test in . The sizes of the corresponding equivalence classes of faults are used as a basis for comparing the diagnostic capability of two given test sets. A novel algorithm, called multiway list splitting, for computing the Equivalence Classes of stuck-at faults, in combinational (full scan) circuits, with respect to a given test set is presented. Experimental results presented show the algorithm to be more efficient than previously known algorithms based on decision diagrams and diagnosibility matrix.Portions of this work were presented in [1].Research Supported by NFS Grant No. MIP9102509.Research Supported by SRC Grant 93-DP-109.     

Avoiding linear dependencies in LFSR test pattern generators

Linear Feedback Shift Registers (LFSRs) constitute a very efficient mechanism for generating pseudoexhaustive or pseudo-random test sets for the built-in self-testing of digital circuits. However, a well-known problem with the use of LFSRs is the occurrence of linear dependencies in the generated patterns. In this paper, we show for the first time that the amount of linear dependencies can be controlled by selecting appropriate characteristic polynomials and reordering the LFSR cells. We identify two classes of such polynomials which, by appropriate LFSR cell ordering, guarantee that a large ratio of linear dependencies cannot occur. Experimental results show significant enhancements on the fault coverage for pseudo-random testing and support the theoretical relation between minimization of linear dependencies and effective fault coverage.This work was partially supported by NSF grant MIP-9409905, a 1993–94 ACM/IEEE Design Automation Scholarship and a grant from Nissan Corporation. A preliminary version of this work appeared in A Class of Good Characteristic Polynomials for LFSR Test Pattern Generators, in Proc. of IEEE International Conference on Computer Design, Oct. 1994, pp. 292–295, where it received the ICCD'94 Best Paper Award.     

CMOS Front-end Amplifier Dedicated to Monitor Very Low Amplitude Signal from Implantable Sensors

We describe in this paper a low-noise, low-power and low-voltage analog front-end amplifier dedicated to very low amplitude signal recording and processing applications. Our main focus is acquiring action potentials from peripheral nerves to recuperate lost functions in paralyzed patients. Low noise and low DC offset are realized using Chopper stabilization (CHS) technique. In addition, due to the use of a rail-to-rail input stage, low power supply (1.8 V) and wide common mode input range (0–1.8 V) are achieved also. It features a gain of 51 dB and a bandwidth of 4.5 kHz. The equivalent input noise is about 56 nV/ . The proposed preamplifier includes a matching clock generator, a 4th order continuous-time low-pass filter and an instrumentation amplifier. The proposed design has been implemented in 0.35 m double-poly n-well CMOS process with an active die area of 450 × 1150 m². The total data acquisition device consumes only 775 W.