Multiplicative Inverse Research Articles

Bayesian Estimation of Density via Multiple Sequential Inversions of Two-Dimensional Images With Application to Electron Microscopy

We present a new Bayesian methodology to learn the unknown material density of a given sample by inverting its two-dimensional images that are taken with a scanning electron microscope. An image results from a sequence of projections of the convolution of the density function with the unknown microscopy correction function that we also learn from the data; thus, learning of the unknowns demands multiple inversions. We invoke a novel design of experiment, involving imaging at multiple values of the parameter that controls the subsurface depth from which information about the density structure is carried, to result in the image. Real-life material density functions are characterized by high-density contrasts and are highly discontinuous, implying that they exhibit correlation structures that do not vary smoothly. In the absence of training data, modeling such correlation structures of real material density functions is not possible. So we discretize the material sample and treat values of the density function at chosen locations inside it as independent and distribution-free parameters. Resolution of the available image dictates the discretization length of the model; three models pertaining to distinct resolution classes (at micrometer to nanometer scale lengths) are developed. We develop priors on the material density, such that these priors adapt to the sparsity inherent in the density function. The likelihood is defined in terms of the distance between the convolution of the unknown functions and the image data. The posterior probability density of the unknowns given the data is expressed using the developed priors on the density and priors on the microscopy correction function as elicited from the microscopy literature. We achieve posterior samples using an adaptive Metropolis-within-Gibbs inference scheme. The method is applied to learn the material density of a three-dimensional sample of a nano-structure, using real image data. Illustrations on simulated image data of alloy samples are also included.

PERFIDI FILTERS VALIDATION: FROM NUCLEAR MAGNETIC RESONANCE RELAXOMETRY TO MAGNETIC RESONANCE IMAGING

In order to well distinguish different tissues of the human body by magnetic resonance imaging (MRI), it is of great importance to find procedures to improve the image contrast. In particular, a valuable feature is to image only specific parts of organs and/or tissues while ignoring all the others. Dedicated MRI sequences able to filter the 1 H nuclei signals based on the different longitudinal relaxation times (T1) of the tissues have been developed. Standard signal selection/attenuation sequences, such as the Short Time Inversion Recovery and Multiple Inversion Recovery, have the effect to zero the signal for a discrete number of T1 values. Parametrically Enabled Relaxation Filters with Double and multiple Inversion (PERFIDI) sequences act on a range of T1 values and behave as an electronic band-pass or high-pass or low-pass filters. PERFIDI filters are therefore primarily focused on the components that pass through, rather than on those that are blocked. These filters have been developed and tested by nuclear magnetic resonance relaxometry. Here, these sequences have been validated for MRI on phantom samples to mimic T1 distributions present in tissues. Preliminary applications show that PERFIDI filters can effectively work on a range of T1 values to give well contrasted images.

MR-based attenuation correction for PET/MRI neurological studies with continuous-valued attenuation coefficients for bone through a conversion from R2* to CT-Hounsfield units

AimMR-based correction for photon attenuation in PET/MRI remains challenging, particularly for neurological applications requiring quantitation of data. Existing methods are either not sufficiently accurate or are limited by the computation time required. The goal of this study was to develop an MR-based attenuation correction method that accurately separates bone tissue from air and provides continuous-valued attenuation coefficients for bone. Materials and methodsPET/MRI and CT datasets were obtained from 98 subjects (mean age [±SD]: 66yrs [±9.8], 57 females) using an IRB-approved protocol and with informed consent. Subjects were injected with 352±29MBq of 18F-Florbetapir tracer, and PET acquisitions were begun either immediately or 50min after injection. CT images of the head were acquired separately using a PET/CT system. Dual echo ultrashort echo-time (UTE) images and two-point Dixon images were acquired. Regions of air were segmented via a threshold of the voxel-wise multiplicative inverse of the UTE echo 1 image. Regions of bone were segmented via a threshold of the R2* image computed from the UTE echo 1 and UTE echo 2 images. Regions of fat and soft tissue were segmented using fat and water images decomposed from the Dixon images. Air, fat, and soft tissue were assigned linear attenuation coefficients (LACs) of 0, 0.092, and 0.1cm−1, respectively. LACs for bone were derived from a regression analysis between corresponding R2* and CT values. PET images were reconstructed using the gold standard CT method and the proposed CAR-RiDR method. ResultsThe RiDR segmentation method produces mean Dice coefficient±SD across subjects of 0.75±0.05 for bone and 0.60±0.08 for air. The CAR model for bone LACs greatly improves accuracy in estimating CT values (28.2%±3.0 mean error) compared to the use of a constant CT value (46.9%±5.8, p<10−6). Finally, the CAR-RiDR method provides a low whole-brain mean absolute percent-error (MAPE±SD) in PET reconstructions across subjects of 2.55%±0.86. Regional PET errors were also low and ranged from 0.88% to 3.79% in 24 brain ROIs. ConclusionWe propose an MR-based attenuation correction method (CAR-RiDR) for quantitative PET neurological imaging. The proposed method employs UTE and Dixon images and consists of two novel components: 1) accurate segmentation of air and bone using the inverse of the UTE1 image and the R2* image, respectively and 2) estimation of continuous LAC values for bone using a regression between R2* and CT-Hounsfield units. From our analysis, we conclude that the proposed method closely approaches (<3% error) the gold standard CT-scaled method in PET reconstruction accuracy.

Modeling thermal properties of plutonium mononitride

The thermal properties of plutonium mononitride (PuN) were investigated by molecular dynamics method. The interatomic potentials of PuN were fitted by using Chen–Möbius multiple lattice inversion technique. Based on these interatomic potentials, the lattice constant, bulk modulus, compressibility, cohesive energy and heat capacity of PuN were obtained and the results are well consistent with experimental data and previous reports. It indicates that the potentials we build in this study are effective for studying thermal properties of PuN.

Fault slip distribution of the 2014 Iquique, Chile, earthquake estimated from ocean‐wide tsunami waveforms and GPS data

AbstractWe applied a new method to compute tsunami Green's functions for slip inversion of the 1 April 2014 Iquique earthquake using both near‐field and far‐field tsunami waveforms. Inclusion of the effects of the elastic loading of seafloor, compressibility of seawater, and the geopotential variation in the computed Green's functions reproduced the tsunami traveltime delay relative to long‐wave simulation and allowed us to use far‐field records in tsunami waveform inversion. Multiple time window inversion was applied to tsunami waveforms iteratively until the result resembles the stable moment rate function from teleseismic inversion. We also used GPS data for a joint inversion of tsunami waveforms and coseismic crustal deformation. The major slip region with a size of 100 km × 40 km is located downdip the epicenter at depth ~28 km, regardless of assumed rupture velocities. The total seismic moment estimated from the slip distribution is 1.24 × 1021 N m (Mw 8.0).

Image contrast reversals in contact resonance atomic force microscopy

Multiple image contrast inversions are observed along with the increase of modulation frequency for contact resonance atomic force microscopy (CR-AFM) imaging of a highly oriented pyrolytic graphite (HOPG) specimen. Analysis of the contact vibrational spectra indicates that the inversions can be attributed to structure-induced variations of tip-sample contact mechanics. Contact stiffness and damping at HOPG step edges exhibit significant increases relative to those in the flat regions. For quantitative evaluation of mechanical properties in CR-AFM, coupling effects of the surface geometry must be considered.

Multiplicative-Regularized FFT Twofold Subspace-Based Optimization Method for Inverse Scattering Problems

In this paper, we combine two techniques together, i.e., the fast Fourier transform-twofold subspace-based optimiza- tion method (FFT-TSOM) and multiplicative regularization (MR) to solve inverse scattering problems. When applying MR to the objective function in the FFT-TSOM, the new method is referred to as MR-FFT-TSOM. In MR-FFT-TSOM, a new stable and effec- tive strategy of regularization has been proposed. MR-FFT-TSOM inherits not only the advantages of the FFT-TSOM, i.e., lower computational complexity than the TSOM, better stability of the inversion procedure, and better robustness against noise compared with the SOM, but also the edge-preserving ability from the MR. In addition, a more relaxed condition of choosing the number of current bases being used in the optimization can be obtained compared with the FFT-TSOM. Particularly, MR-FFT-TSOM has even better robustness against noise compared with the FFT- TSOM and multiplicative regularized contrast source inversion (MR-CSI). Numerical simulations including both inversion of synthetic data and experimental data from the Fresnel data set validate the efficacy of the proposed algorithm.

An Algorithm to Find the Irreducible Polynomials Over Galois Field GF(pm)

Irreducible Polynomials over GF(p) and the multiplicative inverses under it are important in cryptography. Presently the method of deriving irreducible polynomials of a particular prime modulus is very primitive and time consuming. In this paper, in order to find all irreducible polynomials, be it monic or non-monic, of all prime moduli p with all its order m, a fast deterministic computer algorithm based on an algebraic method producing a (m×m) matrix is proposed. The maximum number of terms in each column of the matrix is 2 where j is the column index. General Terms Algorithms, Irreducible polynomial.

Nonlinear response prediction of cracked rotor based on EMD

The empirical mode decomposition (EMD) method is introduced, to improve the prediction accuracy of cracked rotor׳s nonlinear response during a long-term period. The EMD method was applied to decompose the nonlinear response into series of intrinsic mode functions (IMF). Consequently, the prediction results of IMF were obtained, based on the maximal local Lyapunov exponent (LLE). By adding all the prediction results of IMF, the nonlinear response of cracked rotor can be predicted, called the IMF prediction method. Compared with the response predicted directly by the maximal local Lyapunov exponent, when the forecasting step is less than the maximal prediction time which is calculated by the multiplicative inverse of maximal Lyapunov exponent, the IMF method has the same prediction accuracy. When the forecasting step is greater than maximal prediction time, the IMF prediction method is more advantageous than the Lyapunov prediction method. Bently RK4 rotor test is used to validate the IMF prediction method׳s advantage.

Multiple heat sources with multi-parameter inversion of nondestructive infrared detection

This study deals with the case of multiple internal heat source inversion problem of steady-state based on nondestructive infrared detection. We construct homogeneous and heterogeneous steady heat conduction models of different shapes. Neither the number of heat sources, nor their locations, nor their sizes nor their intensities are known. We use the finite element method (FEM) based on numerical algorithm to analyze the two-dimensional model discretely. The internal heat conduction process of model is analyzed. The resultant temperature field can be decomposed into the temperature field caused by the ambient temperature and those given by internal heat sources. We simplify the finite element matrix equation according to decomposition process above. Finally, the problem boils down to solving the highly underdetermined matrix equation of Ax = b. The unknown x item corresponds to internal thermal heat source field. The piecewise polynomial spectral truncated singular value decomposition (PPTSVD) is applied for the first time to the inverse heat source problem. Its regular operator matrix is changed from the original more order differential operator matrix to regional node weighted matrix. After replacement this solution improves the effect of heat source field tending to the boundary. Results of the solution confirms a real heat source field when there are less heat sources or different heat sources are far from each other. But there also exits a serious superimposed effect between neighboring heat sources. We improve the algorithm to study this problem through using the iterative elimination process which complies with the idea of spreading heat source field and then gathering. The iteration tolerance and number of times belonging to one single PPTSVD solving process are reduced. Through iterating the multiple PPTSVD solving process and reconstructing matrixes A and b in each iteration, we obtain the scatter heat source field distribution surrounding real field. Finally, this scattered distribution solution is gathered again. According to the heat source parameter calculated by algorithm, the temperature field of whole model can be reconstructed using FEM. Comsol numerical simulation and real physical experiments are performed to verify the validity and accuracy of the algorithm in different heat conduction models. The results demonstrate that the algorithm can access each parameter of multiple heat sources. Even in the heterogeneous model, it can still obtain accurate results and reconstruct the temperature field of the two-dimensional model. The algorithm can be applied to non-destructive material detection and human infrared medical imaging fields.

An Algorithm to Generating Inverse S-box for Rijndael Encryption Standard

The S-box transformation is very important step for advanced encryption standard algorithm. The S-box values are generated from the multiplicative inverse of finite field with an affine transform. There are many techniques in literature to generate the multiplicative inverse values. In this paper, a software method of producing the multiplicative inverse values, which is the generator of S-box values will be discussed. The proposed technique is based on the mathematical concept of log and antilog. Xtime function.

Division by zero in non-involutive meadows

Meadows have been proposed as alternatives for fields with a purely equational axiomatization. At the basis of meadows lies the decision to make the multiplicative inverse operation total by imposing that the multiplicative inverse of zero is zero. Thus, the multiplicative inverse operation of a meadow is an involution. In this paper, we study ‘non-involutive meadows’, i.e. variants of meadows in which the multiplicative inverse of zero is not zero, and pay special attention to non-involutive meadows in which the multiplicative inverse of zero is one.

Low-complexity digit-serial systolic Gaussian normal basis multiplier for curve-based cryptosystems

Finite field multiplication over GF(2m) is one of the most important arithmetic operations for curve-based cryptosystems, such as elliptic curve cryptosystems and pairing-based cryptosystems. Gaussian normal basis (GNB) multiplier is very suitable for realizing finite field multiplication in curve-based cryptosystems because GNB multiplier is highly efficient in performing square operations of multiplicative inversion, squaring, and exponentiation operations. Therefore, this study will present a digit-serial GNB multiplier with unidirectional systolic array which is suitable for fault-tolerant design and very large scale integrated circuits implementation. The proposed digit-serial GNB multiplier can save about 22% space complexity and requires almost the same time complexity when compared with the existing similar multipliers.

Adolescents' understanding of inversion and associativity

The present study investigated when an understanding of the mathematical concepts of inversion and associativity matures and whether the application of these concepts to problem solving requires the interruption of computational strategies (e.g., Siegler & Araya, 2005). In the study, 40 adolescent participants per grade from Grades 7, 9, and 11 and 40 adult participants solved multiplication and division inversion and associativity problems and completed a task that measured whether the execution of the inversion shortcut or associativity strategy prevents the execution of competing computational strategies. Inversion shortcut use approached adult levels in Grade 9. Associativity strategy use significantly increase in early adulthood. Also, there was considerable individual variability in strategy use. Finally, the execution of both conceptually-based strategies interrupted computational strategies. Thus, adolescence is an important developmental period for understanding multiplicative concepts and applying conceptual mathematical knowledge to problem solving may require the interruption of procedural knowledge.

Toward Formal Design of Practical Cryptographic Hardware Based on Galois Field Arithmetic

This paper presents a formal method for designing cryptographic processor datapaths on the basis of arithmetic circuits over Galois fields (GFs). The proposed method describes GF arithmetic circuits in the form of hierarchical graph structures, where nodes represent sub-circuits whose functions are defined by arithmetic formulae over GFs, and edges represent data dependency between nodes. In this paper, we first introduce the application of graph representation to arithmetic circuits over extension fields of ${\mbi {GF}}({{\mbi {p}}^{\mbi {m}}})$ $({\mbi {p}} \geq {\bf 2})$ and composite fields, which are commonly used in the design of cryptographic processors. The newly proposed graph representation can be formally verified through symbolic computation techniques based on polynomial reduction and Grobner basis. We then demonstrate the capabilities of the proposed approach through an experimental design of a 128-bit AES (Advanced Encryption Standard) datapath including multiplicative inversion circuits over the composite field ${\mbi {GF}}{(((2^2)^2)^2})$ . The results show that the proposed method can describe such practical datapaths, as well as that complete verification of such a datapath can be carried out within a short period of time.

특수한 정규기저를 이용한 유한체위에서의 역원 계산 알고리즘에 관한 연구

유한체 연산에서 MONB를 사용하면 곱셈 역원 계산시에 대량의 제곱계산이 필요하므로 역원을 계산하는 데에 긴 시간이 필요하게 된다. 이에 본 논문에서는 바탕체 <TEX>$GF(2^{2n})$</TEX> 위의 확대체 <TEX>$GF(2^{2^nm})^*$</TEX>에서 특수한 정규기저를 사용하여 역원을 구하는 저 비용의 알고리즘을 제안한다. 제안하는 알고리즘을 사용하면 곱셈 역원 계산에는 <TEX>$nb(2^nm-1)+w(2^nm-1)-2$</TEX>번의 곱셈과 <TEX>$2^n-1$</TEX>번의 제곱연산이 소요되며, H/W에서 구현한 결과 Itoh 등의 방법 보다 곱셈역원 계산속도가 빠르게 나타났다. Since the computation of a multiplicative inverse using MONB includes many squarings and thus calculating inverse is expensive, we, in this paper, propose a low cost inverse algorithm requiring <TEX>$nb(2^nm-1)+w(2^nm-1)-2$</TEX> multiplications and <TEX>$2^n-1$</TEX> squarings to compute an inverse in <TEX>$GF(2^{2^nm})^*$</TEX> using special normal basis over <TEX>$GF(2^{2^n})$</TEX>, and give some implementation results using the algorithm and, show that the timing results of our implementation is faster than that of Itoh et al.'s method.

Dreamlet-based interpolation using POCS method

Due to incomplete and non-uniform coverage of the acquisition system and dead traces, real seismic data always has some missing traces which affect the performance of a multi-channel algorithm, such as Surface-Related Multiple Elimination (SRME), imaging and inversion. Therefore, it is necessary to interpolate seismic data. Dreamlet transform has been successfully used in the modeling of seismic wave propagation and imaging, and this paper explains the application of dreamlet transform to seismic data interpolation. In order to avoid spatial aliasing in transform domain thus getting arbitrary under-sampling rate, improved Jittered under-sampling strategy is proposed to better control the dataset. With L0 constraint and Projection Onto Convex Sets (POCS) method, performances of dreamlet-based and curvelet-based interpolation are compared in terms of recovered signal to noise ratio (SNR) and convergence rate. Tests on synthetic and real cases demonstrate that dreamlet transform has superior performance to curvelet transform.

Investigating fault reactivation during multiple tectonic inversions through mechanical and numerical modeling: An application to the Central-Northern Apennines of Italy

Within inversion tectonic contexts, the reactivation of pre-existing discontinuities depends upon many structural, stratigraphic and mechanical factors. Among these, a fundamental aspect is represented by the orientation and attitude of pre-existing discontinuities with respect to the new stress field. The aim of this work is to evaluate the stress regimes and the mechanical conditions that controlled the reactivation of segmented pre-thrusting normal faults in the Central-Northern Apennines of Italy. These faults were involved both in positive and negative inversion tectonic events during Tertiary-Quaternary times. Mechanical (slip tendency analysis) and numerical (using COULOMB 3.3 software) models allowed us to reconstruct a 3D stress field for both inversion events and to evaluate the effect of many parameters (orientation of faults, fault dip-angle and mechanical parameters) on fault reactivation. Our results highlighted that the orientation of pre-existing faults with respect to the stress axes represents the main factor controlling inversion tectonic phenomena, where other mechanical and geometric variables exert a subordinate role. The models well reproduce the Neogene-Quaternary deformation history reconstructed from detailed geological and structural analyses for the Central-Northern Apennines of Italy and could be applied to other regions that have experienced inversion tectonic processes.

Near-regional CMT and multiple-point source solution of the September 5, 2012, Nicoya, Costa Rica Mw 7.6 (GCMT) earthquake

We use acceleration data from the Observatorio Vulcanologico y Sismologico, Universidad Nacional de Costa Rica (OVSICORI-UNA) and Laboratorio de Ingenieria Sismica, Universidad de Costa Rica (LIS-UCR) seismic network for the relocation and moment-tensor solution of the September 5, 2012, 14:42:03.35 UTC, Nicoya, Costa Rica earthquake (Mw 7.6 GCMT). Using different relocation methods we found a stable earthquake hypocenter, near the original OVSICORI-UNA location in the Nicoya Peninsula, NW Costa Rica at Lat 9.6943°N, Lon 85.5689°W, depth 15.3 km, associated with the subduction of the Cocos plate under Caribbean plate. Acceleration records at OVSICORI-UNA and LIS-UCR stations (94–171 km), at 0.03 < f < 0.06 Hz were used in the waveform inversion for a single-point centroid moment tensor (CMT). Using spatial grid search the centroid position was found at the depth of 30 km, situated at Lat 10.0559°N, Lon 85.4778°W, i.e. of about 41 km NNE from the epicenter. The centroid time is 14:42:18.89 UTC, i.e. 15.54 s later relative to the location-based origin time. The nodal plane (strike 318°, dip 27° and rake 115°) is the fault plane that agrees with the geometry of the subducted slab at Nicoya, NNW Costa Rica. Increasing the maximum studied frequency from 0.06 to 0.15 Hz, the multiple point source inversion model leads to two subevents. The first one was located near the centroid and the second subevent was situated 20 km along strike and 10 km down dip from the first subevent and 6 s later. The uncertainty of the source model was carefully examined using complementary inversion methods, viz the iterative deconvolution and non-negative least squares.

On Newton–Raphson Iteration for Multiplicative Inverses Modulo Prime Powers

We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over $p$-adic numbers gives a recurrence relation computing modular inverse modulo $p^m$, that is logarithmic in $m$. We solve the recurrence to obtain an explicit formula for the inverse. Then we study different implementation variants of this iteration and show that our explicit formula is interesting for small exponent values but slower or large exponent, say of more than $700$ bits. Overall we thus propose a hybrid combination of our explicit formula and the best asymptotic variants. This hybrid combination yields then a constant factor improvement, also for large exponents.