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NASA NTRS1993

Parallel algorithm for dominant points correspondences in robot binocular stereo vision

This paper presents an algorithm to find the correspondences of points representing dominant feature in robot stereo vision. The algorithm consists of two main steps: dominant point extraction and dominant point matching. In the feature extraction phase, the algorithm utilizes the widely used Moravec Interest Operator and two other operators: the Prewitt Operator and a new operator called Gradient Angle Variance Operator. The Interest Operator in the Moravec algorithm was used to exclude featureless areas and simple edges which are oriented in the vertical, horizontal, and two diagonals. It was incorrectly detecting points on edges which are not on the four main directions (vertical, horizontal, and two diagonals). The new algorithm uses the Prewitt operator to exclude featureless areas, so that the Interest Operator is applied only on the edges to exclude simple edges and to leave interesting points. This modification speeds-up the extraction process by approximately 5 times. The Gradient Angle Variance (GAV), an operator which calculates the variance of the gradient angle in a window around the point under concern, is then applied on the interesting points to exclude the redundant ones and leave the actual dominant ones. The matching phase is performed after the extraction of the dominant points in both stereo images. The matching starts with dominant points in the left image and does a local search, looking for corresponding dominant points in the right image. The search is geometrically constrained the epipolar line of the parallel-axes stereo geometry and the maximum disparity of the application environment. If one dominant point in the right image lies in the search areas, then it is the corresponding point of the reference dominant point in the left image. A parameter provided by the GAV is thresholded and used as a rough similarity measure to select the corresponding dominant point if there is more than one point the search area. The correlation is used as a final decision tool when there is still more than one point in the search area. If there is no dominant point in the search area of if the points in the search area are below a correlation threshold, then the dominant point in the reference image is occluded and can not be corresponded. The algorithm has been modeled, implemented and shown to be fast, robust and parallel. The parallelism is created from three main features: locality of the operators; a memory optimization scheme; and the ability to fully parallelize the extraction phase which is the most computational intensive task in the algorithm. The last feature is achieved by performing the extraction phase on the two images simultaneously.

Al-Tammami, A.↗

NASA NTRS1994

A real-time algorithm for integrating differential satellite and inertial navigation information during helicopter approach

A real-time, high-rate precision navigation Kalman filter algorithm is developed and analyzed. This Navigation algorithm blends various navigation data collected during terminal area approach of an instrumented helicopter. Navigation data collected include helicopter position and velocity from a global position system in differential mode (DGPS) as well as helicopter velocity and attitude from an inertial navigation system (INS). The goal of the Navigation algorithm is to increase the DGPS accuracy while producing navigational data at the 64 Hertz INS update rate. It is important to note that while the data was post flight processed, the Navigation algorithm was designed for real-time analysis. The design of the Navigation algorithm resulted in a nine-state Kalman filter. The Kalman filter's state matrix contains position, velocity, and velocity bias components. The filter updates positional readings with DGPS position, INS velocity, and velocity bias information. In addition, the filter incorporates a sporadic data rejection scheme. This relatively simple model met and exceeded the ten meter absolute positional requirement. The Navigation algorithm results were compared with truth data derived from a laser tracker. The helicopter flight profile included terminal glideslope angles of 3, 6, and 9 degrees. Two flight segments extracted during each terminal approach were used to evaluate the Navigation algorithm. The first segment recorded small dynamic maneuver in the lateral plane while motion in the vertical plane was recorded by the second segment. The longitudinal, lateral, and vertical averaged positional accuracies for all three glideslope approaches are as follows (mean plus or minus two standard deviations in meters): longitudinal (-0.03 plus or minus 1.41), lateral (-1.29 plus or minus 2.36), and vertical (-0.76 plus or minus 2.05).

Hoang, TY↗

NASA NTRS1995

SeaWiFS Technical Report Series. Volume 29: SeaWiFS CZCS-type pigment algorithm

The Sea-viewing Wide Field-of-view Sensor (SeaWiFS) mission will provide operational ocean color that will be superior to the previous Coastal Zone Color Sensor (CZCS) proof-of-concept mission. an algorithm is needed that exploits the full functionality of SeaWiFS whilst remaining compatible in concept with algorithms used for the CZCS. This document describes the theoretical rationale of radiance band-radio methods for determining chlorophyll alpha and other important biogeochemical parameters, and their implementation for the SeaWiFS mission. Pigment interrelationships are examined to explain the success of the CZCS algorithms. In the context where chlorophyll alpha absorbs only weakly at 520 nm, the success of the 520 nm to 550 nm CZCS band ratio needs to be explained. This is explained by showing that in pigment data from a range of oceanic provinces chlorophyll alpha (absorbing at less than 490 nm), carotenoids (absorbing at greater than 460 nm), and total pigment are highly correlated. Correlations within pigment groups particularly photoprotectant and photosynthetic carotenoids are less robust. The sources of variability in optical data re examined using the NIMBUS Experiment Team (NET) bio-optical data set and bio-optical model. In both the model and NET data, the majority of the variance in the optical data is attributed to variability in pigment (chlorophyll alpha, and total particulates, with less than 5% of the variability resulting from pigment assemblage. The relationships between band ratios and chlorophyll is examined analytically, and a new formulation based on a dual hyperbolic model is suggested which gives a better calibration curve than the conventional log-log linear regression fit. The new calibration curve shows that 490:555 ratio is the best single-band ratio and is the recommended CZCS-type pigment algorithm. Using both the model and NET data, a number of multiband algorithms are developed; the best of which is an algorithm based on the 443:555 and 490:555 ratios. From model data, the form of potential algorithms for other products, such as total particulates and dissolved organic matter (DOM), are suggested.

Hooker, Stanford B.↗

NASA NTRS1995

Evaluation of passive microwave precipitation algorithms in wintertime midlatitude situations

The second intercomparison project of the Global Precipitation Climatology Project (GPCP) examined the estimation of midlatitude, cool-season precipitation. As part of that effort, the authors report here on the results of two microwave techniques, the Goddard scattering algorithm and the physical retrieval algorithm of Kummerow. Results from the estimation of instantaneous rain rate for five overpasses of the Special Sensor Microwave/Imager (SSM/I) are presented in a case study mode to illustrate both the strong and weak points of each technique. These five cases represent a sampling of the various types of precipitating systems observed. Results for the complete set of 20 swaths chosen by the United Kingdom Meteorological Office (UKMO) are then categorized by scatterplots and statistics of instantaneous radar versus microwave-estimated rain rate, rain no-rain contingency tables, and scatterplots of areal coverage of rainfall. Neither algorithm produced a good statistical correlation with the radar data, yet in general, both did well at determining rainy areas. Two reasons are suggested for the low correlation coefficients between both algorithms and the radar data. Time differences between the SSM/I overpass and the radar observations can occasionally account for some of the differences. The primary reason for the low correlations, however, appears to be the predominance of very light rain in the area of interest during the winter. Both algorithms are in good spatial agreement with the radar when the radar data are restricted to rates above 1 mm/h. When all radar rain rates are included, the radar areal coverage increases by as much as a factor of 10 in some cases. Because the Kummerow algorithm does not handle such low rain rates over land very well, and because the Goddard scattering algorithm uses 1 mm/h as the minimum reliably detectable rain rate, regimes that contain large areas of very light rain present inherent difficulties for these retrieval methods. Therefore, the proliferation of low rain rates observed during the experiment is the main contributor to low correlation coefficients and high root-mean-square differences. Mididentification of cold surface (e.g., snow cover) as precipitation was also a problem in several instances.

Negri, Andrew J.↗

NASA NTRS1994

Comparison of trend analyses for Umkehr data using new and previous inversion algorithms

Ozone vertical profile Umkehr data for layers 3-9 obtained from 12 stations, using both previous and new inversion algorithms, were analyzed for trends. The trends estimated for the Umkehr data from the two algorithms were compared using two data periods, 1968-1991 and 1977-1991. Both nonseasonal and seasonal trend models were fitted. The overall annual trends are found to be significantly negative, of the order of -5% per decade, for layers 7 and 8 using both inversion algorithms. The largest negative trends occur in these layers under the new algorithm, whereas in the previous algorithm the most negative trend occurs in layer 9. The trend estimates, both annual and seasonal, are substantially different between the two algorithms mainly for layers 3, 4, and 9, where trends from the new algorithm data are about 2% per decade less negative, with less appreciable differences in layers 7 and 8. The trend results from the two data periods are similar, except for layer 3 where trends become more negative, by about -2% per decade, for 1977-1991.

Reinsel, Gregory C.↗

NASA NTRS1984

Phase 2 development of Great Lakes algorithms for Nimbus-7 coastal zone color scanner

A series of experiments have been conducted in the Great Lakes designed to evaluate the application of the NIMBUS-7 Coastal Zone Color Scanner (CZCS). Atmospheric and water optical models were used to relate surface and subsurface measurements to satellite measured radiances. Absorption and scattering measurements were reduced to obtain a preliminary optical model for the Great Lakes. Algorithms were developed for geometric correction, correction for Rayleigh and aerosol path radiance, and prediction of chlorophyll-a pigment and suspended mineral concentrations. The atmospheric algorithm developed compared favorably with existing algorithms and was the only algorithm found to adequately predict the radiance variations in the 670 nm band. The atmospheric correction algorithm developed was designed to extract needed algorithm parameters from the CZCS radiance values. The Gordon/NOAA ocean algorithms could not be demonstrated to work for Great Lakes waters. Predicted values of chlorophyll-a concentration compared favorably with expected and measured data for several areas of the Great Lakes.

Tanis, Fred J.↗

NASA NTRS1995

Local multiplicative Schwarz algorithms for convection-diffusion equations

We develop a new class of overlapping Schwarz type algorithms for solving scalar convection-diffusion equations discretized by finite element or finite difference methods. The preconditioners consist of two components, namely, the usual two-level additive Schwarz preconditioner and the sum of some quadratic terms constructed by using products of ordered neighboring subdomain preconditioners. The ordering of the subdomain preconditioners is determined by considering the direction of the flow. We prove that the algorithms are optimal in the sense that the convergence rates are independent of the mesh size, as well as the number of subdomains. We show by numerical examples that the new algorithms are less sensitive to the direction of the flow than either the classical multiplicative Schwarz algorithms, and converge faster than the additive Schwarz algorithms. Thus, the new algorithms are more suitable for fluid flow applications than the classical additive or multiplicative Schwarz algorithms.

Cai, Xiao-Chuan↗

NASA NTRS1995

Development of Fast Algorithms Using Recursion, Nesting and Iterations for Computational Electromagnetics

In the first phase of our work, we have concentrated on laying the foundation to develop fast algorithms, including the use of recursive structure like the recursive aggregate interaction matrix algorithm (RAIMA), the nested equivalence principle algorithm (NEPAL), the ray-propagation fast multipole algorithm (RPFMA), and the multi-level fast multipole algorithm (MLFMA). We have also investigated the use of curvilinear patches to build a basic method of moments code where these acceleration techniques can be used later. In the second phase, which is mainly reported on here, we have concentrated on implementing three-dimensional NEPAL on a massively parallel machine, the Connection Machine CM-5, and have been able to obtain some 3D scattering results. In order to understand the parallelization of codes on the Connection Machine, we have also studied the parallelization of 3D finite-difference time-domain (FDTD) code with PML material absorbing boundary condition (ABC). We found that simple algorithms like the FDTD with material ABC can be parallelized very well allowing us to solve within a minute a problem of over a million nodes. In addition, we have studied the use of the fast multipole method and the ray-propagation fast multipole algorithm to expedite matrix-vector multiplication in a conjugate-gradient solution to integral equations of scattering. We find that these methods are faster than LU decomposition for one incident angle, but are slower than LU decomposition when many incident angles are needed as in the monostatic RCS calculations.

Chew, W. C.↗

NASA NTRS1995

SeaWiFS Technical Report Series: The SeaWiFS CZCS-Type Pigment Algorithm - Volume 29

The Sea-viewing Wide Field-of-view Sensor (SeaWiFS) mission will provide operational ocean color that will be superior to the previous Coastal Zone Color Sensor (CZCS) proof-of-concept mission. An algorithm is needed that exploits the full functionality of SeaWiFS whilst remaining compatible in concept with algorithms used for the CZCS. This document describes the theoretical rationale of radiance band-ratio methods for determining chlorophyll-a and other important biogeochemical parameters, and their implementation for the SeaWIFS mission. Pigment interrelationships are examined to explain the success of the CZCS algorithms. In the context where chlorophyll-a absorbs only weakly at 520 nm, the success of the 520 nm to 550 nm CZCS band ratio needs to be explained. This is explained by showing that in pigment data from a range of oceanic provinces chlorophyll-a (absorbing at less than 490 nm), carotenoids (absorbing at greater than 460 nm), and total pigment are highly correlated. Correlations within pigment groups particularly photoprotectant and photosynthetic carotenoids are less robust. The sources of variability in optical data are examined using the NIMBUS Experiment Team (NET) bio-optical data set and bio-optical model. In both the model and NET data, the majority of the variance in the optical data is attributed to variability in pigment (chlorophyll-a), and total particulates, with less than 5% of the variability resulting from pigment assemblage. The relationships between band ratios and chlorophyll is examined analytically, and a new formulation based on a dual hyperbolic model is suggested which gives a better calibration curve than the conventional log-log linear regression fit. The new calibration curve shows the 490:555 ratio is the best single-band ratio and is the recommended CZCS-type pigment algorithm. Using both the model and NET data, a number of multiband algorithms are developed; the best of which is an algorithm based on the 443:555 and 490:555 ratios. From model data, the form of potential algorithms for other products, such as total particulates and dissolved organic matter (DOM), are suggested.

Hooker, Stanford B.↗

NASA NTRS1994

A Real-Time Algorithm for Integrating Differential Satellite and Inertial Navigation Information During Helicopter Approach

A real-time, high-rate precision navigation Kalman filter algorithm is developed and analyzed. This navigation algorithm blends various navigation data collected during terminal area approach of an instrumented helicopter. Navigation data collected include helicopter position and velocity from a global position system in differential mode (DGPS) as well as helicopter velocity and attitude from an inertial navigation system (INS). The goal of the navigation algorithm is to increase the DGPS accuracy while producing navigational data at the 64 Hertz INS update rate. It is important to note that while the data was post flight processed, the navigation algorithm was designed for real-time analysis. The design of the navigation algorithm resulted in a nine-state Kalman filter. The Kalman filter's state matrix contains position, velocity, and velocity bias components. The filter updates positional readings with DGPS position, INS velocity, and velocity bias information. In addition, the filter incorporates a sporadic data rejection scheme. This relatively simple model met and exceeded the ten meter absolute positional requirement. The navigation algorithm results were compared with truth data derived from a laser tracker. The helicopter flight profile included terminal glideslope angles of 3, 6, and 9 degrees. Two flight segments extracted during each terminal approach were used to evaluate the navigation algorithm. The first segment recorded small dynamic maneuver in the lateral plane while motion in file vertical plane was recorded by the second segment. The longitudinal, lateral, and vertical averaged positional accuracies for all three glideslope approaches are as follows (mean +/- two standard deviations in meters): longitudinal (-0.03 +/- 1.41), lateral (-1.29 +/- 2.36), and vertical (-0.76 +/- 2.05).

Hoang, Ty↗

NASA NTRS1998

Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes: A Recursive Maximum Likelihood Decoding - Part 3

The Viterbi algorithm is indeed a very simple and efficient method of implementing the maximum likelihood decoding. However, if we take advantage of the structural properties in a trellis section, other efficient trellis-based decoding algorithms can be devised. Recently, an efficient trellis-based recursive maximum likelihood decoding (RMLD) algorithm for linear block codes has been proposed. This algorithm is more efficient than the conventional Viterbi algorithm in both computation and hardware requirements. Most importantly, the implementation of this algorithm does not require the construction of the entire code trellis, only some special one-section trellises of relatively small state and branch complexities are needed for constructing path (or branch) metric tables recursively. At the end, there is only one table which contains only the most likely code-word and its metric for a given received sequence r = (r(sub 1), r(sub 2),...,r(sub n)). This algorithm basically uses the divide and conquer strategy. Furthermore, it allows parallel/pipeline processing of received sequences to speed up decoding.

Lin, Shu↗

NASA NTRS1999

Automated Development of Accurate Algorithms and Efficient Codes for Computational Aeroacoustics

The simulation of sound generation and propagation in three space dimensions with realistic aircraft components is a very large time dependent computation with fine details. Simulations in open domains with embedded objects require accurate and robust algorithms for propagation, for artificial inflow and outflow boundaries, and for the definition of geometrically complex objects. The development, implementation, and validation of methods for solving these demanding problems is being done to support the NASA pillar goals for reducing aircraft noise levels. Our goal is to provide algorithms which are sufficiently accurate and efficient to produce usable results rapidly enough to allow design engineers to study the effects on sound levels of design changes in propulsion systems, and in the integration of propulsion systems with airframes. There is a lack of design tools for these purposes at this time. Our technical approach to this problem combines the development of new, algorithms with the use of Mathematica and Unix utilities to automate the algorithm development, code implementation, and validation. We use explicit methods to ensure effective implementation by domain decomposition for SPMD parallel computing. There are several orders of magnitude difference in the computational efficiencies of the algorithms which we have considered. We currently have new artificial inflow and outflow boundary conditions that are stable, accurate, and unobtrusive, with implementations that match the accuracy and efficiency of the propagation methods. The artificial numerical boundary treatments have been proven to have solutions which converge to the full open domain problems, so that the error from the boundary treatments can be driven as low as is required. The purpose of this paper is to briefly present a method for developing highly accurate algorithms for computational aeroacoustics, the use of computer automation in this process, and a brief survey of the algorithms that have resulted from this work. A review of computational aeroacoustics has recently been given by Lele.

Goodrich, John W.↗

NASA NTRS1998

Attitude Estimation Signal Processing: A First Report on Possible Algorithms and Their Utility

In this brief effort, time has been of the essence. The data had to be acquired from APL/Lincoln Labs, stored, and sorted out to obtain the pertinent streams. This has been a significant part of this effort and hardware and software problems have been addressed with the appropriate solutions to accomplish this part of the task. Passed this, some basic and important algorithms are utilized to improve the performance of the attitude estimation systems. These algorithms are an essential part of the signal processing for the attitude estimation problem as they are utilized to reduce the amount of the additive/multiplicative noise that in general may or may not change its structure and probability density function, pdf, in time. These algorithms are not currently utilized in the processing of the data, at least, we are not aware of their use in this attitude estimation problem. Some of these algorithms, like the variable thresholding, are new conjectures, but one would expect that someone somewhere must have utilized this kind of scheme before. The variable thresholding idea is a straightforward scheme to use in case of a slowly varying pdf, or statistical moments of the unwanted random process. The algorithms here are kept simple but yet effective for processing the data and removing the unwanted noise. For the most part, these algorithms can be arranged so that their consecutive and orderly execution would complement the preceding algorithm and improve the overall performance of the signal processing chain.

Riasati, Vahid R.↗

NASA NTRS1999

A Parallel Compact Multi-Dimensional Numerical Algorithm with Aeroacoustics Applications

In this study we propose a novel method to parallelize high-order compact numerical algorithms for the solution of three-dimensional PDEs (Partial Differential Equations) in a space-time domain. For this numerical integration most of the computer time is spent in computation of spatial derivatives at each stage of the Runge-Kutta temporal update. The most efficient direct method to compute spatial derivatives on a serial computer is a version of Gaussian elimination for narrow linear banded systems known as the Thomas algorithm. In a straightforward pipelined implementation of the Thomas algorithm processors are idle due to the forward and backward recurrences of the Thomas algorithm. To utilize processors during this time, we propose to use them for either non-local data independent computations, solving lines in the next spatial direction, or local data-dependent computations by the Runge-Kutta method. To achieve this goal, control of processor communication and computations by a static schedule is adopted. Thus, our parallel code is driven by a communication and computation schedule instead of the usual "creative, programming" approach. The obtained parallelization speed-up of the novel algorithm is about twice as much as that for the standard pipelined algorithm and close to that for the explicit DRP algorithm.

Povitsky, Alex↗

NASA NTRS2000

Any Two Learning Algorithms Are (Almost) Exactly Identical

This paper shows that if one is provided with a loss function, it can be used in a natural way to specify a distance measure quantifying the similarity of any two supervised learning algorithms, even non-parametric algorithms. Intuitively, this measure gives the fraction of targets and training sets for which the expected performance of the two algorithms differs significantly. Bounds on the value of this distance are calculated for the case of binary outputs and 0-1 loss, indicating that any two learning algorithms are almost exactly identical for such scenarios. As an example, for any two algorithms A and B, even for small input spaces and training sets, for less than 2e(-50) of all targets will the difference between A's and B's generalization performance of exceed 1%. In particular, this is true if B is bagging applied to A, or boosting applied to A. These bounds can be viewed alternatively as telling us, for example, that the simple English phrase 'I expect that algorithm A will generalize from the training set with an accuracy of at least 75% on the rest of the target' conveys 20,000 bytes of information concerning the target. The paper ends by discussing some of the subtleties of extending the distance measure to give a full (non-parametric) differential geometry of the manifold of learning algorithms.

Wolpert, David H.↗

NASA NTRS2001

An Algorithm for Interactive Modeling of Space-Transportation Engine Simulations: A Constraint Satisfaction Approach

In this research we have developed an algorithm for the purpose of constraint processing by utilizing relational algebraic operators. Van Beek and others have investigated in the past this type of constraint processing from within a relational algebraic framework, producing some unique results. Apart from providing new theoretical angles, this approach also gives the opportunity to use the existing efficient implementations of relational database management systems as the underlying data structures for any relevant algorithm. Our algorithm here enhances that framework. The algorithm is quite general in its current form. Weak heuristics (like forward checking) developed within the Constraint-satisfaction problem (CSP) area could be also plugged easily within this algorithm for further enhancements of efficiency. The algorithm as developed here is targeted toward a component-oriented modeling problem that we are currently working on, namely, the problem of interactive modeling for batch-simulation of engineering systems (IMBSES). However, it could be adopted for many other CSP problems as well. The research addresses the algorithm and many aspects of the problem IMBSES that we are currently handling.

Mitra, Debasis↗

NASA NTRS2001

Comparative Analysis of Daytime Fire Detection Algorithms, Using AVHRR Data for the 1995 Fire Season in Canda: Perspective for MODIS

Two fixed-threshold Canada Centre for Remote Sensing and European Space Agency (CCRS and ESA) and three contextual GIGLIO, International Geosphere and Biosphere Project, and Moderate Resolution Imaging Spectroradiometer (GIGLIO, IGBP, and MODIS) algorithms were used for fire detection with Advanced Very High Resolution Radiometer (AVHRR) data acquired over Canada during the 1995 fire season. The CCRS algorithm was developed for the boreal ecosystem, while the other four are for global application. The MODIS algorithm, although developed specifically for use with the MODIS sensor data, was applied to AVHRR in this study for comparative purposes. Fire detection accuracy assessment for the algorithms was based on comparisons with available 1995 burned area ground survey maps covering five Canadian provinces. Overall accuracy estimations in terms of omission (CCRS=46%, ESA=81%, GIGLIO=75%, IGBP=51%, MODIS=81%) and commission (CCRS=0.35%, ESA=0.08%, GIGLIO=0.56%, IGBP=0.75%, MODIS=0.08%) errors over forested areas revealed large differences in performance between the algorithms, with no relevance to type (fixed-threshold or contextual). CCRS performed best in detecting real forest fires, with the least omission error, while ESA and MODIS produced the highest omission error, probably because of their relatively high threshold values designed for global application. The commission error values appear small because the area of pixels falsely identified by each algorithm was expressed as a ratio of the vast unburned forest area. More detailed study shows that most commission errors in all the algorithms were incurred in nonforest agricultural areas, especially on days with very high surface temperatures. The advantage of the high thresholds in ESA and MODIS was that they incurred the least commission errors.

Ichoku, Charles↗

NASA NTRS2003

Description of a Normal-Force In-Situ Turbulence Algorithm for Airplanes

A normal-force in-situ turbulence algorithm for potential use on commercial airliners is described. The algorithm can produce information that can be used to predict hazardous accelerations of airplanes or to aid meteorologists in forecasting weather patterns. The algorithm uses normal acceleration and other measures of the airplane state to approximate the vertical gust velocity. That is, the fundamental, yet simple, relationship between normal acceleration and the change in normal force coefficient is exploited to produce an estimate of the vertical gust velocity. This simple approach is robust and produces a time history of the vertical gust velocity that would be intuitively useful to pilots. With proper processing, the time history can be transformed into the eddy dissipation rate that would be useful to meteorologists. Flight data for a simplified research implementation of the algorithm are presented for a severe turbulence encounter of the NASA ARIES Boeing 757 research airplane. The results indicate that the algorithm has potential for producing accurate in-situ turbulence measurements. However, more extensive tests and analysis are needed with an operational implementation of the algorithm to make comparisons with other algorithms or methods.

Stewart, Eric C.↗