This paper looks at techniques to simplify data analysis of large multivariate military sensor systems. The approach is illustrated using representative raw data from a videoscene analyzer. First, develop fuzzy neural net relations using Matlab. This represents the best fidelity fit to the data and will be used as reference for comparison. The data is then converted to Boolean, and using Boolean Decision Diagrams (BDD) techniques, to find similar relations between input vectors and output parameter. It will be shown that such Boolean techniques offer dramatic improvement in system analysis time, and with minor loss of fidelity. To further this study, Boolean Neural Net techniques (BNN) were employed to bridge the Fuzzy Neural Network (FNN) to BDD representations of the data. Neural network approaches give an estimation method for the complexity of Boolean Decision Diagrams, and this can be used to predict the complexity of digital circuits. The neural network model can be used for complexity estimation over a set of BDDs derived from Boolean logic expressions. Experimental results show good correlation with theoretical results and give insights to the complexity. The BNN representations can be useful as a means to FPGA implementation of the system relationships and can be used in embedded processor based multivariate situations.
Modern radar systems trace their roots to efforts in several countries to refine existing concepts and theories just prior to WWII. This was the culmination of theory and systems dating back to the 1860s. Simplistically, a transmitted electromagnetic signal reflects from an object back to an antenna where the characteristics of the signal determine properties such as location, speed and direction of the target. We take this simple problem, a show how variable complexity grows, and how the corresponding computational problem also grows.
Consider a simple TimeofFlight range detector. For a stationary object, the range is simply determined by
For a moving target, radar systems use the Doppler Effect where the relative motion between source and observer produces a frequency shift
Both equations assume colocated transmit and receive
The paper describes an environment for the rapid analysis, synthesis and optimization of embedded systems. The simplification strategy involves structure reduction and is carried out through an iterative algorithm aiming at selecting a minimal number of rules for the problem at hand. The selection algorithm allows manipulation of the neurofuzzy model or binary decision diagram to minimize complexity and to preserve a good level of accuracy. Since the implementation of these systems is rather complicated, we propose a methodology which automates the entire design flow. Flexibility is achieved by allowing manual intervention which is realized via a modular implementation of algorithms which are being provided.
An extension of the TimeofFlight system is a typical trackwhilescan (TWS) radar system, which gets a discrete (temporally separated) sequence of detections (“hits”) on a given target. Each detection is of a finite, typically quite short, duration. From each detection a given collection of attributes about that hit is extracted. The attributes typically include detection time, position, and reflected intensity, as well as various statistical moments and other mathematical features of the detection. The goal of these radar detections is typically to identify (detect and classify) the target of interest and its trajectory. By successfully tracking the target in motion, knowledge of the target can usually be substantially improved, since repeated detections of the moving target provide a greater volume of information and allow aggregate, trackbased, statistics to also be used.
If we expand the problem to a multidimensional target identification problem, let us consider an example of a cluttered video image, and we need to find and classify the target in the scene. The image attributes include distance (to target it is called
Data reproduced from [
Target no.  Distance  Aspect  Vert  Area  Target lum  Dark area lum  Surround lum  Edgepts  Search time 

Type  m  ass (sin)  Pixels  (Pixels)  Scene  Dark  Grass  Pts  Search time (s) 
1  4007  0.71  10  141  14  17  29  9571  14.6 
1  2998  0.82  11  225  21  10  27  8927  15.2 
2  3974  0.71  13  173  20  24  28  9138  12.4 
3  5377  0.05  5  49  18  23  30  8970  29.8 
2  1013  0.52  50  2708  19  5  34  8706  2.8 
4  3052  0.00  11  100  12  18  30  8755  6.4 
5  5188  0.41  9  76  18  23  28  9053  26.7 
6  3679  0.12  10  96  12  20  26  8620  10.0 
2  860  1.00  54  3425  9  1.5  40  8961  2.7 
4  1951  0.85  16  332  15  11  27  8572  2.8 
3  3992  0.79  11  154  20  19  26  9194  11.9 
6  1041  0.74  24  1645  11  4  35  9074  2.5 
7  2145  0.98  17  553  8  5  18  8280  3.7 
3  1998  0.76  19  659  20  10  22  8739  8.1 
2  4410  0.00  11  101  22  18  29  9404  12.4 
1  2893  0.42  16  320  12  7  23  8670  2.5 
5  1933  0.98  13  368  15  12  23  8606  4.8 
1  1850  0.96  28  876  3  4  9  8464  2.8 
8  1045  0.09  26  985  19  10  12  8613  12.3 
2  1933  0.95  22  867  16  11  27  8376  2.8 
7  4206  0.00  9  79  26  29  38  9506  15.1 
1  5722  0.88  7  73  38  40  46  9044  25.6 
4  4920  0.42  8  61  20  21  36  8618  12.1 
6  4206  0.81  9  142  18  12  21  9152  8.0 
5  2348  0.94  9  198  18  21  30  8504  5.5 
1  3992  0.88  11  217  15  14  26  9078  7.8 
9  4410  0.96  11  247  16  8  19  9397  9.6 
8  2321  0.83  15  458  22  21  47  8365  5.1 
5  3661  0.76  9  84  17  25  23  8807  7.5 
3  3670  0.00  13  192  14  15  27  8483  6.1 
7  1671  1.00  19  893  15  13  31  8959  3.5 
4  4345  0.81  8  63  15  12  20  9021  12.3 
2  3662  0.57  10  203  26  25  44  8702  5.4 
5  633  0.71  50  4403  20  5  39  8741  2.5 
3  492  0.07  57  3045  20  16  23  8992  2.2 
4  1497  0.78  16  560  10  7  20  9014  5.8 
5  1041  1.00  33  1613  17  5  32  8486  2.6 
1  2891  0.99  19  486  12  12  35  9021  12.1 
7  5147  0.93  5  81  18  27  34  9075  34.9 
6  1648  0.59  18  648  23  7  37  9070  2.7 
8  948  0.73  35  1463  18  5  38  8790  3.7 
7  3662  0.41  12  188  19  25  39  8524  5.8 
6  2900  0.00  17  340  20  10  49  8791  4.1 
2  5136  0.00  10  79  25  16  27  8941  10.6 
And, as is shown [
Looking at Table
Using the Verimax analysis in MiniTab, we calculate, Table
Reproduced from [
Distance  Aspect  Vert  Area  Target lum  Dark area  Surround  Edgepts  

Distance  1  −0.24  −0.79  −0.72  0.37  0.72  0.06  0.41 
Aspect  −0.24  1  0.08  0.12  −0.24  −0.25  −0.08  −0.10 
Vert  −0.79  0.08  1  0.95  −0.19  −0.58  0.07  −0.19 
Area  −0.72  0.12 

1  −0.14  −0.53  0.15  −0.13 
Target lum  0.37  −0.24  −0.19  −0.14  1  0.62  0.52  0.25 
Dark area  0.72  −0.25  −0.58  −0.53  0.62  1  0.32  0.22 
Surround  0.06  −0.08  0.07  0.15  0.52  0.32  1  0.04 
Edgepts  0.41  −0.10  −0.19  −0.13  0.25  0.22  0.04  1 
There are other inferences that can also be made. But, none that allows a simple determination of logical connections. Employing the Neurofuzzy ANFIS function from the MatLab toolkit, we see in Table
Correlation and other analysis of data from [
CORR  TRMS  STD  MAD  EWI  ERR  

2 factor  0.77  5.73  3.60  5.35  14.92  13.44 
3 factor  0.56  6.65  5.81  6.21  19.11  20.52 
4 factor  0.46  7.59  14.39  7.09  29.62  16.55 
5 factor  0.22  21.11  32.05  19.70  73.63  58.87 
6 factor  0.71  6.92  10.46  6.46  24.13  18.03 
7 factor  0.84  3.31  4.83  3.09  11.39  8.05 
Original 

2.84  4.40  2.65  10.02  7.38 
Where the terms as used by MiniTab are [
Correlation between the original output and the estimated output from the fuzzy neural system using the data from each method.
total Root mean square for the distance between the original output and the estimated output using the same testing data through the fuzzy neural system
where
Standard deviation for the distances between the original output and the estimated output using the same testing data through the fuzzy neural system.
Mean of the absolute distances between the original output and the estimated output using the same testing data through the fuzzy neural system.
The index value from the summation of the values with multiplying the statistical estimation value by its equally weighted potential value for each field.
The error rate is
where
This forms our baseline. We now compare this analysis to what we can get if the data in Table
This Boolean data is presented in Table
Target no.  Distance  Aspect  Vert  Area  Target lum  Dark area lum  Surround lum  Edgepts  Search time 

Type  m  ass (sin)  Pixels  (Pixels)  Scene  Dark  Grass  Pts  Search time (s) 
1  1  1  0  0  0  0  0  1  0 
1  1  1  0  0  1  0  0  0  0 
2  1  1  0  0  1  1  0  1  0 
3  1  0  0  0  0  1  1  1  1 
2  0  1  1  1  1  0  1  0  0 
4  1  0  0  0  0  0  1  0  0 
5  1  0  0  0  0  1  0  1  1 
6  1  0  0  0  0  1  0  0  0 
2  0  1  1  1  0  0  1  1  0 
4  0  1  0  0  0  0  0  0  0 
3  1  1  0  0  1  0  0  1  0 
6  0  1  0  0  0  0  1  1  0 
7  0  1  0  0  0  0  0  0  0 
3  0  1  0  0  1  0  0  0  0 
2  1  0  0  0  1  0  0  1  0 
1  0  0  0  0  0  0  0  0  0 
5  0  1  0  0  0  0  0  0  0 
1  0  1  0  0  0  0  0  0  0 
8  0  0  0  0  1  0  0  0  0 
2  0  1  0  0  0  0  0  0  0 
7  1  0  0  0  1  1  1  1  0 
1  1  1  0  0  1  1  1  1  1 
4  1  0  0  0  1  1  1  0  0 
6  1  1  0  0  0  0  0  1  0 
5  0  1  0  0  0  1  1  0  0 
1  1  1  0  0  0  0  0  1  0 
9  1  1  0  0  0  0  0  1  0 
8  0  1  0  0  1  1  1  0  0 
5  1  1  0  0  0  1  0  0  0 
3  1  0  0  0  0  0  0  0  0 
7  0  1  0  0  0  0  1  1  0 
4  1  1  0  0  0  0  0  1  0 
2  1  1  0  0  1  1  1  0  0 
5  0  1  1  1  1  0  1  0  0 
3  0  0  1  1  1  0  0  1  0 
4  0  1  0  0  0  0  0  1  0 
5  0  1  1  0  0  0  1  0  0 
1  0  1  0  0  0  0  1  1  0 
7  1  1  0  0  0  1  1  1  1 
6  0  1  0  0  1  0  1  1  0 
8  0  1  1  0  0  0  1  0  0 
7  1  0  0  0  1  1  1  0  0 
6  1  0  0  0  1  0  1  0  0 
2  1  0  0  0  1  0  0  1  0 
Verimax analysis when we employ Boolean relationships to determine the correlation functions shows correlation between the original output and the estimated output to be 82%. This compares well with the FNN analysis and is a reasonable approximation of the original data. It is this acceptable degradation that provides an opportunity to use Binary Decision Diagrams and Boolean techniques, to reduce the number of variables and to simplify the computational problem. The general correlation similarities between vectors are approximately preserved. And, again,
The analysis techniques used above show that reasonable computation reduction can be achieved by data quantization implying that using fullprecision data as baseline, we still achieve approximately 82% fidelity in the correlation data using Boolean quantization of the data. This opens the door to using Boolean Decision tools for rapid, reasonable fidelity and analysis of the system.
Let us again start with the data in Table
Table
Original visual scene data, after Boolean quantization, and after factor reduction.
Target type 




Output 

1  1  1  0  0  1 
1  0  1  0  1  1 
2  1  1  1  0  1 
3  1  0  1  1  1 
2  0  1  0  1  0 
4  1  0  0  1  1 
5  1  0  1  0  1 
6  1  0  1  0  1 
2  0  1  0  1  0 
4  0  1  1  1  0 
3  1  1  0  0  1 
6  0  1  0  1  0 
7  0  1  0  1  0 
3  0  1  0  1  1 
2  1  0  1  0  1 
1  0  1  0  1  0 
Using Boolean Decision Diagram techniques, this is represented as
From this, we can create the Kmap (Figure
Kmap of data represented by Table
And, we can rewrite
The complexity of circuit and systems design increases rapidly. Therefore, in seeking efficient algorithms and data structures, we choose, binary decision diagrams (BDDs). These have been used in a wide variety of applications and were intensively studied. And, the Boolean Decision Diagram as Figure
Boolean Decision Diagram for data represented by Table
Having this Boolean Decision Diagram, we can apply the analysis tools [
The proposed algorithm to implement the above technique is illustrated in this section. The input space is reduced and the number of rules is decreased, and the simulation results illustrate the approach is practicable, simple, and effective and the performance index is not significantly degraded. The video scenes, in this illustrative example, are first converted recognition attributes as required by the classification algorithm. And, using FNN and ANFIS techniques, a baseline answer fidelity index is generated. Then, the data streams are converted to Boolean and checked to see that resultant fidelity degradation is acceptable. This assessment can be ruleset or can rely on expert apriori knowledge to judge when the quantization is acceptable. Then, using Factor and Cluster techniques, the problem size is reduced first by combining or eliminating the input vectors, and then, by selecting orthogonal data rows, such that a full set of outcomes is represented. Finally, use BDD techniques, to draw Kmap diagrams and to develop a Boolean equation to solve the problem.
The synthesis, optimization, and implementation of embedded systems is rather complicated, and we propose a methodology which automates the entire design flow. Flexibility is achieved by allowing manual intervention which is realized via a modular implementation of algorithms which are being provided. This is represented in the following flow (see Figure
Flowchart for the proposed algorithm.
Assume there are
Using the raw data, and FNN, ANFIS function from the MatLab toolkit, calculate the correlation matrix,
Review input vector correlation data, to determine which variables can be reduced (or combined).
Use iterative convergence to reduce/combine input vectors. Each time verifying that the resultant data is reasonably correlated to the original data.
Convert remaining data stream to Boolean. Typically this binary representation is of a reduced set of input variables.
Then, using Factor and Cluster techniques, the problem size is reduced first by combining or eliminating the input vectors, and then, by selecting orthogonal data rows, such that a full set of outcomes is represented. This can be performed, based on heuristic, apriori, or correlation analysis.
Repeat MiniTab/VeriMax analysis to validate that the resultant correlation value is within acceptable range as compared to the fullfidelity FNN correlation value. This step can be automated via a rule set, or can include judgment, depending on confidence in the factoring algorithms.
Next, use BDD techniques, to draw Kmap diagrams and to develop a Boolean equation to solve the problem.
Create the BDD to represent the Boolean relationship.
Verify the diagram with representative cases.
Use FPGA design techniques, to convert the BDD to code.
The system is now ready to take a raw stream of input data and yield a high efficiency solution to the target classification problem.
Binary Decision Diagrams are generated by applying Shannon expansion [
In summary, a high number of input parameters can result in a large number of rules that are computationally difficult to handle. Hence, it is imperative to develop the techniques, which can reduce the input parameters such that the original system and reduced systems have approximately the same behavior. In this paper, we have taken multivariate streams of asavailable data from a video scene analyzer and have created a technique for order reduction, and of digitization, such that the problem can reasonably be reduced to a Boolean problem, and, that this Boolean problem can then be solved using wellknown techniques for Boolean Decision Diagrams. In the context of the example chosen for this paper, data streams from a video recorder are selected and analyzed for target classification efficiency. It is shown that an original 8 inputparameter data set with an 87% confidence full fuzzy neural net analysis, the reduced order problem was converted to a 4variable problem, and the technique was able to generate a reasonably accurate 82% accurate solution. And, that such a reducedorder problem can then be converted to a Boolean relationship. Existing research [
From this study, we may decide the best heuristic and fuzzy techniques and also when the usage of fuzzy or heuristic method is better than the other. These are of significant advantage in satellite sensors where power and compute capacities are limited, and where such approximations can then cue more detailed analysis if warranted.