A method is presented for feature extraction of multispectral scanner data. Non-training data is used to demonstrate the reduction in processing time that can be obtained by using feature extraction rather than feature selection

Crane, R. B.

Crimmins, T.

Reyer, J. F.

English

Purdue E-Pubs

Purdue UniversityPurdue e-PubsLARS Symposia Laboratory for Applications of Remote Sensing10-1-1973Feature Extraction of Multispectral DataR. B. CraneEnvironmental Research Institute of MichiganT. CrimminsEnvironmental Research Institute of MichiganJ. F. ReyerEnvironmental Research Institute of MichiganFollow this and additional works at: http://docs.lib.purdue.edu/lars_sympThis document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.Crane, R. B.; Crimmins, T.; and Reyer, J. F., "Feature Extraction of Multispectral Data" (1973). LARS Symposia. Paper 29.http://docs.lib.purdue.edu/lars_symp/29Conference onMachine Processing ofRemotely Sensed DataOctober 16 - 18, 1973The Laboratory for Applications ofRemote SensingPurdue UniversityWest LafayetteIndianaCopyright © 1973Purdue Research FoundationThis paper is provided for personal educational use only,under permission from Purdue Research Foundation.-rFEATURE EXTRACTION OF MULTISPECI'RAL DATAR. B. Crane, T. Crimmins, J. F. ReyerEnvironmental Research Institute of Michigan (ERIM)*Ann Arbor, MichiganI. ABSTRACTA method is presented for feature extraction of multispectralscanner data. Non-training data is used to demonstrate the re-duction in processing time that can be obtained by using featureextraction rather than feature selection.II. INTRODUCTIONThe data gathered by a multispectral scanner must be processed before much of the USeful:irU'ormation is available. The amount of data can be inconveniently large when the groundarea to be surveyed ~s large and many spectral channels of data are recorded. This is par-ticularly true when using general purpose, sequential computers. One method of circumventingthis problem, is to limit the number of spectra channels in the scanner itself,., This methodhas the disadvantage of requiring that the spectral channels be chosen before or during thegathering of the data. The choice of spectral regions is dependent on the atmospheric con-ditions and the state of the various ground covers. A problem with this approach is that thefactors needed to choose the spectral channels for' a particular mission may not be accuratelylmown in advance.Another method of spectral band selection, now being used with most aircraft multispectralscanners, is to record a reasonably large number of spectral channels of data, and determiningfran the training data during processing which subset of channels should be used. Previousexperience (Crane, 1972) has shown that 4 to 6 spectral channels provide aJmost as much recognitioninformation on test data as do all of the channels available.The experiments that will be described in this paper were designed to show an alternativemethod of data.reduction. Subsets of linear combinations of recorded channels (Crane, to bepublished; Crane, 1972; Jegewski, 1973; Hsia, 1973; Quirein, 1972) rather than a subset of purechannels, were tested to see if a fewer number of linear combinations of channels could be usedwithout sacrificing recognition accuracy. If the computation giving the linear combinations ofchannels can be carried out in less time than the difference in time in using for recognition alarger number of pure channels and a subset of linear combinations then a very practical advantageis achieved for sequentially organized computers. The practical impact may be that many applica-tions may be reduced to using 2 or 3 channels (linear combinations) giving nearly the same re-cognition information as if all pure channels were used but requiring less time (cost) andstorage.Of course a pure channel is a special type of linear combination, with the advantage thatthe pure channel is available without additional processing. The additional processing requiredto form the linear canbinations may be a problem. With analog data available, the linear combina-tions can be fonned at the time the data is digitized. For data in digital format, it may be con-venient to form the linear canbinations when the data is converted into a format suitable for re-cognition, or during the preprocessing operation.tThis work was supported under NASA Contract NAS9-9784.*Formerly Willow Run laboratories of 'lbe University of Michigan.4B-6A s imp Le example can illustrate the formation of linear canbinations and the performancethat is possible. Consider the problem of recognizing one of four possible classes using onelinear combinat~on of two channel data. The data channels are assumed to be independent eachwith variance CJ. The location of the mean values for the four classes is shown in Fi~e 1.The calculation of the average probability of misclassification for this geometry was made byassuming that a linear decision rule would be used, that the data were normal and described bythe means and variance, (1. e. the covariance terms are zero) and that the a priori probabilitiesof occurrence of each class are equal. The calculations were limited to pairwise evaluations,whereby the probabilities of misclassification possible for each pair of classes were not affectedby the presence of the remaining class.The calculation results are shown in Figure 2, Where the average probability of misclassifica-tion is shown as a function of the angle between the abcissa and projection line that determinesthe linear canbination. The performance that would be obtained by using either pure channel isfound for angles of 0° and 90°. The lowest average probability of misclassification occurs for alinear combination described by an angle of approximately 15°. Thus for this particular example,a linear combination would be more desirable for recognition than a subset (i. e. a subset ofone) of pure channels.A procedure for finding the best linear combination in this case might be to start with anycombination, and compute the average probability of misclassification. Then, repeat the calcula-tion for a linear combination described by an angle close to the first angle. This procedurewould then be repeated, alWays using an angle close to that angle which in the previous computa-tion had provided the ~owest average probability of misclassification, until a minimum was found.For this example, where there are two minima, the lowest would be the one that would be found mostof the time.A slight variation of the geometry previously described is shown in Figure 3. ~e onlydifference is the change in the variance, CJ. Figure 4 depicts the calculated average probabilityof misclassification. The averages are all lower than those shown previously, a result of thereduced variance. In addition, there are now 5 minima, rather than 2, so that a mmtmm seekingtechnique is more dependent on the starting linear combination. The number of easily detectableminima can be reduced by artificially increasing the variance. This phenomena may lead to anImpr-oved minimum seeking technique.III. FORMING LINEAR COMBINATIONSThe problem of finding a good method of choosing linear combinations is primarily one offinding a workable algoritlln in three distinct steps: (1) develop a measure of performance;(2) develop a minimum seeking technique; and (3) find suitable ~tarting points for initiatingthe minimum seeking technique. In addition, the algorithn should not require an excessive amountof canputational time.The performance measure used is similar to that employed to find a subset of pure channelsand is derived from the linear decision rule now used routinely in this laboratory. The measurecan be expressed as:M l:i,j(1)-where the summation is for all signatures, the i-th class is distributed normally with meanvector u and covariance matrix Ri,and 4>(X) is the normal distribution function. The rowvectors ~f the mxn matrix A represent the linear combinations in question. (n is the number ofpure channels and m is the number of linear canbinations.) An advantage of Eq , (1) is that itcan be developed directly from the maxiII1UIIl likelihood decision rule, so the approximations usedcan be enumerated and evaluated. In fact, Eq. (1) is approximately proportional to a constantminus the average probability of misclassification that would be measured.A method has been developed to find a local minimum of a function of several variables bystarting at a point and following a path of steepest descent by steps of variable but con-trollable size. Both the local gradient and the local curvature are used to estimatt; the pathof steepest descent. .48-7Finding starting points, the third step, is more difficult. The following are suggestedstarting points.Best Subset of Channels Starting PointEach individual channel can be thought of as a linear ccmbination of channels. (The vectorrepresenting OOis ccmbination has a 1 in the appropriate coordinate and 0' s elsewhere.) There-fore, a subset of m channels can be thought of as a set of m linear combinations. Since therecan be a very large number of subsets of m channels, rather than check through all of them to findthe best one, we use a stepwise procedure to find a "good" one. Experience has shown that thisgood subset is usually either the best or second best one. This stepwise procedure successivelyadds the one channel which gives the lowest average probability of misclassification when usedwith the channels alr-eady selected. The linear combinations represented by this subset are thenused as a starting point.Norm Squared Starting PointBy replacing each covariance matrix by the average of all of them, the problem is reduced tominimizing the functionM(P) = L <l>(1/21IPw.II)i lwhere the w. are a fixed set of vectors and P ranges over all orthogonal projectionsThe number ~f vectors w. is the total number of pairwise combinations of signatures.jection P correspoinds in a simple fashion to a matrix A in the original formulationThe projection P which maximizesis found analytically and the corresponding A is used as a starting point.Principal Eigenvector Starting Point(2)of rank m,Each pro-(see [lJ).First calculate the average of all the covariance matrices. Then transform the data so thatthe average covariance matrix is the identity matrix. Let N(~,R) be the distribution of all thetransformed data in the training area lumped together. This distribution can be calculat~fromthe distributions of the various materials if we can estimate the frequency of occurrence of eachmaterial. The starting point A is then taken as the matrix whose row vectors are the m orthogonaleigenvectors correspoinding to the m largest eigenvalues of the cavariance rratrix R.Clustered Starting PointThis method is based on the fact that if there are only two signatures and we are usinglinear discrimination, then there always exists a single linear combination channel which dis-tinguishes exactly as well as all n chRnnels no rratter what the value of n. If there are manysignatures, then for each pair S., S. (i I j) let v .. be the unit vector corresponding to thisl J lJbest single linear combination. In general, the number of vectors v.. , will be greater than rn,l,JThe v.. are then clustered into m clusters. For each cluster Ck, k=l, ... ,m, a weighted average,l,Jwk' of the v.. in that cluster is corrputed , The starting point A is formed from the wk as rowl,Jvectors. The weights can be rrade to reflect the sensitivity of the recognition accuracy to thedecision rule.There is one additional problem concerning the determination of A that should be mentioned.If A is an m x n matrix, there are mn components to be determined. This number of components canbe reduced to men - m) by the choice of a suitable canonical form for A. A canonical form ispossible because the value of Mobtained for any A is not changed if PA is substituted for A,where P is any nonsingular matr-Lx . We actually use PA, where P is chosen to scale the averagecovariance matrix and the mean vectors of the materials to our data f'ormat , The canonical formwe chose is:4B-8rA = (4)where 1m is the identity rmtrix with rank m, (For a specific example see Figure 1, where m = 3,n = 10, and the 10 pure channels have been rearranged in order of the wavelength).The canonical form with the ei j has two advantages. °The first is that, in general, aminimum number of unknown scalars must be found. The second advantage is that the minimizationprocess can be acconplished by varying the ei j with a nearly uniform step size. It is not necessaryto have large jumps in the values of the unknown scalars, which occur if the tan eij are con-sidered to be the unknown scalars.N. EXPERIMENTAL RESULTSAn experiment was devised to conpare subsets of linear combinations with subsets of purechannels. The data used were one of the data sets previously employed to test our linear de-cision rule (Ref. 1). This particular set of data was chosen because of the difficulty we havenoticed in obtaining satisfactory recognition with it. We felt that with relatively poor re-o cognition accuracy, the test results would represent greater statistical accuracy. If only afew data points were incorrectly recognized, the test results would be too dependent on thosefew points.The test procedure we used was to first select data that corresponded to 20 training fields.From these fields we developed statistics (means and covariance) for each of the 7 classes ofmaterials. °The statistics or signatures were then used to develop the decision rules which wereapplied to data that corresponded to 23 test fields different from the training fields. We thenfound the average correct recognition for each field, and then the average for each rmterial.Finally we averaged recognition accuracies for the rmterials to obtain an average recognitionaccuracy for the data set. The computer programs were merely functional, not optimized forminimum computation time, so meaningful comparisons were not made.The material classes consisted of bare soil and six vegetative species; alfalfa, barley,lettuce, sugar, safflower, and rye. The bare soil data tended to be atypical, because three ormore pure channels of data provided almost perfect recognition, whereas all of the subsets of 3linear combinations of': channels provided reduced accuracy. Note that the various subsets oflinear combinations were chosen to optimize over all the species, therefore it is not surprizingthat they did less well for one of them. There is some evidence that one infrared channel orratio of channels can be used to separate vegetative and non-vegetative mater-Lal,s . Thus forscme applicati-ons of layered or sequential classifiers, bare soil may not be considered as aclass to be recognized when discriminating among vegetation types. However, for this study,we retained bare soil as a class.The test results are shown in Figure 5. The top line of each bar indicates the averagerecognition accuracy obtained for the 7 classes. The next line indicates accuracies for the6 vegetative classes. Note that the recognition accuracy for the subset of 3 linear combina-tions was better than that obtained when subsets of either 3 or 4 best pure channels were used.In fact, the accuracy approached that obtained using all 10 pure channels, especially when onlythe 6 vegetative materials are considered.Figure 5 shows the average recognition accuracy obtained for one subset of 3 linear com-binations only. We actually tested three subsets of 3 linear combinations. '!'wo of the subsetsresulted frcm minimizing our measure function with two different starting points, (the first ofthese was used for Figure 5) and the third subset was an unweighted addition of channels. Weobtained approxirmtely the same average recognition accuracy for each of the subsets of linearcCillbinatiorts.In Figure 6, the average recognition accuracies we obtained from the 3 subsets of 3 linear4B-9cc:mbinations are compared with each other and with subsets of pure channels. Note thecorrespondence with the predicted accuracies, especially for subsets of pure channels withoutthe bare soil class.The first subset of linear combinations is shown in Figure 7, each row represents one com-bination. This matrix is not determined uniquely, because premultiplication by any nonsingularmatrix result~ in a new set of linear combinations which would produce identical recognitionperformance.The starting point for this set was the subset of 3 pure channels that we used for com-parison.The tentative conclusion that we drew from our test program was that the use of linearcc:mbinations may be a feasible method of spectral feature extraction to reduce overall pro-cessing time. The tests should be extended to include more data sets and different startingpoints.AcknowledgementsThe authors received valuable assistance through discussions with Dr. Quinten Holmes ofNASA-MSC and with staff members H. Horwitz, R. Kauth, and J. Erickson of ERIM.REFERENCESCrane, R. B. and W. Richardson, "Performance Evaluation of Multispectral Scanner ClassificationMethods," Proceedings of the Eighth International SYmPosium on Remote Sensing of Environ-ment, The University'of Michigan, Ann Arbor, 2-6 October 1972.Crane, R. B., W. Richardson and R. Hieber, "A Study of Techniques for Processing MultispectralScanner Data," Environmental Research Institute of Michigan Technical Report 3l650-l55-T(to be published).Crane, R. B., "Linear Combinations," presented at Quarterly Supporting Research and TechnologyReview, 26-28 September 1972, and reported in Environmental Research Institute of MichiganProgress Report 3l650-l47-L, September 15 to October 15, 1972.Hsia, W. S. and J. P. de Figueiredo, "Optimal Feature Extraction - The Two Class Case,"Institute for Computer Services and Applications, Rice University, Houston, Texas,ICSA-275-025-0l0, May 1973.Jegewski, D. J., "Optimal Feature Extraction by a Linear Transformation, Revision 1," MSCInternal Note No. 73-FM-19, 5 March 1973.Quirein, J. A., "An Interactive Approach to the Feature Selection Classification Problem,"TRW Systems, Earth Resources Technology, Houston Operations, Technical Note 99900-H019-RO-OO,December 1972.4B-IO•• •r-CT-1•IIiII1.3I1.21.11.0Figure 1. Location of Means for Exanple with Large VarianceFigure 2. Calculated Recognition Accuracy for Different ProjectionAngles for Example with Large Variance4B-11Figure 3. Location of Means for Exanple with Small Variance0.50.40.30.20.1•••0.0 ~,,--_..a..-_..a...._"""_"""_""""_--"_........I~_"""'_"-'-o 20 40 60 80 100 120 140 160 180Figure 4. Calculated Recognition Accuracy for Different ProjectionAngles for Exanple with Small Var1iIDce4B-12Figure 5. Corrparison of Linear Combinations with Subsets ofChannels Test Fields4B-13dsning1969. I V II- mpena a ey,20 fields, 7 classes for tra i-23 fields for testing- No 1 choice of training fiel.-- enc.2- ..,caen en Jt!c- - .-- G) G) .QC C G) Ec c ·cca ca c 0- ~ ~ ca (J(J (J ~ ~(J ca- (W) ~ G).., 0 c.., ... .-en en - ...- G) G) -m m « (W)10090z 80~0-Z ~ 70-w Z(J CJ 60a: 0w (JQ.W 50w a:CJ~« (J 40a: ww a: 30> a:« 0(J 20100Figure 6. Percentage Recognition Accuracy Obtained by UsingLinear Combinations, Subsets of Channels, and byAnalytic PredictionWithout Soil 66 66 64 51 59 68With Soil 67 70 69 58 65 73Predicted 71 75 51 62 70LINEAR COMBINATIONSNo.1 No.2 No.3SUBSETS3 4 104B-141 .29 .45 -.74 -.42 -.06 .07 0 0 -.200 .23 .65 -.02 .60 .76 .82 1 0 -.010 .18 .57 -.34 .20 .31 .11 0 1 -.09Figure 7. Matrix Describing 3 Linear Channels When theStarting Point is the Best Subset of 3Channels4B-15

Feature Extraction of Multispectral Data

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Feature Extraction of Multispectral Data

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