Detecting, classifying and localizing minor amounts of an element within a sample of material

A fast and efficient method to detect, locate and classify minor amounts of unknown elements that may exist in material under test (MUT), the material being of unknown exact composition having characteristics represented by a first data set from an interferogram, and in which a pre-specified number of control samples of material similar to the MUT but known to be free of said minor amounts of unknown elements have been prepared in advance and characterized in at least one second data set from an interferogram, comprising:

• developing a first best tree using a MUT wavelet-packet (WP) best tree operation to decompose said interferogram of said MUT using a pre-specified family of wavelets in which corresponding WPs are generated;
• employing an entropy criterion, providing a first output as a set of WPs, MUTWP1, MUTWP2, . . . , MUTWPN, where N is a positive whole number;
• developing a second best tree employing a non-contaminated sample (NCS) WP best tree operation to decompose said interferogram of a first of said control samples, designated NCS1;
• employing a specified family of wavelets in which corresponding WPs are generated using an entropy criterion,
• wherein said corresponding WPS are provided in a second output as a set of WPs, NCS1WP1, NCS1WP2, NCS1WP3, . . . , NCS1WPK, where K is a positive whole number;
• performing a third operation on said first and second outputs with a branching/non-branching WP operation,
• wherein corresponding WPs of said best trees generated by said MUT WP best tree operation and said control sample WP best tree operation are compared to determine if both said MUT WP nodes and said sample WP nodes either branch or do not branch;
• selecting those pairs, one each of said MUT and said control sample WP nodes, that are not matched as mismatched pairs,
• wherein mismatch is determined when one said WP node branches and its corresponding said WP node does not branch;
• for each said control sample, outputting said mismatched pairs as a set of mismatched WP pairs:
• [(control sample number one's mismatched WP 1 (NCS1MMWP1), MUT's mismatched WP 1 (MUTMMWP1)], (NCS1MMWP2, MUTMMWP2), . . . (NCS1MMWPR, MUTMMWPR), where R is a positive whole number;
• performing a fourth operation by reusing results from said first operation and repeating said second and third operations for each of said available control samples,
• wherein corresponding mismatched WP pairs are generated for each available said control sample;
• employing said corresponding mismatched WP pairs to populate a table of values of said MUT Mismatched WPs (MMWP) vs. Corresponding said Control Sample Mismatched WPs (NCSMMWP);
• performing a fifth operation on each said available mismatched control sample WP, NCS1MMWP1, NCS2MMWP1, . . . NCSJMMWP1, using a Sample Principal Component Analysis (PCA) Operation (SPCAO),
• wherein a first covariance matrix is generated;
• further employing said first covariance matrix to generate matrices of eigenvectors, eigenvalues and explained vectors to permit output of mismatched packet eigenvectors, mismatched packet eigenvalues, and mismatched packet explained values;
• performing a sixth operation on said output of said fifth operation with a Sample Q-Limit Operation, thus computing a Q-Limit, NCSQLO,
• wherein for a first said control sample, said output of said sixth operation is an NCS1Q-Limit, for a second said control sample, as available, said output of said sixth operation is an NCS2Q-Limit, continuing through said available number of said control samples until said pre-specified number of control samples is reached;
• performing a seventh operation on said mismatched pairs of said first WP of said first control sample for each said control samples, MUTMMWP1 and NCS1MMWP1, MUTMMWP1 and NCS2MMWP1, . . . , and MUTMMWP1 and NCSJMMWP1 with a MUT PCA Operation (MUTPCAO);
• generating a second covariance matrix from said seventh operation,
• wherein from said second covariance matrix, corresponding second matrices of second eigenvectors, second eigenvalues and second explained value vectors are generated;
• outputting a set of MUT principal components (MUTMMWP1PC);
• and
• performing an eighth operation on said MUTMMWPIPC with a MUT Q-Value Operation (MUTQVO),
• wherein MUTMMWP1PC-Q-Values are provided as output;
• generating residuals from said MUTMMWP1PC-Q-Values;
• comparing said residuals from said MUTMMWP1PC-Q-Values,
• wherein MUTMMWP1PC-Q-Values greater than said NCS1Q-Limit indicate localized presence of said elements in said MUT corresponding to row 1 of said populated table of values;
• and
• performing a ninth operation that repeats said eighth operation for information in said 2nd column-2nd row, 2nd column-3rd row, . . . , and 2nd column-Rth row of said populated table of values,
• wherein corresponding NCS2Q-Limit, NCS3Q-Limit, NCS4Q-Limit, . . . , NCSRQ-Limit are generated;
• repeating said ninth operation for information in said 2nd row, 3rd row, 4th row., and Rth row of said table of values,
• wherein said MUTMMWP2PC-Q-Values, MUTMMWP3PC-Q-Values, MUTMMWP4PC-Q-Values, . . . , MUTMMWPRPC-Q-Values are generated;
• and
• comparing said MUTMMWPXPC-Q-Values with corresponding NCSXQ-Limits, where X=2, 3, 4, . . . , R,
• wherein at least some corresponding localized appearances of even minor amounts of an element in said MUT are identified, classified, and localized.

A method for detecting, identifying, and localizing particular elements in a sample of material from which a first data set is collected involving comparing said first data set to at least one second data set collected from at least one sample of said material known not to contain measurable amounts of said particular elements, each said data set suitable for characterizing components of said samples at least spectrally and spatially, comprising:

• decomposing said first data set;
• decomposing said at least one second data set,
• wherein said decomposing of said first and second data sets is accomplished by applying to each said data sets at least one pre-specified wavelet family;
• generating corresponding wavelet-packets for each said decomposed first and second data sets using an entropy criterion;
• generating a best tree having nodes each representing one said corresponding wavelet packets for each said decomposed first and second data sets;
• generating a branching/non-branching operation on each said best trees to establish a basis for comparison of nodes of said best trees,
• wherein said branching/non-branching operation is performed at each said node, and
• wherein an individual said node is used to form each said best tree if branching appears at one said node in a first said best tree and not at the other said equivalent node in a second said best tree, and
• wherein only said nodes present in both said first and second best trees are used in subsequent analysis;
• employing a principal component analysis, for each said mismatched pair:
• placing said data from said second data set corresponding to a same said node, but taken from a different sample, in a first data file;
• generating a covariance matrix such that the data points of a first interferogram represent a first subset and that of a second interferogram represent a second subset,
• wherein elements of said covariance matrix are the variance of said first subset, the covariance of said first subset, the variance of said second subset, and the covariance of said second subset, and
• wherein first values of said interferograms are values of a first observation and nth values of said interferograms are values of an nth observation;
• generating eigenvectors, eigenvalues, and explained vectors,
• wherein said generated eigenvectors, eigenvalues, and explained vectors characterize said covariance matrix;
• employing said eigenvectors, eigenvalues and explained vectors, generating a Q-Limit,
• wherein all said eigenvalues are added, and
• wherein elements of said explained vector are used to establish a value on the impact of a particular measurement on the observed data as a whole, and
• wherein sums of said eigenvalues, squares of said eigenvalues, cubes of said eigenvalues together with a variance parameter that is normally distributed with zero mean and unit value are used to generate said Q-limit, and
• wherein said Q-limit is used for comparison of said first and second data files to a pre-specified threshold;
• adding said data associated with a same node from said first data set to said first data file to obtain a second data file;
• finding principal components of said second data file,
• wherein said principal components are generated using said eigenvectors of said covariance matrix, said interferograms and corresponding mean values of said interferograms;
• and
• using said principal components, computing corresponding Q-Values,
• wherein said Q-values are generated from the sums of the squares of said corresponding principal components, and
• wherein said corresponding Q-values higher than said calculated Q-Limit indicate existence of said particular matter in said tested sample of material at each location at which said particular material is observed.