Correlation Pattern Recognition by B. V. K. Vijaya Kumar

By B. V. K. Vijaya Kumar

Correlation is a strong and common procedure for trend popularity and is utilized in many functions, similar to automated objective reputation, biometric popularity and optical personality attractiveness. The layout, research and use of correlation development attractiveness algorithms calls for historical past details, together with linear structures concept, random variables and strategies, matrix/vector equipment, detection and estimation conception, electronic sign processing and optical processing. This publication offers a wanted overview of this assorted history fabric and develops the sign processing idea, the trend reputation metrics, and the sensible program knowledge from easy premises. It indicates either electronic and optical implementations. It additionally includes know-how offered by way of the crew that constructed it and contains case stories of vital curiosity, resembling face and fingerprint reputation. compatible for graduate scholars taking classes in development acceptance thought, when attaining technical degrees of curiosity to the pro practitioner.

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The nth moment of the RV X is defined as follows: 36 Mathematical background Z1 mn ¼ E fX g ¼ xn f ðxÞdx n (2:84) À1 A smooth PDF can be reconstructed if we know all its moments. But sometimes, all we need are the first two moments (namely the mean and the variance) of the RV. They tell us where the PDF is centered and how broad it is. Also, the popular Gaussian PDF in Eq. 76) is completely characterized by its mean m and variance 2. The variance 2 of an RV is defined as follows: n o Z 2 ¼ E ðX À mÞ2 ¼ ðx À mÞ2 f ðxÞdx ¼ Z x2 f ðxÞdx À 2m Z xf ðxÞdx þ m2 Z f ðxÞdx È É È É ¼ E X 2 À 2m2 þ m2 ¼ E X 2 À m2 (2:85) We have omitted the integration limits in the above expression since they are from À1 to þ1, as should be obvious from the context.

If the correlation coefficient is close to +1, then the two RVs track each other closely. If the correlation coefficient is close to À1, the two RVs track each other closely but one is the negative of the other. The larger the absolute value of the correlation coefficient, the more we can tell about one RV from our knowledge of the other RV. Bivariate Gaussian Two RVs are said to be jointly Gaussian provided that their joint PDF takes on the following form: 40 Mathematical background 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2pX Y 1 À 2 " ( )# 1 ðx À mX Þ2 ðy À mY Þ2 ðx À mX Þðy À mY Þ þ À 2 Â exp À 2ð1 À 2 Þ X Y 2X 2Y fX;Y ðx; yÞ ¼ (2:94) where  is the correlation coefficient of X and Y.

Suppose we wish to find h maximizing the following ratio where matrices A and B are assumed to be real and symmetric: J ðhÞ ¼ hþ Ah hþ Bh (2:63) The ratio in Eq. 63) is known as the Rayleigh quotient, and to maximize J(h) with respect to h, we set the gradient rhJ(h) to zero as shown below: 28 Mathematical background   hþ Ah rh J ðhÞ ¼ rh þ h Bh þ 2ðh BhÞAh À 2ðhþ AhÞBh ¼0 ¼ 2 ðhþ BhÞ (2:64) Simple manipulations yield À1 B Ah ¼   hþ Ah h hþ Bh (2:65) where B is assumed to be invertible. Since JðhÞ ¼ hþ Ah=hþ Bh, we see from Eq.

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