Chapter 12 Object Recognition - SJTUbcmi.sjtu.edu.cn/~luhongtao/dip/res/Chapter12.pdf · Patterns...

Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com

Chapter 12Object Recognition

Patterns and Pattern Classes

• A pattern is an arrangement of descriptors, such as those discussed in Chapter 11.

• The name feature is used often in the pattern recognition literature to denote a descriptor.

• A pattern class is a family of patterns that share some common properties.

Recognition Based on Decision-Theoretic Methods

• In other words, an unknown pattern x is said to belong to the ith pattern class if, upon substitution of x into all decision functions,

yields the largest numerical value.

( )( ) ( ) ijWjxdxd

≠=>=

;,...,2,1 ,,, classespattern for W ,,,Let W2121 ωωω KK

( )xdi

MatchingMinimum distance classifier

• Suppose that we define the prototype of each pattern class to be the mean vector of the patterns of that class:

• We then assign x to class ωi if Di(x) is the smallest distance.

j ,,2,1 1K== ∑

( ) jj mxxD −=

Minimum distance classifier

• It is not difficult to show (Problem 12.2) that selecting the smallest distance is equivalent to evaluating the functions

• assign x to class ωi if di(x) is the largest numerical value.

• This formulation agrees with the concept of a decision function, as defined in Eq. (12.2-1).

( ) Wjmmmxxd jTjj

Tj ,...,2,1

Matching by correlation

• correlation coefficient, which is defined as( ) ( ) ( )∑∑ ++=

s ttysxwtsfyxc ,,,

( )( ) ( )[ ] ( )[ ]

( ) ( )[ ] ( )[ ] 21

⎭⎬⎫

⎩⎨⎧

−++−

−++−=

∑∑∑∑

∑∑

s ts t

wtysxwtsftsf

wtysxwtsftsfyxγ

Optimum Statistical Classifiers

• the equation often called the conditional average risk or loss in decision-theory terminology.

• The classifier that minimizes the total average loss is called the Bayes classifier.

• Thus the Bayes classifier assigns an unknown pattern x to class ωi ifin other words, x is assigned to class ωi if

( ) ( )∑=

kkkjj xpLxr

( ) ( ) ijWjforxrxr ji ≠=< ;,...,2,1

( ) ( ) ( ) ( )q

kkki PxpLPxpL ωωωω ∑∑

Bayes classifier for Gaussian pattern classes

( ) ( ) ( )( )

( ) 2,1 2

2 ===−

jPePxpxd j

ωσπ

ωω σ

( )( )

( ) ( )jjT

j mxCmx

−−− −

ω { }xEm jj =

( )( ){ }Tjjjj mxmxEC −−= ∑

−=jx

jj mmxx

Bayes decision function for class ( ) ( ) ( )jjjj Pxpxd ωωω / is =

( ) ( ) jTjj

Tjj mCmmCxPxd 11

21ln −− −+= ω

Neural Network

( ) ( ) ( )( )kwww

wJkwkw=

⎥⎦⎤

⎢⎣⎡∂

∂−=+ α1

Structural Method

String Matching

Automatic as string recognizers

Syntactic Recognition of Trees

Chapter 12 Object Recognition - SJTUbcmi.sjtu.edu.cn/~luhongtao/dip/res/Chapter12.pdf · Patterns...

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