OpenVision LibraryGaussian MixtureImplementation

Pi19404

March 2, 2014

Contents

Contents

OpenVision Library Gaussian Mixture Implementation 3

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.1.1 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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OpenVision Library Gaussian Mixture Implementation

OpenVision Library Gaussian

Mixture Implementation

0.1 Introduction

� In this article we will look at gaussian mixture model.

� Mixture Models are a type of density model which comprise a

number of component functions, usually Gaussian.

� These component functions are combined to provide a multi-

modal density.

� GMM is a weighted average of gaussians where gaussian by its

own mean can covariance matrix matrix

� The mixture density is given by

P (Xj�;�; w) =P

kwkN (X;�k;�k)

� To compute the likelyhood that a random variable is sampled from

the mixture model we just need to compute the probability of

individual gaussian component and then find the weighted aver-

age.

� Instead of probability we can compute the log likelyhood of

gaussian components which would simply be the log of sum of

weighted exponentials.

L(Xj�;�; w) = logP

kexp(logwk + logN (X;�k;�k))

� A gaussian mixture model is characterized by

� Number of mixture components

� Weights of mixtures

� Mean and covariances of individual mixture components

� Thus given the model parameters computation of probability

that a vector is sampled from the gaussian mixture model is

fairly simple

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OpenVision Library Gaussian Mixture Implementation

vector<Gaussian> components;

/**

* @brief ProbMix:method returns the probability that

* vector X belongs to each of mixture gaussian

* @param X : input vector X

* @return :output vector of probabilities

*/

vector<float> ProbMix(Mat X)

{

vector<float> res;

res.resize(_nmix);

for(int i=0;i<_nmix;i++)

{

// cerr << _weights[i] << endl;

res[i]=(_components[i].Prob(X));

}

return res;

}

/**

* @brief Prob : method computes the probability that vector X

* is drawn from the gaussian mixture model

* @param X : input vector X

* @return

*/

float Prob(Mat X)

{

float res=0;

for(int i=0;i<_nmix;i++)

{

// cerr << _weights[i] << endl;

res=res+_weights[i]*(_components[i].Prob(X));

}

return res;

}

� For example if we consider the following mixture models

MU1 = [1 2];

SIGMA1 = [2 0; 0 .5];

MU2 = [-3 -5];

SIGMA2 = [1 0; 0 1];

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OpenVision Library Gaussian Mixture Implementation

priors=[0.4 0.6];

X=[1.1 2.3];

� The probability that vector X belongs to GMM is computed as

0:0580374 using the method Prob

� The probability that vector X belongs to individual mixture com-

ponents is computed as 0:145093 : 9:5455e� 17

� Both the above results can be verified by manual calculation or

other packages etc.

0.1.1 Code

� The code for the same can be found in OpenVision git repos-

itory https://github.com/pi19404/OpenVision in files ImgMl/gaus-

sianmixture.cpp and ImgMl/gaussianmixture.hpp files.

� OpenVision is attempt at a opensource C/C++ Library developed

in C/C++ using OpenCV,Eigen etc providing modular interface

for image processing,computer vision,machine learning applica-

tions.

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