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Maximum likelihood and Bayesian Parameter

Estimation

Overview

- Bayes formula
- Bayes decision rule
- Decide w1 if P(w1x)gtP(W2x) otherwise

decide w2

Overview

- Parameter estimation
- ---Maximum likelihood estimation
- ---Bayesian estimation
- The parameters in MLE are fixed but unknown in

BE the parameters are random variables having

some known prior distribution.

Overview

- Maximum likelihood estimation

Bayesian estimation

- Prior density p(?) ? posterior density p(?D) ?

class-conditional density p(xD)

Application Example

- Parameter estimation in Bayesian High-Resolution

Image Reconstruction with Multisensors, IEEE

transactions on Image Processing by Rafael

Molina,2003.

Introduction

- Due to hardware and cost limitations, image

systems often provide us with only multiple low

resolution images, such as in remote sensing,

surveillance and astronomy. - Our goal is to reconstruct a high resolution

image from multiple undersampled, shifted,

degraded frames with sub-pixel displacement

errors.

Problem Formulation

- Consider a sensor array with L1L2 sensors, where

each sensor has N1N2 pixels and the size of each

sensor element is T1T2. - Our goal is to reconstruct an M1M2 high

resolution image, where M1L1N1, M2L2N2, from

L1L2 low resolution images. - Here we consider the case L1L2L.

Problem Formulation

Problem Formulation

- In ideal case, the low resolution sensors are

shifted with respect to each other by a value

proportional to (T1/L)(T2/L). - In practice, there can be small perturbations

around these ideal locations, thus the horizontal

and vertical displacements dl1,l2x and dl1,l2y of

the l1,l2-th sensor with respect to the

0,0-th reference sensor are given by

Problem Formulation

Image and degradation models

- Let f be the (M1M2)1 high resolution image and

gl1,l2 be the (N1N2)1 observed low resolution

image from l1,l2-th sensor. - We want to construct f from gl1,l2 using Bayesian

paradigm.

Image and degradation models

- The first step with this paradigm is the

definition of a prior distribution over high

resolution image f, p(fa), parameter a measures

the smoothness of the high resolution image f.

Image and degradation models

- We also need to specify p(gl1,l2f, ßl1,l2), the

probability distribution of the observed low

resolution image gl1,l2 if f is the true high

resolution image. vl1,l2 is modeled as

independent white noise with variance ßl1,l2-1.

Image and degradation models

Bayesian Analysis

- Step 1 estimation of the parameters a and ß are

selected as - where

Bayesian Analysis

- Step 2 estimation of original high resolution

image fa,ß is selected as the image satisfying

Bayesian Analysis

Bayesian Analysis

Experiment Result

My Conclusion

- These two parameter estimation techniques can be

used in many applications - ---the evaluation of employees job

performance thus design an employee schedule - ---the classifier of the scenery of SF into

tree, street, building and ground etc - --- Bayesian Methods for Cosmological Parameter

Estimation from Cosmic Microwave Background

Measurements.

My Conclusion

- They are classic parameter estimation techniques

and their feature is using prior distribution

information to determine unknown parameters in

problems with known function form. - In practice many problems will give us general

knowledge about the situation, that is, we know

the function form and prior distribution

information, then we can use these techniques to

solve the problems such as in image processing

field.

references

- 1. Rafael Molina, Miguel Vega, Javier Abad and

Aggelos K.Katsaggelos, Parameter Estimation in

Bayesian High-Resolution Image Reconstruction

With Multisensors, IEEE Trans. Image Processing,

vol.12, No.12, December 2003. - 2. Anze Slosar, Pedro Carreira, Kieran Cleary,

etc, Cosmological parameter estimation and

Bayesian model comparison using VSA data, Mon.

Not. R. Astron. Soc. 000,1-6(2002). - 3. Miguel Vega, Javier Mateos, Rafael Molina and

Aggelos K.Katsaggelos, Bayesian Parameter

Estimation in image construction from subsampled

blurred observations, IEEE Trans.Image

Processing,2003. - 4. D.P. Morton and E. Popova, "A Bayesian

stochastic programming approach to an employee

scheduling problem," IIE Transactions on

Operations Engineering 36 155-167 (2003).5.Nir

Friedman and Yoram Singer, Efficient Bayesian

Parameter Estimation in large discrete domains,

NIPS, 1998.

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