Spring 2012 BioE 2630 (Pitt) : 16-725 (CMU RI) 18-791 (CMU ECE) : 42-735 (CMU BME)

The content of these slides by John Galeotti, © 2012 Carnegie Mellon University (CMU), was made possible in part by NIH NLM contract# HHSN276201000580P, and is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/3.0/ or send a letter to Creative Commons, 171 2nd Street, Suite 300, San Francisco, California, 94105, USA. Permissions beyond the scope of this license may be available either from CMU or by emailing itk@galeotti.net.The most recent version of these slides may be accessed online via http://itk.galeotti.net/

Lecture 3Math & Probability Background

ch. 1-2 of Machine Vision by Wesley E. Snyder & Hairong Qi

Spring 2012 BioE 2630 (Pitt) : 16-725 (CMU RI)

18-791 (CMU ECE) : 42-735 (CMU BME)

Dr. John Galeotti

General notes about the bookThe book is an overview of many conceptsTop quality design requires:

Reading the cited literatureReading more literatureExperimentation & validation

2

Two themes

ConsistencyA conceptual tool implemented in many/most

algorithmsOften must fuse information from local

measurements to make global conclusions about the image

OptimizationMathematical mechanismThe “workhorse” of machine vision

3

Image Processing Topics

EnhancementCoding

CompressionRestoration

“Fix” an imageRequires model of image degradation

Reconstruction

4

Machine Vision Topics

AKA: Computer vision Image analysis Image understanding

Pattern recognition:1. Measurement of features

Features characterize the image, or some part of it

2. Pattern classificationRequires knowledge about the possible classes

5

Our Focus

Feature Extraction

Classification & Further Analysis

Original Image

Feature measurement

6

Noise removal

Segmentation

Original Image

Shape Analysis Consistency Analysis

Matching

Features

RestorationCh. 6-7

Ch. 8

Ch. 10-11

Ch. 12-16

Ch. 9

Probability

Probability of an event a occurring:Pr(a)

IndependencePr(a) does not depend on the outcome of event b, and

vice-versaJoint probability

Pr(a,b) = Prob. of both a and b occurringConditional probability

Pr(a|b) = Prob. of a if we already know the outcome of event b

Read “probability of a given b”

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Probability for continuously-valued functionsProbability distribution function:

P(x) = Pr(z<x)Probability density function:

8

Linear algebra

Unit vector: |x| = 1Orthogonal vectors: xTy = 0Orthonormal: orthogonal unit vectorsInner product of continuous functions

Orthogonality & orthonormality apply here too

9

Linear independence

No one vector is a linear combination of the othersxj aixi for any ai across all i j

Any linearly independent set of d vectors {xi=1…d} is a basis set that spans the space d

Any other vector in d may be written as a linear combination of {xi}

Often convenient to use orthonormal basis setsProjection: if y=aixi then ai=yTxi

10

Linear transforms

= a matrix, denoted e.g. AQuadratic form:

Positive definite:Applies to A if

11

More derivativesOf a scalar function of x:

Called the gradientReally important!

Of a vector function of x

Called the JacobianHessian = matrix of 2nd derivatives of a scalar function

12

Misc. linear algebra

Derivative operatorsEigenvalues & eigenvectors

Translates “most important vectors” Of a linear transform (e.g., the matrix A)

Characteristic equation:A maps x onto itself with only a change in length is an eigenvaluex is its corresponding eigenvector

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Function minimization

Find the vector x which produces a minimum of some function f (x)x is a parameter vector f(x) is a scalar function of x The “objective function”

The minimum value of f is denoted:

The minimizing value of x is denoted:

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Numerical minimization

Gradient descentThe derivative points away from the minimumTake small steps, each one in the “down-hill” direction

Local vs. global minimaCombinatorial optimization:

Use simulated annealingImage optimization:

Use mean field annealing

15

Markov models

For temporal processes:The probability of something happening is dependent on

a thing that just recently happened.For spatial processes

The probability of something being in a certain state is dependent on the state of something nearby.

Example: The value of a pixel is dependent on the values of its neighboring pixels.

16

Markov chain

Simplest Markov modelExample: symbols transmitted one at a time

What is the probability that the next symbol will be w?For a Markov chain:

“The probability conditioned on all of history is identical to the probability conditioned on the last symbol received.”

17

Hidden Markov models (HMMs)

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1st MarkovProcess

2nd MarkovProcess

f (t)

f (t)

HMM switching

Governed by a finite state machine (FSM)

19

Output 1st

Process

Output 2nd

Process

The HMM Task

Given only the output f (t), determine:1. The most likely state sequence of the switching FSM

Use the Viterbi algorithm Computational complexity =

(# state changes) * (# state values)2

Much better than brute force, which =(# state values)(# state changes)

2. The parameters of each hidden Markov model Use the iterative process in the book Better, use someone else’s debugged code that they’ve shared

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