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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

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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

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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

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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.

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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.”

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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)

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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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