Chapter 20 Accuracy

20.1 Introduction

accurately characterizing performance: important aspect of vision system

20.2 Mensuration Quantizing Error

position on digital grid: has inherent quantizing error due to discreteness

: coordinate of line's right endpoint

: spacing between pixel centers

: uniform random variable,

relationship between the line segment end and the digital grid

=====Fig. 20.1=====

: digital coordinate of the line's rightmost pixel

natural quantizing model:

letting be a random variable where

restate the quantizing model:

note: and

the mean of digital coordinate :

thus unbiased estimator for position is

the variance of digital coordinate :

: line's left endpoint handled in a similar way

20.3 Automated Position Inspection: False-Alarm and Misdetection Rates

in industrial position inspection: mechanism machines part to specification

inspection: ensures machining or part placement is correct

automated inspector: consists of machine identifying critical object points

: known number for relative position

: actual position

: Gaussian distribution with mean and standard deviation

: tolerance interval centered around position

: position is good

: position is bad

actual position : not known

measurement : obtained by observing actual position and measuring it

measurement : noisy and not equal to

given : Gaussian distribution with mean and standard deviation

: acceptance interval for decision that actual position in tolerance

: inspection system decides the position is good

: inspection system decides the position is bad

false alarm: good position falsely called bad

misdetection: bad position missed and incorrectly called good

false-alarm rate is the conditional probability:

misdetection rate is the conditional probability:

entire probability model: characterized by five parameters

problem: how to compute false-alarm and misdetection probabilities

20.3.1 Analysis

: probability density function for actual position

: conditional probability density function for given

with Gaussian distribution assumption:

conditional probability closely related to false-alarm probability:

now

inherent invariance of false-alarm and misdetection probabilities to the scale

define relative precision of the measurement:

=====Garfield 17:67=====

20.3.2 Discussion

when
large: acceptance interval large

large: all good positions are accepted

large: false-alarm rate small

large: bad positions will also be accepted

large: high rate of misdetection

small: acceptance interval relatively small

small: all bad positions expected not to be accepted

small: misdetection rate small

small: good positions will also not be accepted

small: high rate of false alarm

false alarm rate and misdetection rate approximately inverse proportional

three operating curves for a fixed failure rate of 0.05

top operating curve: relative precision of 0.1

middle operating curve: relative precision of 0.065

bottom operating curve: relative precision of 0.05

=====Fig. 20.2=====

three operating curves for a fixed failure rate of 0.01

top operating curve: relative precision of 0.1

middle operating curve: relative precision of 0.075

bottom operating curve: relative precision of 0.05

=====Fig. 20.3=====

fix failure rate and misdetection rate: as relative precision better,

tolerance interval i.e. st. dev. of measurements smaller

operating curves for smaller values of relative precision below larger ones

fix relative precision and misidentification rate: as failure rate increases

false-alarm rate increases

three operating curves for a fixed relative precision of 0.075

top operating curve: failure rate of 0.02

middle operating curve: failure rate of 0.01

bottom operating curve: failure rate of 0.005

=====Fig. 20.4=====

operating curves for larger failure rates uniformly above smaller ones

for failure rate to increase when relative precision fixed, tolerance interval

must remain the same while st. dev. of actual position increase

if acceptance interval does not change, misidentification rate decreases

20.4 Experimental Protocol

controlled experiments: important component of computer vision

experimental protocol: so experiment can be repeated and evidence verified by

another researcher

experiment protocol states:

- quantity (or quantities) to be measured
- accuracy of measurement
- population of scenes/images or artificially generated data

The experimental design describes how a suitably random, independent, and
representative set of images from the specified population is to be sampled,
generated, or acquired.

accuracy criterion: how comparison between true, measured values evaluated

experimental data analysis plan: how hypothesis meets specified requirement

experimental data analysis plan: how observed data analyzed

experimental data analysis plan: detailed enough for another researcher

analysis plan: supported by theoretically developed statistical analysis

20.5 Determining the Repeatability of Vision Sensor Measuring Positions

vision sensors: measure position or location in 1D, 2D, 3D

to determine repeatability of vision sensor: some number of points, times

20.5.1 The Model

: number of points to be measured

: actual but unknown positions of these points

: number of times each point is measured

: each point is -dimensional

: th measurement of the th point

assumption: measurements independent

assumption: difference between actual and measured positions

: standard deviation describing repeatability of vision sensor

20.5.2 Derivation

mean observed positions:

sum of norms squared of differences between observed positions and mean:

We need to determine the relationship between and .

20.6 Determining the Positional Accuracy of Vision Sensors

vision sensors: may measure position in 1D, 2D, 3D

To determine the accuracy of the vision sensor (after it has been suitably
calibrated), an experiment must be performed in which some number of points
in known positions are exposed to the sensor, the measured positions
are compared with the known positions, and the accuracy is computed in terms
of the degree to which the actual and measured positions agree.

positions of points: random and not follow regular pattern

number of points measured large enough: variance of accuracy small

20.6.1 The Model

: number of points to be measured

: actual but unknown positions of these points

: unknown expected positions of these points

points: independent

points: deviations between actual and nominal position

: number of times each point is measured

: each point is -dimensional

: th measurement of th point

assumption: measurements independent

difference between :

bias vector :

positional accuracy of vision sensor: described by

The purpose of the experiment is to estimate
by using a
large enough number of samples so that the unbiased estimate
of
is guaranteed to be sufficiently close to
.

20.6.2 Derivation

sum of norms squared of differences between observed and known positions:

We need to determine the relationship between .

20.7 Performance Assessment of Near-Perfect Machines

machines in recognition and defect inspection: required to be nearly flawless

error rate: fraction of time that machine's judgment incorrect

error rate: contains false detection and misdetection errors

false-detection rate: false-alarm rate: unflawed part judged flawed

misdetection rate: flawed part judged unflawed

20.7.1 Derivation

consider false-alarm errors; misdetection errors similar

: sampling size, total number of parts observed

: number of false-alarm judgements observed to occur in acceptance test

: machine performance specification of false-alarm fraction

maximum likelihood estimate of based on :

: machine passes acceptance test

: machine fails acceptance test

: true error rate

: random variable taking value 1 for false alarm, 0 otherwise

in maximum-likelihood technique, compute estimate as maximizing:

20.7.2 Balancing the Acceptance Test

If the buyer and seller balance their own self-interests exactly in a middle
compromise, the operating point chosen for the acceptance test will be the one
for which the false-acceptance rate (which the buyer wants to be small) equals
the missed-acceptance rate (which the seller wants to be small).

20.7.3 Lot Assessment

In the usual lot inspection approach, a quality control inspector makes
a complete inspection on a randomly chosen small sample from each lot.

reason for not inspecting all of the lot: cost

more than specified number of defective products found: entire lot rejected

20.8 Summary

mensuration quantizing error model: computes variance due to random error

=====Oldie 33:120=====

2002-02-26