Perception: objects in the environment · (sub-populations) a new observation belongs, on the basis of a training set of data containing observations (or instances) whose category

Perception: objects in the environment

Zsolt Vizi, Ph.D.

Robert Bosch Kft.

2018

Perception: objects in the environment Robert Bosch Kft.

Self-driving cars


Sensor fusion: one categorization

I Type 1: low-level/raw data fusioncombining several sources of raw data to produce new datathat is expected to be more informative than the inputs.

I Type 2: intermediate-level/feature level fusioncombining various features (e.g. positions) into a featuremap, which can be used for higher level decisions

I Type 3: high-level/decision fusioncombining decisions from several experts (e.g. voting,fuzzy-logic, statistical methods)


Perception: topics of this lecture

I Object tracking

I Object type classification


Tracking problems

I The basic tracking problem is to estimate the positionand velocity of the target(s), using the available sensordata (from a sequence of scans).

I The multi-target tracking problem is not simply atracking problem when there are more than one target →problem of associating measurements with targets.


Tracking problems vs. typical estimation problems

I Strong temporal component is involved.

I Estimation of quantities, which are expected to change overtime.

I Current state is interested.

I Current state is computed from previous states.


Bayesian inference

I Inference methods consist of estimating the current valuesfor a set of parameters based on a set of observations ormeasurements.

I Bayesian estimation: the parameters are randomvariables that have a prior probability and the observationsare noisy as well


Recursive Bayesian estimation

Source: http://slidedeck.io/robertsy/ensemble-da


Bayes theorem

p(x|z) =p(z|x)p(x)

p(z)

I X: target state (random vector variable);

I Z: observation (random vector variable);

I p(x) (probability density function of X): prior density;

I p(x|z): posterior density;

I p(z|x): likelihood function;

I p(z): normalization constant

p(z) =

∫Rnx

p(z|x)p(x) dx

I Notation: p(x, z) = joint density


Point estimators

In summary: we estimate densities at each steps... How do weuse it in a specific tracking problem? → ’Mapping’ the densityinto real world: point estimator xProcedure

1. Define cost function L(x, x), which defines a penalty for anerroneous estimate x 6= x. Typical choice:

L(x, x) = (x− x)TM(x− x).

2. Bayesian risk R:

R = E (L(x, x)) =

∫Rnx

L(x, x)p(x)dx

Note: x = x(z). Optimal choice:

x(z) = argminx∗(z)

∫Rnx+nz

L(x∗(z), x)p(x, z) dx dz


Point estimators

In summary: we estimate densities at each steps... How do weuse it in a specific tracking problem? → ’Mapping’ the densityinto real world: point estimator xProcedure

3. Using first derivative, we get [homework]

x(z) = E(x|z)

4. Uncertainity for this estimation:

Pxx = E((x− x)(x− x)T |z)


Dynamic and sensor model

We focus on first-order Markov processes (current state isdependent only on the previous state).

1. System dynamic model:

xn = fn−1(xn−1) + un + vn−1

I xn: state vector at time tnI fn−1: deterministic transition functionI un: known deterministic controlI vn−1: additive noise


Dynamic and sensor model

We focus on first-order Markov processes (current state isdependent only on the previous state).

2. Sensor model:zn = hn(xn) + wn

I zn: current observation vectorI hn: deterministic observation functionI wn: additive noise

Simplifying assumption: fn and hn are adiabatic (changing veryslowly in time) → fn = f , hn = h


Recursive Bayesian filtering

Notation: z1:n = {z1, z2, . . . , zn} (observations up to nth step)Goal: estimating p(xn|z1:n) → applying Bayes theorem:

p(xn|z1:n) =p(z1:n|xn)p(xn)

p(z1:n)

After some calculations, we obtain

p(xn|z1:n) =p (zn|xn) p (xn|z1:n−1)

p (zn|z1:n−1)

Using Chapman-Kolmogorov equation for p (xn|z1:n−1), wederive

p (xn|z1:n−1) =

∫Rnx

p(xn|xn−1)p(xn−1|z1:n−1) dxn−1


Recursive Bayesian filtering


Back to the point estimators: a notation

State estimation:xn|p = E{xn|z1:p}

Uncertainity estimation:

Pxxn|p = E

{(xn − xn|p)(xn − xn|p)T |z1:p

}


Back to the point estimators: prediction

More algebraic manipulation (combining previous formulae,system dynamic model) give:

xn|n−1 =

∫Rnx

{fn−1(xn−1) + un + vn−1} p(xn−1|z1:n−1)dxn−1

and

Pxxn|n−1 =

∫Rnx

{fn−1(xn−1) + un − xn|n−1

}×{

fn−1(xn−1) + un − xn|n−1}T ×

p(xn−1|z1:n−1)dxn−1 +Q,

where Q is the covariance matrix of the system noise:

Q =

∫Rnx

vn−1vTn−1p(xn−1|z1:n−1)dxn−1


Back to the point estimators: update [by Kalman]

xn|n = Azobsn + b

Assumption:I E(xn − xn|z1:n−1) = 0I E((xn − xn)zobsn |z1:n−1) = 0

Goal #1: Determine A, bSome calculations give

b = xn|n−1 −Azn|n−1and

A = Kn = P xzn|n−1

(P zzn|n−1

)−1which implies

xn|n = xn|n−1 +Kn(zobsn − zn|n−1)P xxn|n = KnP

zzn|n−1K

Tn


Back to the point estimators: update [by Kalman]

Goal #2: Determine zn|n, Pxzn|n−1, P

zzn|n−1

zn|n =

∫Rnx

{hn(xn) + wn}p(xn|z1:n−1)dxn,

P xzn|n−1 =

∫Rnx

{xn − xn|n−1}{hn(xn)− zn|n−1}T p(xn|z1:n−1)dxn,

P zzn|n−1 =

∫Rnx

{hn(xn)− zn|n−1}{hn(xn)− zn|n−1}T×

p(xn|z1:n−1)dxn +R

where covariance matrix of observation noise is

R =

∫Rnx

wnwTn p(xn|z1:n−1)dxn


Recursive point estimation process


Gaussian densities: Kalman filters

I Linear KF: f , h linear

I Extended KF: f , h nonlinear + Taylor approximation

I Finite Difference KF: f , h nonlinear + Stirlingapproximation

I Unscented KF: f , h nonlinear + sigma points: onhypersphere in all directions

I Spherical Simplex KF: f , h nonlinear + sigma points:on intersection of simplex & hypersphere

I Gaussian-Hermite KF: f , h, nonlinear + sigma points:vertices of hypercube

I Monte Carlo KF: f , h nonlinear + MC sampling


Kalman Filters

Source: https://www.mathworks.com/videos/understanding-kalman-filters-part-1-why-use-kalman-filters–1485813028675.html


LKF

System dynamic model:

xn = Fxn−1 + vn−1

Sensor model:zn = Hxn + wn

All densities are Gaussian:

N (x;m,C) =1√

(2π)ndet(C)exp

{−1

2(x−m)C−1(x−m)T )

}

I p(xn|z1:n) = N (xn; xn|n, Pxxn|n)

I p(qn|z1:n−1) = N (qn; qn|n, Pqqn|n−1)

I p(zn) = N (zn; zn|n, Pzzn|n−1)


LKF

Using these assumptions, we obtain:

1. Prediction:xn|n−1 = Fxn−1|n−1,

P xxn|n−1 = FP xx

n−1|n−1FT +Q

2. Observation prediction/likelihood:

zn|n−1 = Hxn|n−1,

P zzn|n−1 = HP xx

n|n−1HT +R,

P xzn|n−1 = P xx

n|n−1HT


LKF

Using these assumptions, we obtain:

3. Update:

Kn = P xzn|n−1

(P zzn|n−1

)−1xn|n = xn|n−1 +Kn

(zobsn − zn|n−1

),

P xxn|n = P xx

n|n−1 −KnPzzn|n−1K

Tn ,


An example [finally...]

State vector:

xn =[pn vn

]=[position velocity

]Constant velocity model:

F =

[1 dt0 1

]System noise:

vn−1 ∼ N (0, Q)

Observation model (sensor output: noisy measurement forposition):

H =[1 0

]Measurement noise:

wn−1 ∼ N (0, σ2)


An example [finally...]

(http://david.wf/kalmanfilter/)


Another example

Simo Sarkka, Bayesian Filtering and Smoothing, CambridgeUnversity Press, 2013

Figure: Chapter 4.3, Example 4.3


Some words about multi-target tracking problem

I Nearest Neighbors FilterI Probabilistic Data Association FilterI Multihypothesis Filter

More details: http://ais.informatik.uni-freiburg.de/teaching/ws10/robotics2/pdfs/rob2-15-dataassociation.pdf


Non-Gaussian densities: Particle Filters

Explanation without equations:https://www.youtube.com/watch?v=aUkBa1zMKv4

Tutorial with a bunch of equations:Doucet A., Johansen A.M., A Tutorial on Particle Filtering andSmoothing: Fifteen years later


References

1. Anton J. Haug, Bayesian Estimation and Tracking: APractical Guide, Wiley, 2012

2. Sudha Challa et. al, Fundamentals of object tracking,Cambridge University Press, 2011


Classification problems

The problem of identifying to which of a set of categories(sub-populations) a new observation belongs, on the basis of atraining set of data containing observations (or instances)whose category membership is known.


Classification problems

I Supervised learning: learning where a training set ofcorrectly identified observations is available (MachineLearning)

I Feature/Explanatory variable/Independentvariable: quantifiable property, which is used in therepresentation of the observation

I Category/Outcome/Dependent variable/Targetvariable/Class/Response variable: e.g.spam/not-spam

I Classifier: classification algorithm

I Binary and multiclass classification: two and moreclasses are involved


Classification vs. ’estimation’

I Many classifiers output simply the ’best class’ as an answer

I Probabilistic classifiers/estimators return a probability ofthe instance being a member of each of the possible classes→ ’best class’ is naturally the one with the highestprobability

I Advantages of probabilistic methods:I probability = confidence valueI efficient in large-scale problems, because error-propagation

can be avoided


General Linear Models

GLMLarge class of models, where the response variable is assumed tofollow an exponential family distribution and this variable is a(typically nonlinear) function of the linear combination offeature values.



Exponential family

p(x) = h(x) exp{ηTT (x)−A(η)

}More details:https://people.eecs.berkeley.edu/ jordan/courses/260-spring10/other-readings/chapter8.pdf

Examples for family members

I Bernoulli distribution: X ∈ {0, 1};P (X = 1) = π

p(x) = πx(1− π)1−x, x ∈ {0, 1}

I Normal distribution



General equation

E(Y ) = g−1(βTX),

where

I X: explanatory variables;

I Y : response variable from an exponentatial familydistribution;

I β: parameter vector;

I g: link function



An example for GLM: linear regression

Model:E(Y ) = β0 + β1x1 + · · ·+ βkxk

Y is normally distributedLink function: g−1(E(Y )) = E(Y ) [identity]


Logistic regression

I Response variable Y is Bernoulli distributed withparameter π, i.e. the observations can correspond to twoclasses (binary case)→ if Y ∼ Bernoulli(π), then E(Y ) = π.

I X = {X1, X2, . . . , Xk} is the feature vector.I Odds ratio: ratio of π and 1− π, which gives a measure

for comparing class membershipsI Link function: logarithm of odds ratio

g(π) = log

(π

1− π

)I Model:

log

(π

1− π

)= β0 + β1X1 + · · ·+ βkXk


Logistic regression

Probably more familiar with mean function g−1(βTX):

π =1

1 + exp(−βTX)

Source: https://rcompanion.org/rcompanion/e06.html


Logistic regression

I We use a training set{X(i), Y (i)

}Ni=1

.

I Goodness of parameter vector = how correctly it works onthe training set ⇐⇒ gap/error is small

I Cost function/Error function/Loss function:measures the error of the estimate, the goal is to minimize

I Classical choice: mean squared error

MSE(β) =1

N

N∑i=1

(Yi − Yi)2


Logistic regression

I A more convenient cost function now:

CE(β) =− 1

N

N∑i=1

Y (i) log(g−1

(βTX(i)

))− 1

N

N∑i=1

(1− Y (i)) log(

1− g−1(βTX(i)

))Finding optimal parameter vector = minimizing cost function


Some notes on optimization

Gradient descent

Source: http://charlesfranzen.com/posts/multiple-regression-in-python-gradient-descent/


Some notes on optimization

Stochastic gradient descent

Source: Andrew Ng, Machine Learning (online course, Coursera)


Multiclass problem

Possible modifications:

I 1 vs. rest [K classifiers]

I 1 vs. 1 [K(K − 1)/2 classifiers]

I multinominal extension: for logistic regression, introducesoftmax function

πj =exp

(βTj X

)∑K

i=1 exp(βTi X

) ; j = 1, 2, . . . ,K


Evaluation

Confusion matrix

Source:https://www.packtpub.com/books/content/supervised-learning-classification-and-regressionLiu et. al, Learning accurate and interpretable models based on regularized random forestsregression, doi:10.1186/1752-0509-8-S3-S5


Evaluation

ROC curve

Source:http://jxieeducation.com/2016-09-27/Attractive-Mathematical-Properties-Of-The-ROC-Curve/


Naive Bayes classifier

Probabilistic model: the goal is to calculate the probability

p(Cj |x),

where Cj is the correspodence to class j, x = (x1, x2, . . . , xk) isan instance of the feature vector.Approach: Bayes theorem!

p(Ck|x) =p(x|Ck)p(Ck)

p(x)

Condition of ’naivety’: features are conditionallyindependent

p(xi|xi+1, . . . , xk, Cj) = p(xi|Cj)


Naive Bayes classifier

Using definition of conditional PDFs, law of total probabilityand chain rule, we can derive

p(Cj |x) =p(Cj)

∏ki=1 p(xi|Cj)∑k

i=1 p(Ci)p(x|Cj)

How to classify:

y = argmaxj∈{1,2,...,K}p(Cj)

k∏i=1

p(xi|Cj)


Naive Bayes-Gauss classifier


More general topics (not covered now)

I Bias, variance

I Problem of underfitting/overfitting

I Regularization

I Curse of dimensionality

I Feature engineering

I Model selection (e.g. elimination techniques)


More basic classification algorithms

I Decision Tree, Random Forest

I K-nearest neighbors

I Support Vector Machine


Deep Learning

Artificial Neural Networks

Source: https://www.digitaltrends.com/cool-tech/what-is-an-artificial-neural-network/


Deep Learning


Source: https://www.mathworks.com/discovery/convolutional-neural-network.html


Deep Learning


Source: http://www.flytxt.com/deep-learning-with-recurrent-neural-networks-rnn/


References

Online courses

I Kirill Eremenko, Machine Learning (udemy.com)

I Kirill Eremenko, Deep Learning (udemy.com)

I Andrew Ng, Machine Learning (coursera.org)

I Pennsylvania State University, Statistics online(https://onlinecourses.science.psu.edu/statprogram/)

Books

I Christopher Bishop, Pattern Recognition and MachineLearning, Springer, 2011

I Goodfellow et. al, Deep Learning, MIT Press, 2016

Useful links

I https://www.kaggle.com

I https://www.analyticsvidhya.com/


Thank You For YourAttention!


Documents

Perception: objects in the environment · (sub-populations) a new observation belongs, on the basis of a training set of data containing observations (or instances) whose category