Background Subtraction Algorithms

8/22/2019 Background Subtraction Algorithms

1/26

Background

Subtraction AlgorithmsGuassian Mixture Model for Foreground Segmentation

ViBe: A Universal Background Subtraction Algorithm


2/26


3/26

Guassian Mixture Model

Problem Statement

At each sample time t, estimate the mostlikely state k (Background or Foreground)from a set of observations sampled frompixel values X which are samples of somerandom variable X.

The pixel value process X is modelled by amixture of K guassian densities withparameter set (k)


4/26

Modelling of Pixel Value Processusing K Guassian Densities

Mean MatrixCovariance MatrixPixel Value Process

K States

(k) is parameter set defined as = {(k),(k)}


5/26

Distribution of Pixel Value ProcessX

Distribution of Xfor all states

Probability of EachState K

Distribution of pixelvalue process foreach state K

is the total parameter set = * P(k1),. P(kn), 1,2,.n}

Where P(k) = w(k) is probability of each state and

(k) = {(k),(k)}


6/26

Estimation of Current StateStauffer-Grimson Algorithm

Bayess Theorem: Probability of Current state givenCurrent Pixel Sample and Parameter Set.

The State k which maximizes P(k|X,) is called Match in, it isthe Maximum A Posteriori Estimate

k = argmax(k) w(k) f X|k(X|k,(k))


7/26

Segmentation of Foreground

The procedure of demarcation ofbackground and foreground starts with

the relation :w(k)/(k) is proportional to w(k)fX|k

Where w(k) is the probability of occurrence ofstate k, which will be greater for Background

states and (k) will be lower as Background doesnot vary much. The first B of ranked states whoseaccumulated Probability is greater than T aredeemed as background


8/26

Estimating The Parameters

These Equations assume stationary processes k and X and a fixednumber of observations N.


9/26

Practical Algorithm


10/26

Practical Algorithm

Instead of calculating

Stauffer and Grimson Algorithm defines apixel match with the background if the pixellies between = 2.5 of the mean of one ofGuassian distributions. Which can be stated as :


11/26

Practical Algorithm

FasterImplementation


12/26

ViBe Background

Subtraction Algorithm


13/26

Pixel Model and Classification

This algorithm doesnot opt for a particulartype of Pdf.

Classifies each new pixel wrt itsimmediate neighbourhood.

Models each background pixel with a setof samples. The current pixel value iscompared to its closest samples within thecollection of samples.


14/26

Each Background pixel x is modeled by acollection of N background sample values takenin Previous Frames

In order to classify a new pixel v(x) theEuclidean distance is calculated between v(x)

and samples of model m(x). If this distance isless for a number of samples lying in modelsample m(x) set as a threshold (cardinality = 2),then the pixel belongs to background.



15/26


Cardinality = 2


16/26

Background Initialization fromSingle Frame

Assumption : Neighbouring pixels sharesimilar temporal distribution.

Populating the pixel models with values foundin the spatial neighbourhood of each pixeltaken randomly.

Random selection of samples in 8-connectedneighbourhood of each pixel.

Assuming t = 0 the first frame and NG(x) thespatial neighborhood of a pixel location x


17/26

Background Model Initializationfrom a single Frame

Here, if the cardinality of m(x) is greater than theneighborhood, there is a possibility that a given Pixel v(y) isselected several times.


18/26

Updating the Background ModelOver Time

How to incorporate Foreground Pixel in aBackground Model over period of time.

Two distinct Update policies are:-

Conservative Policy: Never includes a

Foreground pixel in Background Blind Update Policy: Samples added to

background whether they belong toForeground or not.


19/26

Updating the Background ModelOver Time

Adopted Conservative Policy, exploitspatial information to inject information

regarding background evolution intobackground pixel models masked locallyby foreground.


20/26

Important Features of UpdateScheme

Memory-less Update Policy which ensuressmooth decaying lifespan of samples

stored in background pixel models Random Time Sub sampling to extend the

time window covered by the backgroundpixel models

A Mechanism that propagates backgroundpixel samples spatially to ensure spatialconsistency.


21/26

Memoryless Update Policy

Randomly discard pixel sample from Modelbased on Uniform Distribution.

In this way the systematic removal of oldsample first is avoided.

Mathematically, the probability of a samplepresent at time t being preserved afterupdate of pixel model is given by:

( N-1)/NFor a fraction of time dt this probability canbe written as :


22/26

Time SubSampling

It is not necessary to update the eachbackground pixel model for each new

frame. Instead the background is updatedbased on the value of a Random Variable,mostly selected to be 16.

In practice when a pixel value has been

classified as belonging to background, arandom process determines whether thisvalue is used to update the correspondingpixel model or not.


23/26

Spatial Consistency throughBackground Samples Propagation

To provide a way of updating thebackground pixel model that are hidden

by foreground. Count the number of timesa pixel has been classified as foreground.

If this number reaches a given thresholdfor a particular pixel location, the currentpixel value at that location is inserted intobackground.


24/26


Assumption: Neighboring BackgroundPixels share a similar temporaldistribution

A new background sample of a pixel shouldalso update the models of the neighboringpixels.

Background models hidden by foreground willbe updated with background samples fromneighboring pixel locations from time to time.


25/26


For a 4 or 8 connected NG(x), if it isdecided to update the model m(x) set by

inserting pixel v(x), the same pixel is usedto update the Neighborhood NG(x) basedon Uniform Law.


26/26

Background Subtraction Algorithms

Documents

Transcript of Background Subtraction Algorithms