background subtraction algorithms
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Background
Subtraction AlgorithmsGuassian Mixture Model for Foreground Segmentation
ViBe: A Universal Background Subtraction Algorithm
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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)
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Modelling of Pixel Value Processusing K Guassian Densities
Mean MatrixCovariance MatrixPixel Value Process
K States
(k) is parameter set defined as = {(k),(k)}
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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)}
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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))
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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
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Estimating The Parameters
These Equations assume stationary processes k and X and a fixednumber of observations N.
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Practical Algorithm
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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 :
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Practical Algorithm
FasterImplementation
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ViBe Background
Subtraction Algorithm
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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.
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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.
Pixel Model and Classification
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Pixel Model and Classification
Cardinality = 2
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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
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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.
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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.
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Updating the Background ModelOver Time
Adopted Conservative Policy, exploitspatial information to inject information
regarding background evolution intobackground pixel models masked locallyby foreground.
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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.
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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 :
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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.
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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.
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Spatial Consistency throughBackground Samples Propagation
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.
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Spatial Consistency throughBackground Samples Propagation
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.
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