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Compressive Sensing as a tool for Video Analysis
Rhian DaviesIdris Eckley, Lyudmila Mihaylova, Nicos Pavlidis
August 4, 2013
Motivation
“Big Brother is Watching You.”- George Orwell, 1984
What is compressive sensing?
Compressive sensing is a method of reducing the amount of datacollected from a signal without compromising the ability to laterreconstruct the signal accurately.
CS Methodology
Figure: CS measurement process, courtesy of Volkan Cevher.
Restricted Isometry Property (RIP)
A matrix ΦΦΦ satisfies the Restricted Isometry Property (RIP) of order K ifthere exists a δK ∈ (0, 1) such that
(1− δk)||xxx ||22 ≤ ||ΦΦΦxxx ||22 ≤ (1 + δk)||xxx ||22, (1)
for all xxx ∈∑
K = xxx : ||xxx ||0 ≤ K .
Recovery of sparse transforms
I y = Φx
I ∆(y ,Φ) = x
I Infinitely many solutions!
x̂ = argminy=φx ||x ||0
x̂ = argminy=φx ||x ||1Optimisation based on the l1 norm can closely approximate compressiblesignals with high probability.
Orthogonal Matching Pursuit
We shall define the columns of Φ to be ϕ1, ϕ2, . . . , ϕN each of length M.
I Step 1: Find the index for the column of Φ which satisfiesλt = argmax j=1,...,N | < rt−1, ϕj > |
I Step 2: Keeps track of the columns used. Λt = Λt−1 ∪ λt ,Φt = [Φt−1, ψλt ]
I Step 3: Update the estimate of the signal. xt = argminx ||v − Φtx ||2 .
I Step 4: Update the measurement residual. rt = y − Φtxt .
I Output: Estimated sparse vector x̂
Background Subtraction
Figure: The background subtraction process
Foreground sparsity
(a) Original frame (b) Background Model (c) Foreground Mask
Background Subtraction with Compressive Sensing.
1. Initialise a compressed background yb0 .
2. Compressively Sense yt = Φxt .
3. Reconstruct ∆(yt − ybt )
4. Update Background ybt+1 = αyi + (1− α)ybi
Experimentation
Figure: Sanfran test video courtesy of Seth Benton
Further Work
I Choice of Φ and ∆?
I More advanced methods of background subtraction.
I Adapting with varying sparsity.
I Knowing when to reconstruct.
I Exploiting the properties of natural images.
Any Questions?