upgrade2013

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?