Upgrade Presentation

1. Compressive Sensing as a tool for Video Analysis
Rhian Davies
Idris Eckley, Lyudmila Mihaylova, Nicos Pavlidis
August 4, 2013

2. Motivation
“Big Brother is Watching You.”
- George Orwell, 1984

3. What is compressive sensing?
Compressive sensing is a method of reducing the amount of data
collected from a signal without compromising the ability to later
reconstruct the signal accurately.

4. CS Methodology
Figure: CS measurement process, courtesy of Volkan Cevher.

5. Restricted Isometry Property (RIP)
A matrix ΦΦΦ satisfies the Restricted Isometry Property (RIP) of order K if
there exists a δK ∈ (0, 1) such that
(1 − δk)||xxx||2
2 ≤ ||ΦΦΦxxx||2
2 ≤ (1 + δk)||xxx||2
2, (1)
for all xxx ∈ K = xxx : ||xxx||0 ≤ K.

6. Recovery of sparse transforms
y = Φx
∆(y, Φ) = x
Infinitely many solutions!
ˆx = argminy=φx ||x||0
ˆx = argminy=φx ||x||1
Optimisation based on the l1 norm can closely approximate compressible
signals with high probability.

7. Orthogonal Matching Pursuit
We shall define the columns of Φ to be ϕ1, ϕ2, . . . , ϕN each of length M.
Step 1: Find the index for the column of Φ which satisfies
λt = argmax j=1,...,N| < rt−1, ϕj > |
Step 2: Keeps track of the columns used. Λt = Λt−1 ∪ λt,
Φt = [Φt−1, ψλt ]
Step 3: Update the estimate of the signal. xt = argminx ||v − Φtx||2 .
Step 4: Update the measurement residual. rt = y − Φtxt.
Output: Estimated sparse vector ˆx

8. Background Subtraction
Figure: The background subtraction process

9. Foreground sparsity
(a) Original frame (b) Background Model (c) Foreground Mask

10. Background Subtraction with Compressive Sensing.
1. Initialise a compressed background yb
0 .
2. Compressively Sense yt = Φxt.
3. Reconstruct ∆(yt − yb
t )
4. Update Background yb
t+1 = αyi + (1 − α)yb
i

11. Experimentation
Figure: Sanfran test video courtesy of Seth Benton

12. Further Work
Choice of Φ and ∆?
More advanced methods of background subtraction.
Adapting with varying sparsity.
Knowing when to reconstruct.
Exploiting the properties of natural images.

13. Any Questions?