**Research summary and some important papers of Namrata Vaswani**

(somewhat old -- Recent Research Summary
)

**Talk
on Recursive Structured Signals Recovery
and Applications in Bio-Imaging, ECE
dept colloquium at UIUC, December 2013**

Some key papers:** **

**Recursive Recovery of Sparse Signal Sequences**- Kalman
filtered Compressed Sensing (ICIP 2008)
**First studied the problem of recursive reconstruction of sparse signal sequences with time-varying sparsity patterns**and proposed an efficient solution called KF-CS. Introduced the practically valid assumption of slow sparsity pattern change.- LS-CS-residual (IEEE Trans. Sig.
Proc., Aug 2010)
- The LS-CS-residual
idea and extensive simulations
- Bound for CS-residual
error and comparison with simple CS error bound
**Obtained mild conditions under which the support errors and hence the signal reconstruction errors of LS-CS are bounded by time-invariant and small values (algorithm is stable)**- Brief idea of KF-CS
**Modified-CS**- Modified-CS (IEEE Trans. Sig. Proc.,
Sept 2010)
- The modified-CS idea,
extensive simulations and proof-of-concept applications in MRI/video
**Proof of Exact Reconstruction under much weaker assumptions**on the measurement matrix than those needed for simple CS (means: need much fewer measurements)- Regularized
modified-CS idea (use the prior knowledge of signal values along with
support knowledge)
- Time
Invariant Error Bounds for Modified-CS based Sparse Signal Sequence
Recovery (arXiv:1305.0842 [cs.IT], shorter versions in ISIT 2013,
Allerton 2010)
- Obtained mild conditions under which the support
errors and hence the signal reconstruction errors of modified-CS are bounded
by a time-invariant and small value at all times (algorithm is
``stable``). Did this for a weaker and much more practically valid
signal change model than what was assumed in the LS-CS paper. Exact same
idea can be used to obtain similar conditions for LS-CS as well.
**Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise**

o Brian Lois and Namrata Vaswani, A Correctness Result for Online Robust PCA, submitted, on arXiv:1409.3959 [cs.IT].

o Chenlu Qiu, Namrata Vaswani, Brian Lois and Leslie Hogben, Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise, IEEE Trans. Information Theory, August 2014, arXiv:1211.3754v6 [cs.IT]

o Han Guo, Chenlu Qiu and Namrata Vaswani, An Online
Algorithm for Separating Sparse and Low-dimensional Signal Sequences from their
Sum, IEEE Trans. Signal Processing, August 2014, arXiv:1310.4261 [cs.IT] (older
title: Practical ReProCS for Separating Sparse and Low-dimensional Signal
Sequences from their Sum)

§ This work
studies the recursive robust principal components analysis (PCA) problem. If
the outlier is the signal-of-interest, this problem can be interpreted as one
of recursively recovering a time sequence of sparse vectors, $S_t$, in the
presence of large but structured noise, $L_t$. The structure that we assume on
$L_t$ is that $L_t$ is dense and lies in a low dimensional subspace that is
either fixed or changes ``slowly enough". A key application where this
problem occurs is in video surveillance where the goal is to separate a slowly
changing background ($L_t$) from moving foreground objects ($S_t$) on-the-fly.

§
To solve the above problem, in recent
work, we introduced a novel solution called Recursive Projected CS (ReProCS).
In this work we develop a simple modification of the original ReProCS idea and
analyze it. This modification assumes knowledge of a subspace change model on
the $L_t$'s. Under mild assumptions and a denseness assumption on the
unestimated part of the subspace of $L_t$ at various times, we show that, with
high probability (w.h.p.), the proposed approach can exactly recover the
support set of $S_t$ at all times; and the reconstruction errors of both $S_t$
and $L_t$ are upper bounded by a time-invariant and small value. In simulation
experiments, we observe that the last assumption holds as long as there is some
support change of $S_t$ every few frames.

§
We also show extensive experiments for
separating real videos into foreground and background layers on the fly.
ReProCS significantly outperforms existing approaches for robust PCA when the
foreground objects are large and move slowly and also when when the
foreground-background intensities are similar (small magnitude nonzero sparse
part). See webpages of Han Guo or Chenlu Qiu