We study the problem of
reconstructing a sparse signal from a limited number of its linear
projections when a part of its support is known. This may be available
from prior knowledge. Alternatively, in a problem of recursively
reconstructing time sequences of sparse spatial signals, one may use
the support estimate from the previous time instant as the "known"
part of the support. The idea of our solution (modified-CS) is
to solve a convex relaxation of the following problem: find the
sparsest possible signal that satisfies the data constraint and whose
support contains the "known"
part of the support. In other words, we try to find a signal with the
smallest number of new additions to the known support that satisfies
the data constraint.
We derive sufficient conditions for exact
reconstruction using modified-CS. These are much weaker than the
sufficient conditions needed for CS, particularly when the known part
of the support is large compared to the unknown part.
Dynamic MRI
Reconstruction Results Using Modified-CS(Noiseless measurements)
1. Reconstruction of a Larynx
Sequence(Size m=256X256, Sparsity S=7%m) (1)
Measurements n0=0.5m for t=1 and
n=0.19m for t>1
Original
Video
CS Reconstruction CS-residual Reconstruction
Mod-CS
Reconstruction
Reconstruction Error Plot
(2) Measurements
n=0.19m
for all frames and lowest subband in Wavelet domain is
considered as
partly known support at t=1.
Original
Video
CS
Reconstruction CS-residual
Reconstruction Mod-CS Reconstruction
Reconstruction Error Plot
(3)
Measurementsn=0.19mt>1
frames and n0=0.2m at t=1 with lowest
subband in Wavelet domain
is considered as
partly known support at t=1.
Original
video
CS
Reconstruction
CS-residual
Reconstruction
Mod-CS
reconstruction
Reconstruction Error Plot
2. Reconstruction of a Cardiac
Sequence(Size m=128X128, Sparsity S=10%m, Measurements n0=0.5m for t=1
and n=0.19m for t>1)
Functional MRI active regions detection
Brain image size
m=64X64.Sparisty S=7%m. Measurement n=100%m at t=1 and n=0.3m random
kx sampling for t>1
CNR=4 real brain sequence with simulated activation --Activation Map
Full Sampling Activation Map
BPDN Activation Map
Kt-FOCUSS
Activation
Map
Modified-CS-residual(complex)
Activation Map
--ROC plot
Three methods are compared(averaging over 10 realizatioins): BPDN,
Kt-FOCUSS and our method
Modified-CS-residual for complex image
