Reconstruction from Magnitudes of Redundant
Representations
Radu Balan, Department of Mathematics, CSCAMM, ISR and NWC
Problem Formulation
Solution/Approach
In
Analysis
Features:


Processing
Synthesis
x
Relative low complexity O(Nlog(N))
On-line version is possible
c
<·,gi>
Theorem Assume {gk,f}(k,f)∈ZxZF is a frame for l2(Z).
Then
Out
Signal space: l2(Z)
y
Kx
K
The Analysis/Synthesis Components:
x
Main Result:
First observation [BBCE’09]:
Typical signal processing pipeline:
1.
is a frame for its span in HS(l2(Z)) iff for each m∈ZF,
H(ω,m) either vanishes identically in ω, or it is never zero;
2.
is a Riesz basis for its span in HS(l2(Z)) iff for each m∈ZF and ω,
H(ω,m) is never zero.
E=span{Kgk,f}
Kgk,f
nonlinear
embedding
Set:
Under these hypotheses:
Hilbert-Schmidt: HS(l2(Z))
Final step: Signal reconstruction from Qx by solving:
Analysis
Synthesis
Assume {Kg } form a frame for its span, E. Then the projection PE can
be written as:
k,f
Example: Short-Time Fourier Transform
for xt, assuming we already estimated xs for s<t,
Reconstruction scheme:
where {Qk,f} is the canonical dual of {Kg } .
k,f
Frame operator
Problem:
Given the Short-Time Fourier Amplitudes (STFA):
Stage 1
Stage 2
we want an efficient reconstruction algorithm:

Reduced computational complexity

On-line (“on-the-fly”) processing
ck,f
dk,f
|.|
Reconstruction
|ck,F-1|2
…
WL-1
Least
Square
Solver
x
Example:
it follows:
Where is this problem important:
Speech enhancement
Speech separation
Neural speech production
X-Ray Crystallography
Optical Communication
W0
Second Observation: Since
g
–
–
–
–
–
I
F
F
T
…
…
|ck,0|2
Explicitly
t
H0
W0