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A Nonlinear Reconstruction Algorithm from Absolute Value of Frame Coefficients for Low Redundancy Frames. Radu Balan University of Maryland College Park, MD 208742 email: [email protected] SampTA 2009 - Marseille, France. Statement of the Problem . H=E n , where E= R or E= C

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A Nonlinear Reconstruction Algorithm from Absolute Value of Frame Coefficients for Low Redundancy Frames

Radu Balan

University of Maryland

College Park, MD 208742

email: [email protected]

SampTA 2009 - Marseille, France


Statement of the problem
Statement of the Problem Frame Coefficients for Low Redundancy Frames

  • H=En , where E=R or E=C

  • F={f1,f2,...,fm} a spanning set of m>n vectors

  • Assume the map:

    is injective up to a constant phase factor ambiguity

  • The Problem: Given c=N(x) construct a vector y equivalent to x (that is, invert N up to a constant phase factor)


Where is this problem relevant
Where is this problem relevant? Frame Coefficients for Low Redundancy Frames

X-Ray Crystallography

Very thin layer, so that r is 2-D

Problem:

Given I(k) , estimate R(r).


What is known
What is known? Frame Coefficients for Low Redundancy Frames

Theorem [R.B.,Casazza, Edidin, ACHA(2006)]

  • For E = R , m  2n-1, and a generic frame set F, then N is injective.

  • For E = C , m  4n-2, and a generic frame set F, then N is injective.


But how to invert
But how to invert? Frame Coefficients for Low Redundancy Frames

  • Observation [R.B.,Bodman, Casazza, Edidin, SPIE(2007)/JFAA(2009)]

  • Algorithm: Assume {Kfk} is spanning in M(En)

  • Compute the dual set { } to {Kfk}

  • Compute

  • Compute

    Then y~x


  • The algorithm is quasi-linear, but has a drawback: it requires a high redundancy.

  • Specifically: it requires m=O(n2)

    whereas we know that m=O(n) should be sufficient.

    This paper presents a novel algorithm that interpolates between O(n) and O(n2) keeping similar properties to the previous algorithm.


Nonlinear embedding
Nonlinear Embedding requires a high redundancy.

  • Generalize xKx to (d,d) tensors:

  • and embed the frame set F into s.lin func.


Geometry

(E requires a high redundancy.n)

P

x

P

Geometry

En

Key Observation:


Dimension condition for linear reconstruction
Dimension Condition for Linear Reconstruction requires a high redundancy.

Necessary Condition for Linear Reconstruction:


Real Case: requires a high redundancy.E=R

The condition:

is satisfied for

Note the critical case

m=2n-1 !!


Complex Case: requires a high redundancy.E=C

The condition:

is satisfied by the following (suboptimal) choice

Note m=O(n)

but larger than 4n-2


Is this enough
Is this enough? requires a high redundancy.

  • We obtained a necessary condition for the set to be spanning in Ed.

  • However this is not a guarantee that this happens!

  • In our experimental testing the set is spanning in Ed.

  • Open Problem/Conjecture:

    is generically spanning in Ed.


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