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Radu Balan University of Maryland College Park, MD 208742 email: rvbalan@math.umd

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

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Radu Balan University of Maryland College Park, MD 208742 email: rvbalan@math.umd

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  1. A Nonlinear Reconstruction Algorithm from Absolute Value of Frame Coefficients for Low Redundancy Frames Radu Balan University of Maryland College Park, MD 208742 email: rvbalan@math.umd.edu SampTA 2009 - Marseille, France

  2. Statement of the Problem • 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)

  3. Where is this problem relevant? X-Ray Crystallography Very thin layer, so that r is 2-D Problem: Given I(k) , estimate R(r).

  4. What is known? 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.

  5. But how to invert? • 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

  6. 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.

  7. Nonlinear Embedding • Generalize xKx to (d,d) tensors: • and embed the frame set F into s.lin func.

  8. (En) P x P Geometry  En Key Observation:

  9. Dimension Condition for Linear Reconstruction Necessary Condition for Linear Reconstruction:

  10. Real Case: E=R The condition: is satisfied for Note the critical case m=2n-1 !!

  11. Complex Case: E=C The condition: is satisfied by the following (suboptimal) choice Note m=O(n) but larger than 4n-2

  12. Is this enough? • 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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