The Problem of Phase Retrieval in Light and Electron Microscopy of Strong Objects

Drenth, A.J.J.; Huiser, A.M.J.; Ferwerda, H.A.

doi:10.1080/713819083

Cited by 51 publications

(20 citation statements)

References 4 publications

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“…As another example, in an optical imaging system it may be possible to measure the field intensity in two slightly defocused planes. For this case it has also been shown that there is a unique solution to the phase retrieval problem [9,19].…”

Section: 2: the Phase Retrieval Problemmentioning

confidence: 99%

“…Finally, suppose that NŽ2 and consider the sequence xz ( n). Since A X 2 (z) = 2 + 2z-' + 5z -N (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23) has no zeros on the unit circle or in reciprocal pairs, then o,(W,I 2 ) along the line oz=No, is also sufficient to uniquely specify x(n,,n 2 ) to within a scale factor provided NŽ2. This approach of transforming multidimensional sequences into their I-D projections provides only a partial answer to the uniqueness question.…”

Section: Iv3: Uniqueness Via Projectionsmentioning

confidence: 99%

“…and therefore G(z) = G(1/z) (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) From (3-12) it follows that the singularities of G(z) (i. i.e., for each w, o,(w) is either equal to zero or w. However, in order for g(n) to be a zero phase sequence, in addition to being real, G(w) must also be non-negative.…”

Section: 31: Uniqueness In Terms Of Continuous Pjasementioning

confidence: 99%

“…Therefore, note that if g(n) is a zero-phase sequence, then its Fourier transform, G(w), must be real and non-negative. However, if G(~) is real, then g(n) must be conjugate symmetric, g(n) = g*(-n) (3)(4)(5)(6)(7)(8)(9)(10)(11) and therefore G(z) = G*(l/z*) (3-12)…”

Section: 31: Uniqueness In Terms Of Continuous Pjasementioning

confidence: 99%

“…-sin w0-Therefore, the matrix S is singular. Furthermore, any solution to (5)(6)(7)(8)(9)(10) corresponds to a sequence y(n) of the form y(n) = a8(n) + aS8(n-1) + a8(n-2) …”

Section: V211: Time-domain Solutionmentioning

confidence: 99%

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Signal reconstruction from phase or magnitude

Hayes

Lim

Oppenheim³

1980

IEEE Trans. Acoust., Speech, Signal Process.

335

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This thesis addresses two issues related to the problem of reconstructing a one-dimensional or a multidimensional sequence from either the phase or the magnitude of its Fourier transform. The first concerns the development of conditions under which a sequence is uniquely defined in terms of only phase or magnitude information. For example, it is shown that a one-dimensional sequence is, in most cases, uniquely specified by the phase of its Fourier transform if the sequence is finite in length. In the case of magnitude, a condition for uniqueness is presented which is a generalization of the minimum and maximum phase constraints and includes them as a special case. For multidimensional sequences, on the other hand, it is shown that a finite support constraint is sufficient, in most cases, for the sequence to be uniquely defined by either the phase or magnitude of its Fourier transform.The second issue which is addressed concerns the development of algorithms for reconstructing a sequence from either the phase or magnitude of its Fourier transform. In particular, several algorithms are presented for reconstructing a sequence from its phase. These algorithms, which include iterative as well as non-iterative approaches, always lead to the correct solution provided the appropriate uniqueness constraints are fulfilled. An iterative procedure is also described for reconstructing a multidimensional sequence from the magnitude of its Fourier transform. However, the convergence of this algorithm to the desired sequence appears to depend upon the availability of an initial estimate which is sufficiently close to the correct solution. Finally, a number of examples are presented which illustrate the use of these algorithms. -2-

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Section: 2: the Phase Retrieval Problemmentioning

confidence: 99%

Section: Iv3: Uniqueness Via Projectionsmentioning

confidence: 99%

Section: 31: Uniqueness In Terms Of Continuous Pjasementioning

confidence: 99%

Section: 31: Uniqueness In Terms Of Continuous Pjasementioning

confidence: 99%

“…-sin w0-Therefore, the matrix S is singular. Furthermore, any solution to (5)(6)(7)(8)(9)(10) corresponds to a sequence y(n) of the form y(n) = a8(n) + aS8(n-1) + a8(n-2) …”

Section: V211: Time-domain Solutionmentioning

confidence: 99%

See 3 more Smart Citations