#### DMCA

## Phase retrieval by iterated projections (2003)

Venue: | J. Opt. Soc. Amer. A |

Citations: | 29 - 2 self |

### Citations

291 | Phase retrieval algorithms: a comparison
- Fienup
- 1982
(Show Context)
Citation Context ...h respect to a common basis in EN , which we take to be the object domain. Because the Fourier modulus projection is more naturally expressed in the Fourier domain, we write πmod = F −1 · ˜πmod · F , =-=(1)-=- where F is the unitary transformation to the Fourier domain, and ˜πmod is the projection operator which acts componentwise, that is, on each pixel of the (discrete) Fourier transform. Geometrically, ... |

288 |
A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures
- Gerchberg, Saxton
- 1972
(Show Context)
Citation Context ...3. The difference map Given two arbitrary projections π1 and π2, we consider the “difference map” D:E N → E N defined by D = 1 + β∆ , (4) where β is a nonzero real parameter and ∆ = π1 ◦ f2 − π2 ◦ f1 =-=(5)-=- 9is the difference of the two projection operators, each composed with a map fi:E N → E N . The detailed form of the maps fi is secondary to the global behavior of the difference map and is discusse... |

48 |
Fienup, "Phase Retrieval and Image Reconstruction for Astronomy
- Dainty, R
- 1987
(Show Context)
Citation Context ...map onto the subspace Csupp of objects having a specified support S. If ρn is the value of pixel n in the object domain, then support projection is 5the map ⎧ πsupp : ρn ↦→ ρ ′ n = ⎪⎨ ⎪⎩ ρn if n ∈ S =-=(2)-=- 0 if n /∈ S. Since Csupp is a linear subspace it is convex; projection computations require at most N operations. Positivity. For real-valued objects one can impose positivity. The unique, distance m... |

35 |
Image restoration by the method of generalized projections with application to restoration from magnitude
- Levi, Stark
- 1984
(Show Context)
Citation Context ...s secondary to the global behavior of the difference map and is discussed in the next Section. A fixed point of D, ρ ∗ , is characterized by ∆(ρ ∗ ) = 0, or (π1 ◦ f2)(ρ ∗ ) = ρ1∩2 = (π2 ◦ f1)(ρ ∗ ) , =-=(6)-=- where ρ1∩2 is required to lie in the intersection of the corresponding constraint subspaces. If π2 = πmod, say, and π1 represents some object domain constraint, then ρ1∩2 is a solution of that partic... |

22 |
Vector Space Projections
- Stark, Yang
- 1998
(Show Context)
Citation Context ...overdetermined constraints is the exception rather than the rule. 3. The difference map Given two arbitrary projections π1 and π2, we consider the “difference map” D:E N → E N defined by D = 1 + β∆ , =-=(4)-=- where β is a nonzero real parameter and ∆ = π1 ◦ f2 − π2 ◦ f1 (5) 9is the difference of the two projection operators, each composed with a map fi:E N → E N . The detailed form of the maps fi is seco... |

13 |
The question of phase retrieval in optics,” Opt
- Walther
- 1963
(Show Context)
Citation Context ...constraint subspaces so that they lie on the sphere of unit norm. The magnitude of the error estimate is then related to the angular separation of the constraint 10subspaces by e = 2sin ( ) θ1∩2 2 . =-=(9)-=- While small values of θ1∩2 imply a solution is nearby, small values encountered during the early iterates are a cause for concern. This is because it is highly improbable that the random starting poi... |

10 |
Direct Phasing in Crystallography,
- Giacovazzo
- 1998
(Show Context)
Citation Context ...ng H = {h1,h2,...} and also finding an ordering of the pixels, n ↦→ o(n), such that their values, {ρo(1),ρo(2),...} are also sorted. Histogram projection is then given by the map πhist : ρo(n) ↦→ hn. =-=(3)-=- It is straightforward to show that πhist is distance minimizing. Since N log N operations are required to sort N real numbers, histogram projection is easy to compute. For complex valued objects it i... |

10 |
The Squaring Method: A New Method for Phase Determination
- Sayre
- 1952
(Show Context)
Citation Context ...and π2. Let ρ ∈ EN be the current iterate (object). Using projections we can obtain two additional points, π1(ρ) and π2(ρ). Pairs of points determine lines; in particular, fi(ρ) = (1 + γi)πi(ρ) − γiρ =-=(10)-=- is a general point, parametrized by a real number γi, on the line defined by ρ and πi(ρ). Using (10) as a definition of the maps fi might be viewed as taking the first step beyond simply using identi... |

10 |
On the application of the minimal principle to solve unknown structures, Science
- Miller, DeTitta, et al.
- 1993
(Show Context)
Citation Context ...of the distance minimizing property of π1 and π2): π1(x1 + x2 + y) = x1 + a2 + b1 π2(x1 + x2 + y) = a1 + x2 + b2 . (12) One application of G brings us to the fixed point G(x1 + x2 + y) = a1 + a2 + b1 =-=(13)-=- and stagnation occurs. Subsequent applications of the elementary projections simply hop between a1 + a2 + b1 and a1 + a2 + b2, the two points on C1 and C2 with minimum separation. 12The behavior of ... |

8 |
Histogram matching as a new density modification technique for phase refinement and extension of protein molecules, Acta Crystallogr. A
- Zhang, Main
- 1990
(Show Context)
Citation Context ...use (6) to find ρ1∩2. Even if the solution ρ1∩2 is unique (up to translation and inversion), the set of fixed points is in general the large space given by (π1 ◦ f2) −1 (ρ1∩2) ∩ (π2 ◦ f1) −1 (ρ1∩2) . =-=(7)-=- One practical consequence of this is that the fixed point object ρ∗ will appear to be contaminated with noise whose origin, ultimately, is the randomness of the starting point ρ(0). This is illustrat... |

7 |
On the application of phase relationships to complex structures. XXVI. Developments of the Sayre-equation tangent formula
- Debaerdemaeker, Tate, et al.
- 1988
(Show Context)
Citation Context ... X1 and X2, then a general point can be uniquely expressed as ρ = x1 + x2 + y, where xi ∈ Xi and y ∈ Y . The constraint spaces are now explicitly approximated as C1 = X1 + a2 + b1 C2 = a1 + X2 + b2 , =-=(11)-=- where ai ∈ Xi and bi ∈ Y . Orthogonality of the spaces X1, X2 and Y implies the shortest element in C1 − C2 is b1 − b2 ∈ Y . Thus a solution (intersection) corresponds to b1 = b2, while b1 ̸= b2 repr... |

4 |
Crambin: A Direct Solution for a 400 Atom Structure", Acta Cryst
- Weeks, Hauptman, et al.
- 1995
(Show Context)
Citation Context ...behavior of the difference map is quite different, as we now show. A straightforward calculation gives the result D(x1 + x2 + y) = a1 + a2 + y + (1 − βγ2)(x1 − a1) + (1 + βγ1)(x2 − a2) + β(b1 − b2) . =-=(14)-=- First consider the case b1 = b2 = b, corresponding to a true intersection of the subspaces at the solution ρ1∩2 = a1 + a2 + b. From (14) we see that subsequent iterates approach the fixed point ρ∗ = ... |

1 |
Linear time heuristic for the bipartite Euclidean matching problem
- Elser
(Show Context)
Citation Context ...rting point ρ(0). This is illustrated by the numerical experiments in Section 7. The progress of the iterates ρ(i) can be monitored by keeping a record of the norm of the differences ei = ‖∆(ρ(i))‖ , =-=(8)-=- where ‖ · ‖ denotes the Euclidean norm. The “error” ei has the geometrical interpretation as the currently achieved distance between the two constraint subspaces, C1 and C2. When this distance become... |