Inverse problems play an important role in many areas of physics. In an inverse problem we intend to reconstruct physical properties (“scattering potential”) of some object given the result of the interaction (“scattering”) with a well-defined input signal. Typically, propagating wave fields (e.g., acoustic waves, electromagnetic waves, . . .) are used to scan the object. However, for rapidly oscillating electromagnetic wave fields like x-rays, detection devices are only capable of recording the average intensity of the scattered wave field. Its phase information, which also contains important information about the scattering potential, is lost during the measurement process. Despite the lost phase information, x-rays are an important, non-destructive tool for the investigation of structural properties of matter because they exhibit high penetration depth and high spatial resolution [1–3]. Note, that 3^rd generation synchrotrons [4] provide brilliant x-ray beams with a very high degree of coherence and with very high flux, so that even dot- and wire-like nanostructures can be investigated [2,5–11].

In this paper, we introduce a new algorithm – the (HIO+O_R)+ER-algorithm – which is capable of reconstructing the missing phase information in a variety of cases for which the traditionally and commonly used HIO+ER-algorithm [12, 13] (reviewed in sec. 2) fails. Before we introduce this new algorithm in sec. 2, we shortly review the phase retrieval problem in sec. 1. In particular, we want to stress the fact that the (HIO+O_R)+ER-algorithm is based on randomized overrelaxation [14,15] and, thus, does not require additional mathematical constraints [16] compared to the HIO+ER-algorithm. We reformulate our approach using projection polynomials which facilitate a straight forward investigation of randomization of the HIO-algorithm without correlations in the coefficients enforced by overrelaxation. Section 3 discusses numerical aspects how to measure the convergence of phase retrieval. Finally, sec. 4 illustrates the advantages of the (HIO+O_R)+ER-algorithm with respect to the traditional HIO+ER-algorithms in detail. For this purpose, we focus on three numerical examples each with different specific properties and compare the success rate of reconstructions based on our (HIO+O_R)+ER-algorithm and on the HIO+ER-algorithm. Moreover, we investigate the sensitivity of our algorithm to the choice of its parameters. In this paper, we focus on the algorithm itself and its improved convergence properties for specific, well-known problems related to phase retrieval based on the HIO+ER-algorithm. A separate publication will illustrate the improved features of the (HIO+O_R)+ER-algorithm for the reconstruction of the displacement field in inhomogeneously strained nanocrystals.

1. The phase retrieval problem

Consider some unknown object f_in(x⃗) in d-dimensional direct space (“position space”). We assume that (i) the shape 𝕊 of f_in(x⃗) (i.e., the smallest domain in direct space for which f_in(x⃗) ≠ 0) and (ii) the quantity A_k⃗ ≡ |FT{f_in(x⃗)}| (i.e., the amplitudes of its Fourier transform) are known. It has been proven that this information is sufficient to reconstruct the object f_in(x⃗) if the shape 𝕊 is finite and the dimension d of direct space is at least two [2, 17–19]. Physically, the second condition is fulfilled in scattering experiments which can be described in lowest order Born-approximation (also called kinematical approximation). This is typically true for nanostructures illuminated by coherent x-ray beams.

Suppose we discretize a rectangular region 𝔻 ⊂ ℝ^d in direct space, which fully contains our object f_in(x⃗) (i.e., 𝕊 ⊂ 𝔻), with N_i, i ∈ {1, 2,...,d}, equidistant points. The discrete Fourier transform (DFT) maps this set of points to a mesh 𝕄 in reciprocal space which is related to the Fourier transform of our continuous direct space object f_in(x⃗) [20]. Hence, we have

In direct space, every point x⃗ inside the object domain 𝕊 corresponds to (up to) two real unknowns (magnitude and phase). However, outside the domain 𝕊, f_in(x⃗) is precisely zero for all positions x⃗. Hence, each point outside 𝕊 defines two real equations for the determination of the N real phases Φ_k⃗, if we require f_in(x⃗) = 0 ∀ x⃗ ∉ 𝕊. Therefore, we can hope to reconstruct the missing phases Φ_k⃗ on the sampled grid 𝕄 once

With that knowledge, we restate the phase retrieval problem more precisely: Given the shape 𝕊 of some object f_in(x⃗) and the modulus A_k⃗ of the Fourier transform of f_in(x⃗) on a mesh 𝕄 that satisfies Eq. (2), we try to reconstruct the phases Φ_k⃗ and, thus, the object f_in(x⃗).

Note, that the reconstruction is only unique up to the unavoidable inherent symmetries of the Fourier transform [2, 17, 18], i.e., a global phase shift (typically irrelevant), a plane wave modulation e^ik⃗·x⃗ in reciprocal space (fixed by the position of the shape 𝕊 inside 𝔻) and the degeneracy between the object f_x⃗ and its complex conjugated, inversion symmetric object $f_{\overrightarrow{x}}^{*}$. This last ambiguity constitutes a severe problem if the shape 𝕊 is inversion symmetric, but the object f_in does not possess this symmetry [13,23]. For the remainder of this paper, we restrict to domains 𝕊 which are not inversion symmetric. However, even if we exclude objects with inversion symmetric shape, it is highly non-trivial to formulate a phase retrieval algorithm that works automatically without manual supervision and tuning of parameters during the reconstruction. The (HIO+O_R)+ER-algorithm which we propose in the next section is capable of performing this task for a large class of objects f_in.

2. Reconstruction algorithms

In this section, we will shortly review the traditional combination of the hybrid input output and error reduction (HIO+ER) algorithm [12, 13] and propose our extension to this algorithm. Throughout the entire reconstruction process, we assume that we know the exact geometrical shape 𝕊 of the object which we reconstruct.

2.1. Hybrid input output and error reduction

A fast and efficient iterative phase retrieval algorithm is based on alternating projections of a trial solution onto the constraints in direct space and in reciprocal space. Hence, a single iterative step (i) of this algorithm, which is called error reduction (ER) algorithm [12], is defined by

The projection operator P_A is non-linear, non-convex and non-unique [24]. The values $\left|g_{\overrightarrow{k}}^{(i)}\right|$ are important for determining the difference to the (experimentally accessible) input data A_k⃗ (see Eq. (18)). The ER-algorithm is a local minimizer of a suitable chosen error metrics [12,13]. However, in practice, phase retrieval problems involve many local minima. Hence, the ER-algorithm will in general not converge to the correct solution f_in(x⃗), but to some local minimum. In addition, the error metric may stagnate for many iterations before decreasing further. More information on stagnation problems and the ER-algorithm can be found in [12,13,23].

Due to these problems, further algorithms, which try to avoid stagnation and convergence to local minima, have been developed. A very important algorithm is the hybrid input output (HIO) algorithm proposed by Fienup [12, 13, 23]. This algorithm is also an iterative procedure which is defined by the map

The convergence properties of the HIO-algorithm have been investigated in more detail in [22,25,26].

This algorithm is much stronger in avoiding stagnation and local minima. However, the HIO-algorithm shows its full potential only if it is combined with the ER-algorithm. One step of this combined HIO+ER-algorithm consists of N_HIO iterations of the HIO-algorithm followed by N_ER iterations of the ER-algorithm. This combination is more successful than both algorithms on their own as it was observed in practice [13, 23]. Although this algorithm is already very powerful, it is not yet satisfactory for many objects as we will show in sec. 4.

2.2. Hybrid input output with randomized overrelaxation ((HIO+O_R)+ER)

The HIO+ER-algorithm can be further improved by including randomized overrelaxation [14, 15,27]. Basically, overrelaxation corresponds to replacing a projection operator P_μ by

Overrelaxation (without any randomization) has been investigated for convex problems [15] and in connection with the ER-algorithm for phase retrieval [14]. Moreover, overrelaxation is also included in the difference map algorithm proposed by Elser in [27]. In the difference map algorithm with overrelaxation, the iterative step for our set of constraints is given by [28]

We propose to replace the (non-linear and non-convex) projection operator P_A in reciprocal space in the HIO-algorithm itself by its relaxed expression

The deviation β (λ_A − 1) from the identity operator in the first term can neither be represented by the traditional HIO-algorithm for any value of β (see Eq. (6)) nor by the difference map algorithm for any (β, λ_A, λ_𝕊) (see Eq. (8)). In both cases, the previous iterative result $f_{\overrightarrow{x}}^{(i)}$ is weighted with 1 or projected at least once in either direct (P_𝕊) or reciprocal space (P_A) before being included in the calculation for $f_{\overrightarrow{x}}^{(i+1)}$.

Finally, we have to choose suitable values for the additional parameter λ_A in Eq. (9). If we restrict to a fixed set of values λ_A for all iterations, we risk simply exchanging one trap by another or one local minimum by another. Trying to overcome local minima and stagnation, we propose a new approach: For each iteration, we randomly select the parameter λ_A. More precisely, for the remainder of this paper, we investigate a uniform random distribution in the range [1 − ν, 1 +ν], ν ≥ 0, for λ_A which is reassigned each iteration. Unless stated otherwise, we choose ν = 0.5. Note, that a fixed value λ_A ≡ 1 for all iterations corresponds to the usual HIO-algorithm. In order to distinguish our new extension from the traditional HIO-algorithm, we call our extension HIO+O_R-algorithm. The power of this extension is strongly supported by the fact, that the HIO+O_R-algorithm is capable of reconstructing objects without including the ER-algorithm with significant success rates. If the HIO+O_R-algorithm is combined with ER, we call the algorithm (HIO+O_R)+ER-algorithm (see Fig. 1).

 

Fig. 1 Graphical illustration of the (HIO+O_R)+ER-algorithm and its building blocks.

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If we look at Eq. (11) from a bird’s eye view, the most important difference of the operator Ĥ_HIO+OR (β, λ_A) to the other algorithms which have been proposed [28] is the fact, that the traps and tunnels [14] of the operator in general depend on λ_A and, therefore, should change from iteration to iteration as a consequence of randomization. Thus, stagnation in a specific trap or tunnel is strongly reduced, as most traps and tunnels are no longer persistent throughout the iterative reconstruction process. However, the true solution f_in(x⃗) is a fixed point of the iterative procedure defined in Eq. (11) for all values of λ_A.

2.3. Projection polynomials

The extension of the HIO+ER-algorithm to the (HIO+O_R)+ER-algorithm is based on two ingredients: overrelaxation and randomization. In order to gain a deeper understanding of the (HIO+O_R)+ER-algorithm, it is useful to investigate both ingredients separately. Studying the HIO-algorithm extended by overrelaxation of P_A, but without randomization is straight forward: we simply keep a fixed, predefined overrelaxation parameter λ_A throughout the entire iterative reconstruction process. In this section, we present a framework for randomization of iterative projection algorithms performed in a way which is not based on overrelaxation of P_A. Although this framework can be applied for studying randomization of any iterative projection algorithm, we limit our numerical simulations (in section 4) to parameter values whose deterministic contribution coincides the traditional HIO-algorithm.

For this purpose, consider the projection polynomial operator

This operator exploits the fact, that any combination of projection operators ${\mathbf{P}}_{{\xi }_1}^{n_1}{\mathbf{P}}_{{\xi }_2}^{n_2}\dots {\mathbf{P}}_{{\xi }_m}^{n_m}$ of two different kinds ξ_j ∈ {𝕊,A}, j ∈ {1,...,m}, reduces to one of the four building blocks P_𝕊 (P_AP_𝕊)ⁿ, P_A (P_𝕊P_A)ⁿ, (P_𝕊P_A)ⁿ and (P_AP_𝕊)ⁿ for some integer n ≥ 0. This is a consequence of the defining property ${\mathbf{P}}_{\xi }^2={\mathbf{P}}_{\xi }$ (idempotence) of projection operators (which implies ${\mathbf{P}}_{\xi }^n={\mathbf{P}}_{\xi }\forall n\ge 1$). Note, that the product of two non-commutative idempotent operators (like P_A and P_𝕊) is no longer idempotent. The special case of the identity operator 1 and a single projection operator is included for the case n = 0 in those building blocks. However, the identity operator is included twice, namely for n = 0 for (P_𝕊P_A)ⁿ and (P_AP_𝕊)ⁿ. Therefore, one spurious coefficient in the projection polynomial is introduced if both building blocks are included for n = 0. This is avoided by restricting those two building blocks to n ≥ 1 and including the identity operator separately. Therefore, Eq. (12a) represents the most general polynomial expression which contains a maximum of q successive projections of two different kinds, if we incorporate the idempotence of projection operators.

In order to guarantee, that the true solution f_in(x⃗) is a fixed point of Ĥ_Proj for any choice of its parameters (b, c⃗_𝕊, c⃗_A), it is necessary to enforce one constraint on the coefficients: If the input for the projection operators P_𝕊 and P_A coincides with the true solution f_in(x⃗), the projection operators P_𝕊 and P_A reduce to the identity operator 1. Therefore, given the true solution f_in(x⃗) as input, the operator Ĥ_Proj simplifies to the identity operator 1 for any choice of its parameters (b, c⃗_𝕊, c⃗_A) if and only if the additional constraint

Given this fix point property for the true solution f_in(x⃗), we can perform proper randomization of this operator. For this purpose, we split our coefficients c_ξ,n, ξ ∈ {𝕊, A}, n ≥ 1, in a deterministic part $c_{\xi ,n}^{(D)}$ and a random part $r_{\xi ,n}{c}_{\xi ,n}^{(R)}$ and assume r_ξ,n to be uniformly distributed in the range [−1, 1], i.e.,

In order to obtain the traditional HIO-algorithm in the limit $c_{\xi ,n}^{(R)}\to 0$ for all ξ and n, we have to choose (at least) q = 2 (see Eq. (6)). Therefore, Ĥ_Proj reduces to

Consequently, we can apply the framework of projection polynomials to investigate whether the benefits of randomization rely on the precise correlations evident in Eq. (16b) or if randomization of the traditional HIO-algorithm implies significant benefits also in absence of these correlations.

3. Monitoring convergence of the reconstruction

In order to investigate the success and convergence properties of the reconstruction, we monitor three parameters. First of all, we monitor the change of the reconstructed object f⁽ⁱ⁾ from iteration (i − 1) to iteration (i). Second, we measure the mathematical distance to the (sampled) true solution f_in(x⃗). And last, but not least, we measure the deviation in reciprocal space between the a priori given reciprocal input data A_k⃗ and the magnitude of the Fourier transform of the reconstructed object. All three parameters yield different information:

The distance of the reconstructed object f⁽ⁱ⁾ after iteration (i) to the true direct space object f_in is, of course, the best measure for the quality of the reconstructed image. We define this mathematical distance by an angle

A dimensionless, normalized error measure ε⁽ⁱ⁾ which depends only on the a priori known (measured) amplitudes A_k⃗ and not on the true direct space solution f_in(x⃗) is

In addition, from a numerical point of view, the change of the reconstructed object f⁽ⁱ⁾ from iteration to iteration is very important, too. This must not be confused with the change of the non-invertible map ε⁽ⁱ⁾ from iteration to iteration: Moving along a connected line of constant error metric does not change the error metric itself, but the reconstructed object f⁽ⁱ⁾ may change arbitrarily. Nevertheless, the algorithm may converge (i.e. no change from iteration to iteration), but not to the true solution f_in of the problem [14, 16]. For simulated data, we can judge if the algorithm converged to the true solution f_in by Eq. (17). A suitable choice for observing the change from iteration to iteration is given by the angle

4. Numerical examples

In this section, we demonstrate the improvements resulting from randomized overrelaxation for three carefully chosen numerical examples: First, we consider two pure phase objects

 

Fig. 2 Pure phase objects used for the investigation of the convergence of the (HIO+O_R)+ER-algorithm. The magnitude of both objects is constant. The phase is plotted using a HSV color-bar, however, the region outside the shape 𝕊 has been set to black. The oversampling ratio is σ = 8.456.

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Fig. 3 Investigation of the (HIO+O_R)+ER-algorithm for a purely real object f_in with strong variation of its magnitude over short length scales (oversampling ratio σ = 5.06). In Fig. (b), continuous lines represent the (HIO+O_R)+ER-algorithm, isolated dots the HIO+ER-algorithm. A pure HIO+O_R-calculation without ER is included as black, dash-dotted curve.

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Fig. 4 Comparison of the success rate of reconstructions of pure phase objects (see Eq. (20), (22) and (23)) with the HIO+ER- and the (HIO+O_R)+ER-algorithm. The parameter β was fixed to 0.85. Continuous lines represent results of the (HIO+O_R)+ER-algorithm, isolated dots of the HIO+ER-algorithm. A pure HIO+O_R-calculation without ER is included as black, dash-dotted curve.

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The first pure phase objects which we used in our numerical simulations is given by

The second object is defined by

A reconstruction is considered successful once the angle φ⁽ⁱ⁾ to the input object (see Eq. (17)) falls below some given limit φ_Converged (=0.05° unless stated otherwise). We repeat the reconstruction process for N_Real initial trials (random phases at each k⃗-point, amplitudes $A_{\overrightarrow{k}}^{(0)}$ equal to given amplitudes A_k⃗). Finally, we calculate the success rate which tells us the percentage of reconstruction processes that have been successful up to iteration (i). A good reconstruction algorithm

Each iteration in the HIO+ER-algorithm and its (HIO+O_R)+ER-extension can be implemented very efficiently exploiting the good scaling properties of the FFT-algorithm. Therefore, as long as the number of iterations is not too high, the third requirement is clearly met by both, the HIO+ER and (HIO+O_R)+ER-algorithm.

In order to investigate the other points, we plotted the respective success rates for the HIO+ER- and (HIO+O_R)+ER-algorithm for different internal parameters in Fig. 4. We normalized the computational effort, i.e., one iteration of the (HIO+O_R)+ER-algorithm with N_HIO = 130 and N_ER = 10 corresponds to two successive iterations with N_HIO = 50 and N_ER = 20. This prefactor of 2 can be found in the legend of the graph in front of the square brackets for each calculation.

The inclusion of overrelaxation improves the success rate for the first phase object defined by Eq. (22) for each combination of N_HIO and N_ER: in contrast to the HIO based calculations, it quickly converged to 100%. In fact, overrelaxation is so powerful that stagnation in the HIO-algorithm is eliminated entirely and intermediate ER-iterations are no longer necessary (see black dash-dotted curve in Fig. 4). The success rate of the traditional HIO+ER-algorithm stagnates on some level far below 100%. This stagnation is significant in the long term behavior, as we illustrate in Fig. 5 by plotting the success rate of the traditional HIO+ER-algorithm for 50 times as many iterations as in Fig. 4(a). Note, that all trials that did not converge after 2500 iterative steps using the traditional HIO+ER-algorithm with N_HIO = 130 and N_ER = 10 in Fig. 5 are separated from the true solution by an angle greater than φ = 55°.

 

Fig. 5 Long-term stagnation of the success rate of the traditional HIO-algorithm (without overrelaxation and without randomization) for the first phase object (see Eq. 22) for different choices of the internal parameters (N_HIO, N_ER), but fixed β = 0.85.

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If we consider the second phase object defined by Eq. (23), the traditional HIO+ER-algorithm performs better here than for the first phase object. For the one specific set of parameters N_HIO = 130 and N_ER = 10, the success rate of the traditional HIO+ER-algorithm is even slightly superior to the (HIO+O_R)+ER-extension, but for this specific set of parameters, the success rate converges to 100% very quickly for both algorithms. However, for the set of parameters (N_HIO = 50, N_ER = 20) and (N_HIO = 40, N_ER = 30), the success rate quickly converges to 100% only for the (HIO+O_R)+ER-algorithm, but not for the HIO+ER-algorithm. Hence, the (HIO+O_R)+ER-algorithm is much more stable with respect to the particular choice of the internal parameters N_HIO and N_ER including, in particular, the case N_ER = 0.

The traditional HIO+ER-algorithm also has severe problems performing a successful reconstruction for the real object depicted in Fig. 3(a): The success rate for the reconstruction process is very poor if we apply our criterion φ_Converged ≤ 0.05° to determine successful reconstructions. The (HIO+O_R)+ER-algorithm is much more successful and reaches success rates of 100% with few iterations (see Fig. 3(b)). Furthermore, reaching the success rate of 100% does not sensitively depend on the internal parameters N_HIO and N_ER. Again, keeping N_ER small compared to N_HIO or eliminating ER-iterations completely yields the best results. Keep in mind, that neither the reality constraint nor the positivity constraint of the object have been exploited during reconstruction.

For the second and third test object, all trials that failed to converge to the true solution f_in basically evolved towards the correct direction. However, at some point, stagnation in form of stripes close to the true solution f_in appeared. These stripes are very persistent and prevent the usual HIO+ER-algorithm from further convergence to the true solution f_in. Typically, the mathematical distance φ of the objects containing stripes to the true solution f_in is in the order of 0.05 to 5.0 degrees. In case of the purely real test object, after 150 iterations all reconstructed objects reached a distance φ ≤ 0.5° (= 10 · φ_Converged) for HIO₁₃₀ − ER₁₀ and 31% for HIO₅₀ − ER₂₀. The remaining 71% of the objects for HIO₅₀ − ER₂₀ reached a distance φ ≤ 5° (= 100 · φ_Converged) using traditional HIO+ER-algorithm and 150 iterations. Note, that the change χ⁽ⁱ⁾ (see Eq. (19)) from iteration to iteration does not vanish, i.e. numerically, the algorithm did not converge within the given number of iterations.

Figure 6(a) shows the behavior of the success rate of the (HIO+O_R)+ER-algorithm for different values of the parameter ν. In addition, Fig. 6(b) depicts the success rate for different values of the parameter β. Within the range β ∈ [0.5, 1.0], which is typically used in the traditional HIO+ER-algorithm, the (HIO+O_R)+ER-extension does not sensitively depend on the particular choice of β. Therefore, the (HIO+O_R)+ER-algorithm does not depend sensitively on any of its internal parameters N_HIO, N_ER, β and ν. Hence, it fulfills all four requirements on a good reconstruction algorithm which we stated above.

 

Fig. 6 Investigation of the sensitivity of the (HIO+O_R)+ER-algorithm with N_HIO = 50 and N_ER = 20 on the choice of the parameter β and on the range of the uniform distribution determining the relaxation parameter λ_A for the first phase object (see Eq. 22).

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Two concepts (overrelaxation and randomization) have been exploited to modify the HIO-algorithm: Fig. 7 depicts the success rate, if overrelaxation is performed with a static parameter λ_A. We see extremely sensitive behavior of the success rate on the precise value of λ_A if it is kept fixed. Whereas increasing λ_A up to 1.02 improved the success rate for the first object, it completely vanished for λ_A = 1.01 for the second phase object. For a deviation from 1.00 greater than ten percent, almost no successful reconstructions have been observed for any of the three test objects. In addition, we do not a priori know the (non-universal) optimum value for λ_A which depends on the object f_in. Consequently, static overrelaxation does not fulfill the requirements for a good reconstruction algorithm stated above.

 

Fig. 7 Success rate of the HIO+ER-algorithm for fixed overrelaxation λ_A (no randomization). Parameters are chosen as (N_HIO = 130, N_ER = 10) and β = 0.85.

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Based on the discussion in sec. 2.3, we test randomization of the HIO-algorithm for β = 0.85 without employing overrelaxation by setting the amplitudes $c_{\xi ,n}^{(R)}$ for the random contribution to the coefficients c_ξ,n in the operator Ĥ_HIOR defined in Eq. (14) equal to $c_{𝕊,1}^{(R)}=c_{A,1}^{(R)}=c_{𝕊,2}^{(R)}=c_{A,2}^{(R)}=R=0.2$. The deterministic contribution $c_{\xi ,n}^{(D)}$ is set according to Eq. (15). The result for all three test objects is illustrated in Fig. 8 and compared to the result for the HIO+O_R-algorithm. Clearly, pure randomization without overrelaxation performs equally well as the HIO+O_R-algorithm for all three test objects. However, in our opinion, the HIO+O_R-algorithm should be preferred over randomization without overrelaxation, because (i) it requires only two Fourier transformations for each iteration (lower computational effort) and (ii) it has less degrees of freedom (one uniform random distributions instead of four). Note, however, that due to the large number of degrees of freedom of projection polynomials, further optimization of this approach is possible (including more advanced statistical correlations in the coefficients of the projection polynomial). In addition, more elaborate random distributions might improve both approaches further.

 

Fig. 8 Comparison of the success rate of both frameworks which provide a generalization of the traditional HIO-algorithm based on randomization. Continuous lines illustrate the behavior for randomized overrelaxation of P_A (see Eq. (10)), whereas dots represent the behavior of the success rate resulting from independent randomization of the coefficients in a projection polynomial (see Eq. (13) and Eq. (14)). In both cases, the deterministic contribution is equivalent to the traditional HIO-algorithm with parameters N_HIO = 140 and β = 0.85. No ER has been performed.

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In summary, we succeeded in overcoming stagnation of the traditional HIO+ER-algorithm for various objects. For this purpose, the concept of overrelaxation was applied to the reciprocal space projection in the HIO-algorithm which produced additional degrees of freedom. Randomization of these additional degrees of freedom from iteration to iteration was exploited to prevent stagnation in local minima, especially in traps or tunnels. In particular, this was demonstrated for two pure phase objects with smooth phase variation (with different behavior with respect to the traditional HIO+ER-algorithm) and a purely real object with strong amplitude variations over short length scales. The improved algorithm is far less sensitive to the specific choice of the internal parameters N_HIO and N_ER than the traditional HIO+ER-algorithm. Moreover, the (HIO+O_R)+ER-algorithm remains almost insensitive to the particular value of the internal parameter β in the range β ∈ [0.5, 1.0]. Furthermore, the the good scaling properties in terms of computational time and memory consumption of the HIO+ER-algorithm are fully maintained in the (HIO+O_R)+ER-algorithm. In conclusion, a reconstruction based on the (HIO+O_R)+ER-algorithm exhibits significantly enhanced stability and success probability in comparison with the traditional and commonly used HIO+ER-algorithm.