Nanoscale Photonic Imaging pp 377403  Cite as
Constrained Reconstructions in Xray Phase Contrast Imaging: Uniqueness, Stability and Algorithms
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Abstract
This chapter considers the inverse problem of Xray phase contrast imaging (XPCI), as introduced in Chap. 2. It is analyzed how physical a priori knowledge, e.g. of the approximate size of the imaged sample (support knowledge), affects the inverse problem: uniqueness and—for a linearized model—even wellposedness are shown to hold under support constraints, ensuring stability of reconstruction from realworld noisy data. In order to exploit these theoretical insights, regularized Newton methods are proposed as a class of reconstruction algorithms that flexibly incorporate constraints and account for the inherent nonlinearity of XPCI. A Kaczmarztype variant of the approach is considered for 3D imagerecovery in tomographic XPCI, which remains applicable for largescale data. The relevance of constraints and the capabilities of the proposed algorithms are demonstrated by numerical reconstruction examples.
2010 Mathematics Subject Classification:
65R10 65R32 78A45 78A46 92C55 94A0814.1 Forward Models
We aim to describe (propagationbased) Xray phase contrast imaging (XPCI) in the language of inverse problems. To this end, we deduce forward operators \(F: X \rightarrow Y\), that model the dependence of the measured nearfield diffraction patterns (called holograms) \(I \in Y\) from the samplecharacterizing parameters \(f\in X\) (the sought image). Different models F are obtained for various settings of practical interest, including Xray phase contrast tomography (XPCT).
14.1.1 Physical Model and Preliminaries
The basic physical model of XPCI is detailed in Chap. 2 and summarized by Fig. 14.1: incident monochromatic Xrays, modeled by plane waves, are scattered by the imaged sample, that is parametrized by its spatially varying refractive index \(n(\varvec{x}, z) = 1  \delta (\varvec{x}, z) + \text {i}\beta (\varvec{x}, z)\) (\(\delta , \beta \): refractive and absorption decrement). By the scatteringinteraction, a perturbation (the image) is imprinted upon the transmitted Xray wavefield. The intensity I of the perturbed wavefield is recorded by a detector placed at a finite distance \(d >0\) behind the sample.
XPCIexperiments provide intensity data I of the form (14.1) (up to data errors), whereas the images \(\phi \), \(\mu \) are the quantities of interest. Hence, the following principal inverse problem has to be solved:
Inverse Problem 1
(XPCI) For some set A, reconstruct a 2Dimage \(h = \mu + \text {i}\phi \in A \) from measured holograms I of the form (14.1).
By rotating the object in Fig. 14.1, holograms \(I_{\varvec{\theta }_j}\) may be acquired for different incident directions \(\varvec{\theta }_j \in \mathbb {S}^2 = \{\varvec{x}\in \mathbb {R}^3: \varvec{x} = 1\}\) of the Xrays onto the sample (in Fig. 14.1, the incident direction coincides with the zaxis). This is the setting of Xray phase contrast tomography (XPCT). A mathematical model will be provided in Sect. 14.1.3. XPCT allows to probe 3Dvariations of the parameters \(\delta ,\beta \) beyond mere projections \(\phi ,\mu \).
Inverse Problem 2
(XPCT) For some set A, recover a 3Dimage \(f = k\beta + \text {i}k\delta \in A \) from holograms \(\{ I_{\varvec{\theta }_j} \}\) measured under different incident directions \(\{\varvec{\theta }_j\} \subset \mathbb {S}^2\).
14.1.1.1 A Priori Constraints

Support constraints: realworld samples are of finite size. This implies that the functions \(f \in \{ \phi , \mu , \delta , \beta \}: \mathbb {R}^m \rightarrow \mathbb {R}\) have a compact support, i.e. are identically zero outside some bounded objectdomain \(\varOmega \subset \mathbb {R}^m\).

Nonnegativity: by the physics of hard Xrays, the decrements \(\delta , \beta \)—and thus also \(\phi , \mu \)—are always nonnegative.

Pure phase object: especially for biological samples, \(\beta \) and \(\mu \) are typically orders of magnitude smaller than \(\delta \) and \( \phi \). Assuming a purely shifting, i.e. nonabsorbing object \(\beta ,\mu = 0\), is then a good approximation.

Homogeneous objects: as is rigorously true for samples composed of a single material, proportionality of \(\delta \) and \(\beta \) [\(\phi \) and \(\mu \)] may often be assumed.

Regularity: realistic images \(\phi , \mu , \delta , \beta \) are not arbitrarily singular functions, but typically have some characteristic smoothness properties.

Tomographic consistency: Images \(\phi \) and \(\mu \) that arise as tomographic projections of one object under different incident directions are correlated.
Focussing on supportknowledge, we study the role of such constraints on inverse Problems 1 and 2 and outline how to exploit them algorithmically.
14.1.1.2 Additional Notation
14.1.2 Forward Operators for XPCI
Based on Sect. 14.1.1, we introduce forward maps \(F: X\rightarrow Y\) modeling different settings of XPCI. Note that we define the maps in arbitrary dimensions \(m\in \{1,2,3,\ldots \}\) although the natural case are images and holograms in \(m = 2\) dimensions. The benefit of this will be seen in Sect. 14.3.4.1.
14.1.2.1 General Nonlinear Forward Operator
14.1.2.2 Linearized Forward Map and ContrastTransferFunctions
As \(s_0(\varvec{\xi }), c_0(\varvec{\xi }) \le 1\) for all \(\varvec{\xi }\in \mathbb {R}^m\), \(\mathscr {T}: L^2(\mathbb {R}^m) \rightarrow L^2 (\mathbb {R}^m)\) is a bounded \((\mathbb {R})\)linear operator with \(\Vert \mathscr {T}(h) \Vert _{L^2} \le 2 \Vert h \Vert _{L^2}\) for all \(h \in L^2(\mathbb {R}^m)\).
14.1.2.3 Homogeneous Objects and Pure Phase Objects
The cases of homogeneous objects and pure phase objects, see Sect. 14.1.1.1, may be treated in a unified manner, by expressing the complexvalued image \(h = \mu + \text {i}\phi = \text {i}\text {e}^{\text {i}\nu } \varphi \) in terms of a single realvalued function \(\varphi \) and a parameter \(\nu = \arctan (\beta /\delta ) \in [0; \pi /2)\) (\(\nu = 0\): pure phase object).
14.1.2.4 Multiple Holograms
14.1.3 Forward Operators for XPCT
14.2 Uniqueness Theory
 1.
The richness of the data, i.e. the size of the dataspace Y: for example, it is commonly argued that measuring several holograms \(I_1, I_2, \ldots \) at different Fresnelnumbers (see Sect. 14.1.2.4) helps to ensure uniqueness in XPCI.
 2.
Available a priori knowledge, i.e. the size of the objectspace X: the smaller X the more likely it is that any two images \(h_1, h_2 \in X\) with \(h_1 \ne h_2\) induce distinguishable data \(F(h_1) \ne F(h_2)\).
In addition, it may happen that the nonlinear forward model is unique but its linearization is nonunique or vice verser. Accordingly, the different forward models from Sect. 14.1.2 have to be investigated individually.
14.2.1 Preliminary Results and CounterExamples
 Linearized model: \(\mathscr {T}: L^2(\mathbb {R}^m) \rightarrow L^2(\mathbb {R}^m); \; h \mapsto  2\text {Re}(h) \) has a huge nullspace composed of all h for which \(\mathcal {D}(h)\) is purely imaginaryvalued:$$\begin{aligned} \text {kern }(\mathscr {T}) := \{ h \in L^2(\mathbb {R}^m) : \mathscr {T}(h) = 0 \} = \mathcal {D}^{1} \left( \text {i}L^2(\mathbb {R}^m, \mathbb {R}) \right) \end{aligned}$$(14.18)
 Nonlinear model (example from [8]): Images \(h_\pm : \mathbb {R}^2\setminus \{0\} \rightarrow \mathbb {C}; \; \varvec{x}\mapsto a(\varvec{x}) \pm \text {i}\nu \arctan \!2(\varvec{x})\) for \(\nu \in \mathbb {N}\) and smooth functions \(a: \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}\) give rise to socalled phasevortices in the wavefield. The sign of the vortex is not determined by Fresnelintensities (\(A:= \exp (a)\)):$$\begin{aligned} \left \mathcal {D}( \exp (h_+) ) \right ^2&= \left \mathcal {D}(A \cdot \exp (\text {i}\nu \arctan \!2(\cdot ) ) \right ^2 \nonumber \\&= \left \mathcal {D}(A \cdot \exp ( \text {i}\nu \arctan \!2(\cdot ) ) \right ^2 = \left \mathcal {D}( \exp (h_) ) \right ^2 \end{aligned}$$(14.19)

Uniqueness under homogeneityconstraints (linear): the operator \(\mathscr {S}_{\nu }: L^2(\mathbb {R}^m) \rightarrow L^2(\mathbb {R}^m)\) from Sect. 14.1.2.3 is injective, as the zeromanifolds of the Fouriermultiplier \(s_\nu \) are sets of the Lebesguemeasure 0 in \(\mathbb {R}^m\).

Uniqueness for two holograms (linear): in [9], it is shown by a similar argument based on the CTFrepresentation (14.11) that also the operator \(\mathscr {T}^{(\mathfrak {f}_{1}, \mathfrak {f}_{2})}: L^2(\mathbb {R}^m) \rightarrow L^2(\mathbb {R}^m)^2\) (see Sect. 14.1.2.4) is injective for \(\mathfrak {f}_{1} \ne \mathfrak {f}_{2}\).
Moreover, it is argued in [9] that both results carry over to the nonlinear model, provided that the image h is compactly supported. Indeed, a much stronger uniqueness result holds true under such an assumption, as will be seen in the following.
14.2.2 Sources of Nonuniqueness—The Phase Problem

Phasewrapping: The exponential is \(2\pi \)periodic in the imaginarypart of its argument. Hence, the phaseimage \(\phi = \text {Im}(h)\) may only be determined by the data up to increments by multiples of \(2\pi \).

Phase problem: The squared modulus, arising from the restriction of Xray detectors to measuring intensities, eliminates the phaseinformation.
The first aspect is simpler to analyze and often turns out to be of lesser practical impact in XPCI: for moderately strongly scattering samples, \(\phi \) is a priori known to assume values within \([0; 2\pi )\), so that nonuniqueness due to phasewrapping is not an issue. In the following, we therefore focus on possible ambiguities due to the phase problem.
14.2.3 Relation to Classical Phase Retrieval Problems
Up to possibly remaining phasewrapping ambiguities, the image reconstruction problem in XPCI may be rephrased as follows:
Given data \(I = \left \mathcal {D}(O) \right ^2\), reconstruct the objecttransmissionfunction (OTF) \(O := \exp (h) \in \tilde{A}\) from some admissible set \(\tilde{A}\).
Such settings are known as phase retrieval problems as recovering O is equivalent to retrieving the missing phase of \(\mathcal {D}(O)\) (and then inverting \(\mathcal {D}\)). Uniqueness of phase retrieval has been extensively studied ever since the pioneering works of Walther [10] and Akutowicz [11, 12], primarily for the case where \(\mathcal {D}\) is replaced by the Fouriertransform \(\mathcal {F}\), i.e. for the reconstruction from phaseless Fourierdata. We refer to [13, 14, 15, 16, 17] for reviews.
The OTF O is not a compactly supported function in any realistic setting. Only the contrast \(o := O1\) typically has compact support.
14.2.4 Holographic Nature of Phase Retrieval in XPCI
Formula (14.22) places the inverse problem of XPCI in the realm of holographic phase retrieval problems, i.e. reconstruction in the presence of a reference signal—here provided by the unscattered part of the incident Xrays. Several theoretical and practical works have shown that such a holographic reference facilitates phase retrieval, see e.g. [18, 19, 20, 21].
14.2.5 General Uniqueness Under Support Constraints
According to (14.22), image reconstruction in XPCI is equivalent to retrieving \(o = \exp (h) 1\) from data of the form (14.22) (up to possible phasewrapping). By invertibility of the Fresnelpropagator \(\mathcal {D}\), uniqueness thus holds if it is possible to disentangle the summands \(\mathcal {D}(o)\), \(\overline{\mathcal {D}(o)}\), and \(\left \mathcal {D}(o) \right ^2\). As shown in [22] using the theory of entire functions, the latter is indeed possible whenever o is known to have compact support, which is true for any sample of finite size. The principal result reads as follows:
Theorem 14.1
(Uniqueness of XPCI [22]) Let o (\(= \exp (h)1\)) be a compactly supported function (or distribution).
Then o is uniquely determined by XPCIdata \(I = \left \mathcal {D}(1 + o) \right ^2\). Furthermore, uniqueness is retained if only restricted data \(I_K\) is available, measured for any detectiondomain \(K \subset \mathbb {R}^m\) that contains an open set.
Importantly, Theorem 14.1 establishes uniqueness in the most challenging setting of XPCI: single hologram, no homogeneityconstraint. The result trivially extends to every less difficult case with more data or additional constraints. However, note that the extension of uniqueness to restricted measurements \(I_K\) is based on analytic continuation of the data—a very unstable procedure in practice.
14.2.5.1 Uniqueness for the Linearized Model
Uniqueness for the linearized XPCImodel has to be shown individually. According to Sect. 14.1.2.2, it corresponds to data of the form \(I_{\text {lin}} = 1  \mathcal {D}(h )  \overline{\mathcal {D}(h)}\). Compared to (14.22), merely the quadratic term \(\left \mathcal {D}(o) \right ^2\) is omitted and \(o = \exp (h)  1\) is replaced by \(h\) (note that this rules out phasewrapping!). Hence, the principal uniqueness argument from [22] remains valid: the summands \(\mathcal {D}(h )\) and \(\overline{\mathcal {D}(h )}\) may be disentangled owing to their different “fingerprints” as entire functions:
Corollary 14.1
(Uniqueness of linearized XPCI [22]) For any bounded domain \(\varOmega \subset \mathbb {R}^m\) and any \(K \subset \mathbb {R}^m\) that contains an open set, the linearized forward operator \(\mathscr {T}_K: L^2(\varOmega ) \rightarrow L^2(\mathbb {R}^m);\; h \mapsto \mathscr {T}(h)_K\) is injective.
14.2.5.2 Uniqueness for XPCT
By combining with standard results on uniqueness of tomographic reconstruction described by the theory of the Radon transform, the uniqueness theorems may be easily extended to XPCT. We refer to [22] for details.
14.3 Stability Theory
The uniqueness results of the preceding Sect. 14.2, suprisingly strong though they are, do not guarantee that accurate images may actually be reconstructed from holograms acquired in realworld XPCIsetups. Experimental data always contains errors due to noise and/or inaccuracies of the physical model. As detailed in Chap. 5 such data errors may lead to arbitrarily strongly corrupted images due to the phenomenon of illposedness: even if a forward model \(F: X\rightarrow Y\) is injective, its inverse \(F^{1}: F(X)\rightarrow X\) may be discontinuous such that small perturbations in the data \(g^{{\text {obs}}} = F(f) + \varvec{\epsilon }\) may be arbitrarily amplified in the reconstruction \(F^{1}(F(f) + \varvec{\epsilon })\).
The aim of this section is thus to supplement the uniqueness results with an analysis of stability, exploring how susceptible image reconstruction is to data errors. Thereby, it sheds a light on the question which reconstructions are feasible in practice. Due to difficulties arising from nonlinearity, the stability analysis is restricted to the linearized forward models.
14.3.1 LipschitzStability and its Meaning
14.3.2 Stability for General Objects and one Hologram
14.3.2.1 Analytical Approach
14.3.2.2 Stability Bound
Since \(\Vert \tilde{h} \Vert _{L^2} = \Vert h \Vert _{L^2}\), the bound (14.32) can be regarded as a relative stability estimate: recovering an image \(h \in L^2(\varOmega )\) from XPCIdata \(\mathscr {T}(h)\) is at least as stable as the reconstruction of \(\tilde{h} \in L^2(\varOmega )\) from Fourierdata outside the domain \(\varOmega _{\mathfrak {f}}\). Reconstruction from incomplete Fourierdata in turn is a wellstudied problem: an uncertainty principle from [27] implies that Lipschitzstability holds, \(\Vert \mathcal {F}( \tilde{h} )_{\varOmega _{\mathfrak {f}}^c } \Vert \ge C_{\text {stab}}^{\text {gen}}\Vert \tilde{h} \Vert \) for some \(C_{\text {stab}}^{\text {gen}}> 0\), provided that \(\varOmega \) is bounded along at least one dimension. For rectangular domains \(\varOmega \), the stabilityconstant \(C_{\text {stab}}^{\text {gen}}\) may be expressed in terms of the principal eigenvalue of a compact selfadjoint operator, for which asymptotics are derived in [28]. Via (14.32), these results yield stability estimates for linearized XPCI:
Theorem 14.2
While Theorem 14.2 only gives a worstcase bound on the datacontrast \(\Vert \mathscr {T}(h) \Vert /\Vert h \Vert \) over all images h, the result may be sharpened considerably, as detailed in [24]: for any \(h \in L^2(\varOmega )\), an individual lower bound for \(\Vert \mathscr {T}(h) \Vert \) may be given based on the eigenvalues from [28] and the images that minimize \(\Vert \mathscr {T}(h) \Vert /\Vert h \Vert \) may be characterized in terms of the associated eigenmodes.
14.3.2.3 Stability in a Practical Sense?
Numerical computations in [24] indicate that the bound (14.34) is quite sharp. While this is good news for a (pure) mathematician, it is bad news from an applied perspective: the predicted (quasi) exponential decay \(C_{\text {stab}}^{\text {gen}}(\varOmega ,\mathfrak {f}) \sim \exp (\mathfrak {f}/8)\) implies that the constant quickly becomes very small for larger values of \(\mathfrak {f}\), e.g. \(C_{\text {stab}}^{\text {gen}}(\varOmega ,\mathfrak {f}) \lesssim 10^{5}\), for \(\mathfrak {f}\gtrsim 100\). Notably, \(\mathfrak {f}= kb^2/d\) is the modified Fresnelnumber associated with the width of the supportdomain \(\varOmega \), i.e. with the diameter of the imaged sample.^{4} In typical XPCIexperiments at synchrotrons, one has \(10^2 \lesssim \mathfrak {f}\lesssim 10^5\), so that Theorem 14.2 only guarantees stability in practice for imaging settings at the lower end of typical Fresnelnumbers.
Notably, this is in line with empirics: after all, independent reconstruction of phase and absorptionimage \(\phi \) and \(\mu \) from a single hologram, as analyzed here, is widely considered as infeasible by practioners. It is thus highly surprising in the first place that the problem is technically wellposed at all.
14.3.2.4 Extension to Other Domains

Translation and rotationinvariance: As the map \(\mathscr {T}\) is invariant under shifts and/or rotations of the coordinates, it holds that \(C_{\text {stab}}^{\text {gen}}(\tilde{\varOmega },\mathfrak {f}) = C_{\text {stab}}^{\text {gen}}(\varOmega ,\mathfrak {f})\) whenever \(\tilde{\varOmega }\) is a shifted and/or rotated version of \(\varOmega \subset \mathbb {R}^m\).

Monotonicity: \(C_{\text {stab}}^{\text {gen}}(\varOmega _1,\mathfrak {f}) \ge C_{\text {stab}}^{\text {gen}}(\varOmega _2,\mathfrak {f})\) for any \(\varOmega _1 \subset \varOmega _2 \subset \mathbb {R}^m\).

Scaling: \(C_{\text {stab}}^{\text {gen}}(r \cdot \varOmega ,\mathfrak {f}) \ge C_{\text {stab}}^{\text {gen}}(\varOmega ,r^2\mathfrak {f})\) for any \(\varOmega \subset \mathbb {R}^m\) and \(r > 0\).
Analogous properties hold for the stability constants in Sect. 14.3.3.
14.3.3 Homogeneous Objects and Multiple Holograms
According to Sect. 14.2.1, uniqueness then also holds without support constraints, but image reconstruction is still illposed in general: the associated forward maps \(\mathscr {S}_{\nu }: L^2(\mathbb {R}^m) \rightarrow L^2(\mathbb {R}^m) \) and \(\mathscr {T}^{(\mathfrak {f}_{1}, \ldots , \mathfrak {f}_{\ell })}: L^2(\mathbb {R}^m) \rightarrow L^2(\mathbb {R}^m)^\ell \) do not have a bounded inverse due to zeros of the CTFs.
When both homogeneity and support constraints can be assumed, wellposedness holds true with an improved stability constant compared to (14.34):
Theorem 14.3
By Theorem 14.3, the original decay \(C_{\text {stab}}^{\text {gen}}\sim \exp (\mathfrak {f}/8)\) of the stability constant as \(\mathfrak {f}\rightarrow \infty \) improves to \(C_{\text {stab}}^{\text {hom}}(\varOmega ,\mathfrak {f}, \nu ) \sim \mathfrak {f}^{\gamma }\) with \(\gamma = 1\) for \(\nu = 0\) and \(\gamma = 1/2\) for \(\nu > 0\). This ensures practical stability also at larger Fresnelnumbers.
A similar improvement applies for the reconstruction of general objects (no homogeneityconstraint) from two holograms:
Theorem 14.4
Note that the r.h.s. of the stability bound (14.37) increases with the difference \(\mathfrak {f}_{}^{1}\) between the reciprocal Fresnelnumbers \(\mathfrak {f}_{1}^{1}, \mathfrak {f}_{2}^{1}\). Improved stability is thus guaranteed only if \(\mathfrak {f}_{1}\) and \(\mathfrak {f}_{2}\) differ strongly, i.e. if the two holograms are acquired in significantly different experimental setups.
14.3.3.1 OrderOptimality
14.3.3.2 Numerical Stability Computations
Other than for the setting in Theorem 14.2, the prediction (14.35) for the stability constant \(C_{\text {stab}}^{\text {hom}}\) (and thus for \(C_{\text {stab}}^{\text {two}}\)) is far from sharp if the analytical bounds on the constants \(c_j\) from [24] are inserted. Sharp values of \(C_{\text {stab}}^{\text {hom}}\) may however be computed numerically by approximating the minimum singular value of the operator \(\mathscr {S}_{\nu }\) via techniques presented in [29, Sect. 3.4].
14.3.4 Extensions
14.3.4.1 Phase Contrast Tomography
As detailed in [29, Sect. 3.3], the relation (14.40) allows to express stability of linearized XPCT via known results for tomographic reconstruction, combined with stability bounds for \(\mathscr {T}^{\text {(3d)}} , \mathscr {S}_{\nu }^{\text {(3d)}}: L^2(\varOmega ) \rightarrow L^2(\mathbb {R}^3)\) where \(\varOmega \subset \mathbb {R}^3\). Stability then depends on a threedimensional support constraint \({{\,\mathrm{supp}\,}}(\beta + \text {i}\delta ) \subset \varOmega \subset \mathbb {R}^3\) for the imaged sample’s refractive index.
14.3.4.2 Imaging with Finite Detectors
There are a number of idealizing assumptions underlying to the obtained stability estimates: in addition to the neglected nonlinearity and idealizations in the basic physical model such as full coherence, it has also been assumed that the hologram I is measured in the whole detectorplane in Fig. 14.1. Due to the finite size of realworld detectors (and—more fundamentally—the finite width of illuminating Xray beams), however, only restrictions \(I_K\) to some bounded domain K are available in practice.
According to Theorem 14.1, such restricted data has no impact on uniqueness (if K contains an open set). The situation is quite different in terms of stability, as analyzed in [32]: for any bounded \(K \subset \mathbb {R}^m\)—however large—the inverse problem of XPCI becomes severely illposed, i.e. Lipschitzstability is lost so that data errors may severely corrupt the reconstructed images. Yet, it is also proven in [32] that the situation may be repaired by restricting to images \(h = \mu + \text {i}\phi \) of finite resolution (smoothness constraint in the sense of Sect. 14.1.1.1): by imposing that the h are Bsplines on a Cartesian grid of sufficiently large spacing \(r(\varOmega , K, \mathfrak {f}) > 0\) (i.e. pixelated images in some sense), Lipschitzstability can be restored in the finitedetector setting. Physically, the necessity of such a restriction corresponds to a resolution limit that arises due to the finite numerical aperture associated with the detector size.
14.4 Regularized Newton Methods for XPCI
The following section considers regularized Newtontype methods for image reconstruction in XPCI. The proposed algorithm is motivated by the theoretical insights gained from Sects. 14.2 and 14.3.
14.4.1 Motivation
14.4.1.1 Significance of Constraints
The stability results of Sect. 14.3 heavily rely on support constraints—without such, XPCI is illposed or even nonunique. To guarantee stability in practice, image reconstruction methods must thus be able to exploit supportknowledge. Also other types of a priori knowledge (see Sect. 14.1.1.1) are known to be beneficial. In particular, imposing nonnegativity often has a similar stabilizing effect as support constraints.
14.4.1.2 Necessity of Iterative Methods
By far the most commonly used reconstruction method for XPCI at synchrotrons is direct CTFinversion, as presented in Sect. 2.3. Within the notation of this chapter, the approach corresponds to quadratic Tikhonov regularization applied to the linearized forward maps \(\mathscr {S}_{\nu }^{(\mathfrak {f}_1, \ldots , \mathfrak {f}_\ell )}\) or \(\mathscr {T}^{(\mathfrak {f}_1, \ldots , \mathfrak {f}_\ell )}\). Owing to the linearity and translationinvariance of these maps, the reconstruction may be implemented via a multiplication in Fourierspace (deconvolution), which renders the approach computationally fast.

Support constraints \({{\,\mathrm{supp}\,}}(h) \subset \varOmega \) for \(\varOmega \subsetneq \mathbb {R}^m\) break translationinvariance

Nonnegativity is a nonlinear constraint: any reconstruction imposing it depends nonlinearly on the data \(I1\)—even for linear forward models!
In either case, reconstruction may thus no longer be achieved by deconvolution. Thus, iterative algorithms have to be applied to impose support and/or nonnegativityconstraints in lack of efficient direct reconstruction formulas.
14.4.1.3 XPCI Beyond Linear Models
Although the linear CTFmodel of XPCI has a surprisingly large regimeofvalidity, there are settings where linear image reconstruction induces severe artifacts arising from the neglected nonlinearity, as demonstrated in Sect. 13.3. Reconstruction algorithms based on the full nonlinear XPCImodel are thus preferable in principle. The main obstacle in using such is that direct inversion formulas for the nonlinear model are not known. However, when iterative methods are needed anyway (Sect. 14.4.1.2), nonlinear forward maps cause little additional difficulty.
14.4.2 Reconstruction Method
In the following, we propose a reconstruction algorithm that meets the requirements discussed in Sect. 14.4.1. Details can be found in [33].
Note that we use a standard squared \(L^2\)norm as a datafidelity term in (14.41), in lack of an accurate model for the data error statistics in flatfield corrected holograms. The squared Sobolevterm \(\alpha _k \Vert h  h_0 \Vert _{H^s}^2\) (\(\Vert f \Vert _{H^s}^2 := \Vert (1 + \varvec{\xi }^2)^{s/2} \cdot \mathcal {F}(f) \Vert _{H^s}^2\)) imposes tunable (by the choice of \(s \ge 0\)) smoothness of the iterates \(h_k\) and acts as a regularizer. Finally, \(\mathcal {R}_{\ge 0}(h, h_k)\) is a quadratic penalty term that is designed to correct negative values of \(\text {Re}(h_k)\) or \(\text {Im}(h_k)\) in the subsequent iterate \(h_{k+1}\), see [33] for details.
In the numerical algorithm, a discretized analogue of the quadratic minimization problem in (14.41) is solved for images \(\varvec{h}_*\in \mathbb {C}^N\), data \(\varvec{I}^{\text {obs}}\in \mathbb {R}^M\) and forward map \(F_{\text {dis}}: \mathbb {C}^N \rightarrow \mathbb {R}^M\), via a conjugategradient method. The \(\alpha _k\) and \(k_{\text {stop}}\) are chosen in a widely automated fashion, as detailed in [33].
14.4.3 Reconstruction Example
The true phaseimage \(\phi \) (Fig. 14.3b) is given by a bulk disk of magnitude 0.2, whereas the true absorptionimage \(0 \le \mu \le 0.02\) shows a logostructure (Fig. 14.3c). Accordingly, no homogeneityconstraint is applicable so that the testcase is situated in the most challenging, unstable setting of XPCI, which has been analyzed in Sect. 14.3.2. In particular, recall that image reconstruction is nonunique without exploiting further constraints.
The data is reconstructed using the regularized Newton method from Sect. 14.4.2, imposing nonnegativity of \(\phi \) and \(\mu \) as well as support constraint, allowing nonzero values of \( \phi , \mu \) only within the circular region marked by the blue dashed line in Fig. 14.3b, c. The reconstructed images in Fig. 14.3d, e show that the proposed method correctly attributes the diskstructure to the phaseimage \(\phi \) and the logopattern to \(\mu \), without visible signs of “mixing things up”. The overall lower reconstructionquality in \(\mu \) compared to \(\phi \) is due to the lower signaltonoise in this parameter, as a realistically low absorptionrefractionratio \(\beta /\delta \le 0.1\) has been assumed in the test case.
Now why does reconstruction of both \(\phi \) and \(\mu \) from a single hologram work here, contrary to the usual experience? The diameter of the circular support corresponds to a relatively low (modified) Fresnelnumber \(\mathfrak {f}\approx 87\). According to the analysis in Sect. 14.3.2, this ensures stability of image reconstruction, as is discussed to greater detail in [33] and [24, Sect. 6]. By its ability to impose support constraints (and nonnegativity), the proposed Newtontype method allows to exploit this theoretical stability in practice.
14.5 Regularized NewtonKaczmarzSART for XPCT
In the final section, we present a Newtontype reconstruction method for Xray phase contrast tomography (XPCT) that is a compromise between flexibility w.r.t. a priori constraints and computational performance. We note that the method is an allatonce approach, as also proposed in [30, 31, 34]: the 3Dobject parameters \(\delta ,\beta \) are recovered directly from the full tomographic hologramseries, instead of first reconstructing 2Dimages \(\phi ,\mu \) for each hologram individually. Thereby, tomographic consistency is imposed as an additional constraint in image reconstruction, compare Sect. 14.1.1.1.
Iterations of the form (14.43) are known as regularized NewtonKaczmarz [35]. The advantage compared to bulk (i.e. nonKaczmarz)methods is that the operatorblocks \(f \mapsto F({\mathscr {P}}_{\varvec{\theta }_{j}}(f))\) require much less computations to evaluate than the total XPCT operator \(F_{\text {PCT}}\), which permits efficient computation of the iterates (14.41). Moreover, Kaczmarztype methods often exhibit fast initial convergence, typically reaching a good reconstruction already after one or two cycles over the data, i.e. for \(n_{\text {stop}}\in \{1,2\}\). To promote convergence, the processing order \(\{j_1,j_2,\ldots \} \subset \{1,\ldots , t\}\) of the datablocks should be chosen such that subsequently fitted directions \(\varvec{\theta }_{j_{k}}, \varvec{\theta }_{j_{k+1}}\) differ as strongly as possible, which we achieve by following a “multilevelscheme” from [36].
14.5.1 Efficient Computation by Generalized SART
Although the processed datasize is reduced by the Kaczmarzstrategy, the iterates (14.41) still involve a minimization problem on a highdimensional space of 3Dobjects f. Moreover, if the minimization is performed iteratively, each iteration requires evaluations of the (discretized) projector \({\mathscr {P}}_{\varvec{\theta }_{j_k}}\) and its adjoint \({\mathscr {P}}^*_{\varvec{\theta }_{j_k}}\), the backprojector, both of which typically amount to much higher computational costs than evaluating the XPCI forward map F.^{5}
Both computational issues can be resolved by computing the iterates (14.41) via a generalized SART^{6} (GenSART) scheme, as introduced in [38] for a much more general class of tomographic Kaczmarziterations:
 1.
Forwardprojection: \(p_k := {\mathscr {P}}_{\varvec{\theta }_{j_k}}(f_k)\)
 2.Optimization in projectionspace (\(u_{j} = {\mathscr {P}}_{\varvec{\theta }_j}(\varvec{1}_\varOmega )\)):$$\begin{aligned} \varDelta p _k&\in \!{{\,\mathrm{argmin\,}\,}}_{p \in L^2(\mathbb {R}^2, (\mathbb {R}))}\! \Vert F(p_k) + F'[ p_k ]( u_{j_k} \cdot p )  (I^{{\text {obs}}}_{\varvec{\theta }_{j_k}}1) \Vert _{L^2}^2\nonumber \\&\qquad \qquad \quad + \alpha \big ( (1\gamma ) \Vert u_{j_k}^{1/2} \cdot p \Vert _{L^2}^2 + \gamma \Vert u_{j_k}^{1/2} \cdot \nabla p \Vert _{L^2}^2 \big ) \end{aligned}$$(14.44)
 3.
Backprojection update: \(f_{k + 1} = f_k + {\mathscr {P}}^*_{\varvec{\theta }_{j_k}} ( \varDelta p _k )\)
The main benefit of the approach is that the required minimization is cast to projectionspace, i.e. no longer needs to be solved on a highdimensional space of 3Dobjects but merely on 2Dimages. Moreover, the whole scheme requires only a single evaluation of \({\mathscr {P}}_{\varvec{\theta }_{j_k}}\) (1.) and its adjoint \({\mathscr {P}}^*_{\varvec{\theta }_{j_k}}\) (3.), whereas the optimization (2.) does not involve any of these costly operations anymore.
As is standard for Kaczmarztype methods, nonnegativity of the iterates \(f_{k+1}\) (in real and imaginary part) may be imposed by adding a final step to the GenSARTscheme: \(f_{k+1,\ge 0} = \max \{0, \text {Re}(f_{k+1})\} + \text {i}\max \{0, \text {Im}(f_{k+1})\}\).
14.5.2 Parallelization and LargeScale Implementation

Low memory requirements: if the backprojection update (3.) (as well as the optional nonnegativity projection) is implemented as an inplace operation, only a single 3Darray (storing \(f_0, f_1, (f_{1,\ge 0},) f_2, \ldots \)) needs to be kept in memory throughout the whole NewtonKaczmarzreconstruction.

Parallelized optimization: as the optimizationstep (2.) works on 2Dimages only, its memoryrequirements are low enough to be performed on a single graphical processing unit (GPU) even for largescale data. This permits efficient parallized implementation of this step.

Parallelized 3Dcomputations: The only operations on the 3Dobjects \(f_k\) are forward and backprojections \({\mathscr {P}}_{\varvec{\theta }_{j_k}}, {\mathscr {P}}^*_{\varvec{\theta }_{j_k}}\) and pointwise arithmetics. All of these can be easily parallelized at low communication requirements between the different processors. In fact, it is possible to implement GenSARTschemes in a distributed manner: the objectiterates \(f_k\) may be split into chunks, that are stored and managed by dedicated machines throughout the whole reconstruction. This property allows to run NewtonKaczmarz reconstructions efficiently on multiple GPUs.
14.5.3 Reconstruction Example
We assess the NewtonKaczmarz method for XPCTdata of freezedried Deinococcus radiodurans bacteria. The experimental data set, acquired with the GINIX setup from Chap. 3, is composed of 641 holograms of size \(2048 \times 2048\) at tomographic incident angles \(\theta = 0^\circ , 0.25^\circ , \ldots , 119^\circ , 139^\circ , 139.25^\circ , \ldots , 180^\circ \) (one hologram per angle). 2D orthoslices of the 3D tomographic data (two spatial and one angular dimension) are shown in Fig. 14.4a–c, emphasizing the missing data between \(\theta = 119^\circ \) and \(\theta = 139^\circ \).
 1.
CTF+FBP: direct CTFinversion for each hologram, followed by filtered backprojection applied to the recovered projections of \(\delta \).
 2.
Linear Kaczmarz: reconstruction by (14.43) over a single cycle \(n_{\text {stop}} = 1\), using the linearized XPCImodel \(F = \mathscr {S}_{0}\). Nonnegativity of the reconstructed \(\delta \) and support in a centered cube of \(512^3\) voxels is imposed.
 3.
NewtonKaczmarz: same as (2.), but with the nonlinear model \(F = \mathscr {N}_{0}\).

The additional constraints exploited in “Linear Kaczmarz” compared to “CTF+FBP” widely eliminate lowfrequency backgroundartifacts (compare Fig. 14.4e–h) and thereby enable quantitatively correct reconstructions \(\delta \).

Though the sampleinduced phase shifts are moderate, \(\phi _{\varvec{\theta }} = k {\mathscr {P}}_{\varvec{\theta }} (\delta ) \lesssim 1\), going over to the nonlinear XPCImodel has significant effects: especially in Fig. 14.4h, it can be seen that using the linearized model causes artificial distortions in the recovered objectdensity compared to the nonlinear NewtonKaczmarzreconstruction in Fig. 14.4i–l.
Accordingly, both the nonlinearity and the ability to exploit a priori constraints of the proposed NewtonKaczmarz method turn out to be vital here to accurately reconstruct the anticipated 3D structure of the imaged bacteria^{7}: cytoplasm with blobshaped inclusions containing the DNA, where each of the two compounds is of approximately uniform density.
Footnotes
 1.
The classical Fresnelnumber is given by Open image in new window . However, using the parameter \(\mathfrak {f}\) is notationally more convenient as it avoids excessive occurence of \(2\pi \)factors.
 2.
To ensure welldefinedness on the whole space \(L^2(\varOmega )\), the exponential has to be suitably truncated for the physically irrelevant case of negative absorption \(\text {Re}(f) = \mu < 0\).
 3.
Note that this behavior of \(\mathcal {D}\) is fundamentally different from that of the Fouriertransform \(\mathcal {F}\), which maps constants to Diracdeltas centered at the origin.
 4.
The lateral lengthscale b associated with \(\mathfrak {f}\) is implicitly fixed to the width of \(\varOmega \) by assuming the latter to be 1 in Theorem 14.2, as will also be done in all subsequent results.
 5.
For images of size \(N\times N\), the discretized forward maps \(F = \mathscr {N}^{(\mathfrak {f}_{1}, \ldots , \mathfrak {f}_{\ell })}\) may be evaluated in \(\mathcal {O}(\ell N^2\log N)\) operations, while (back)projecting 3Darrays of size \(N\times N\times N\) is \(\mathcal {O}(N^3)\).
 6.
“SART” refers to the simultaneous algebraic reconstruction technique from [37].
 7.
The additional object in the topleft of Fig. 14.4e, h, k is a contaminant particle.
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