Directional dark-field implicit x-ray speckle tracking using an anisotropic-diffusion Fokker–Planck equation

Consider the x-ray speckle-tracking setup that is sketched in Fig. 1. Here, we see a quasi-monochromatic x-ray source that paraxially illuminates a thin spatially-random speckle-generating mask, to produce a reference speckle image $I_{\textrm{R}}(x,y)$, over the planar surface of a position-sensitive detector which is perpendicular to the optical axis $z$. The reference speckle image may be produced via either or both of the following mechanisms: (i) attenuation contrast due to the position-dependent absorption of x rays as they traverse the spatially-random mask, (ii) phase contrast due to the position-dependent phase shifts imparted upon the x rays as they traverse the mask, with these phase shifts leading upon free-space propagation to intensity variations over the surface of the position-sensitive detector [63]. Having recorded the reference speckle image, we may then perform a second measurement, in which a thin non-absorbing object is placed in between the speckle-generating mask and the detector. The distance $\Delta$ from the sample to the detector should be sufficiently large that the transverse deflections, induced by the refractive profile of the sample, lead to measurable transverse shifts in the reference speckles measured in the absence of the sample. Conversely, $\Delta$ should be sufficiently small that the transverse shifts, of the reference speckles, should not be more than around two detector pixels in magnitude. The speckle image $I_{\textrm{S}}(x,y)$, measured in the presence of both the sample and the mask, will then have the following property: every x-ray photon that strikes a given location $(x,y)$ in $I_{\textrm{S}}$ will have struck a nearby position $(x+\delta x,y+\delta y)$ in $I_{\textrm{R}}$, and conversely. This property, which arises from the previously articulated assumption that $\Delta$ be sufficiently small, implies $I_{\textrm{R}}$ and $I_{\textrm{S}}$ to be connected via a conserved current that locally preserves the number of detected photons. Stated differently, local conservation of energy implies there to be a geometric flow—which may also be termed an optical flow [64, 65] or a Noether conserved current [66]—that can be used to smoothly deform $I_{\textrm{R}}$ into $I_{\textrm{S}}$ [56].

It is natural to assume that this flow, which deforms $I_{\textrm{R}}$ into $I_{\textrm{S}}$, will have both coherent and diffuse components. The coherent component will be due to the near-monochromatic nature of the illuminating radiation, together with the position-dependent phase and amplitude shifts that are coherently imparted by both the sample and the speckle-generating mask. The diffuse component will be due to several mechanisms: (i) the partially-coherent nature of the illumination, (ii) the presence of spatially-unresolved random micro-structure in the sample and mask, and (iii) the presence of sharp edges in both sample and mask. Spatially-unresolved random micro-structure leads to diffusive x-ray energy flow on account of position-dependent small-angle x-ray scattering (SAXS) [44]. Sharp edges lead to diffusive flow via Young–Maggi–Rubinowicz boundary-diffraction waves (edge scattering, edge-diffracted rays) [67, 68, 69, 70, 71].

All of the above considerations—namely a locally energy-preserving optical flow, possessing both coherent and diffuse components, that maps $I_{\textrm{R}}$ into $I_{\textrm{S}}$—may be quantified using a forward-finite-difference Fokker–Planck equation [60, 61, 57, 72]. In particular, Eq. (55) in Ref. [57] generalizes the optical-flow method for x-ray speckle tracking [56], to give:

Here ${\bf{r}}_{\perp}\equiv(x,y)$, $I_{\textrm{R}}({\bf{r}}_{\perp})$ and $I_{\textrm{S}}({\bf{r}}_{\perp})$ are the intensities of a reference speckle image acquired in the absence and presence of a sample, respectively, $\Delta$ is the sample-to-detector distance, $k=2\pi/\lambda$ is the wavenumber of the x rays, $\lambda$ is the wavelength, $\phi_{\textrm{ob}}$ is the phase shift caused by the sample, and $\nabla_{\perp}=(\partial/\partial x,\partial/\partial y)$ denotes the transverse gradient operator. The first term, on the right-hand side of Eq. (1), quantifies the coherent energy flow that was described earlier. An identical term appears in the transport-of-intensity equation (TIE) [59]. Indeed, the entire first line of Eq. (1) is mathematically identical in form to a first-order finite-difference form of the TIE. This correspondence exists because local energy conservation, under transverse energy flow, is the underpinning principle in both the TIE and the first line of Eq. (1). The difference lies in the fact that (i) the transverse flow in the TIE is induced by free-space propagation from plane to parallel downstream plane in vacuo, whereas (ii) in the first line of Eq. (1) the transverse flow occurs in a fixed plane, with the flow induced by a phase object whose presence smoothly perturbs $I_{\textrm{R}}({\bf{r}}_{\perp})$ into $I_{\textrm{S}}({\bf{r}}_{\perp})$ [56]. The second term, on the right-hand side of Eq. (1), describes the diffusive flow via the effective diffusion coefficient $D_{\textrm{eff}}({\bf{r}}_{\perp};\Delta)$. If we assume the mask to be spatially statistically stationary, and the source to be an extended incoherent source, then the diffusive effects due to source-size blur will be describable via a position-independent additive constant term in $D_{\textrm{eff}}({\bf{r}}_{\perp};\Delta)$. With this understanding in place, we henceforth consider $D_{\textrm{eff}}({\bf{r}}_{\perp};\Delta)$ to quantify only sample-induced contributions, that are due to both (i) local position-dependent SAXS fans emerging from each point on the exit surface of the sample, as well as (ii) edge-diffracted x rays. Note, also, that the position-dependent sample-induced SAXS fans, as described by the above formalism, are considered to be rotationally symmetric (by assumption) at each transverse location over the exit surface of the thin sample.

To generalize Eq. (1) to the case of directional dark-field imaging [52, 53], in which the position-dependent SAXS fans have an elliptical transverse profile [53, 73], we introduce the symmetric rank-two diffusion tensor field [61, 57] :

This enables us to write down a directional-dark-field generalization of Eq. (1), namely the following anisotropic-diffusion forward-finite-difference Fokker–Planck speckle-tracking equation due to Pavlov et al. [58]:

Let us now assume that an attenuating object is placed in the well-resolved reference speckle field. This causes variations in the registered speckle image, which can be described by (cf. Eq. (9) in Ref. [74], with incorporation of Eq. (4) in Ref. [61] and Eq. (51) in Ref. [57]):

Here $I_{\textrm{o}b}({\bf{r}}_{\perp})$ is the object’s attenuation term. We also assume that the components of the diffusion tensor, $D^{(xx,yy,xy)}_{\textrm{eff}}$, are slowly-varying functions of the transverse position (i.e., we can neglect their transverse spatial derivatives, which are small compared to the retained terms).

The second-order transverse spatial derivatives, applied to the diffuse-scatter terms on the right side of Eq. (4), yield several components. For example, the first of these three diffusely-scattering components gives:

We have neglected the following terms in the right hand side of Eq. (5): $I_{\textrm{R}}({\bf{r}}_{\perp})I_{\textrm{o}b}({\bf{r}}_{\perp})\frac{\partial^{2}}{\partial x^{2}}D^{(xx)}_{\textrm{eff}}({\bf{r}}_{\perp};\Delta)$ and $2I_{\textrm{R}}({\bf{r}}_{\perp})\frac{\partial}{\partial x}[D^{(xx)}_{\textrm{eff}}({\bf{r}}_{\perp};\Delta)]\frac{\partial}{\partial x}[I_{\textrm{o}b}({\bf{r}}_{\perp})]$, because we have assumed that $D^{(xx)}_{\textrm{eff}}$ is a slowly-varying function everywhere. We have also neglected the term $2\frac{\partial}{\partial x}[D^{(xx)}_{\textrm{eff}}({\bf{r}}_{\perp};\Delta)I_{\textrm{o}b}({\bf{r}}_{\perp})]\frac{\partial}{\partial x}[I_{\textrm{R}}({\bf{r}}_{\perp})]$, for the following reason. The intensity of a reference speckle image $I_{\textrm{R}}({\bf{r}}_{\perp})$, acquired in the absence of a sample, is produced by a spatially random mask. Therefore, the gradient of such an intensity map will be a vector field that is rapidly changing in both direction and magnitude. Thus the scalar product, of such a random vector field with a more slowly changing gradient of the product of two functions, can be neglected. A similar approximation was previously employed in Refs. [74, 58]. Bearing all of the above points in mind, we can simplify Eq. (4) as follows:

Above, we have used an additional approximation, similar to that used earlier, namely the neglect of the term $\nabla_{\perp}[I_{\textrm{R}}({\bf{r}}_{\perp})]\cdot[I_{\textrm{o}b}({\bf{r}}_{\perp})\nabla_{\perp}\phi_{\textrm{o}b}({\bf{r}}_{\perp})]$ (see also Refs. [74, 58]). Here we are again making use of the fact that the intensity of a reference speckle image, acquired in the absence of a sample, $I_{\textrm{R}}$, is produced by a random mask. Therefore, the gradient of such a field is again a vector field that is rapidly changing in both direction and magnitude. Thus, the scalar product of such a random vector field with a more slowly changing phase gradient can be neglected. It is also worth noting that the terms on the right side of Eq. (6), containing the second-order derivatives of the object’s attenuation term, become more prominent at the object’s internal and external boundaries.

A further modification of of Eq. (6) can be achieved by dividing both sides by $I_{\textrm{R}}$ and rearranging, to give:

This equation describes a general case of speckle-based imaging for an attenuating object, where several terms incorporating the diffusion tensor field are taken into account. This forward-finite-difference anisotropic-diffusion Fokker–Planck equation completes our description of the forward problem associated with image formation and subsequent data collection, in the context of x-ray speckle tracking.

The formulation of the forward problem in Eq. (7) establishes the model upon which the associated inverse problem [62], of recovering sample properties based on one or more pairs of reference-only and reference-plus-sample speckle images, may be based. Here we consider two such inverse problems, corresponding to two different limit cases of Eq. (7). These two special cases are: (i) the sample is assumed to be a phase object, i.e. it is non-attenuating; (ii) the sample is taken to be monomorphous, i.e. its phase shifts are proportional to the logarithm of the associated attenuation, as is the case e.g. for a thin object that is composed of a single material.

Here we consider the relationship between the diffusion-tensor field (Eq. (2)) and the associated position-dependent elliptical SAXS fans emanating from each point on the exit surface of the sample.

If we consider the transverse location $(x_{0},y_{0})$ at the nominally-planar exit surface of the sample, then the resulting anisotropic blurring due to the locally-elliptical SAXS fan will correspond to an ellipse of $(x,y)$ coordinates in the plane at distance $\Delta>0$ downstream of the sample, with these $(x,y)$ coordinates obeying:

The above ellipse field, in which we have a different ellipse for each transverse location $(x_{0},y_{0})$, incorporates the effects of two distinct but related factors: (i) the opening angles $\theta(x_{0},y_{0})$ of the elliptical SAXS fans that emanate from each point $(x_{0},y_{0})$ over the nominally-planar exit surface of the sample; (ii) the corresponding dimensionless fractions $F(x_{0},y_{0})$ of the incident x-ray photons that are converted to the diffusely-scattering channel, on account of unresolved micro-structure and edge scatter from the sample.

See Fig. 3 of Paganin & Morgan [57] for the relations connecting the following quantities: (i) the effective diffusion coefficients, (ii) the position-dependent SAXS-fan opening angles, and (iii) the diffuse-scatter fractions. For an example of such connecting relations, for the ‘$xx$’ diffuse-scatter channel we have

Here, $F^{(xx)}(x_{0},y_{0})$ is a dimensionless quantity taking values between zero and unity, that denotes the fraction of the incident x rays (at the specified energy) that are converted to ‘$xx$’ diffuse scatter by the sample at the transverse location $(x_{0},y_{0})$, with $\theta^{(xx)}(x_{0},y_{0})$ being the corresponding local-SAXS-fan opening angle. Similar expressions may be written down, by replacing $(xx)$ with either $(yy)$ or $(xy)$. Note, also, that these three opening angles may be converted to (i) an angle for the SAXS-ellipse semi-major axis, (ii) an angle for the SAXS-ellipse semi-minor axis, and (iii) an angle denoting the angular orientation of the semi-major ellipse axis.

The form of the right side of Eq. (23) implies that the effective diffusion coefficients are invariant under the concurrent mappings

Here, $\alpha(x_{0},y_{0})$ is a real positive function which may assume otherwise-arbitrary values, provided that, both before and after the above mapping, the scattering fractions obey $F(x_{0},y_{0})\ll 1$ and the scattering angles obey $\theta(x_{0},y_{0})\ll 1$. The physical origin of this invariance is as follows. Decreasing $F(x_{0},y_{0})$ at any fixed transverse location will decrease the degree of local-SAXS blurring in the detection plane. Conversely, increasing the corresponding SAXS-fan opening angle at the same sample location will increase the degree of such blurring. These two opposing influences can be chosen to exactly balance one another (in the sense of providing an identical measured map of radiant exposure), in a continuous infinity of different ways, corresponding to all of the different choices that can be made for the scalar field $\alpha(x_{0},y_{0})$. This is a fundamental ambiguity in the Fokker–Planck formalism that underpins the present paper.

The above ambiguity implies that, rather than recovering the SAXS-fan ellipse at any transverse location $(x_{0},y_{0})$, the method is only able to recover a family of similar concentric ellipses at each $(x_{0},y_{0})$ location. Each member of this family is similar to the actual SAXS ellipse at each transverse location $(x_{0},y_{0})$, in the sense of having the same eccentricity and orientation angle, but which member of the family is the actual SAXS ellipse remains undetermined.

Fortunately, there are two absolute quantities that may be extracted, since they are both independent of $\alpha(x_{0},y_{0})$ and are therefore the same for all ellipses in the previously mentioned family of similar ellipses. These two invariant directional-dark-field quantities are:

• •

  the eccentricity $\epsilon(x_{0},y_{0})$ of the SAXS ellipse at each transverse location $(x_{0},y_{0})$;

• •

  the angular orientation $\psi(x_{0},y_{0})$ of each SAXS ellipse (note that these angular orientations are only meaningful modulo $\pi$ radians, since the major axes of the ellipses form a director field rather than a vector field, i.e. they are ‘arrowless vectors’).

Both $\epsilon(x_{0},y_{0})$ and $\psi(x_{0},y_{0})$ (positive direction is counterclockwise) may be extracted from the symmetrical quadratic form (see Eq. 2.4-1 in [76]) of an ellipse, where we assume that the center of this ellipse is at $(x_{0},y_{0})$:

Hence

where

which corresponds to Eq. (22).

Subsequently, using Eq. (26), both $\epsilon(x_{0},y_{0})$ and $\psi(x_{0},y_{0})$ may be extracted directly from the effective diffusion tensor in Eq. (2), via the following relations:

Thus, while this symmetric rank-two diffusion tensor contains three independent components at each transverse location $(x_{0},y_{0})$, the invariance under the mapping of Eq. (24) implies that only two independent invariant quantities may be extracted using our Fokker–Planck speckle-tracking formalism.

Evidently, we have four independent recoverable channels of information in total, corresponding to four different scalar fields at the exit surface of the sample. The first pair of scalar fields, which is associated with the coherent channel for x-ray energy flow, is the intensity $I_{\textrm{o}b}(x_{0},y_{0})$ and phase $\phi_{\textrm{o}b}(x_{0},y_{0})$ at each point $(x_{0},y_{0})$ on the exit surface of the sample. The other pair of scalar fields, which is associated with the diffuse channel for x-ray energy flow, is the eccentricity $\epsilon(x_{0},y_{0})$ and ellipse-orientation $\psi(x_{0},y_{0})$ of the local position-dependent SAXS ellipses emerging from each point on the sample’s exit surface. A fifth scalar field, namely the dimensionless scattering fraction $F(x_{0},y_{0})$, is not recovered by the formalism developed in the present paper.

The recovered ellipse-eccentricity and ellipse-orientation fields, namely $\epsilon(x_{0},y_{0})$ and $\psi(x_{0},y_{0})$ respectively, can be represented in various ways, including:

• •

  A color representation in which the brightness of each pixel is proportional to $\epsilon(x_{0},y_{0})$ and the color is a function of $\psi(x_{0},y_{0})$ (modulo $\pi$ radians);

• •

  A grey-scale representation where the brightness of each pixel is proportional to $\epsilon(x_{0},y_{0})$;

• •

  A color representation in which the color of the pixel is a linear combination of three different hues (e.g. red, green and blue) for the three components of the diffusion tensor. Other choices of color space may also be employed.

Our Fokker–Planck speckle-tracking model, for the combined coherent-flow and diffuse-flow channels of x rays traversing a non-crystalline sample, enables us to extract information pertaining to both channels. This, in turn, relates directly to different aspects of the distribution of the complex refractive index within the sample. The coherent channel corresponds to a coarse-grained form of the complex refractive index, with the coarse-graining being induced by the finite size of the detector pixels as well as the lack of perfect spatial coherence in the illuminating photon field. Conversely, the diffuse channel corresponds to a fine-grained form of the complex refractive index, relating to structures that are not directly spatially resolved by the system, but which nevertheless have a measurable influence that may be extracted from recorded maps of radiant exposure. In this sense, the method may be aptly described using the language of Kagias et al. [81], as simultaneously extracting both real-space and reciprocal-space information regarding the sample.

We have given particular emphasis to the recovery of information associated with the diffuse channel, and in particular to the directional-darkfield aspects of this channel. The closed-form analytic solutions, developed in solving the inverse problem associated with our model, enable computationally rapid darkfield reconstructions using a relatively small number of images. We formulated two different sample models: either a pure phase object or a single-material object. These two models were tested on two different experimental x-ray beamlines with two different samples. The obtained results already seem to show interesting features on the composition of the bone and calcified cartilage that needs to be confirmed with histology.

Next, we briefly discuss our use of the term ‘darkfield’ in the context of our paper. This term has a more general usage, relating to any form of imaging in which unscattered photons—or other imaging quanta, such as electrons or neutrons—make no contribution to the output image [82]. This imaging-system property causes samples, which incompletely cover the field of view, to appear with positive contrast against a dark background field. Thus, when we speak of the darkfield signal in our paper, we are actually referring to a particular form of darkfield signal, namely that which is associated with the diffusion tensors appearing in the anisotropic Fokker–Planck equation (Eq. (3)) for speckle tracking. For a different form of darkfield signal in an x-ray imaging setting, see e.g. the technique for dark-field x-ray microscopy that is reported in Simons et al. [83].

It is also worth pointing out a partial analogy that exists, between our technique based on Eq. (3), and the non-equilibrium dynamics of Brownian motion. In this analogy, consider a thin sheet of ($2+1$)-dimensional fluid in non-uniform flow, at two closely-spaced times $t_{1}$ and $t_{2}>t_{1}$. By assumption, this fluid has a position-dependent temperature $T(x,y,t)$, in addition to being anisotropic at a microscopic level, as would be the case e.g. in a flowing liquid crystal [84]. Suppose that small clusters of pollen grains have been randomly positioned over the surface of the flowing anisotropic fluid, at time $t_{1}$. As the fluid flows, each pollen cluster will be advected. The timestep $t_{2}-t_{1}$ is sufficiently small, by assumption, that the pollen-grain clusters move by distances no larger than their diameter, in evolving from $t_{1}$ to $t_{2}$. The local displacement of the center of mass of each pollen cluster will be a direct measure of the local velocity of the fluid. This pollen-cluster advection is analogous to the ‘speckle tracking’ first term on the right side of Eq. (3), if one replaces the randomly-positioned pollen clusters with the randomly-positioned illuminating x-ray speckles, and considers the flow to be induced by placing a transparent sample in the x-ray beam, rather than letting the pollen-laden anisotropic fluid evolve from $t_{1}$ to $t_{2}$. In this analogy, $\Delta\propto(t_{2}-t_{1})$ and $D\propto T$, with the latter fluctuation-dissipation proportionality arising from the Sutherland-Einstein-Smoluchowski relation. In addition to being advected, the pollen clusters will diffuse during the small time interval from $t_{1}$ to $t_{2}$, on account of Brownian motion associated with the position-dependent temperature of the fluid. The temperature distribution, together with the microscopically-anisotropic nature of the fluid film, imply locally-elliptical diffusion of the pollen-grain clusters over small timesteps $t_{2}-t_{1}$. This is analogous to the second and subsequent ‘anisotropic speckle diffusion’ terms on the right side of Eq. (3).

Here we list several possible avenues for future work.

1. 1.

   While the formalism and experiment of the present paper was developed in the specific context of x-ray radiation, its Fokker–Planck approach is more broadly applicable to directional dark-field speckle tracking using other parts of the electromagnetic spectrum such as already under study with visible light [85, 86]. Imaging techniques based on alternative particles providing different imaging contrasts like electrons or neutrons [87] could benefit as well from these developments to probe smaller or bulkier samples since one can readily imagine the possibility of developing randomly structured membranes with a broad range of characteristics.

2. 2.

   Following Refs. [61, 57], the symmetric rank-two diffusion tensor in Eq. (2) may be considered as the first member in an infinite hierarchy of progressively higher-rank tensors, which can be used to extend the speckle-tracking Fokker–Planck equation (Eq. (3)) into a Kramers–Moyal form [60]. Higher-order diffusion-tensor fields relate to progressively higher-order moments of the position-dependent SAXS fans emanating from each point on the exit surface of the sample. Reference [57] gives explicit expressions for these higher-order diffusion tensors. See, also, the earlier papers of Modregger et al. [88, 89], regarding the role of higher-order SAXS-fan moments in directional dark-field imaging.

3. 3.

   Explicit expressions have been derived for the local-SAXS contribution to the diffusion tensor in Eq. (2), together with its higher-order Kramers–Moyal generalizations [56]. It would be interesting to obtain corresponding edge-scatter-induced diffusion tensors. Recall, in this context, papers which show DDF arising from sample edges, with the semi-major axis of the diffuse-scatter ellipse being tangential to sample edges [53]. This edge-induced DDF signal may be formulated in at least three different ways: (i) Keller’s concept of diffracted rays in the geometric theory of diffraction [90]; (ii) the Young–Maggi–Rubinowicz boundary-diffraction wave [67, 68, 69, 70, 71]; (iii) critical points of the second kind, resulting from sharp sample edges in asymptotic approximations to diffraction integrals [91]. It is also worth noting, in the context of edge-induced diffuse scatter, that the parabolic equation of paraxial optics is a complex diffusion equation with purely imaginary diffusion coefficient. Another approach to better isolate and quantify the edge dark-field effect would be to utilize the approach in Groenendijk et al. [80] to remove the propagation-based phase contrast edge fringes from the sample image before applying the algorithm described here. Further computation work could look at whether the local stretching of a speckle due to a strong phase gradient can appear as a dark-field signal (see e.g. discussions in Morgan & Paganin [61]).

4. 4.

   Our two-dimensional directional-dark-field reconstructions could be extended to three-dimensional reconstructions, i.e. tensor dark-field tomography, in an analogous manner to what has already been achieved using periodic-grating methods [92, 93, 94, 95, 96, 97].

5. 5.

   Statistical dynamical diffraction theory [98, 99, 100, 101, 102] would form an interesting perspective from which one might extend the results of the present paper.

6. 6.

   The temporal-coherence requirements on x-ray speckle tracking are lax [103]. Hence our results might be extended to paraxial polychromatic radiation from sufficiently spatially coherent sources, although the model would likely need in that case to correct for some artifacts observed for instance in Ref. [104].

7. 7.

   As previously mentioned, XSVT and UMPA minimize suitable functionals, with MIST instead solving a particular partial differential equation. A link between all three speckle-tracking approaches might be explored by recalling the Lagrangian formulation of classical field theory [66]. Here, minimization of an action-integral functional leads directly to an associated partial-differential equation, namely the Euler–Lagrange equation. Similarly, minimization of the XSVT and UMPA functionals might lead to a partial differential equation for directional-dark-field x-ray speckle tracking.