# Fluorescent X-ray Scan Image Quality Prediction

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## Abstract

This paper summarizes approaches to image quality prediction in support of an effort under the IARPA RAVEN program to demonstrate a non-destructive, tabletop X-ray microscope for high-resolution 3D imaging of integrated circuits (ICs). The fluorescent X-rays are generated by scanning an electron beam along an appropriately patterned target layer placed in front of the sample and are then detected after passing through the sample by a high-resolution (in both solid angle and energy) backside sensor array. The images are created by way of a model-based tomographic inversion algorithm, with image resolution depending critically on the electron beam scan density and diversity of sample orientations. We derive image quality metrics that quantify the image point spread function and noise sensitivity for any proposed experiment design. Application of these metrics will guide final system design when physical data are not yet available.

## Keywords

Image quality X-ray tomography Scanning electron microscope (SEM) Transition edge sensor (TES) Integrated circuit (IC)## 1 Introduction

Image quality (IQ) assessment is an important element of system design, optimization, and quality control [1, 2]. A complete assessment evaluates the entire imaging chain including both data acquisition and image reconstruction stages. *Retrospective* IQ measurement is based on careful analysis of images taken in a series of laboratory measurements, often using specialized phantoms. In contrast, *prospective* image quality prediction is based on a high-quality end-to-end simulation of the system data product, and subsequent image reconstruction algorithm applied to that data. A key application is to guide system design in advance of hardware construction when physical data are not yet available. Similarly, it allows one to tune or customize an existing system to a particular set of specialized tasks under conditions where the retrospective approach is time-consuming or infeasible for other reasons (e.g., target sensitivity to, or potential degradation under, multiple measurements).

Given an accurate system forward model, IQ assessment is relatively straightforward in cases of “direct” image reconstruction algorithms, in which a relatively simple (though perhaps numerically intensive) forward algorithm is applied to the data to obtain an image. An example is back projection image formation, which is also linear in the data. Various Fourier transform methods (e.g., SAR or range-Doppler processing) are special cases. Construction of various IQ metrics, such as point spread functions, are equally direct.

This paper is intended to provide a detailed theoretical basis for a general set of IQ metrics. Specific illustrative applications to the RAVEN IC tomography problem will be presented separately.

## 2 System Forward Model

In the X-ray microscope system (Fig. 1), the collected fluorescent photon count data forms the basis for a tomographic inversion of the sample structure. We begin by accounting for the physics and geometry of the measurement, assuming ideal sample stage (translation and rotation) operation, perfect knowledge of the electron beam location, and ideal photon detector behavior. Later, we will include effects of sample stage and electron beam uncertainty, as well as non-ideal detector effects such as pixel cross-talk and noise properties.

### 2.1 Measurement Physics

In what follows, we choose a 3D coordinate system that is fixed in the sample. Of course, it is generally the sample that is translated and rotated while the electron beam and detector apparatus remain fixed. However, it is the detailed sample structure that is the desired end-product of the tomographic inversion, and these take the form of fixed functions of position within the sample volume. In this frame, the relative positions and orientations of the source, receiver, and electron beam must be carefully tracked for each measurement.

**x**

_{S}and point receiver

**x**

_{R}, the mean count rate for a chosen fluorescent photon with sharp energy

*E*is modeled in the form

*n*

_{0}is the raw fluorescent photon production rate,

*n*

_{B}is the background rate (due to bremsstrahlung, multiple scattering, etc.) of continuum photons deposited in the energy bin Δ

*E*about

*E*. The first term represents the photon intensity derived from the physical optics approximate solution to the 3D wave equation, including geometric spreading and energy dissipation. For the latter, fluorescent photons are absorbed (or scattered to lower energy) according to the line integral (valid in the weak absorption limit characteristic of higher energy X-rays)

*s*≤|

**x**

_{R}−

**x**

_{S}|, connecting the two endpoints. The function

*μ*(

**x**,

*E*) is the local absorption rate as a function of 3D location

**x**, commonly modeled as a linear superposition

*A*ranges over atomic elements (Au, Al, Si, Cu, etc.) and

*n*

_{A}(

**x**) is the element number density.

_{S}is normalized to unit integral in the first argument, and is now explicitly parameterized as well by the electron beam energy

*E*(through fluorescent production efficiency, electron multiple scattering, etc.), the beam center point \(\textbf {x}_{S}^{0}\), and the incident direction \(\hat {\textbf {e}}_{S}^{0}\) of that beam. Thus,

_{S}to define an average of any function

*F*(

**x**

_{S},

**x**

_{R}) of the source and receiver points by

*A*

_{R}. The result is defined by the entry center point \(\textbf {x}_{S}^{0}\) of the electron beam on the target surface, the receiver center coordinate \(\textbf {x}_{R}^{0}\), and the corresponding mean incident X-ray flux direction \(\hat {\textbf {e}}_{{SR}}^{0} = \frac {\textbf {x}_{R}^{0} - \textbf {x}_{S}^{0}}{|\textbf {x}_{R}^{0} - \textbf {x}_{S}^{0}|}\). The receiver area integral is normalized by the total projected area \(\textbf {A}_{R} \cdot \hat {\textbf {e}}_{{SR}}^{0}\) along the incident X-ray flux direction. More generally, one could also incorporate a receiver sensitivity profile that is not uniform over the pixel.

*E*, which can be fit and subtracted directly from the data. Although in principle present, any residual dependence of the background on

*is neglected.*

**μ**The function Φ_{S} encompasses the entire fluorescent X-ray creation process, including incident electron beam shape and energy spectrum, fluorescing layer thickness, and orientation. All of these need to be included in the system model. For a horizontally homogeneous target, the extra dependence of Φ_{S} on \(\textbf {x}_{S}^{0}\) drops out (with only the difference variable \(\textbf {x}_{S} - \textbf {x}_{S}^{0}\) surviving). However, target inhomogeneity will in general be present (e.g., through varying deposited film thickness).

The source region may be crudely thought of as a cylinder with axis oriented along \(\hat {\textbf {e}}_{S}^{0}\) and diameter determined by electron beam scattering characteristics. More generally, Φ_{S} would either need to be determined via prior calibration measurements in the absence of the sample, or modeled through Monte Carlo simulations of the electron beam, with given energy and geometry, interacting with the target with given atomic composition and geometry [4]. Clearly, detailed calibration measurements would be most desirable, but these are unlikely to be available given that the target film is most likely directly applied to the sample in our setup (Fig. 1). In absence of this, an effort should be made to produce as uniform a film as possible. With sufficient measurement diversity, it turns out that although it is essential that the electron beam position be tracked to high precision, the source intensity (hence film thickness) may to a certain extent be included as part of the inversion. Thus, given a sufficiently large combination of source and receiver pixel positions, there may be sufficient redundancy in the geometry of rays passing through the sample to refine the estimate for the amplitude \(\bar n_{0}\) in Eq. 8 based on the sample-present measurements only (see, e.g., [5] and references therein).

Of course, for tomographic purposes, the critical dependence of Eq. 8 is on the variation of *M* with the precise fluorescent photon path, which could be quite complex due to discontinuities of *μ* across sample constituent boundaries. As alluded to above, the coordinate system is defined by the fixed function *μ*(**x**), while all other parameters, including the detailed geometry of the function Φ_{S}, vary with the measurement.

The aim is to reconstruct *μ*(**x**) from an appropriately large set of line integral estimates \(\{M(\textbf {x}_{S}^{0},\textbf {x}_{R}^{0};E) \}\). These are extracted from \(n_{\text {meas}}(\textbf {x}_{S}^{0},\textbf {x}_{R}^{0};E)\), after subtraction of the continuous background, and properly accounting for photon Poisson statistics in the low count regime (see Section 3).

### 2.2 Simplified form

*n*

_{0}(

**x**

_{S},

*E*) source photons make it through the sample, generating a correspondingly reasonable fluorescence count rate, distinguishable from the background. Thus,

*M*(

**x**

_{S},

**x**

_{R};

*E*) =

*O*(1), and we further assume that, for most source-receiver pairs \((\textbf {x}_{S}^{0},\textbf {x}_{R}^{0})\), the variation of

*M*with \(\textbf {x}_{S} - \textbf {x}_{S}^{0}\) and \(\textbf {x}_{R} - \textbf {x}_{R}^{0}\) is weak:

*δ*

*M*

^{2}〉

_{0}are neglected. Equation 13 conveniently allows one to treat the measured photon count as a single Poisson process involving a “cone volume” average of rays, rather than a complex superposition of Poisson processes for each individual ray in the cone. This greatly simplifies the inversion scheme. For an actual microchip, there will be structure on many length scales, requiring a multi-scale approach to estimate

*μ*(or, more specifically, the atomic densities

*n*

_{A}) constrained by available prior information.

### 2.3 Measurement Imperfections

In the above discussion, we have assumed perfect knowledge of the source positions **x**_{S} and the associated function Φ_{S}. These will be impacted by electron beam wandering and by sample stage translation and rotation uncertainties. We have also not yet accounted for detector uncertainties, such as pixel cross-talk and multiple photon absorption within a detection time window, that will generate, respectively, photon count and energy estimate errors.

For typical CT scans which use scintillator detectors, rather than being directly absorbed the X-ray photon creates a cascade of visible light photons that are then converted to an electronic signal. Scattering of the photons between pixels can be an important blurring effect.

*E*, would be less than

*E*, hence would appear as part of the background Bremsstrahlung. This leads to a fluorescent X-ray undercount rather than a spreading of the count [2]. Such processes might then be modeled in the form [1, 2]

*ε*by a matrix

**B**that mixes the count among nearby pixels [2].

## 3 Tomographic Inversion

*μ*(

**x**) is based on minimization of an appropriate objective function

*now represents the absorption function values over an*

**μ***M*element sample grid, and the array

**n**represents the (integer photon count) data. The function Φ takes a penalized likelihood (PL) form

*L*, quantifying the photon count Poisson statistics, and regularization term

*R*which encompasses all sample prior knowledge. The latter includes prior constraints on the decomposition (4), such as material stepwise uniformity (with sharp, flat interfaces), metal interconnect geometry, and expected metal types in various sample regions. The relative weight

*β*controls the balance between data fidelity and regularization and its optimal choice must be part of an investigation using various known test samples.

*τ*

_{l}is the dwell time, and we define the following:

*-dependence targeted by the inversion. For simplicity, additional dependence on beam incident direction \(\hat {\textbf {e}}_{S}\), beam geometry, etc., has been suppressed from the notation, and extensions to problems where \(\bar n_{0}\) is also uncertain [5] will not be treated here.*

**μ***n*

_{l}being the actual measured (integer) photon count. Recall here that the dependence on

*resides, via (1) and (2), in the mean count \(\bar n_{l} = \bar n_{l}[{\boldsymbol {\mu }}]\).*

**μ**### 3.1 Simplified Form Likelihood

*-dependence of \(\bar n_{l}\) is complicated (being a superposition of exponentials of linear functionals of*

**μ***), but with the approximation (10), Eq. (13) allows one to simplify (17) to the form*

**μ***now enters through the*

**μ***single*linear functional \(\bar x_{l} = \bar x_{l}[{\boldsymbol {\mu }}]\). Explicitly, upon discretizing the sample one may write an individual line integral in the form

*w*

_{j}(

**x**

_{S},

**x**

_{R}) is the length of the intersection of the line segment connecting

**x**

_{S}to

**x**

_{R}with voxel

*j*. It follows that one may write

*l*(characterized by source spot center \(\textbf {x}_{S,l}^{0}\) and receiver pixel center \(\textbf {x}_{R,l}^{0}\)).

*x*if

*r*

_{l}> 0—unfortunately ruling out certain numerically efficient objective function minimization algorithms. The result (25) simply states that the most likely absorption model

*x*is such that the measured photon count is precisely the expected count: \(n_{l} = \bar n_{l}(x)\). Except for statistical outliers, the measured photon count will typically lie in the range

*b*

_{l}+

*r*

_{l}>

*n*

_{l}>

*r*

_{l}, yielding the physically required positive minimum,

*x*> 0.

*b*

_{l},

*r*

_{l},

**x**

_{S, l},

**x**

_{R, l}}, along with the associated fluorescence weighting functions {Φ

_{S, l}} (which depend on the measurement through the local fluorescent layer structure and orientation) are accurately known, the blur-aware model should produce an unblurred image from the blurred data, whereas the mismatched model will produce a blurred image from blurred data.

## 4 Image Quality Metrics: Smooth Reconstructions

We consider first tomographic reconstructions based on smoothly varying targets, controlled by *L*^{2}-type regularizations. This rules out piecewise constant type prior constraints on which (4) is based. We will generalize the theory to include the latter in subsequent sections.

### 4.1 Spatial Resolution Metrics

In what follows, we assume the existence of a rapidly converging algorithm that produces \(\hat {\boldsymbol {\mu }}[\textbf {n}]\) for given fixed data array **n** and underlying statistical signal forward model defined by Eqs. 17–20.

*not*\(\textbf {n} = \bar {\textbf {n}}(\hat {\boldsymbol {\mu }})\), which would entail a minimization that includes the

*-dependence of \(\bar {\textbf {n}}\)). Since*

**μ***n*

_{i}appears linearly in the

*-dependent terms in Eq. 20 (the*

**μ***n*

_{i}! term is a trivial normalization that plays no role in the minimization (15)), the evaluation of Eq. 27 for continuous \(\bar {\textbf {n}}\) is straightforward.

*. An obvious measure of image quality is therefore how closely the former reproduces the latter. A way to quantify this is through the set of point spread functions*

**μ***j*. These measure the change in the estimated \(\hat \mu \) associated with a change in the actual

*μ*(under the assumption of noise-free data). In a perfect world, \(\hat {\boldsymbol {\mu }} = {\boldsymbol {\mu }}\), and one obtains [

**P**

_{j}]

_{i}=

*δ*

_{ij}—the inversion result precisely tracks the input value. More realistically, the support of

**P**

_{j}will spread to nearby voxels, and optimal measurement design is aimed at minimizing this spread.

**x**

_{j}, with (b) a matrix quantifying the sensitivity of the inversion to changes in the data. Using Eqs. 17 and 18, one obtains for term (a):

*j*within the support of the average (24).

*j*. Resolution will improve as one increases the number of line integrals passing through

**x**

_{j}from as diverse a set of directions as possible. An explicit expression is obtained from the minimum condition

*δ*

*= 0,*

**μ***δ*

**n**= 0. We emphasize here again that ∇

_{μ}operates only on the first argument of Φ, not on the implicit

*-dependence of \(\bar {\textbf {n}}\). To linear order, one obtains*

**μ***δ*

*= 0,*

**μ***δ*

**n**= 0. The smooth reconstruction assumption being made in this section is equivalent here to the existence of the derivatives in Eq. 31, and as a consequence one obtains

_{μ}∇

_{μ}Φ

_{0}is a square matrix, hence with nominally well-defined inverse, while ∇

_{μ}∇

_{n}Φ

_{0}is in general not square. From Eqs. 16–20, one obtains

*l*, while

**R**that defines a smooth reconstruction [1, 2]. A typical choice is a quadratic form for

*R*, giving rise to a constant, positive definite matrix

**R**. The mean photon count second derivatives are given by

*x*

_{l}in Eqs. 29 and 35.

*. In this way, under conditions where the image is expected to be reasonably accurate, correspondingly accurate image quality metrics may be derived from the forward model alone, without actually needing to first derive the tomographic inversion \(\hat {\boldsymbol {\mu }}\).*

**μ**### 4.2 Local Noise Metrics

*spreads to the neighborhood of the corresponding source-receiver ray.*

**μ**### 4.3 Simplified Form Image Quality Metrics

*l*to voxels

*i*,

*j*and the overall expected non-background photon count \((\bar n_{l} - r_{l})\).

Once again, with high-quality data, one may use \(\bar n_{l}[\hat {\boldsymbol {\mu }}]\) interchangeably with \(\bar n_{l}[{\boldsymbol {\mu }}]\), avoiding the need to first reconstruct \(\hat {\boldsymbol {\mu }}\). Moreover, if the photon counts are sufficiently high, one will have \(n_{l}/\bar n_{l} \approx 1\), and the quality metrics may then be computed directly from the data with the replacement \(\bar {\textbf {n}} \to \textbf {n}\) [2]: the simplified forms require only the blur-averaged geometrical parameters \(\bar {\textbf {w}}_{l}\), measured photon counts *n*_{l}, and data-derived background counts *r*_{l} without any direct reference to * μ* or \(\hat {\boldsymbol {\mu }}\).

Correspondingly, from a simulation point of view, one may investigate the quality of a proposed experimental setup using the forward model alone to generate the values \(\bar {\textbf {n}}({\boldsymbol {\mu }})\) (along with background count estimates) from a given target model * μ*. One may either use these values for

**n**, or add further realism by generating simulated Poisson-distributed counts from these computed average counts. One is not required to go through the more complicated inversion step of first deriving \(\hat {\boldsymbol {\mu }}[\bar {\textbf {n}}({\boldsymbol {\mu }})]\) from

*.*

**μ**### 4.4 Numerical Considerations

*r*

_{0},

*A*

_{ij}→ 0 for |

**x**

_{i}−

**x**

_{j}| >

*r*

_{0}, while essentially constant over this same range in the second argument.

**x**, consider a box of diameter 2

*r*

_{0}, and define the local Fourier transforms

*M*is the number of points in the box, and the prime on the sums indicates restriction to the corresponding box centered at the origin. The wave vector

**q**ranges over the box domain reciprocal lattice, and describes the more rapid variation of

*A*and

*B*on scales smaller than

*r*

_{0}. Note that since \(\tilde A\) is short range in its first argument, the sum restriction is essentially redundant.

**x**

_{i}−

**x**

_{j}dependence of the second argument of \(\tilde A\) and thereby succeeded in proving a local version of the convolution theorem. The matrix inversion now reduces to an algebraic division of both sides by \(\hat A\) and a subsequent inverse Fourier transform, and one obtains the desired result

**x**. The global solution is obtained by varying

**x**over the appropriate global set of box centers.

#### 4.4.1 Consistency Conditions

It remains to discuss the conditions under which one indeed expects \(\tilde A\) to obey the desired conditions.

First, the regularization term *R* is typically local, e.g., depending on near-neighbor differences *μ*(**x**_{i}) − *μ*(**x**_{j}). Often a quadratic regularization is used [2], in which case **R** is a constant, near-diagonal matrix (or perhaps a slowly varying set of near-diagonal quadratic coefficients, depending on prior knowledge of the target), and the desired conditions indeed hold because such a regularization biases the solution toward relatively smooth \(\hat \mu (\textbf {x})\).

Similarly, if the bias is toward smooth, slowly varying \(\hat {\boldsymbol {\mu }}(\textbf {x})\), the likelihood term (19) will be a smooth function of the mean counts \(\bar {\textbf {n}}(\hat {\boldsymbol {\mu }})\). However, each independent count \(\bar n_{l}\) depends on a narrow cylinder of voxels connecting source and receiver and is hence strongly nonlocal. In particular, the second derivative (35) will be nonzero for any pair **x**_{i}, **x**_{j} lying within the cylinder defined by \(\bar n_{l}\).

On the other hand, the sum over *l* in Eq. 19 contains many cylinders, and one expects high-resolution images to emerge only when each voxel is intersected by many cylinders. In this case for widely separated **x**_{i}, **x**_{j}, one expects a small number of cylinders to pass through both, and a correspondingly small number of terms will contribute to the *l*-sum in the first equalities in Eqs. 36 or 37. In contrast, for **x**_{i}, **x**_{j} within a cylinder diameter, many terms will contribute. It follows that *∂*^{2}*L*/*∂**μ*_{i}*∂**μ*_{j} will be strongly peaked about the diagonal **x**_{i} = **x**_{j}, and the locality property emerges.

## 5 Generalized Image Quality Metrics

We next consider more general classes of regularization terms *R*. The structure of *L* remains the same as before, and we continue to assume that many *l*-cylinders pass through each voxel. Thus, *∂*^{2}*L*/*∂**μ*_{i}*∂**μ*_{j} continues to be near diagonal. More problematical is the structure (4) in which the densities *n*_{A}(**x**) are piecewise slowly varying, but with sharp interfaces between, further biased, e.g., toward metal interconnect rectilinear geometry. The regularization term will be relatively agnostic to the position of the interface, but sharp interfaces dictate an *L*^{1} rather than *L*^{2} regularization. The second derivative of *R* will therefore be very singular when evaluated at \(\hat \mu (\textbf {x})\). Moreover, strong rectilinear constraints may lead to global shifts in an interconnect position with small change in **n**. For example, for cylinders aligned along a preferred interconnect direction, the value of *n*_{l} will have a large jump as the cylinder crosses from one side of a metal interface to the other.

### 5.1 Illustrative Regularization

*K*different materials/compounds with absorption \(\{\mu (\kappa ) \}_{\kappa = 1}^{K}\). Each target voxel is then assigned a material index

*M*

_{i}∈{1,2,…,

*K*} with absorption

*μ*

_{i}=

*μ*(

*M*

_{i}). A simple regularization term might take the Potts model form

*h*

_{κ}are used to control the fractional area of each material type, and the coupling constants

*J*

_{κ}> 0 favor nearest neighbor voxels 〈

*i*,

*j*〉 being of the same material type. Both could also be made slow functions of position in order to encode prior information on material types in different regions of the target.

*a*,

*b*,

*c*,

*d*are counter-clockwise (starting from the first quadrant) labels of a 2 × 2 plaquette centered on some point

*i*, with corresponding material labels

*M*

_{a}(

*i*),

*M*

_{b}(

*i*),

*M*

_{c}(

*i*),

*M*

_{d}(

*i*), then the terms in

*a*,

*b*are both the same material but different from

*c*,

*d*, or vice versa, hence favoring horizontal interfaces. Additional terms, for example, could be introduced favoring certain interface types (e.g., metal–insulator).

### 5.2 Properly Designed Image Quality Metrics

*L*against this alternative type of regularization term, characteristic material solutions \(\hat {\textbf {M}}\) will emerge with certain types of enforced geometries (e.g., piecewise constant, preferred shape and orientation)—notional geometries are illustrated in Fig. 2. The size of the coefficients

*h*

_{κ},

*J*

_{κ},

*L*

_{κ},… are chosen to emphasize different target features in proportion to one’s degree of confidence in such prior information. Of course, the more high-quality data one has (the larger

*L*is relative to

*R*), the less impact these coefficients will have. They are most useful under data-starved conditions in which

*R*is able to resolve ambiguities in

*L*in favor of the desired geometry. For example, some violations of geometry rules may in fact be real (due to target imperfections, unrecognized manufacturing specifications, etc.), and sufficiently high-quality data must be permitted to resolve the discrepancy in favor of

*L*rather than

*R*. One must therefore ensure that the coefficients in

*R*are chosen in such a way that the regularization does not completely dominate the data. Thus, the difference in the likelihood terms, \(|L[{\boldsymbol {\mu }}(\hat {\textbf {M}}_{0}), \bar {\textbf {n}}_{0}] - L[{\boldsymbol {\mu }}(\textbf {M}_{0}), \bar {\textbf {n}}_{0}]|\) should be small: the adjustments in

**M**

_{0}that significantly reduce

*R*should remain consistent with the data.

The results here are therefore quite different from the continuously varying solutions favored by the *L*^{2}-type regularizations discussed in Section 4. In particular, it no longer makes sense to define the objective function Φ as a continuous function of * μ*, and to subsequently formulate the sensitivity of the minimum \(\hat {\boldsymbol {\mu }}\) in terms of gradients—see Eqs. 28 and 30–32. The regularization smoothing process is unlikely to be sensitive to changes limited to a single site.

*, as in Eq. 28, one should probe the effects of more general changes consistent with*

**μ***R*. Specifically, let

**M**

_{0}be a representative physically consistent material geometry (in the sense that

*R*(

**M**

_{0}) is small), with corresponding mean photon counts \(\bar n_{0} = \bar n[{\boldsymbol {\mu }}(\textbf {M}_{0})]\), and let

*R*. If the data is so starved that such changes have too small an effect on

*L*, then the tomography experiment cannot be expected to yield robust results. To quantify this, consider the class of changes

*δ*

**M**

_{0}such that \(|R(\hat {\textbf {M}}_{0} + \delta \textbf {M}_{0}) - R(\hat {\textbf {M}}_{0})|\) is small. We then demand that

*L*be able sense such changes—the measurement design must be capable of disambiguating nearby target geometries that are both consistent with

*R*. Defining the reconstruction change \(\delta \hat {\textbf {M}}_{0}\) by

*δ*

**M**

_{0}(also in the sense of differing on a small number of target pixels). In the present case,

*δ*

**M**

_{0}might correspond to a small, smooth translation of a material boundary, and the desire would be that \(\delta \hat {\textbf {M}}_{0}\) substantially matches this change.

#### 5.2.1 Forward-Model-Only Approximate Formulation

As discussed below (37), it would be desirable to derive reasonably accurate image quality metrics from the forward model alone. Thus, at the end of Section 4.1, we succeeded in estimating the key second derivative matrices appearing in Eq. 32 in terms of the forward photon data \(\bar {\textbf {n}}({\boldsymbol {\mu }})\) alone without reference to the optimal model \(\hat {\boldsymbol {\mu }}\). Here, the presence of *R*(**M**), which is deliberately sensitive to the discrete nature of **M** in order to help select models with desirable piecewise constant geometries, is a complication because the assumption of continuous second derivatives no longer make sense.

An alternative approach taken here is to consider only models **M**_{0} consistent with the geometrical constraints, hence with relatively small values of *R*[**M**_{0}], and to consider only perturbations *δ***M**_{0} such that *R*(**M**_{0} + *δ***M**_{0}) − *R*(**M**_{0}) is very small, i.e., such that the change in the tomographic estimate \(\delta \hat {\textbf {M}}_{0}\) is driven by the change \(\bar {\textbf {n}}[{\boldsymbol {\mu }}(\textbf {M}_{0} + \delta \hat {\textbf {M}}_{0})] - \bar {\textbf {n}}[{\boldsymbol {\mu }}(\textbf {M}_{0})]\) in the photon count data. Thus, we suppose that the measurement protocol is such that the change in Φ is dominated by the change in *L*, which remains continuous in * μ*.

*(both appearing, via (19), through their respective mean photon counts \(\bar {\textbf {n}}(\hat {\boldsymbol {\mu }})\), \(\bar {\textbf {n}}({\boldsymbol {\mu }})\)). One notes as well that a model perturbation*

**μ****M**

_{0}→

**M**

_{0}+

*δ*

**M**generates an absorption field change

*δ*

**M**. With this notation, the minimum condition for the reconstruction \(\hat {\boldsymbol {\mu }}_{0}({\boldsymbol {\mu }}_{0})\) now takes the form

*, and that one expects only small changes if the support of*

**μ***δ*

*is small (except for singular cases, which we neglect, where the support of a long narrow change happens to line up exactly with a ray trajectory), one may approximate (59) by*

**μ**_{1,2}will refer to

*-derivatives with respect to the first and second arguments.*

**μ**

**μ**_{0}→

**μ**_{0}+

*δ*

**μ**_{0}to the true profile, the generalization of the minimum condition (31) is

*δ*

**μ**_{0}. The second derivatives of \(\tilde L\) are computed as in Section 4.1. As observed below (37), they may be expressed entirely in terms of the forward model parameters alone—one need not perform the tomographic inversion in order to evaluate the image quality.

*δ*

**μ**_{0}from the same set. Thus, if \(\{\delta {\boldsymbol {\mu }}_{a} \}_{a = 1}^{L}\) are the permitted common set of small

*R*perturbations, and one defines the

*L*×

*L*matrices

*a*, and for some value

*c*for each choice of

*b*. The use of “\(\simeq \)” here indicates that perfect equality is unlikely to be achieved when only discrete values of the absorption on a discrete lattice are permitted (compared to the continuous values permitted in the corresponding matrix (31)). Instead, one will likely only obtain approximate correspondence, e.g., of displacements of various interfaces between Potts domains of uniform material. For a perfect measurement, one expects

*c*=

*b*, but in general, there will be an imperfect correspondence between the true and reconstructed perturbations.

## Notes

### Acknowledgments

This material is based upon work supported by The United States Air Force and Air Force Research Laboratory under Contract FA8650-17-C-9114.

### Compliance with Ethical Standards

### **Dislcaimer**

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the view of The United States Air Force and Air Force Research Laboratory.

## References

- 1.Schmitt SM, Goodsitt MM, Fessler JA (2017) Fast variance prediction for iteratively reconstructed CT images with locally quadratic regularization. IEEE Trans Med Imaging 36(1):17–26CrossRefGoogle Scholar
- 2.Wang W, Gang GJ, Siewerdsen JH, Stayman JW (2019) Predicting image properties in penalized-likelihood reconstructions of flat-panel CBCT. Med Phys 46(1):65–80CrossRefGoogle Scholar
- 3.Irwin KD, Hilton GC (2005) Transition-edge sensors Cryogenic Particle Detection. Springer, pp 81–97Google Scholar
- 4.Salvat F (2015) Penelope-2014: A code system for Monte Carlo simulation of electron and photon transport. In: OECD Workshop Proceedings, BarcelonaGoogle Scholar
- 5.Rezaei A, Defrise M, Nuyts J (2014) ML-Reconstruction for TOF-PET with simultaneous estimation of the attenuation factors. IEEE Trans Med Imaging 33:1563–1572CrossRefGoogle Scholar
- 6.Ullom JN, Bennett DA (2015) Review of superconducting transition-edge sensors for x-ray and gamma-ray spectroscopy. Supercond Sci Technol 28(8):84003–84038CrossRefGoogle Scholar

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