The ability to visualize x-ray and neutron emission from fusion plasmas in 3D is critical to understand the origin of the complex shapes of the plasmas in experiments. Unfortunately, this remains challenging in experiments that study a fusion concept known as Magnetized Liner Inertial Fusion (MagLIF) due to a small number of available diagnostic views. Here, we present a basis function-expansion approach to reconstruct MagLIF stagnation plasmas from a sparse set of x-ray emission images. A set of natural basis functions is “learned” from training volumes containing quasi-helical structures whose projections are qualitatively similar to those observed in experimental images. Tests on several known volumes demonstrate that the learned basis outperforms both a cylindrical harmonic basis and a simple voxel basis with additional regularization, according to several metrics. Two-view reconstructions with the learned basis can estimate emission volumes to within 11% and those with three views recover morphology to a high degree of accuracy. The technique is applied to experimental data, producing the first 3D reconstruction of a MagLIF stagnation column from multiple views, providing additional indications of liner instabilities imprinting onto the emitting plasma.

Three-dimensional reconstructions of x-ray and neutron volumes have proven invaluable for understanding 3D structure in inertial confinement fusion (ICF) plasmas.1–7 However, ICF experiments on large facilities such as the National Ignition Facility (NIF), OMEGA Laser, and the Z Pulsed Power Facility suffer from severe spatial constraints, where often only two or three 2D projection images of x-ray and/or neutron self-emission are available. Reconstruction of a 3D profile from such a sparse data set is a highly ill-posed inverse problem, requiring additional prior information about the fusion plasmas to obtain a reliable, unique solution.

Here, we consider sparse-view tomographic reconstruction of the quasi-cylindrical ICF plasmas produced in Magnetized-Liner Inertial Fusion (MagLIF) experiments.8,9 In MagLIF, preheated, pre-magnetized fusion fuel is compressed by a surrounding, metallic cylinder imploding radially inward under the J z × B θ force generated by a large ( > 15 MA) electrical current [Fig. 1(a)]. As the fuel compresses, it eventually stagnates in a narrow column roughly 100 μm wide by 6–8 mm in height.10 Time-integrated, high-resolution images of x-ray self-emission using spherical crystal x-ray microscopes11 have shown highly 3D stagnation columns, exhibiting, e.g., apparent helical structures, as seen in Figs. 1(b) and 1(c). The 3D structures may originate from several processes: Magneto-Rayleigh-Taylor instabilities (MRTIs) on the exterior of the liner can grow to large amplitudes and lead to nonuniform compression.12 Feedthrough from the MRTI and other processes13,14 can generate non-uniform mix of liner material into the fuel, resulting in overwhelming radiative losses that degrade fusion yield. Diagnosing 3D structure in both the fuel and the surrounding liner material may help quantify the impact of instabilities and mix on target performance and inform new target designs to reduce their impact.

FIG. 1.

(a) Diagram of the MagLIF concept, showing a stagnating D 2 plasma surrounded by a thick beryllium liner. (b) and (c) Quasi-orthogonal stagnation images at azimuthal angles θ = 73 ° and 33 °, respectively, created by spherical crystal imagers on Z shot z3719.

FIG. 1.

(a) Diagram of the MagLIF concept, showing a stagnating D 2 plasma surrounded by a thick beryllium liner. (b) and (c) Quasi-orthogonal stagnation images at azimuthal angles θ = 73 ° and 33 °, respectively, created by spherical crystal imagers on Z shot z3719.

Close modal

While 2D x-ray projection images of stagnation emission at two near-orthogonal views and three 120 °-spaced views have recently been obtained on MagLIF experiments, these still represent highly incomplete data sets. Prior information can be added to the sparse projection measurements in a variety of ways. These can include the enforcement of positive solutions (to be physically meaningful in the case of self-emission) and penalizing those that are not sufficiently smooth.2,15 Additional approaches come in the form of parameterized models,16 strongly encoded symmetry assumptions,17 and finite-order basis expansions that impose geometric constraints on the solution space.3,7,18,19 Unfortunately, no single approach is suitable for all problems. For example, the high-aspect ratio (height width) MagLIF stagnation columns would be poorly represented by spherical harmonic basis functions3 compared to the spherical hot-spots in laser-based ICF.

In this paper, we introduce a novel 3D basis, “learned” from a set of quasi-helical training volumes to reconstruct MagLIF stagnation volumes from an extremely sparse set of projection data—as few as two or three azimuthally spaced views. The learned basis represents MagLIF volumes more parsimoniously compared to a cylindrical harmonic basis and a simple Cartesian voxel basis. Reconstructions with the learned basis outperform the other two bases according to several metrics. In some cases, our approach even provides equivalent or superior performance with two views to that from the other bases with three views. Our reconstructions from two orthogonal views using the learned basis can estimate emission volumes to < 11 % and tests with three views show excellent reproduction of emission morphology, insensitive to the choice of data. The technique is applied to a pair of quasi-orthogonal stagnation images from a MagLIF experiment, showing the first instance of a 3D reconstruction of a MagLIF stagnation column from multiple views. A key result is that reconstructed column shows a dominant helical handedness consistent with that expected on the exterior of the liner from MRTI. While the proposed technique of a learned basis for 3D reconstruction is developed and discussed in the context of MagLIF, the approach is general: given appropriate training data, it can be adapted to other tomography problems, such as reconstruction of x-ray and neutron emission volumes in laser-driven ICF or other plasmas.3,4,20,21

We focus on reconstructing 6-keV continuum self-emission generated by free-bound and free-free electronic transitions in a fully ionized, mostly deuterium plasma. This emission is expected to be optically thin (negligible self-reabsorption) and isotropic within the emitting plasma itself for the ion densities 10 23/cc obtained in MagLIF.22 Therefore, self-emission x-ray images g ( x ; ^ ) represent projections of a 3D source of x-rays f ( r ) (e.g., number of x-rays emitting into 4 π per unit volume) along a line-of-sight coordinate ^ and may be computed as
g ( x ; ^ ) = f ( r ) d .
(1)
Here, r is the source coordinate, x is the projection or image coordinate, and ^ is the line-of-sight direction vector. We ignore attenuation of the x-rays through any non-emitting material along the line-of-sight, namely, the exterior Be liner, which would appear as an additional x-dependent extinction factor outside the integral. While non-negligible in current imaging modalities, we provide an approach for removing this factor’s effect under certain assumptions in Sec. V.

In this work, f ( r ) is time-integrated since images are currently recorded on image plate detectors. Furthermore, since spherical crystal imaging is inherently narrowband (bandwidth 10 eV), we consider the emission to be from a fixed energy (or a superposition of energies from multiple-order crystal reflections).

Because images are ultimately recorded in the form of discrete pixels, we rewrite Eq. (1) for a given LOS ^ v in discretized form,
g i v = j N p i , j v f j .
(2)
In the above, g i v = g ( x i , ^ v ) is the i th pixel of the projection, f j is the j th voxel of the source with value f ( r j ), and p i , j v is the path length through voxel f j along ^ v at projection coordinate x i. This can be written more concisely as a matrix equation,
g v = P v f ,
(3)
where f = { f j , j = 1 , , N } is a 1D array of source voxels, g v = { g i v , i = 1 , , N p } is a 1D array of image pixel values, and P v is the N p × N projection matrix for line-of-sight v containing elements p i , j v. When N v 2 views are present, the entire set of measurements can still be described as a single matrix equation,
g = P f ,
(4)
where g = [ g 1 g v g N v ] R N v N p × 1 and P = [ P 1 P v P N v ] R N v N p × N are stacked versions of g v and P v in Eq. (3). Here, denotes the matrix transpose operation. In reality, the measurements will contain some finite amount of noise described by a vector ϵ R N v N p × 1, where in the case of additive Gaussian noise,
g = P f + ϵ .
(5)
In the above model, we ignore the finite spatial resolution of the imaging system, aside from the discretized nature of the detector; we simply reconstruct objects with feature sizes limited to the spatial resolution of the imaging sytem, which is 15–60  μm in the case of spherical crystal imaging on the Z Machine.11 

Tomographic reconstruction consists of inverting Eq. (5) for f, which can be stated as the minimization WRT f of an objective function L ( g , f ) describing how well the projections of f match the actual measurements. This inverse problem is severely ill-posed in the case of sparse tomography, where projection data are only available from a small number of views whose information content does not sufficiently describe the original object. Because N N v N p, the solution is non-unique and sensitive to which projection data are used. The sparse sampling in projection space means that arbitrary functions can be added to the reconstruction whose projections at the specified lines of sight are zero (in the null space of P).23,24 In practice, this can lead to amplification of high-frequency noise and artifacts in the reconstruction.25 Obtaining a reliable solution, therefore, requires incorporation of prior information as additional constraints.

Prior information and constraints may be added without loss of generality in the acceptable solution space. For instance, the reconstructed emissivity profile must be non-negative to be physically valid. Furthermore, smooth solutions may be encouraged in cases where large gradients and high-frequency features are not expected or when inference of global structure is desired, often achieved by adding terms to L ( g , f ) that penalize large gradients in f.15,25 Additional physically relevant constraints may be applied, but require expert knowledge of the system under study. Furthermore, all the above rely on some level of user-input to judge how strongly each constraint should be imposed.

A common approach to creating a more well-posed problem in sparse view tomographic reconstruction is to assume that the object or underlying “signal” can be represented accurately using a relatively sparse basis expansion,18,
f ( r ) = b = 1 B a b d b ( r ) ,
(6)
where B N. In discretized form, Eq. (6) becomes a simple matrix equation,
f = D a ,
(7)
where the columns of the “dictionary” matrix D R N × B are made up of the basis functions or “atoms” d b = { d b ( r j ) , j = 1 , , N } evaluated at all voxel positions r j, and a = { a b } is a vector of basis coefficients. (We note that a simple voxel basis is just the case where D = I is the N × N identity matrix.) This object representation can reduce the number of unknowns by several orders of magnitude from a large number of voxel intensity values (e.g., 10 4 to > 10 5) to perhaps several hundred basis coefficients. A proper basis can also help constrain the reconstruction by imposing specific geometries and symmetries on the problem. The basis functions can also be used to limit the range of spatial frequencies in the solution; i.e., smooth basis functions will lead to smooth reconstructions without the need to explicitly penalize large gradients in f. Alternatively, a well-constructed sparse basis can recover sub-resolution features without significant noise amplification, particularly when these features reside in more global functions.26 

One caveat to the basis expansion approach is that not all bases are equally appropriate for a particular problem. Some may describe the objects being reconstructed accurately with just a few modes, while others require nearly as many modes as voxels. For example, while spherical harmonics are a natural choice for reconstructing hot spots deviating from nominal spherical symmetry in spherical capsule-based fusion,3,7 they are a poor basis for the very high-aspect ratio stagnation columns generated in MagLIF experiments. Cylindrical harmonics18,19 discussed in Sec. II C appear more suitable based on the cylindrical geometry of MagLIF, but may be suboptimal. For instance, the volumes that generate the complex, even multi-stranded, helical features observed in stagnation images still likely represent high-mode structures in a cylindrical harmonic basis.

Presumably, in certain problems, a sparse basis exists that would sufficiently constrain a reconstruction with a very limited number of projections, even as few as two or three. Discovery of such a basis may occur through exhaustive inspection, e.g., testing several sets of orthogonal polynomials in cylindrical coordinates of different maximum order, and solving the optimization problem with relevant test volumes. Alternatively, we propose to efficiently “learn” a more natural basis from a relevant set of training volumes that represent structures similar to those observed in the experiment. The technique of discovering a natural basis within the broader context of “Dictionary Learning” has been applied in a variety of applications, including medical image reconstruction, image denoising, and deblurring and recently for reconstruction of high-resolution images of black holes.26–29 Here, we use “learn” in an unsupervised sense, where we solely extract features (basis functions) from a training data set, rather than use the data to explicitly train a model that maps inputs to outputs. Our fundamental assumption is that the solution f lies approximately in the subspace of R N spanned by the set of basis vectors that also span some set of representative training volumes. Such a basis should exhibit the trade-off of more readily describing expected structures while losing the ability to represent more general structures, which is simply a form of regularization.

Given a set of volumes { f k | k = 1 , , K }, f k R N, we can construct a data matrix X = [ f 1 f k f K ] R N × K. Finding a basis that represents the entire set of training volumes consists of calculating a set of vectors that span the column space of X. Here, we use well-known principal component analysis (PCA) by computing a Singular Value Decomposition (SVD) of X,
X = U Σ V = n = 1 N u n ( σ n v n ) .
(8)
The resulting matrix U contains columns (known as left singular vectors) u n that are orthonormal and span the column space of X; these column vectors become basis functions for tomographic reconstruction (i.e., set D U). A convenience of the SVD is that it can naturally order the basis vectors in terms of the fractional variance they account for across the data set, ranked by the singular values σ n that make up the diagonal matrix Σ R N × K. This enables, for example, truncation of the basis to include only those vectors with sufficiently large singular values, lowering the rank of the system matrix P D and improving uniqueness in the reconstruction as well as reducing sensitivity to noise by lowering the condition number. Typically, this corresponds to removing basis vectors dominated by high-frequency features, i.e., limiting the reconstruction to a lower-mode solution. Furthermore, truncation with the SVD is straightforward, whereas for analytical bases, one must decide between the number of modes along each spatial coordinate (e.g., azimuthal, radial, and axial for 3D cylindrical harmonics) in achieving a given rank.

Learning basis functions requires a large number (potentially several hundred or thousands) of representative volumes/objects. Ideally, such volumes would come from experiments themselves. However, no 3D MagLIF stagnation column has been reconstructed reliably to-date to provide such data. 3D radiation magnetohydrodynamic (rMHD) simulations would be the most promising candidate for producing physics-informed synthetic 3D emission volumes, but generation of hundreds of these from simulations is prohibitively expensive at-present. Here, we instead use an analytic model that can efficiently generate hundreds or thousands of artificial volumes.

In addition to the observation of apparent helical-like structures in stagnation images, early-time radiography30 and self-emission31 in related experiments have demonstrated helical MRT on the outside of the liner, which is expected to imprint a helical pressure profile on the fuel through stagnation. The analytic volume model therefore consists of a superposition of several quasi-helical strands (similar to in other works22,32,33). Each emissivity volume is defined in 3D Cartesian coordinates as
f k ( x , y , z ) = s = 1 S k c s H ( Φ ( r ) , ϑ s ) Z ( z ) ,
(9)
where each of the S k strands (ranging from two to four) has an amplitude c s and super-Gaussian radial envelope
H ( Φ , ϑ s ) = exp ( Φ ϑ s ) ,
(10)
of order ϑ s. The envelope’s phase function, Φ ( r ), describes the distance from the strand’s centroid (normalized by a strand width parameter, w 0 , s), which rotates about the z-axis in a quasi-helical fashion with average radius r 0 , s. The radius and phase of the centroid vary along the axis to preclude purely helical behavior. The function Z ( z ) modulates the axial intensity along the strand. Both Φ ( r ) and Z ( z ) and their associated geometric parameters for a given strand are described in detail in the  Appendix. In total, each strand in a given volume is described by 31 parameters.

For each volume f k, the model parameters are sampled from uniform distributions (discrete in the case of S k) denoted by U [ a , b ], where a and b denote the lower and upper bounds of the distribution. Table I in the  Appendix summarizes the distributions and geometric parameter interpretations. At present, the distribution bounds are qualitatively chosen such that resulting helical strands can have widths, off-axis structure, etc. that appear both larger and smaller than have been observed in experimental images to-date. As in Lewis et al.,33 the width w 0 , s and mean helical radius r 0 , s of individual strands within a single volume are constrained such that the ratio of the maximum to minimum value of each is 2 (i.e., max s [ w 0 , s ] / min s [ w 0 , s ] 2). We also only permit the strands to have a single (right) helical handedness within each volume. This turns out to be an acceptable constraint since a dominant handedness appears in MagLIF, discussed further in Secs. IV and V.

TABLE I.

Geometric parameter summary for helical strand model in Sec. II B.

Training data
Parameter labelPhysical descriptionparameter distribution
Sk Number of helical strands  U { 2 , 4 } 
cs Strand intensity amplitude  U [ 0.25 , 1 ] 
ϑ s Order of super-Gaussian envelope  U [ 1.5 , 4 ] 
(x0,s, y0,sPosition of strand axis in the (x, y) plane  U [ 33 μ m , 33 μ m ] 
w0,s Strand width parameter  U [ 25 μ m , 90 μ m ]a 
r0,s Helical radius  U [ 20 μ m , 120 μ m ]a 
λs Strand primary helical wavelength  U [ 0.8 mm , 4 mm ] 
β 1 , s r , β 2 , s r Radial excursion function relative amplitudes  U [ 0 , 0.3 ] 
ζ 1 , s r , ζ 2 , s r Radial excursion function relative wavelengths  U [ 0.4 mm , 4 mm ] 
β 1 , s ϕ , z , β 2 , s ϕ , z Phase/axial intensity excursion function relative amplitudes  U [ 0 , 0.5 ] 
ζ 1 , s ϕ , ζ 2 , s ϕ Phase excursion function relative wavelengths  U [ 1 mm , 4 mm ] 
ζ 1 , s z , ζ 2 , s z Axial intensity modulation excursion function relative wavelengths  U [ 0.2 mm , 1 mm ] 
β 1 , s w , β 2 , s w Width excursion function relative amplitudes  U [ 0 , 0.2 ] 
ζ 1 , s w , ζ 2 , s w Width excursion function relative wavelengths  U [ 0.4 mm , 1.4 mm ] 
φ 1 , s r , ϕ , z , φ 2 , s r , ϕ , z Radial, phase, and axial intensity modulation excursion function phases  U [ 0 , 2 π ] 
Az,s Axial intensity modulation primary amplitude  U [ 0 , 0.95 ] 
Λs Axial intensity modulation primary wavelength  U [ 0.8 mm , 4 mm ] 
ϕz,s Axial intensity modulation phase  U [ 0 , 2 π ] 
αx,s, αy,s, αz,s Strand rotation angles about x, y, and z axes  U [ 1 ° , 1 ° ] 
Training data
Parameter labelPhysical descriptionparameter distribution
Sk Number of helical strands  U { 2 , 4 } 
cs Strand intensity amplitude  U [ 0.25 , 1 ] 
ϑ s Order of super-Gaussian envelope  U [ 1.5 , 4 ] 
(x0,s, y0,sPosition of strand axis in the (x, y) plane  U [ 33 μ m , 33 μ m ] 
w0,s Strand width parameter  U [ 25 μ m , 90 μ m ]a 
r0,s Helical radius  U [ 20 μ m , 120 μ m ]a 
λs Strand primary helical wavelength  U [ 0.8 mm , 4 mm ] 
β 1 , s r , β 2 , s r Radial excursion function relative amplitudes  U [ 0 , 0.3 ] 
ζ 1 , s r , ζ 2 , s r Radial excursion function relative wavelengths  U [ 0.4 mm , 4 mm ] 
β 1 , s ϕ , z , β 2 , s ϕ , z Phase/axial intensity excursion function relative amplitudes  U [ 0 , 0.5 ] 
ζ 1 , s ϕ , ζ 2 , s ϕ Phase excursion function relative wavelengths  U [ 1 mm , 4 mm ] 
ζ 1 , s z , ζ 2 , s z Axial intensity modulation excursion function relative wavelengths  U [ 0.2 mm , 1 mm ] 
β 1 , s w , β 2 , s w Width excursion function relative amplitudes  U [ 0 , 0.2 ] 
ζ 1 , s w , ζ 2 , s w Width excursion function relative wavelengths  U [ 0.4 mm , 1.4 mm ] 
φ 1 , s r , ϕ , z , φ 2 , s r , ϕ , z Radial, phase, and axial intensity modulation excursion function phases  U [ 0 , 2 π ] 
Az,s Axial intensity modulation primary amplitude  U [ 0 , 0.95 ] 
Λs Axial intensity modulation primary wavelength  U [ 0.8 mm , 4 mm ] 
ϕz,s Axial intensity modulation phase  U [ 0 , 2 π ] 
αx,s, αy,s, αz,s Strand rotation angles about x, y, and z axes  U [ 1 ° , 1 ° ] 
a

[25 μm,  90 μm] are the absolute bounds for w0,s. To ensure max s [ w 0 , s ] / min s [ w 0 , s ] 2, an average strand width, ⟨w0,s⟩ is first sampled from a narrower distribution, U [ 37.5 μ m , 67.5 μ m ] and then individual strand widths w0,s are actually sampled from the distribution U [ 2 w 0 , s / 3 , 4 w 0 , s / 3 ] and scaled such that S k 1 s = 1 S k w 0 , s = w 0 , s . An analogous sampling procedure is used for r0,s to ensure max s [ r 0 , s ] / min s [ r 0 , s ] 2.

Figure 2(a) compares a sample of 2D projections of 8-mm-tall, synthetic volumes to a measured stagnation image from the MagLIF experiment z3208. The left two synthetic projections show qualitatively similar features to those observed in the measured image, whereas the right-most synthetic projection is notably different, with larger helical radii and separation between strands. Future work will utilize studied image metrics34 to improve and validate our training data models by ensuring that the distribution of training volume projections overlaps with the distribution of experimental projections.

FIG. 2.

(a) Projections of several full-height sample volumes from the helical strand model (right), compared to a stagnation image from MagLIF experiment z3208. (b) Sample volumes plotted in 3D with semitransparent isocontours, each with a corresponding pair of orthogonal projection images and ( x , y ) slices at heights represented by the tick marks along the z-axis in the 3D volume. The outermost isocontour corresponds to the 12.5% intensity threshold. The black cross in each slice represents the ( x , y ) = ( 0 , 0 ) position for reference.

FIG. 2.

(a) Projections of several full-height sample volumes from the helical strand model (right), compared to a stagnation image from MagLIF experiment z3208. (b) Sample volumes plotted in 3D with semitransparent isocontours, each with a corresponding pair of orthogonal projection images and ( x , y ) slices at heights represented by the tick marks along the z-axis in the 3D volume. The outermost isocontour corresponds to the 12.5% intensity threshold. The black cross in each slice represents the ( x , y ) = ( 0 , 0 ) position for reference.

Close modal

In order to reduce computational complexity in later reconstructions, 3D training volumes and resulting basis functions are only generated with a 2-mm column height and 0.5 mm widths in both x and y. Here, we use a n x × n y × n z = 64 × 64 × 64 Cartesian grid, leading to a total of 262 144 voxels/volume, each 32  μm in z by 7.9  μm in x and y (roughly a single axial × 3 radial image resolution elements). Figure 2(b) shows a sample of six volumes from this model plotted using a series of eight semi-transparent, 3D isocontours. Each volume in Fig. 2(b) is accompanied by a pair of orthogonal projections and several ( x , y ) slices along the z-axis. The slices demonstrate that the volumes contain a variety of structures, ranging from fully separated lobes to noncircular blobs in regions where multiple strands overlap.

Training volumes created with the helical strand model above are used to generate 3D basis functions. We first generate a set of K = 600 volumes, 8 mm in height, each with a unique set of model parameters. Each of these “full-height” volumes is divided into four 2-mm-tall volumes. The final set of 2400 volumes comprises the full training volume data set. Each training volume is first normalized to have j f j = 1, flattened into a 64 3 × 1 column vector and arranged column-wise into a training data matrix X R 64 3 × 2400. 3D basis functions are then calculated from X using the SVD [Eq. (8)], which we call the “SVD basis.” Figure 3 shows a sample of 12 of these basis functions. Even though the training volumes contain only positive intensity values, the basis functions clearly consist of both positive and negative values, denoted by red and blue in the colorscale, respectively. Inclusion of a positivity constraint in the reconstruction (discussed in Sec. II) will therefore be important to ensure these basis functions do not lead to negative intensities in the solutions. The n = 1 basis function represents a single, nominally cylindrical structure. The next, n = 2, mode shows a positive-negative pair of helical strands that rotate approximately about the z-axis. As n increases, an increasing number of positive and negative strands quickly appear, showing highly structured cross sections containing features < 70 μm by the time n = 100. Figure 4 shows the spectrum of singular values σ n in terms of their fractional explained variance σ n 2 / n K σ n 2. At n > 120, each subsequent mode explains less than 0.1 % additional variance. Summing the individual variances from n = 1 to B K, over 90 % of the total variance in X is represented with B = 177 modes and 99 % with B = 705 modes.

FIG. 3.

A sample of several 3D basis functions calculated from a set of training volumes 2 mm in height and ( x , y ) slices spaced 0.667 mm apart, at z positions corresponding to the vertical tick marks.

FIG. 3.

A sample of several 3D basis functions calculated from a set of training volumes 2 mm in height and ( x , y ) slices spaced 0.667 mm apart, at z positions corresponding to the vertical tick marks.

Close modal
FIG. 4.

Spectrum of singular values for each mode in the 3D learned basis in terms of their fractional variance within the entire training volume data set.

FIG. 4.

Spectrum of singular values for each mode in the 3D learned basis in terms of their fractional variance within the entire training volume data set.

Close modal
In Sec. IV, we compare the performance of reconstructions using a learned basis to those using other bases. The first of these is a simple voxel basis in Cartesian coordinates with additional constraints and regularization used in the reconstruction, described in Sec. III A. We also perform reconstructions using a helical, cylindrical harmonics basis, having azimuthal and axial dependence exp ( i m θ i k q z ), where m is the azimuthal mode number and k q = 2 π q / λ z , 0 is the axial wavenumber with some fundamental axial wavelength, λ z , 0, taken to be the 2-mm patch height. Here, the goal is to still incorporate helical structure relevant to systems under the influence of the MRT instability. For example, structures with m = 0 and m = 1 in this basis correspond to the well-known “sausage” and “kink” instabilities, respectively.35 The expansion of the object in this basis in cylindrical coordinates is
f ( ρ , θ , z ) = q = 0 Q m = 0 M l = 0 L [ a l m q cos ( m θ k q z ) + b l m q sin ( m θ k q z ) ] R l m ( ρ ) ,
(11)
where ρ = x 2 + y 2 and θ = arctan ( y / x ). The total number of unique 3D basis functions is B = ( L + 1 ) [ 2 ( M + 1 ) ( Q + 1 ) 1 ], accounting for both cosine and sine helical terms for real objects and setting all b l 00 = 0. Many candidates exist for the radial basis functions R l m ( ρ ), including Bessel functions, Zernike polynomials, etc. Here, we choose Laguerre–Gauss polynomials36 
R l m ( ρ ) = ( 1 ) l 2 ( | m | + 1 ) / 2 π | m | / 2 [ l ! ( | m | + l ) ! ] 1 / 2 × ρ ~ | m | L l ( m ) ( 2 π ρ ~ 2 ) e π ρ ~ 2 ,
(12)
where L l ( m ) are generalized Laguerre polynomials and ρ ~ ρ / R is the radius normalized by the radius of the domain R = 0.25 mm. The set of Laguerre–Gauss helical cylindrical harmonic functions, e.g., R m l ( ρ ) cos ( m θ k q z ), subsequently referred to only as CHs, are orthonormal. This radial basis has the desirable properties that (i) modes go to zero quickly toward the edge of the domain ( ρ R) and (ii) modes with m > 0 go to zero at r = 0, which avoids high-frequency features near the axis (e.g., gradients even reaching infinity for odd values of m). Figure 5 shows a sample of several 3D CH basis functions used for reconstruction.
FIG. 5.

A sample of 3D modes from a Laguerre-Gauss helical cylindrical harmonic basis, labeled with (m,l,q) indices.

FIG. 5.

A sample of 3D modes from a Laguerre-Gauss helical cylindrical harmonic basis, labeled with (m,l,q) indices.

Close modal
We can reconstruct the 3D object by minimizing the likelihood function quantifying the difference between measurements and model [Eq. (5)] with respect to the object, f. The projection measurements considered here are assumed to have high enough photon statistics that the noise is approximately normally distributed. Then, the negative of the log-likelihood function is simply the weighted sum of squares,
L ( g , f ) = 1 2 ( g P f ) W ( g P f ) = 1 2 g P f W 1 / 2 2 ,
(13)
where W is a diagonal matrix with w i i = 1 / ϵ i 2 and ϵ i is the measurement noise in the i th projection pixel.
When using the voxel basis, we reconstruct each value of f j directly by solving the minimization problem,
minimize f 1 2 g P f W 1 / 2 2 + β ψ ( f ) subject \; to  f 0 , C f 0.
(14)
Here, we only perform voxel basis reconstructions on 2D slices, to limit the algorithm run-time and size of P; 3D volumes are generated by stacking individual 2D slice reconstructions. In Eq. (14), we also add the penalty term ψ ( f ) to the likelihood function to regularize the problem. We use Tikhonov regularization15,20 to limit large gradients and noise amplification in the solution,
ψ ( f ) = j [ D x f ] j 2 + [ D y f ] j 2 ,
(15)
where the matrix D x ( y ) is a finite difference operator in the x ( y ) direction. The parameter β determines how strongly the regularization is enforced relative to the data fit term. We fix β = 0.1, determined empirically as the value at which high-frequency structures below 20  μm are suppressed. Alternative methods (e.g., cross-validation) of determining β can be explored in future work.37 

The first constraint in Eq. (14) ensures non-negativity for physically relevant solutions. The second constraint uses a masking operator C that forces the reconstruction to go to zero outside a region determined by tracing the left and right edges of each 1D projection back through the ( x , y ) plane. The region of overlapping edges forms a polygon whose number of pairs of parallel sides equals the number of views, illustrated by an example in the supplemental material. Left and right edge positions in the projections are defined as where the intensity reaches 5 % the maximum in a given view. The constraint was found to be useful for the voxel basis to avoid solutions with artifacts at the edges of the domain that would only be constrained with additional views.

Equation (14) is solved using a constrained Nesterov fast gradient method (FGM) algorithm38–40 shown in Algorithm 1. The algorithm is initialized with a weighted least-squares estimate.41 

ALGORITHM 1.

Constrained Nesterov fast gradient method algorithm with Tikhonov regularization for voxel basis. The parameter L = 2 λ {max} ( P W P ) is the Lipschitz constant of the gradient of the likelihood function where λ {max} ( Y ) is the largest eigenvalue of a matrix Y.

Step 0: 
f 0 = ( P W P + β 0 I ) 1 P W g 
z 0 = f 0 
τ 0 = 1 
Step k: 
f k + 1 = z k 1 L P W ( P f k g ) β f ψ ( f k ) 
f k + 1 = min ( C max ( f k + 1 , 0 ) , 0 ) 
τ k + 1 = 1 2 ( 1 + 1 + 4 τ k 2 ) 
z k + 1 = f k + 1 + τ k 1 τ k + 1 ( f k + 1 f k ) 
Step 0: 
f 0 = ( P W P + β 0 I ) 1 P W g 
z 0 = f 0 
τ 0 = 1 
Step k: 
f k + 1 = z k 1 L P W ( P f k g ) β f ψ ( f k ) 
f k + 1 = min ( C max ( f k + 1 , 0 ) , 0 ) 
τ k + 1 = 1 2 ( 1 + 1 + 4 τ k 2 ) 
z k + 1 = f k + 1 + τ k 1 τ k + 1 ( f k + 1 f k ) 

For reconstructions using the global basis expansion model, we substitute f = D a, giving the minimization equation for the coefficient vector a,
minimize a 1 2 g P D a W 1 / 2 2 subject \; to  f = D a 0.
(16)
As described in Sec. II, regularization is achieved in this case through the choice of basis, D, both in terms of which specific basis is used (i.e., SVD or cylindrical harmonics) and by truncating that basis to exclude high-frequency spatial structure. Note that we must use the inequality constraint D a 0 since the chosen basis functions contain negative values and simply constraining a 0 will not ensure a positive solution.

We use a modified version of the FGM algorithm (Algorithm 2) to solve Eq. (16), where we have substituted A P D to simplify notation. The algorithm is similarly initialized with a weighted least-squares estimate. The non-negativity constraint D a 0 is imposed approximately in each iteration by making use of the Moore-Penrose pseudoinverse matrix,42  D = ( D D + ε I ) 1 D , where a D f and ε is a small number (chosen to be 10 8) that helps condition D D.

Algorithms 1 and 2 terminate when a stopping criterion is met. Here, we use a simple step convergence criterion N 1 ( f k f k 1 ) / f k 1 < 10 3 in the case of the voxel basis and B 1 ( a k a k 1 ) / a k 1 < 10 3 in the case of the SVD or CH basis.

ALGORITHM 2.

Constrained Nesterov fast gradient method algorithm for global basis where f = D a and A = P D and L = 2 λ {max} ( A W A ).

Step 0: 
a 0 = ( A W A + β 0 I ) 1 A W g 
z 0 = a 0 
τ 0 = 1 
Step k: 
a k + 1 = z k 1 L A W ( A a k g ) 
a k + 1 = D max ( D a k + 1 , 0 ) 
τ k + 1 = 1 2 ( 1 + 1 + 4 τ k 2 ) 
z k + 1 = a k + 1 + τ k 1 τ k + 1 ( a k + 1 a k ) 
Step 0: 
a 0 = ( A W A + β 0 I ) 1 A W g 
z 0 = a 0 
τ 0 = 1 
Step k: 
a k + 1 = z k 1 L A W ( A a k g ) 
a k + 1 = D max ( D a k + 1 , 0 ) 
τ k + 1 = 1 2 ( 1 + 1 + 4 τ k 2 ) 
z k + 1 = a k + 1 + τ k 1 τ k + 1 ( a k + 1 a k ) 

In this section, we evaluate the performance of the proposed reconstruction techniques using a number of known test volumes. We employ multiple metrics to quantify performance, each offering a unique measure of the similarity between a reconstruction, f and its ground truth f t r u e. These metrics include

  • Relative Root Mean Square Error (RRMSE):

    The RRMSE provides a general measure of similarity on a per-voxel basis, in a normalized sense,
    R R M S E = 1 N f t r u e j ( f j f t r u e , j ) 2 ,
    (17)
    where f t r u e = N 1 j N f t r u e , j is the mean value of the ground truth. While values of the RRMSE 1 represent high reconstruction accuracy, this metric lacks specific insight into morphology or voxel inter-dependency.
  • Relative Volume ( R V t):

    In 3D, the R V t is equivalent to the ratio of volumes enclosed by isocontours at the same threshold intensity t between reconstruction and ground truth. It has direct application to MagLIF stagnation plasmas, describing the accuracy with which the reconstruction techniques may estimate stagnated fuel volume and, relatedly, convergence ratio, where the former is important for inferences of stagnation pressures. Formally,
    R V t ( f , f t r u e ) = | { f ~ } t | | { f ~ t r u e } t | ,
    (18)
    which for a perfect match is unity. Here, { y } t indicates the subset of voxels in the array y, { y i | y i t max [ y ] }, | { y } t | is the number of elements in { y } t and 0 < t 1. Here, we first normalize the reconstruction and ground truth by the maximum intensity at each slice, denoted by f ~ and f ~ true, to weight individual slices equally.
  • Intersection over Union ( I o U):

    In 2D, the I o U t describes, in a relative sense, the overlapping area between slices f i thresholded at a particular intensity, t,
    I o U t ( f i , f true , i ) = | { f i } t | | { f true , i } t | | { f i } t { f true , i } t | ,
    (19)
    which for a perfect match is also unity. Here, we use a common threshold intensity ( = t max [ f true , i ]) between the two slices. For the 3D volumes, we use an axially averaged quantity I o U t, which requires an I o U t for each pair of slices. Compared to the other metrics, the I o U better quantifies morphological accuracy since a non-zero intersection requires the sets of voxels from each pair of slices to overlap in space.

We first look at the case where we expect a reasonable match between the learned basis and ground truth by calculating reconstructions on a new set of 10 volumes generated with the helical strand model from Sec. II B. These “in-sample”43 volumes include additional complexity in an attempt to reduce model-dependent bias for the case of the learned basis: in addition to the radius, phase, and modulation intensity, we now allow the width of the strand to vary axially by up to ± 40 %.

To assess the stability of the solutions to the sparse data, we perform several reconstructions for each of the 10 test volumes, each using a unique set of projections. For a given number of views, N v, projections are spaced uniformly by angles π / N v. We use four unique sets of projections shifted by angles ν π / ( 4 N v ) with ν = { 0 , 1 , 2 , 3 }, resulting in a total of 40 reconstructions to evaluate in-sample performance. Intensity-dependent Gaussian noise is added to each projection with a signal-to-noise ratio of 10, and noise floor 5 % the maximum intensity in the data.

Figure 6 shows 3D reconstructions of one of the 10 test volumes from two orthogonal projections using the SVD, CH, and voxel bases. The SVD and CH bases each use B = 1002 basis functions [for CH, ( M , L , Q ) = ( 5 , 5 , 13 )]. The top row uses orthogonal projections at angles 0 ° / 90 ° and the bottom row uses projections at 45 ° / 135 °. All three bases can localize the emission accurately for both sets of projections, with the observation of a dominant helical strand in each. However, the voxel basis shows significant variation in detailed morphology between the two sets of projection data (top vs bottom row); while the reconstruction can accurately distinguish two separate strands with θ = 45 ° / 135 ° in the bottom half of the volume observed in the bottom two slices, these features are much more blended together and show evidence of “streaking” for the θ = 0 ° / 90 ° case. Conversely, both the SVD and CH bases more clearly preserve the double-strandedness for both sets of projections. Furthermore, the SVD basis can better reproduce the individual strand intensities for both sets of projection data compared to the CH basis, comparing the bottom two slices between the top and bottom rows in Fig. 6. This ability to accurately distinguish separated strands is important for MagLIF, where instabilities feeding through the liner may lead to bifurcation of the stagnation column.

FIG. 6.

Two-view reconstructions of a volume generated by the helical strand model using different bases, with ( x , y ) slices for each taken at the tick marks on the z-axis. The top row uses orthogonal projections at angles 0 ° / 90 ° and the bottom row uses projections at 45 ° / 135 °. All plots share a common color scale and the outer-most isocontour represents 5% of the maximum ground truth intensity.

FIG. 6.

Two-view reconstructions of a volume generated by the helical strand model using different bases, with ( x , y ) slices for each taken at the tick marks on the z-axis. The top row uses orthogonal projections at angles 0 ° / 90 ° and the bottom row uses projections at 45 ° / 135 °. All plots share a common color scale and the outer-most isocontour represents 5% of the maximum ground truth intensity.

Close modal

Figure 7(a) shows the pair of calculated projections from the reconstructions for each basis for the θ = 45 ° / 135 ° case. Figure 7(b) shows the corresponding residuals, plotted as percent differences, where at each axial postion, the difference is normalized by the maximum intensity of the ground truth projection. Generally, each basis matches the observed projection morphology. The SVD basis residuals show minimal spatial correlations, indicating a good fit to the data with the chosen number of basis functions. Notably, the projections from the CH basis using the same number of basis functions lack some high-frequency structure compared to the other two bases, evidenced by the correlated structure in the residuals. Better fitting to the data is possible with more basis functions, which may also improve the reconstruction performance. The residuals for the voxel basis also show faint correlated structures at low intensity near the axis, which—combined with high-frequency features in the projections themselves—suggest this basis with chosen level of regularization is still overfitting to the noise within masked regions.

FIG. 7.

(a) Calculated projections from two-view reconstructions ( θ = 45 ° / 135 °) of the test volume shown in Fig. 6 for each basis. (b) Residuals plotted as percent differences, where at each height the difference is normalized by the maximum intensity from the ground truth projection.

FIG. 7.

(a) Calculated projections from two-view reconstructions ( θ = 45 ° / 135 °) of the test volume shown in Fig. 6 for each basis. (b) Residuals plotted as percent differences, where at each height the difference is normalized by the maximum intensity from the ground truth projection.

Close modal

To assess overall performance vs number of projections, we calculate the RRMSE, R V t, and I o U t and the standard deviation in each over the in-sample data set for different numbers of views. For the R V, we use a relatively low threshold value t = 0.2 to encompass a large fraction of the total x-ray volume. For the I o U , we use a higher threshold value of t = 0.6 to focus on morphology in higher-intensity regions, which we expect to better correlate with complex structures such as separated strands. The results are shown in Fig. 8(a), where each point is averaged over the 10 volumes and 4 sets of unique projections for each volume. Horizontal dashed lines represent the ideal values for each metric (=1 for R V and I o U , and 0 for RRMSE). Unsurprisingly for this set of volumes, the SVD basis outperforms the other bases across all three metrics at each number of views, both in terms of the metric values and their lower variation [Fig. 8(b)] across the test volume set. More remarkably, however, the SVD basis provides a slightly higher I o U and lower RRMSE with two views than either of the other bases does with three views. This behavior is even more evident comparing three views to four views. The lower standard deviations using the SVD basis—in particular for the I o U —show that generally this basis is more robust across a range of morphologies and possible projection data for the set of test volumes. These key observations demonstrate that proper matching between a basis and the reconstructed objects can add more value than additional data in certain cases, as a result of good priors. This powerful result can greatly benefit ICF experiments by lessening the need for additional views that would take up space in an already crowded diagnostic layout in the laboratory.

FIG. 8.

Comparison of performance metrics (a) and their standard deviations (b) vs the number of views for reconstructions of 10 volumes generated from the helical strand model. Each metric is averaged over the 10 volumes and 4 sets of projections. Results are shown for the three bases: SVD (triangles), CH (circles), and voxel ( ×’s). RRMSE is shown in gray, R V 0.2 in blue, and I o U 0.6 in red. (c) R V t and (d) I o U t vs relative threshold intensity value t for each of the three bases for the case with N v = 2. In (a), (b), and (d), ideal values for each metric are shown as blue, red, and black dashed lines for R V, I o U , and RRMSE, respectively.

FIG. 8.

Comparison of performance metrics (a) and their standard deviations (b) vs the number of views for reconstructions of 10 volumes generated from the helical strand model. Each metric is averaged over the 10 volumes and 4 sets of projections. Results are shown for the three bases: SVD (triangles), CH (circles), and voxel ( ×’s). RRMSE is shown in gray, R V 0.2 in blue, and I o U 0.6 in red. (c) R V t and (d) I o U t vs relative threshold intensity value t for each of the three bases for the case with N v = 2. In (a), (b), and (d), ideal values for each metric are shown as blue, red, and black dashed lines for R V, I o U , and RRMSE, respectively.

Close modal

Figures 8(c) and 8(d) show the dependence of the R V t and I o U t, respectively, on threshold intensity t for each basis using two views in the reconstruction. Both the SVD and CH bases show excellent ability to calculate volume from even just two views, with R V t = 1.05 ± 0.06 and 1.12 ± 0.03 over 0.1 t 0.5 using SVD and CH bases, respectively, for this number of basis functions. The voxel basis significantly overpredicts the volume for all values of t. The SVD basis leads to higher values of the I o U t compared to the other bases across all threshold intensities. For all bases, I o U t decreases nearly monotonically with t, evidence that each basis limits the spatial resolution in the solution, precluding features at the highest intensities. The voxel basis demonstrates the extreme of this behavior, e.g., significantly undershooting unity at higher threshold intensities, suggesting that the solutions are being over-smoothed by Tikhonov regularization. Smaller values of the regularization parameter, β could lead to improved estimates of relative volume at the risk of more noise amplification in the solution. The slower fall-off of each metric with t for the SVD and CH bases points to their overall higher performance than the voxel basis when using two views. Increasing the number of views flattens each of the curves, though only marginally for the SVD and CH basis without the addition of more modes.

Both the SVD and CH bases have a prescribed helical handedness (right-handed by default), which may overconstrain the solution without prior knowledge of the true handedness (if any). We performed additional two-view reconstructions using a combined right- and left-handed SVD basis, where the left-handed basis functions were generated simply by flipping the right-handed basis across the y-axis.44 When reconstructing the right-handed volumes used previously, the calculated coefficients of the right-handed basis vectors dominated over the corresponding coefficients of the left-handed basis vectors by a factor of 3–18. This suggests that handedness information itself could be recovered from two views of MagLIF stagnation columns.

In addition to the above set of analytically calculated volumes, we look at out-of-sample performance with a stagnation emissivity volume [Fig. 9(a)] generated from a 3D, fully integrated MagLIF simulation using the HYDRA rMHD code.45,46 Recent MagLIF experiments have mainly used the High-Resolution Continuum X-ray Imager11 (HRCXI) to record time-integrated stagnation images on image plate detectors. Because the Ge 220 crystal used in this configuration reflects in multiple orders, n at energies n × 3.1 keV, the resulting image consists of a superposition of emission from energies 6.2, 9.3, and 12.4 keV,47 weighted by the relative energy sensitivities (0.31, 0.61, and 0.08, respectively48) that result from the combination of energy-dependent crystal reflectivity, filtering, and image plate response. We therefore calculate a total emissivity volume with the 3D HYDRA temperature and density profiles as the sum of emissivities at these energies, weighted by their relative sensitivities. We only calculate the emissivity from bound-free + free-free continuum transitions within the D 2 fuel, using the models presented in Epstein et al.49 This emission is optically thin within the D 2 and we ignore attenuation in the Be liner for simplicity. The volume is then blurred using a 3D Gaussian with 25- μm FWHM, to account for the imager’s resolution ( 18–20  μm) and some motion blurring during stagnation since the simulated profiles are from a single time slice near peak neutron production, whereas experimental images are time-integrated over the 2-ns burn duration. Noise is added to the projections in the same manner as in the in-sample data described in Sec. IV A. The 8.8-mm tall column is divided into five 2-mm tall volume patches and corresponding projections that overlap in the axial direction by 320  μm (10 voxels). A full-height column is generated by first reconstructing each patch individually, and then adding the patches together, weighting them by smooth window functions that equal unity in the non-overlapping regions and sum to unity in the overlapping regions.

FIG. 9.

(a) Full-height stagnation column from HYDRA. (b) Two-view reconstructions of a single 2-mm-tall patch from different bases using projections at angles θ = 45 ° / 135 °, with accompanying ( x , y ) slices at the vertical tick marks. (c) Comparison of slices at z = 0.68 mm using different sets of projection data: top row uses θ = 0 ° / 90 ° and bottom row uses θ = 45 ° / 135 °. Black contours are at the 60% intensity, relative to the maximum in the ground truth slice (45% in entire volume); dashed lines are the ground truth contour overlaid for each reconstructed slice.

FIG. 9.

(a) Full-height stagnation column from HYDRA. (b) Two-view reconstructions of a single 2-mm-tall patch from different bases using projections at angles θ = 45 ° / 135 °, with accompanying ( x , y ) slices at the vertical tick marks. (c) Comparison of slices at z = 0.68 mm using different sets of projection data: top row uses θ = 0 ° / 90 ° and bottom row uses θ = 45 ° / 135 °. Black contours are at the 60% intensity, relative to the maximum in the ground truth slice (45% in entire volume); dashed lines are the ground truth contour overlaid for each reconstructed slice.

Close modal

Figure 9(b) shows reconstructions of a single 2-mm patch of the synthetic emission volume from two orthogonal views at θ = 45 ° / 135 ° using the SVD, CH, and voxel bases. The shapes of the outermost 3D isocontours show reasonable agreement with those of the ground truth, as do several of the ( x , y ) slices. However, we note a significant variation in the solutions with projection data, seen in Fig. 9(c) comparing the top slice at z = 0.68 mm between recontructions with projections at θ = 0 ° / 90 ° (top row) and θ = 45 ° / 135 ° (bottom row). The crescent feature is reproduced more clearly in the bottom row than in the top row. The black contours represent the 60% intensity threshold relative to the maximum in the ground truth. Across the three different bases, the SVD basis creates the highest overlap in these contours (solid lines) and the ground truth (dashed).

The calculated projections and residuals [Figs. 10(a) and 10(b)] for the volume patch in Fig. 9(b) show similar trends between the three bases as in the in-sample example: underfitting with the CH basis (though less-so than in the in-sample volume), overfitting with the voxel basis, and more balanced behavior with the SVD basis. We observe faint structure in the residuals with the SVD basis, indicating that more basis functions could improve the fit.

FIG. 10.

(a) Calculated projections from two-view reconstructions of the synthetic emission volume, corresponding to the top row in Fig. 9. (b) Residuals plotted as percent differences, where at each height the difference is normalized by the maximum intensity from the corresponding ground truth data.

FIG. 10.

(a) Calculated projections from two-view reconstructions of the synthetic emission volume, corresponding to the top row in Fig. 9. (b) Residuals plotted as percent differences, where at each height the difference is normalized by the maximum intensity from the corresponding ground truth data.

Close modal

Figure 11 shows the full-height column after piecing together individually reconstructed patches from two views. The reconstruction shows generally similar structure along the axis. Comparison of ( z , x ) slices at y = 0 shows similar positions of breaks in the emission along the axis, with no obvious seams between patches.

FIG. 11.

(a) Full-height reconstruction of HYDRA-calculated stagnation column from two views, using an SVD basis. The full column is generated from five overlapping patches. (b) ( x , z ) slices from each at y = 0.

FIG. 11.

(a) Full-height reconstruction of HYDRA-calculated stagnation column from two views, using an SVD basis. The full column is generated from five overlapping patches. (b) ( x , z ) slices from each at y = 0.

Close modal

Figure 12(a) shows how performance improves with the number of views, comparing the three bases, analogous to Fig. 8(a). Here, each metric shown is calculated from reconstructions of the synthetic MagLIF emission volume from HYDRA, averaged over five axial patches and four unique sets of uniformly spaced projections. We note the overall lower performance on this out-of-sample data relative to the in-sample data from Sec. IV A. For the SVD and CH bases, this can be partially explained by a poorer match between the volume and basis geometry. However, the lower performance from the voxel basis as well suggests the HYDRA volume contains more complex or higher frequency structures than those in the in-sample volume set. Regardless, the SVD basis still outperforms the other bases for a given number of views across all three metrics. While the RRMSE decreases more by increasing the number of views than by using the SVD basis over the voxel basis, the SVD basis again achieves a slightly higher I o U 0.6 with two views than do the other bases with three views (0.56 vs 0.50–0.52)—the same can be said going from three to four views. Therefore, the SVD basis constrains the morphology of MagLIF stagnation columns equally or better than would an additional view using the other bases. As a result, we interpret that the axially correlated, helical structures contained in the training volumes (used to calculate the SVD basis) are good priors for MagLIF stagnation columns.

FIG. 12.

Comparison of performance metrics (a) and their standard deviations (b) vs number of views for reconstructions of the synthetic emission volume from HYDRA, averaged over five patches and four sets of projections. Results are shown for the three bases: SVD (triangles), CH (circles), and voxel ( ×’s). RRMSE is shown in gray, R V 0.2 in blue, and I o U 0.6 in red. (c) R V t and (d) I o U t, respectively, vs threshold intensity from all three basis using two views, as well as from single images (orange squares). In (a), (b), and (d), ideal values for each metric are shown as dashed lines with colors matching the data points.

FIG. 12.

Comparison of performance metrics (a) and their standard deviations (b) vs number of views for reconstructions of the synthetic emission volume from HYDRA, averaged over five patches and four sets of projections. Results are shown for the three bases: SVD (triangles), CH (circles), and voxel ( ×’s). RRMSE is shown in gray, R V 0.2 in blue, and I o U 0.6 in red. (c) R V t and (d) I o U t, respectively, vs threshold intensity from all three basis using two views, as well as from single images (orange squares). In (a), (b), and (d), ideal values for each metric are shown as dashed lines with colors matching the data points.

Close modal

For two views, the sensitivity of R V t to threshold intensity [Fig. 12(c)] from reconstructions using the SVD and CH bases is significantly higher for the HYDRA volumes than for the in-sample volumes [Fig. 8(c)]. However, the SVD basis still provides accurate volume estimates, with 1.15 R V t 0.93 over 0.2 t 0.4, and a standard deviation of 11 % across the patches and sets of projections, which we take to represent the uncertainty on the volume inference. The maximum in the I o U t [Fig. 12(d)] also between 0.2 t 0.4 suggests assessments of morphology from MagLIF experimental data using two views are likely to be most accurate over this threshold range.

We also compare the relative volume estimates from two views to those using just a single projection, since only the latter have been used in physics analyses10,22 until now. For the latter, individual images are first processed using the method described in Sec. IV A of Knapp et al.,22 and then Abel-inverted for radial profiles at each axial position. Volume is estimated by extracting a radius at each height at which the inverted radial profile reaches the specified intensity threshold t enclosing the volume. The resulting estimates in terms of the R V t [orange squares50 in Fig. 12(c)] significantly underpredict the volume for most threshold intensities compared to the two-view reconstructions with SVD and CH bases. As a result, we expect our reconstructions with multiple views to enable more accurate inferences of stagnation pressure and Lawson ignition criteria22,51 (that rely on volume estimates) than were previously achievable using single projections. Furthermore, the two-view reconstructions naturally provide more accurate assessments of morphology, demonstrated in Fig. 12(d) by the much higher values of I o U t compared to when using Abel-inverted profiles from single views.

Figure 13(a) compares reconstructions of the volume patch shown in Fig. 9(b) from three views using the SVD basis ( B = 1002) between two different sets of projection data separated 30 ° with respect to one another. In contrast to when using two views, the cross-sectional structure within each ( x , y ) slice is now reproduced for both sets of projections, demonstrating that adding a third view can significantly reduce sensitivity to the available projection data. Figure 13(b) shows this in more detail for the slice at z = 0.68 mm, where the solid black contours at the 60% threshold within this slice better overlap the ground truth contour (dashed) and vary less between the two sets of projections compared to the case with two views [Fig. 9(c), second column, top and bottom rows]. This insensitivity to projection data is critical to enable detailed assessments of morphology in MagLIF experiments where the orientation of the object WRT a given set of views cannot be chosen ahead of time.

FIG. 13.

(a) Three-view ( N v = 3) reconstructions of a single 2-mm-tall portion of a stagnation column from HYDRA using the SVD basis with B = 1002 and corresponding ( x , y ) slices for two different sets of projections, with θ = 0 ° / 60 ° / 120 ° and θ = 30 ° / 90 ° / 150 °. (b) Comparison of ( x , y ) slices at z = 0.68 mm with 60% (relative to the maximum in this slice) intensity contours overlaid in black. The dashed contours correspond to the ground truth.

FIG. 13.

(a) Three-view ( N v = 3) reconstructions of a single 2-mm-tall portion of a stagnation column from HYDRA using the SVD basis with B = 1002 and corresponding ( x , y ) slices for two different sets of projections, with θ = 0 ° / 60 ° / 120 ° and θ = 30 ° / 90 ° / 150 °. (b) Comparison of ( x , y ) slices at z = 0.68 mm with 60% (relative to the maximum in this slice) intensity contours overlaid in black. The dashed contours correspond to the ground truth.

Close modal

In addition to the higher performance of the SVD basis relative to the other bases, it also reconstructs volumes much faster and with fewer iterations than do the other bases, requiring 1 min per 2–mm patch on a personal laptop—a factor of 2.8 less than the CH basis for the same number of basis functions ( B = 1002) and a factor of 6 less than the voxel basis.

In this section, we reconstruct a 3D MagLIF stagnation volume with our learned basis approach from a pair of quasi-orthogonal ( Δ θ = 106 °) HRCXI images recorded on MagLIF experiment z371952 [see Figs. 1(b) and 1(c)]. As described in Sec. IV B, these images result from the superposition of plasma continuum emission at x-ray energies 6.2, 9.3, and 12.4 keV, which is optically thin in the D 2 fuel; therefore, the reconstructed 3D emissivity will be a superposition of emissivities at these energies.

We acknowledge that several complexities still exist that limit application to real data. In particular, variations in areal density, ρ R (the density-thickness product) of the external, Be liner from the MRTI can lead to significant intensity modulations in the projection images from x-rays with energies 10 keV. Based on simulations and radiography data, we expect the helical pitch of the liner MRT structures to be < 35 °, meaning they traverse 60 μm in the axial direction over the 100- μm width of the stagnation column. This is significantly below the scale of most observed axial intensity modulations ( 150 μm). Therefore, we assume that the azimuthal ρ R variation is slow enough to ignore its effect over the width of the column. However, we do observe significant differences in axial profiles between views, indicating that azimuthal ρ R variations are significant over large enough angular spacing. To ensure that both projections conserve the integrated emissivity in a given axial slice, we normalize one projection WRT to the other, via
g v ( x ) g v ( x ) g ¯ 0 ( z ) g ¯ v ( z ) ,
(20)
where g ¯ v ( z ) is the horizontally integrated, axial intensity profile at the v th view. In Fig. 15(b), first and third columns show the equivalent projections (using the same 7.9 μ m × 32 μ m pixel size as above), where the θ = 33 ° projection has been normalized to the θ = 73 ° projection. Future work will investigate the sensitivities to small-scale horizontal intensity modulations in detail.
FIG. 14.

(a) Normalized spectrum of basis coefficients for n 2, each taken as the L 2-norm across the five patches, from a two-view reconstruction of the z3719 stagnation column, using an SVD basis with B = 1002. (b) Relative cumulative signal energy, summing coefficients in plot (a). (c) Relative cumulative energies for a reconstruction with B = 100 modes each of right- and left-handed SVD bases, for n 2.

FIG. 14.

(a) Normalized spectrum of basis coefficients for n 2, each taken as the L 2-norm across the five patches, from a two-view reconstruction of the z3719 stagnation column, using an SVD basis with B = 1002. (b) Relative cumulative signal energy, summing coefficients in plot (a). (c) Relative cumulative energies for a reconstruction with B = 100 modes each of right- and left-handed SVD bases, for n 2.

Close modal
FIG. 15.

(a) Reconstruction of z3719 stagnation column from two views, using an SVD basis with B = 1002 right-handed basis functions. The column has been normalized by the peak emissivity at each axial position for improved visualization. Labeled projection directions are shown as magenta and teal arrows and ( x , y ) slices corresponding to the axial tick marks are shown on the right. (b) Comparison of the pair of projections calculated from a reconstruction to the experimental data and (c) residuals, where at each height the difference is normalized by the maximum intensity from the corresponding ground truth projection.

FIG. 15.

(a) Reconstruction of z3719 stagnation column from two views, using an SVD basis with B = 1002 right-handed basis functions. The column has been normalized by the peak emissivity at each axial position for improved visualization. Labeled projection directions are shown as magenta and teal arrows and ( x , y ) slices corresponding to the axial tick marks are shown on the right. (b) Comparison of the pair of projections calculated from a reconstruction to the experimental data and (c) residuals, where at each height the difference is normalized by the maximum intensity from the corresponding ground truth projection.

Close modal

Just as in Sec. IV B, we reconstruct the full 8.8-mm tall stagnation column from five overlapping, 2-mm-tall patches, using the right-handed SVD basis with the same B = 1002 modes employed previously. Figure 14(a) shows the normalized spectral density of basis coefficients (excluding the n = 1 mean offset or “DC” mode), where at each mode number we calculate the L 2-norm over the five patches, | a n | z 2 = p = 1 5 | a p , n | 2. The spectrum decays quickly with mode number, falling by 100 × by n 250. Modes with n 870 contribute < 1 % to the cumulative signal energy, defined as n = 1 B | a n | z 2 [Fig. 14(b)], suggesting that enough modes are present to generate the key morphological structures in the column.

The full-height, reconstructed stagnation column from z3719 is shown in Fig. 15(a), normalized by the integrated emissivity at each height to improve visualization. The column appears bifurcated into two strands between axial positions z = 1.4 0.4 mm, with FWHMs 40–80  μm, shown clearly with the color-coded ( x , y ) slices in this range. This is unsurprising given the double-strand appearance in the θ = 33 ° self-emission image/projection. The inferred shapes of the strands likely are not entirely accurate and would be improved with additional views. However, tests sliding the patch limits in the axial direction demonstrate that this bifurcated feature is fairly insensitive to how the projection data are divided up, lending further credence to its overall morphology.

We estimate a total x-ray emission volume of 0.153 mm 3 from the reconstruction using a threshold intensity t = 0.25, the value that gives R V t 1 and maximizes the I o U t in the tests on HYDRA data in Sec. IV B. To calculate volume, we first normalize the reconstructed column by the maximum intensity at each axial position as with the R V t, which mitigates the impact of axial intensity variations from liner ρ R variations. While this x-ray continuum emission volume is not necessarily equal to the neutron emission volume, simulations can help create a mapping between the two for inferences of fuel pressure and Lawson criteria in the future.

Calculated projections from the reconstructions and residuals are shown in Figs. 15(b) and 15(c), respectively. The calculated projections match the data to within 5 % over the majority of pixels, and the residuals show minimal correlations within the stagnation column itself. High frequency features in the data (e.g., at z = 0.1 mm in the 33 ° view and z = 2.6 mm in the 73 ° view) have large residuals, indicating that additional modes are needed to represent these details. We also note the large residuals in the θ = 73 ° view from z = 1.9 mm to z = 0.5, where detailed structure is present and, coincidentally, data were absent in the θ = 33 ° view from debris damage to the detector.

We used a right-handed basis above to be consistent with the expected right-handed helical nature of the MRTI on this experiment from a clockwise B θ and negative B z (i.e., B z / B θ 0). To test this assumption, we also performed a low-mode reconstruction using a combined right-handed + left-handed helical basis, each with B = 100 (total of 200 basis functions). Figure 14(c) shows the relative contribution of the cumulative signal energy vs mode number for the right- and left-handed portions of the basis, excluding the n = 1 mean offset term for each. The right-handed basis contributes > 2 × more energy to the signal for modes n 35 (that dominate the large-scale structures) and 1.6 × more energy for n > 35, relative to the left-handed basis. The confirmed overall right handedness of the column supports the hypothesis that MRT feedthrough imprints onto the fuel through stagnation.

Three-dimensional reconstruction of MagLIF stagnation columns is critical to understand how 3D structures are related to target performance and is challenging due to limited numbers of available views for self-emission imaging. In this paper, we presented a novel approach to reconstruct MagLIF stagnation columns in 3D from an extremely sparse number (as few as two or three) of views using a global basis “learned” from representative training volumes via principal component analysis. Reconstructions of test volumes with the learned basis outperform those using either a standard voxel basis or a helical, cylindrical harmonic (CH) basis according to several image metrics. Reconstructions from two views using the learned basis can be used to estimate the volume of x-ray emission enclosed by an isocontour to within 11% and with more accurate morphology than when using the other bases. At a given number of views, the learned basis can even provide equivalent or superior performance to when using a different basis with an additional view—a critical result for space-constrained ICF experiments. The addition of a third view reproduces morphology to a high degree of accuracy, insensitive to the particular set of projection data. Reconstructions with the learned basis take 1 / 3 rd the time as those using a CH basis with the same number of modes and 1 / 6 th as much time as with a voxel basis. The large reduction in dimensionality (e.g., from > 10 5 using a voxel basis to < 10 3 using a learned basis) for the inverse problem makes tractable the calculation of posteriors on the reconstructed voxel intensities. Additionally, it enables future experimental design studies to discover optimal configurations of sparse measurements, e.g., exploring trade-offs between view angles and spatial resolution with multiple imagers of differing resolution and signal/noise.

We attribute the improved performance of the learned basis over the other bases to its possession of correlated, helical structures relevant to MagLIF that are represented in an efficient manner, i.e., with fewer modes compared to other bases. Higher-order CH bases or a voxel basis with different regularization (e.g., smaller values of β and/or smoothing in the axial direction) may exist that perform equally or superior to the learned basis. But fine-tuning either of these hyperparameters could prove an arduous task and/or add significant computational burden. Furthermore, discovery of a higher-performing learned basis may be possible by improving the training model and/or incorporating training volumes calculated from rMHD simulations, as well as by varying the patch height, which determines the maximum length of axial correlations. Advanced Dictionary Learning methods could also be used, e.g., to refine a high-dimensional, overcomplete basis to a more optimal one through promotion of sparsity.39,53

We also presented the first 3D reconstruction of a MagLIF stagnation column from experimental data using two quasi-orthogonal self-emission images. The reconstruction supports the existence of a double-stranded feature apparent in the images, as well as reproduces a helical handedness consistent with that expected on the outside of the Be liner from helical MRTI. The latter result is an additional indication that MRTI is responsible for the helical shape of the observed emission. The reconstructions ignored effects of small horizontal intensity modulations in the images from azimuthal variations in liner ρ R. While we plan to quantify uncertainties from this simplification, new instruments that image x-rays > 10 keV in energy would bypass this issue in the future.

Going forward, we plan to use 3D reconstructions to better diagnose several physical processes at stagnation and how they depend on input conditions. These include feedthrough of instabilities from the liner into the fuel, as well as how drive asymmetries (e.g., from slots in the current return can) impact the fuel morphology and confinement. They will also enable more accurate volume estimates, improving inferences of stagnation pressure and Lawson ignition criteria. Furthermore, reconstruction of both fuel and liner emission in 3D simultaneously using multi-channel imaging11 will inform how liner material is mixing into the fuel and impeding performance. Accurate radial profiles—possibly obtainable with a third view—could elucidate thermal transport and energy balance at stagnation and the influence of the axial magnetic field on these processes. Improved understanding of these processes can, in turn, feed back on the reconstruction, providing additional physics constraints that result in higher quality solutions.

See the supplementary material for an illustrated example of the polygon masking region used in the voxel basis reconstruction [Sec. III A, Eq. (14), and Algorithm 1] for the case of three views.

The authors thank Benjamin R. Galloway and both the MagLIF and data science working groups within the Pulsed Power Sciences Center at SNL for useful discussions relating to this work. They also thank Matthew R. Gomez for his help collecting data used in this manuscript. This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). This work was supported by SNL’s LDRD programs, Project No. 222431. The employee owns all right, title, and interest in and to the article and is solely responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan https://www.energy.gov/downloads/doe-public-access-plan. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

The authors have no conflicts to disclose.

Jeffrey R. Fein: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead). Eric C. Harding: Conceptualization (supporting); Data curation (lead); Methodology (supporting); Writing – review & editing (equal). William E. Lewis: Conceptualization (supporting); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal). Matthew R. Weis: Data curation (equal); Methodology (supporting); Writing – review & editing (equal). Marc-Andre Schaeuble: Data curation (supporting); Formal analysis (supporting); Validation (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

In Eq. (10), the super-Gaussian envelope’s phase function, Φ ( r ) describing the normalized distance from the strand’s centroid is given by
Φ ( r ) = ( ( 2 ln ( 2 ) 1 / ϑ s ) 1 / ϑ s w 0 , s ) 1 / 2 × [ ( x x s ( z ) x 0 , s ) 2 + ( y y s ( z ) y 0 , s ) 2 ] 1 / 2 .
(A1)
The quasi-helical centroid coordinates are
x s ( z ) = r s ( z ) cos ( 2 π z λ s + ϕ s ( z ) ) ,
(A2)
y s ( z ) = r s ( z ) sin ( 2 π z λ s + ϕ s ( z ) ) ,
(A3)
which rotate about ( x 0 , s , y 0 , s ) with primary helical wavelength λ s and full width at half maximum (FWHM) w 0 , s ln ( 2 ) 1 / ϑ s 1 / ϑ s 2 ( 2 ϑ s 1 ) 1 / ϑ s.54 The helical radius r s ( z ) and phase ϕ s ( z ) themselves vary along the z-axis,
r s ( z ) = r 0 , s ( 1 + Δ r ( z ) ) ,
(A4)
ϕ s ( z ) = ϕ 0 , s ( 1 + Δ ϕ ( z ) ) ,
(A5)
with mean values r 0 , s and ϕ 0 , s, and relative excursion for parameter p ( = r, ϕ or z ),
Δ p ( z ) = β 1 , s p sin ( 2 π z ζ 1 , s p + φ 1 , s p ) + β 2 , s p sin ( 2 π z ζ 2 , s p + φ 2 , s p ) ,
(A6)
which ensures that each individual strand is not purely helical with a single wavelength. For the “in-sample” data used in Sec. IV A, we also allow the strand width to vary along the axis, i.e., w 0 , s w 0 , s ( 1 + Δ w ( z ) ) in Eq. (A1). The intensity of each strand is then modulated axially by the function
Z ( z ) = 1 2 ( 1 + A z , s sin ( 2 π z Λ s + ϕ z , s ) )
(A7)
× Δ z ( z ) ,
(A8)
with dominant wavelength Λ s, where Δ z ( z ) is of the same form as in Eq. (A6). Lastly, while not written here explicitly, each strand is rotated about the x, y, and z axes by small angles α x , s, α y , s, and α z , s, respectively. This is accomplished by initially transforming ( x , y , z ) to a rotated coordinate system prior to computing Φ ( r ) and Z ( z ) for a given strand. Table I provides a description of each model parameter and details of their sampling distributions used in generating the set of training volumes for the learned basis in Sec. II B.
1.
J. D.
Lindl
,
P.
Amendt
,
R. L.
Berger
,
S. G.
Glendinning
,
S. H.
Glenzer
,
S. W.
Haan
,
R. L.
Kauffman
,
O. L.
Landen
, and
L. J.
Suter
,
The physics basis for ignition using indirect-drive targets on the National Ignition Facility
,”
Phys. Plasmas
11
,
339
339
(
2015
).
2.
P. L.
Volegov
,
C. R.
Danly
,
F. E.
Merrill
,
R.
Simpson
, and
C. H.
Wilde
, “
On three-dimensional reconstruction of a neutron/x-ray source from very few two-dimensional projections
,”
J. Appl. Phys.
118
(
20
),
205903
(
2015
).
3.
P. L.
Volegov
,
C. R.
Danly
,
D.
Fittinghoff
,
V.
Geppert-Kleinrath
,
G.
Grim
,
F. E.
Merrill
, and
C. H.
Wilde
, “
Three-dimensional reconstruction of neutron, gamma-ray, and x-ray sources using spherical harmonic decomposition
,”
J. Appl. Phys.
122
(
17
),
175901
(
2017
).
4.
K. W.
Wong
and
B.
Bachmann
, “
Three-dimensional electron temperature measurement of inertial confinement fusion hotspots using x-ray emission tomography
,”
Rev. Sci. Instrum.
93
(
7
),
073501
(
2022
).
5.
P. L.
Volegov
,
S. H.
Batha
,
V.
Geppert-Kleinrath
,
C. R.
Danly
,
F. E.
Merrill
,
C. H.
Wilde
,
D. C.
Wilson
,
D. T.
Casey
,
D.
Fittinghoff
,
B.
Appelbe
,
J. P.
Chittenden
,
A. J.
Crilly
, and
K.
McGlinchey
, “
Density determination of the thermonuclear fuel region in inertial confinement fusion implosions
,”
J. Appl. Phys.
127
(
8
),
083301
(
2020
).
6.
K. M.
Woo
,
R.
Betti
,
C. A.
Thomas
,
C.
Stoeckl
,
K.
Churnetski
,
C. J.
Forrest
,
Z. L.
Mohamed
,
B.
Zirps
,
S. P.
Regan
,
T. J.
Collins
,
W.
Theobald
,
R. C.
Shah
,
O. M.
Mannion
,
D.
Patel
,
D.
Cao
,
J. P.
Knauer
,
V. Y.
Glebov
,
V. N.
Goncharov
,
P. B.
Radha
,
H. G.
Rinderknecht
,
R.
Epstein
,
V.
Gopalaswamy
,
F. J.
Marshall
,
S. T.
Ivancic
, and
E. M.
Campbell
, “
Analysis of core asymmetries in inertial confinement fusion implosions using three-dimensional hot-spot reconstruction
,”
Phys. Plasmas
29
(
8
),
082705
(
2022
).
7.
K.
Churnetski
,
K. M.
Woo
,
W.
Theobald
,
P. B.
Radha
,
R.
Betti
,
V.
Gopalaswamy
,
I. V.
Igumenshchev
,
S. T.
Ivancic
,
M.
Michalko
,
R. C.
Shah
,
C.
Stoeckl
,
C. A.
Thomas
, and
S. P.
Regan
, “
Three-dimensional hot-spot X-ray emission tomography from cryogenic deuterium-tritium direct-drive implosions on OMEGA
,”
Rev. Sci. Instrum.
93
(
9
),
093530
(
2022
).
8.
S. A.
Slutz
,
M. C.
Herrmann
,
R. A.
Vesey
,
A. B.
Sefkow
,
D. B.
Sinars
,
D. C.
Rovang
,
K. J.
Peterson
, and
M. E.
Cuneo
, “
Pulsed-power-driven cylindrical liner implosions of laser preheated fuel magnetized with an axial field
,”
Phys. Plasmas
17
(
5
),
056303
(
2010
).
9.
M. R.
Gomez
,
S. A.
Slutz
,
A. B.
Sefkow
,
D. B.
Sinars
,
K. D.
Hahn
,
S. B.
Hansen
,
E. C.
Harding
,
P. F.
Knapp
,
P. F.
Schmit
,
C. A.
Jennings
,
T. J.
Awe
,
M.
Geissel
,
D. C.
Rovang
,
G. A.
Chandler
,
G. W.
Cooper
,
M. E.
Cuneo
,
A. J.
Harvey-Thompson
,
M. C.
Herrmann
,
M. H.
Hess
,
O.
Johns
,
D. C.
Lamppa
,
M. R.
Martin
,
R. D.
McBride
,
K. J.
Peterson
,
J. L.
Porter
,
G. K.
Robertson
,
G. A.
Rochau
,
C. L.
Ruiz
,
M. E.
Savage
,
I. C.
Smith
,
W. A.
Stygar
, and
R. A.
Vesey
, “
Experimental demonstration of fusion-relevant conditions in magnetized liner inertial fusion
,”
Phys. Rev. Lett.
113
(
15
),
155003
(
2014
).
10.
M. R.
Gomez
,
S. A.
Slutz
,
C. A.
Jennings
,
D. J.
Ampleford
,
M. R.
Weis
,
C. E.
Myers
,
D. A.
Yager-Elorriaga
,
K. D.
Hahn
,
S. B.
Hansen
,
E. C.
Harding
,
A. J.
Harvey-Thompson
,
D. C.
Lamppa
,
M.
Mangan
,
P. F.
Knapp
,
T. J.
Awe
,
G. A.
Chandler
,
G. W.
Cooper
,
J. R.
Fein
,
M.
Geissel
,
M. E.
Glinsky
,
W. E.
Lewis
,
C. L.
Ruiz
,
D. E.
Ruiz
,
M. E.
Savage
,
P. F.
Schmit
,
I. C.
Smith
,
J. D.
Styron
,
J. L.
Porter
,
B.
Jones
,
T. R.
Mattsson
,
K. J.
Peterson
,
G. A.
Rochau
, and
D. B.
Sinars
, “
Performance scaling in magnetized liner inertial fusion experiments
,”
Phys. Rev. Lett.
125
(
15
),
155002
(
2020
).
11.
E. C.
Harding
,
G. K.
Robertson
,
G. S.
Dunham
,
M. R.
Gomez
,
J. R.
Fein
,
P. F.
Knapp
,
A. J.
Harvey-Thompson
,
C. S.
Speas
,
D. J.
Ampleford
,
G. A.
Rochau
,
R.
Doron
, and
Y.
Maron
, “
X-ray self-emission imaging with spherically bent Bragg crystals on the Z-machine
,”
Rev. Sci. Instrum.
94
(
8
),
083509
(
2023
).
12.
J.
Ampleford
,
D. A.
Yager-Elorriaga
,
C. A.
Jennings
,
E. C.
Harding
,
M. R.
Gomez
,
A. J.
Harvey-Thompson
,
T. J.
Awe
,
G. A.
Chandler
,
G. S.
Dunham
,
M.
Geissel
,
K. D.
Hahn
,
S. B.
Hansen
,
P. F.
Knapp
,
D. C.
Lamppa
,
W. E.
Lewis
,
L.
Lucero
,
M.
Mangan
,
R.
Paguio
,
L.
Perea
,
G. A.
Robertson
,
C. L.
Ruiz
,
D. E.
Ruiz
,
P. F.
Schmit
,
S. A.
Slutz
,
G. E.
Smith
,
I. C.
Smith
,
C. S.
Speas
,
T. J.
Webb
,
M. R.
Weis
,
K.
Whittemore
,
E. P.
Yu
,
R. D.
McBride
,
K. J.
Peterson
,
B. M.
Jones
,
G. A.
Rochau
, and
D. B.
Sinars
, “
Controlling morphology and improving reproducibility of magnetized liner inertial fusion experiments
,”
Phys. Plasmas
31
(
2
),
022703
(
2024
).
13.
P. F.
Knapp
,
M. R.
Gomez
,
S. B.
Hansen
,
M. E.
Glinsky
,
C. A.
Jennings
,
S. A.
Slutz
,
E. C.
Harding
,
K. D.
Hahn
,
M. R.
Weis
,
M.
Evans
,
M. R.
Martin
,
A. J.
Harvey-Thompson
,
M.
Geissel
,
I. C.
Smith
,
D. E.
Ruiz
,
K. J.
Peterson
,
B. M.
Jones
,
J.
Schwarz
,
G. A.
Rochau
,
D. B.
Sinars
,
R. D.
McBride
, and
P. A.
Gourdain
, “
Origins and effects of mix on magnetized liner inertial fusion target performance
,”
Phys. Plasmas
26
(
1
),
012704
(
2019
).
14.
A. J.
Harvey-Thompson
,
M. R.
Weis
,
E. C.
Harding
,
M.
Geissel
,
D. J.
Ampleford
,
G. A.
Chandler
,
J. R.
Fein
,
M. E.
Glinsky
,
M. R.
Gomez
,
K. D.
Hahn
,
S. B.
Hansen
,
C. A.
Jennings
,
P. F.
Knapp
,
R. R.
Paguio
,
L.
Perea
,
K. J.
Peterson
,
J. L.
Porter
,
P. K.
Rambo
,
G. K.
Robertson
,
G. A.
Rochau
,
D. E.
Ruiz
,
J.
Schwarz
,
J. E.
Shores
,
D. B.
Sinars
,
S. A.
Slutz
,
G. E.
Smith
,
I. C.
Smith
,
C. S.
Speas
, and
K.
Whittemore
, “
Diagnosing and mitigating laser preheat induced mix in MagLIF
,”
Phys. Plasmas
25
(
11
),
112705
(
2018
).
15.
A. K.
Chattopadhyay
,
A.
Anand
, and
C. V.
Rao
, “
Tomography for SST-1 tokamak with pixel method
,”
Rev. Sci. Instrum.
76
(
6
),
063502
(
2005
).
16.
K. M.
Hanson
and
G. W.
Wecksung
, “
Local basis-function approach to computed tomography
,”
Appl. Opt.
24
(
23
),
4028
(
1985
).
17.
M.
Kassubeck
,
S.
Wenger
,
C. A.
Jennings
,
M.
Gomez
,
E.
Harding
,
J.
Schwarz
, and
M.
Magnor
, “Data-driven compressed sensing tomography,” in IS and T International Symposium on Electronic Imaging Science and Technology, No. 1 (Society for Imaging Science and Technology, 2018).
18.
E. W.
Hansen
, “
Theory of circular harmonic image reconstruction
,”
J. Opt. Soc. Am.
71
(
3
),
304
308
(
1981
).
19.
Y.
Nagayama
, “
Tomography of m = 1 mode structure in tokamak plasma using least-square-fitting method and Fourier – Bessel expansions
,”
J. Appl. Phys.
62
(
7
),
2702
2706
(
1987
).
20.
J.
Mlynar
,
V.
Weinzettl
,
G.
Bonheure
, and
A.
Murari
, “
Inversion techniques in the soft-X-ray tomography of fusion plasmas: Toward real-time applications
,”
Fusion Sci. Technol.
58
(
3
),
733
741
(
2010
).
21.
E. S.
Lavine
,
S. V.
Rocco
,
J. T.
Banasek
,
W. M.
Potter
,
J. B.
Greenly
,
H.
Wilhelm
,
N.
Qi
,
D. A.
Hammer
, and
B. R.
Kusse
, “
Implosion dynamics of triple-nozzle gas-puff z pinches on COBRA
,”
Phys. Plasmas
28
(
2
),
022703
(
2021
).
22.
P. F.
Knapp
,
M. E.
Glinsky
,
M. A.
Schaeuble
,
C. A.
Jennings
,
M.
Evans
,
J.
Gunning
,
T. J.
Awe
,
G. A.
Chandler
,
M.
Geissel
,
M. R.
Gomez
,
K. D.
Hahn
,
S. B.
Hansen
,
E. C.
Harding
,
A. J.
Harvey-Thompson
,
S.
Humane
,
B. T.
Klein
,
M.
Mangan
,
T.
Nagayama
,
A. J.
Porwitzky
,
D. E.
Ruiz
,
P. F.
Schmit
,
S. A.
Slutz
,
I. C.
Smith
,
M. R.
Weis
,
D. A.
Yager-Elorriaga
,
D. J.
Ampleford
,
K.
Beckwith
,
T. R.
Mattsson
,
K. J.
Peterson
, and
D. B.
Sinars
, “
Estimation of stagnation performance metrics in magnetized liner inertial fusion experiments using Bayesian data assimilation
,”
Phys. Plasmas
29
(
5
),
052711
(
2022
).
23.
S.
Siltanen
,
V.
Kolehmainen
,
S.
Järvenpää
,
J. P.
Kaipio
,
P.
Koistinen
,
M.
Lassas
,
J.
Pirttilä
, and
E.
Somersalo
, “
Statistical inversion for medical x-ray tomography with few radiographs: I. General theory
,”
Phys. Med. Biol.
48
(
10
),
1437
1463
(
2003
).
24.
K. M.
Hanson
, “Bayesian and related methods in image reconstruction from incomplete data,” in Image Recovery: Theory and Applications (Academic, 1987), Vol. 125, pp. 79–125.
25.
G. T.
Herman
and
R.
Davidi
, “
Image reconstruction from a small number of projections
,”
Inverse Problems
24
(
4
),
045011
(
2008
).
26.
L.
Medeiros
,
D.
Psaltis
,
T. R.
Lauer
, and
F.
Özel
, “
Principal-component interferometric modeling (PRIMO), an algorithm for EHT data. I. Reconstructing images from simulated EHT observations
,”
Astrophys. J.
943
(
2
),
144
(
2023
).
27.
J.
Liu
,
Y.
Hu
,
J.
Yang
,
Y.
Chen
,
H.
Shu
,
L.
Luo
,
Q.
Feng
,
Z.
Gui
, and
G.
Coatrieux
, “
3D feature constrained reconstruction for low-dose CT imaging
,”
IEEE Trans. Circuits Syst. Video Technol.
28
(
5
),
1232
1247
(
2018
).
28.
L.
Shao
,
R.
Yan
,
X.
Li
, and
Y.
Liu
, “
From heuristic optimization to dictionary learning: A review and comprehensive comparison of image denoising algorithms
,”
IEEE Trans. Cyber.
44
(
7
),
1001
1013
(
2014
).
29.
L.
Ma
,
L.
Moisan
,
J.
Yu
, and
T.
Zeng
, “
A dictionary learning approach for Poisson image deblurring
,”
IEEE Trans. Med. Imaging
32
(
7
),
1277
1289
(
2013
).
30.
T. J.
Awe
,
C. A.
Jennings
,
R. D.
McBride
,
M. E.
Cuneo
,
D. C.
Lamppa
,
M. R.
Martin
,
D. C.
Rovang
,
D. B.
Sinars
,
S. A.
Slutz
,
A. C.
Owen
,
K.
Tomlinson
,
M. R.
Gomez
,
S. B.
Hansen
,
M. C.
Herrmann
,
M. C.
Jones
,
J. L.
McKenney
,
G. K.
Robertson
,
G. A.
Rochau
,
M. E.
Savage
,
D. G.
Schroen
, and
W. A.
Stygar
, “
Modified helix-like instability structure on imploding z-pinch liners that are pre-imposed with a uniform axial magnetic field
,”
Phys. Plasmas
21
(
5
),
056303
(
2014
).
31.
D. A.
Yager-Elorriaga
,
Y. Y.
Lau
,
P.
Zhang
,
P. C.
Campbell
,
A. M.
Steiner
,
N. M.
Jordan
,
R. D.
McBride
, and
R. M.
Gilgenbach
, “
Evolution of sausage and helical modes in magnetized thin-foil cylindrical liners driven by a Z-pinch
,”
Phys. Plasmas
25
(
5
),
056307
(
2018
).
32.
M. E.
Glinsky
,
T. W.
Moore
,
W. E.
Lewis
,
M. R.
Weis
,
C. A.
Jennings
,
D. J.
Ampleford
,
P. F.
Knapp
,
E. C.
Harding
,
M. R.
Gomez
, and
A. J.
Harvey-Thompson
, “
Quantification of MagLIF morphology using the Mallat scattering transformation
,”
Phys. Plasmas
27
(
11
),
112703
(
2020
).
33.
W. E.
Lewis
,
P. F.
Knapp
,
E. C.
Harding
, and
K.
Beckwith
, “
Statistical characterization of experimental magnetized liner inertial fusion stagnation images using deep-learning-based fuel-background segmentation
,”
J. Plasma Phys.
88
(
5
),
895880501
(
2022
).
34.
W. E.
Lewis
,
E. C.
Harding
,
D. A.
Yager-Elorriaga
,
J. R.
Fein
,
P. F.
Knapp
,
K.
Beckwith
, and
D. J.
Ampleford
, “A framework for experimental-data-driven assessment of Magnetized Liner Inertial Fusion stagnation image metrics,” arXiv:2303.15680 (2023).
35.
B. B.
Kadomtsev
, “
Hydromagnetic stability of a plasma
,”
Rev. Plasma Phys.
2
,
153
199
(
1966
).
36.
G.
Jacovitti
and
A.
Neri
, “
Multiresolution circular harmonic decomposition
,”
IEEE Trans. Signal Process.
48
(
11
),
3242
3247
(
2000
).
37.
C.
Bishop
,
Pattern Recognition and Machine Learning
, 2nd ed. (
Springer
,
2006
).
38.
Y. E.
Nesterov
, “
A method of solving a convex programming problem with convergence rate O ( 1 / k 2 )
,”
Dokl. Akad. Nauk.
269
(
3
),
543
547
(
1983
).
39.
A.
Beck
and
M.
Teboulle
, “
A fast iterative shrinkage-thresholding algorithm for linear inverse problems
,”
SIAM J. Imaging Sci.
2
(
1
),
183
202
(
2009
).
40.
S.
Wenger
,
M.
Ament
,
S.
Guthe
,
D.
Lorenz
,
A.
Tillmann
,
D.
Weiskopf
, and
M.
Magnor
, “
Visualization of astronomical nebulae via distributed multi-GPU compressed sensing tomography
,”
IEEE Trans. Vis. Comput. Graph.
18
(
12
),
2188
2197
(
2012
).
41.
This least-squares estimate also includes a small term, β 0 I, corresponding to an additional L 2-norm | f | 2 2 in the objective function, to mitigate very large-amplitude fluctuations/noise in this initial step. We only need to use a small value β 0 = 10 5 for this term to be effective.
42.
A. J.
Laub
,
Matrix Analysis for Scientists and Engineers
(
SIAM
,
2004
).
43.
We acknowledge that this designation is a misnomer given the additional complexity in the model, but still use it to distinguish from truly out-of-sample volumes in Sec. IV B.
44.
This test was done with B = 600, using 300 right-handed and 300 left-handed basis vectors.
45.
M. M.
Marinak
,
G. D.
Kerbel
,
N. A.
Gentile
,
O.
Jones
,
D.
Munro
,
S.
Pollaine
,
T. R.
Dittrich
, and
S. W.
Haan
, “
Three-dimensional HYDRA simulations of National Ignition Facility targets
,”
Phys. Plasmas
8
(
5
),
2275
2280
(
2001
).
46.
M. R.
Weis
,
A. J.
Harvey-Thompson
, and
D. E.
Ruiz
, “
Scaling laser preheat for MagLIF with the Z-Beamlet laser
,”
Phys. Plasmas
28
(
1
),
012705
(
2021
).
47.
The n = 1 and n > 4 orders contribute negligibly to HRCXI images due to filtering at low energies and diminishing reflectivity and emissivities at higher energies.
48.
M. R.
Gomez
,
S. A.
Slutz
,
P. F.
Knapp
,
K. D.
Hahn
,
M. R.
Weis
,
E. C.
Harding
,
M.
Geissel
,
J. R.
Fein
,
M. E.
Glinsky
,
S. B.
Hansen
et al., “
Assessing stagnation conditions and identifying trends in magnetized liner inertial fusion
,”
IEEE Trans. Plasma Sci.
47
(5),
2081
2101
(
2019
).
49.
R.
Epstein
,
V. N.
Goncharov
,
F. J.
Marshall
,
R.
Betti
,
R.
Nora
,
A. R.
Christopherson
,
I. E.
Golovkin
, and
J. J.
Macfarlane
, “
X-ray continuum as a measure of pressure and fuel-shell mix in compressed isobaric hydrogen implosion cores
,”
Phys. Plasmas
22
(
2
),
022707
(
2015
).
50.
Each point is averaged over eight images uniformly spaced in azimuth, comprising the same set of projections used in the N v = 2 estimates.
51.
R.
Betti
,
P. Y.
Chang
,
B. K.
Spears
,
K. S.
Anderson
,
J.
Edwards
,
M.
Fatenejad
,
J. D.
Lindl
,
R. L.
Mccrory
,
R.
Nora
, and
D.
Shvarts
, “
Thermonuclear ignition in inertial confinement fusion and comparison with magnetic confinement
,”
Phys. Plasmas
17
(
5
),
058102
(
2010
).
52.
M. R.
Gomez
, “Performance scaling with drive parameters in magnetized liner inertial fusion experiments,” Phys. Plasmas (unpublished).
53.
M.
Aharon
,
M.
Elad
, and
A.
Bruckstein
, “
K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation
,”
IEEE Trans. Signal Process.
54
(
11
),
4311
4322
(
2006
).
54.
The FWHM increases with super-Gaussian order, which helps maintain the integrated intensity within a given radius.