Computed Tomography Imaging Spectroscopy

• Computed Tomography Imaging Spectroscopy (CTIS) is a computational imaging technique that jointly captures spatial and spectral information using diffraction and inversion models.
• CTIS employs snapshot imaging with diffractive optical elements and iterative algorithms (such as EM and CNN-based models) to reconstruct full 3D datacubes from multiplexed measurements.
• CTIS is applied in optical and X-ray imaging for remote sensing, material decomposition, and agro-food analytics, offering enhanced motion resilience, reduced costs, and rapid analysis.

Computed Tomography Imaging Spectroscopy (CTIS) denotes imaging architectures in which spatial and spectral information are acquired jointly and recovered computationally from multiplexed measurements. In snapshot hyperspectral imaging, CTIS captures a full 3D spatial-spectral datacube in a single camera exposure, typically by using a 2D diffraction grating that forms a $3\times3$ array of images on a monochrome sensor; reconstruction is posed as $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$ or, in the noise-free form used for algorithmic analysis, $\mathbf{g}=\mathbf{H}\mathbf{f}$ (Ahlebæk et al., 27 Aug 2025, Peters et al., 2024, White et al., 2020). In X-ray imaging, closely related work uses hyperspectral bright field imaging, multi-energy projections, or multi-spectral computed tomography to obtain per-voxel spectra across energy bins for material differentiation, elemental mapping, and quantitative decomposition (Warr et al., 2021, Luna et al., 2023, Kehl et al., 2018, Ran et al., 2023). This suggests that CTIS functions less as a single hardware design than as a family of computational imaging systems whose common feature is the inversion of a spectral forward model.

1. Forward models and spectral data structures

In optical snapshot CTIS, the detector measurement is modeled as a linear transform of the sought datacube. The standard representation is

$\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$

where $\mathbf{g}$ is the vectorized observed 2D CTIS diffraction image, $\mathbf{f}$ is the vectorized 3D scene datacube, $\mathbf{H}$ is the system matrix encoding diffraction, transmission, efficiency, optics, and illumination, and $\mathbf{n}$ is noise (Ahlebæk et al., 27 Aug 2025). For the HEIMDAL instrument, $\mathbf{f}$ has size $312\times312\times(145\,\text{or}\,236)$ voxels and $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$0 has size $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$1 pixels; in the grape-quality study, reconstructed datacubes were $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$2 for EM reconstruction or $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$3 for U-Net reconstruction (Ahlebæk et al., 27 Aug 2025, Peters et al., 2024).

Because the system is underdetermined, regularized or iterative algorithms must be used. In the optical papers considered here, two reconstruction families dominate: Expectation-Maximization (EM), formulated as an iterative tomographic inversion, and convolutional neural network approaches that are physics-guided and informed by $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$4 (Ahlebæk et al., 27 Aug 2025). The raw detector image is therefore not itself the hyperspectral datacube; it is an encoded projection whose spatial and spectral content overlap across diffraction orders.

In X-ray CTIS-related work, the data structure changes but the same inverse-model logic remains. Hyperspectral bright field imaging records each X-ray photon’s position and energy, filling up to approximately 200 energy channels per pixel, while multi-energy micro-CT and multi-spectral CT represent each voxel by a vector of attenuation values across energy bins (Warr et al., 2021, Luna et al., 2023, Kehl et al., 2018). In the material-decomposition framework, each voxel receives a feature vector $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$5, where $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$6 denotes the mass attenuation value at energy window $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$7 (Luna et al., 2023). This suggests a shared abstraction across modalities: CTIS data are not single-channel intensities but high-dimensional spectral signatures indexed by spatial coordinates.

2. Optical architectures for snapshot hyperspectral CTIS

The optical CTIS systems described in the cited work use a 2D diffractive optical element (DOE) in the optical path. Incoming light from the scene is dispersed by the DOE, forming a $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$8 array of images on a monochrome sensor, consisting of one central undispersed “0th order” and eight first-order diffracted images (Ahlebæk et al., 27 Aug 2025). Each diffraction order contains overlapping information about scene spatial and spectral content, so recovering the original scene at each wavelength requires computational inversion.

The HEIMDAL balloon instrument was designed as a snapshot hyperspectral camera for Earth observation. Its hardware included a monochrome 4 MP GSENSE2020 CMOS sensor, a custom 2D diffractive optical element, two 35 mm Vis-NIR lenses plus an outer 50 mm Vis-NIR lens, and Thorlabs longpass and shortpass filters defining a spectral range of 600–850 nm (Ahlebæk et al., 27 Aug 2025). The reconstructed spatial dimensions were $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n}$9 pixels, with spectral resolution reported as 145 channels for CNN reconstruction or 236 channels for EM reconstruction, and a stated goal of $\mathbf{g}=\mathbf{H}\mathbf{f}$0 nm (Ahlebæk et al., 27 Aug 2025).

The grape-quality system used the same basic snapshot principle: a monochrome 4 MP GSENSE2020 CMOS sensor, a custom DOE, lenses suitable for 600–850 nm, and 600 nm longpass and 850 nm shortpass filters (Peters et al., 2024). The DOE disperses incoming light onto the detector, resulting in a $\mathbf{g}=\mathbf{H}\mathbf{f}$1 diffraction pattern comprising the zeroth order and first orders. In contrast to pushbroom line-scan hyperspectral imaging, CTIS acquires the entire scene $\mathbf{g}=\mathbf{H}\mathbf{f}$2 from a static shot rather than building the 3D datacube through object movement (Peters et al., 2024).

A central engineering consequence of this architecture is the absence of mechanical scanning parts. For high-altitude balloon platforms, this was presented as less susceptible to failure in low pressure and cold environments, while for agricultural measurements it was associated with reduced susceptibility to motion errors, compactness, portability, and lower cost (Ahlebæk et al., 27 Aug 2025, Peters et al., 2024). A plausible implication is that CTIS trades optical simplicity and motion resilience for greater inversion complexity.

3. Reconstruction algorithms, shift-invariance, and acceleration

A core computational bottleneck in CTIS is the repeated application of the system matrix $\mathbf{g}=\mathbf{H}\mathbf{f}$3 and its adjoint during iterative reconstruction. The acceleration paper develops a highly parallelizable algorithm, referred to as the WBH algorithm, that exploits spatial shift-invariance in the CTIS system matrix (White et al., 2020). The measurement model is written as

$\mathbf{g}=\mathbf{H}\mathbf{f}$4

with $\mathbf{g}=\mathbf{H}\mathbf{f}$5 partitioned as

$\mathbf{g}=\mathbf{H}\mathbf{f}$6

and each spectral-band block represented as

$\mathbf{g}=\mathbf{H}\mathbf{f}$7

where $\mathbf{g}=\mathbf{H}\mathbf{f}$8 is an $\mathbf{g}=\mathbf{H}\mathbf{f}$9 circulant matrix and $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$0 is a sparse selection or mapping matrix (White et al., 2020). Because circulant matrices diagonalize under the Fourier transform, forward projection can be evaluated as

$\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$1

with $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$2 precomputed (White et al., 2020).

The iterative solver demonstrated for this framework is Expectation Maximization: $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$3 All key operations involving $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$4 and $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$5 are then performed through FFT-based circulant computations, custom index mapping routines, GPU kernels, and Nvidia’s cuFFT libraries (White et al., 2020). The reported complexity drops from brute-force $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$6 to $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$7 for the WBH algorithm, with improved space complexity as well (White et al., 2020).

The reported performance gains are substantial. For a realistic data size of $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$8 FPA and 75 spectral bands, the method reduced reconstruction time from approximately 265 s for BF+EM+CPU to 3.8 s on a Quadro P4000 desktop GPU and 7.0 s on an Nvidia Jetson AGX Xavier embedded GPU for 25 iterations (White et al., 2020). For 24 spectral bands, times went below 1 second for 10 iterations, and mean pixel-wise relative error was reported as less than $\mathbf{g}=\mathbf{H}\mathbf{f}+\mathbf{n},$9 compared to traditional EM approaches (White et al., 2020).

Alongside EM, optical CTIS has also been reconstructed with neural networks. The HEIMDAL study used a CNN described as a physics-guided neural network, specifically an autodecoder/UNet that approximates the inversion while leveraging knowledge of $\mathbf{g}$0 (Ahlebæk et al., 27 Aug 2025). The grape study used a U-Net trained on paired ground-truth datacubes and synthetic or experimental CTIS images, outputting 45-band hyperspectral datacubes; preprocessing included dark frame subtraction, masking, thresholding, and averaging spectra over grape area (Peters et al., 2024). These results establish that CTIS reconstruction is not algorithmically monolithic: FFT-accelerated iterative solvers, EM, and learned inversions coexist, with different trade-offs in spectral resolution, compute, and generalization.

4. Remote sensing and agro-food deployments

The HEIMDAL mission demonstrated CTIS on a stratospheric high-altitude balloon platform for Earth observation. The payload was integrated into the BEXUS HAB gondola, mechanically robust up to 20 $\mathbf{g}$1 shocks, and accompanied by an RGB camera, on-board computer, and electronics or power box (Ahlebæk et al., 27 Aug 2025). On-board operation used snapshot images at 1 FPS with approximately 10 ms exposure, storage every second, on-board datacube reconstructions for every 10th image, three SSDs for data storage, and selective transmission via E-LINK at 1 image per 90 s (Ahlebæk et al., 27 Aug 2025). Pre-flight environmental testing in the Mars Simulation Laboratory exposed the system to $\mathbf{g}$2 mbar, $\mathbf{g}$3 to $\mathbf{g}$4, and durations exceeding 5 hours; SSDs required multilayer insulation, and the system operated successfully (Ahlebæk et al., 27 Aug 2025). During flight, the instrument reached 28 km altitude in a mission longer than 5 hours, recorded all images successfully, survived recovery with all data intact, and experienced only minor condensation in late-phase images (Ahlebæk et al., 27 Aug 2025).

The practical rationale for CTIS in this setting was motion resilience. Traditional pushbroom HSI requires precise knowledge or control of platform velocity, often impractical for balloons that can rotate and oscillate, whereas CTIS snapshot imaging captures the entire datacube per exposure, greatly reducing artifacts from relative motion (Ahlebæk et al., 27 Aug 2025). The HEIMDAL study reported proof-of-concept land or water classification using Partial Least Squares Discriminant Analysis, with both CNN and EM reconstructions allowing discrimination of water versus land and better accuracy for CNN at approximately 0.8; Variable Importance in Projection metrics indicated the most useful spectral features around 650 nm (Ahlebæk et al., 27 Aug 2025).

In agro-food sensing, CTIS was evaluated for prediction of °Brix and pH in Sheegene 20 table grapes through Partial Least Squares Regression. One hundred grapes were imaged with both a CTIS system and a state-of-the-art line scan HSI system, and reference measurements were obtained directly using a refractometer and a pH meter (Peters et al., 2024). The reported comparative results for the 600–850 nm range were as follows (Peters et al., 2024):

System	°Brix	pH
Line scan, 180 bands	$\mathbf{g}$5, RMSECV $\mathbf{g}$6, MAECV $\mathbf{g}$7	$\mathbf{g}$8, RMSECV $\mathbf{g}$9, MAECV $\mathbf{f}$0
CTIS (EM), 236 bands	$\mathbf{f}$1, RMSECV $\mathbf{f}$2, MAECV $\mathbf{f}$3	$\mathbf{f}$4, RMSECV $\mathbf{f}$5, MAECV $\mathbf{f}$6
CTIS (U-Net), 45 bands	$\mathbf{f}$7, RMSECV $\mathbf{f}$8, MAECV $\mathbf{f}$9	$\mathbf{H}$0, RMSECV $\mathbf{H}$1, MAECV $\mathbf{H}$2
Line scan, full range, 800 bands	$\mathbf{H}$3, RMSECV $\mathbf{H}$4, MAECV $\mathbf{H}$5	$\mathbf{H}$6, RMSECV $\mathbf{H}$7, MAECV $\mathbf{H}$8

The study concluded that, with the same limited spectral range of 600–850 nm, the predictive accuracy of CTIS was comparable to the line scan system for grape °Brix and pH, while the full-range line scan system performed better because it had access to more informative NIR bands (Peters et al., 2024). CTIS was described as less sensitive to movement during acquisition, and the paper explicitly identified “lower cost, portability, and reduced susceptibility to motion errors” as advantages (Peters et al., 2024).

5. X-ray CTIS: hyperspectral tomography and spectral detector designs

In X-ray bioimaging, hyperspectral bright field imaging has been used to collect computed tomographic images with excellent energy resolution, reported as 800 eV, and to map the distribution of stain in a fixed biological sample through its characteristic K-edge (Warr et al., 2021). The HEXITEC detector used in that work was an energy-sensitive CdTe system with $\mathbf{H}$9 pixels and 250 $\mathbf{n}$0m pitch, recording each photon’s position and energy into up to approximately 200 energy channels per pixel (Warr et al., 2021). The central reconstruction problem was low count-rate and poor signal-to-noise ratio in energy-selective images, especially under low-dose or undersampled conditions.

The dedicated iterative reconstruction addressed this by jointly reconstructing all channels through a spatiospectral objective,

$\mathbf{n}$1

where $\mathbf{n}$2 is the reconstructed 4D volume, $\mathbf{n}$3 is the forward projection operator, $\mathbf{n}$4 enforces spatial regularity, and $\mathbf{n}$5 imposes smoothness and edge preservation along the spectral dimension (Warr et al., 2021). The optimization was solved with a Primal-Dual Hybrid Gradient algorithm in the open-source Core Imaging Library. For a multi-phase phantom, a 36 times reduction in scan time was demonstrated, and for an iodine-stained lizard head the method preserved the sharp K-edge step required for chemical mapping (Warr et al., 2021). Spectral analysis included K-edge subtraction and absorption step-size fitting, yielding spatial maps of iodine uptake in soft tissues and separation from bone (Warr et al., 2021).

A distinct detector strategy for multi-energy CT employed side-illuminated X-ray scintillation using Cs$\mathbf{n}$6Cu$\mathbf{n}$7I$\mathbf{n}$8 metal halide. In this geometry, the scintillator is arranged parallel to the X-ray beam, so lower-energy photons are absorbed near the entry face while higher-energy photons penetrate deeper, producing a non-uniform scintillation intensity profile that encodes the incident spectrum (Ran et al., 2023). The method is governed by Beer’s law,

$\mathbf{n}$9

and the discretized spectral reconstruction is written as

$\mathbf{f}$0

A proof-of-concept multi-energy CT imaging system featuring eight energy channels was implemented, and the relative error between reconstructed and measured X-ray spectra was reported as less than $\mathbf{f}$1 (Ran et al., 2023). The study emphasized that this energy-resolving capability was obtained without extra hardware components beyond reorientation and processing, and that multi-energy CT images provided material discrimination not available from conventional energy-integration approaches (Ran et al., 2023).

Taken together, these X-ray studies show that CTIS can refer not only to snapshot optical diffraction encoding but also to hyperspectral tomographic acquisition and reconstruction in which each voxel carries an energy-resolved signature. This suggests that the operative criterion is computational spectroscopy within tomographic imaging rather than any single sensing geometry.

6. Material decomposition, segmentation, and persistent challenges

Quantitative material decomposition is a major theme in X-ray CTIS-related research. The Iterative Clustering Material Decomposition (ICMD) pipeline combines empirical spectral correction, cluster analysis, and multi-step iterative material decomposition for high-resolution photon-counting detectors (Luna et al., 2023). Using a CdTe-based Medipix3RX photon counting detector with 55 $\mathbf{f}$2m pixel pitch in a tiled WidePix array, the work addressed inter-pixel variations, charge sharing, fluorescence, K-escape, pulse pileup, incomplete charge collection, and energy-dependent detector responses (Luna et al., 2023). The empirical signal-to-thickness calibration does not require detector response models and uses

$\mathbf{f}$3

followed by voxel-wise correction to mass attenuation coefficients (Luna et al., 2023). Clustering is then performed with a Gaussian Mixture Model,

$\mathbf{f}$4

and material decomposition uses the linear mixing model

$\mathbf{f}$5

In a phantom with potassium iodide, gadopentetic acid, hydroxyapatite, and water, the method quantitatively separated all four materials using five energy bins from 20–60 keV; reported iodine and gadolinium mass fractions were within approximately 4–9% error of nominal values, and the corrected attenuation values closely matched NIST standards (Luna et al., 2023). The same procedure separated bone, muscle, and adipose tissue in ex vivo mouse imaging (Luna et al., 2023).

For unsupervised material segmentation, the MUSIC work introduced an open-access multi-spectral CT dataset and compared Fast Adaptive Mean Shift with unconstrained graph cuts (Kehl et al., 2018). MUSIC2D contains 32 spectral images, including 11 single-material reference scans and 21 multi-object scans, while MUSIC3D contains 7 volumetric spectral scans acquired with a CdTe detector spanning 20–160 keV and up to 128 energy bins (Kehl et al., 2018). Reconstructions used algebraic reconstruction technique with total variation, and adaptive, anisotropic spectral binning was proposed to concentrate representation in the most informative energy ranges (Kehl et al., 2018). Under isotropic binning, graph cuts produced statistically and visually acceptable segmentations with Dice coefficients of approximately 0.65 on MUSIC2D and approximately 0.54 on MUSIC3D; with adaptive binning, Dice coefficients improved to approximately 0.66 and approximately 0.73, respectively (Kehl et al., 2018). The same study emphasized that acquisition noise, tomographic reconstruction artefacts, and scanning setup application constraints remain serious obstacles, and that metal artefacts can make spectral material separation unfeasible, making Metal Artefact Reduction necessary (Kehl et al., 2018).

Across the optical and X-ray literature, several misconceptions are explicitly contradicted by the reported evidence. Snapshot acquisition does not remove the need for heavy post-processing: accurate datacube extraction from CTIS images remains a significant computational challenge, vulnerable to reconstruction errors and reliant on advanced algorithms such as EM, CNNs, U-Net variants, PDHG, TV, and TGV (Peters et al., 2024, Warr et al., 2021). Spectral information does not automatically guarantee reliable material differentiation: detector nonuniformities, charge sharing, low count-rates, undersampled projections, and metal artefacts can dominate performance unless calibration, correction, or regularization are carefully designed (Luna et al., 2023, Kehl et al., 2018). Conversely, the studies also show that these limitations are not merely theoretical. Real-time or embedded optical CTIS on commodity GPUs, stratospheric balloon deployment, quantitative decomposition of more than three materials, and hyperspectral tomography with 36 times shorter scan time have all been demonstrated within the constraints stated by the respective papers (White et al., 2020, Ahlebæk et al., 27 Aug 2025, Luna et al., 2023, Warr et al., 2021).