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Informed Sinogram Interpolation for Sparse View Reconstruction

Yuejie Liu, Alessandro Lupoli, David Uribe Gallo, Andreas Fischer, Felix Krahmer

arXiv:2609.18415v1eess.IV

TL;DR

Micro-CT off-focal radiation can corrupt projections, and collecting a full matching set of secondary scans doubles acquisition time. This paper models secondary sinograms as unions of sinusoidal curves and uses that model to interpolate missing secondary scans from limited observations. On the studied coil dataset, interpolation with N = 25 eliminated the artifact, while reducing N from 5 to 4 noticeably degraded reconstruction quality.

Abstract

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Computed tomography (CT) has been widely used in medical examinations and non-destructive testing. Micro-CT (microfocus X-ray CT system) is an advanced version that can observe the internal structures of small objects. However, secondary radiation can prevent micro-CT from imaging the full structure of the object. Given the same amount of secondary scans as primary scans, simple subtraction solves the problem at the price of doubled acquisition time. To reduce the acquisition time, one aims to limit the secondary scans to as few imaging angles as possible and interpolate to recover the missing data. Different interpolation methods have been explored for limited-angle tomography, such as polynomial or spline interpolation, compressive sensing, deep learning, etc. In contrast to many of these scenarios, the secondary imaging setup considered in this paper often exhibits simple sinusoidal structures that are not exploited directly by any of the aforementioned approaches. This paper aims to fill this gap with a continuous mathematical model describing the secondary sinogram only with information from the projection space. Building on this model, we propose an interpolation method that effectively interpolates secondary scans from limited observations.

I. INTRODUCTION

Micro-CT off-focal radiation adds secondary projections that can corrupt primary scans, while acquiring a full matching set of secondary scans doubles acquisition time. The paper proposes acquiring secondary scans at limited angles and interpolating the missing data for artifact correction.

I. INTRODUCTION

  • The Beer-Lambert law describes attenuation of monochromatic X-rays, and the line integral of the attenuation coefficient is the Radon transform used to represent projections as a sinogram.The Radon-transform formulation is introduced for parallel-beam geometry; FBP and FDK are cited as reconstruction algorithms for parallel- and cone-beam geometry, respectively.Source: §I · p. 1

A. Secondary radiation artifacts

  • Micro-CT projections combine intended primary radiation with off-focal secondary radiation, which can produce artifacts in scans and reconstructed images.The paper focuses on transmission microfocus X-ray tubes and describes four possible mechanisms of off-focal X-ray generation.Source: A. Secondary radiation artifacts · pp. 1–2
  • Subtracting a corresponding secondary scan from each primary scan can remove the artifact, but acquiring a full secondary sequence doubles acquisition time.The proposed alternative is to interpolate missing angles from secondary scans acquired at a limited number of angles.Source: A. Secondary radiation artifacts · p. 2
  • Off-focal radiation often forms a radially symmetric, donut-shaped component, and its intensity sinogram resembles parallel-beam X-ray behavior.The specific artifact varies with X-ray tube design and materials.Source: A. Secondary radiation artifacts · pp. 2–3

B. Image processing ingredients

The image-processing steps account for detector background and isolate the useful artifact region before reconstruction. They use background-only images for correction and a thresholded mask with a bounding box to suppress surrounding noise.

B. Image processing ingredients

  • A secondary background-only image approximates the constant intensity component in the secondary sinogram that represents X-ray energy unaffected by the sample.A primary background-only image is denoted separately for the primary scans.Source: B. Image processing ingredients · p. 3
  • Median-based correction rescales the flat-field-corrected image, with median weighting preferred over mean weighting when the data are skewed or contain outliers.The corrected image is then used by FDK to reconstruct the sample’s attenuation coefficient.Source: B. Image processing ingredients · p. 3
  • A mask and bounding box reduce surrounding noise by isolating the useful information around the donut-shaped artifact.The mask is created by normalizing the secondary background-only image to [0, 255] and applying a binary threshold.Source: B. Image processing ingredients · p. 3, Figure 4 · p. 3

C. Modeling fundamentals

The section frames missing secondary scans as an interpolation problem and proposes a continuous model of their sinusoids for recovering unobserved angles. It works with cropped sinograms and processes scan rows sequentially, using prior reconstructions to initialize later rows.

C. Modeling fundamentals

  • The proposed continuous mathematical model describes the secondary intensity sinogram using its empirically observed sinusoidal structure.The model is intended to interpolate missing angles from projection-space information.Source: C. Modeling fundamentals · p. 3
  • Only some secondary scans are available, so the missing angles must be interpolated to support subtraction from primary scans.The paper considers a subsampled set of secondary-scan angles.Source: C. Modeling fundamentals · pp. 3–4
  • Rows are interpolated sequentially, with completed reconstructions used to initialize the following rows.Candidate-row absorption-sinogram interpolations are evaluated to find a good starting point.Source: C. Modeling fundamentals · p. 4
  • Interpolation focuses on the cropped central artifact region and completes all angles for each sinogram row.The scans are cropped to bounding-box dimensions Hcont × (2R); detector-row indices are not subsampled.Source: C. Modeling fundamentals · p. 4

II. PREVIOUS WORK

Previous work spans pixel-wise interpolation, methods that use sinogram structure, variational approaches, compressive sensing, deep learning, and inpainting. The paper notes that key assumptions or data requirements of some approaches do not fit secondary sinograms.

II. PREVIOUS WORK

  • Pixel-wise methods apply standard interpolation or robust smoothing independently at detector pixels, without explicitly using sinogram structure.The cited alternatives include nearest neighbor, linear, spline, Hermite, and iterative robust smoothing methods.Source: §II · p. 4
  • Structure-aware methods include sine-wave grouping and interpolation, weighted candidate sine waves, displacement-function interpolation, and directional interpolation.The cited approaches include,,, and.Source: §II · p. 4
  • Compressive sensing does not fit the described secondary-sinogram setting because its angular Fourier sparsity is absent or obscured by noise, and the angles are equidistant rather than random.The cited approach assumes random angle subsampling and sparse angular Fourier data.Source: §II · p. 4
  • Variational methods fill missing regions with regularization, while deep-learning and inpainting approaches face data or computational constraints for secondary projections.The review describes variational work on simulated and real data, and notes deep learning's dataset dependence, lack of secondary-projection training data, and computational expense [28] [29] [30]; inpainting methods often rely on deep learning,,,,,,,,,,,,,,, [46],.Source: §II · pp. 4–5

III. ALGORITHMIC INGREDIENTS

The initial model removes the secondary background and treats the resulting absorption sinogram as a blurred collection of sinusoidal traces. Estimating those traces from sampled angles can become ill-posed when their peaks are not sufficiently separated.

A. Background separation and initial model

  • Subtracting the secondary background-only image from the secondary intensity sinogram produces the secondary absorption sinogram.This removes the constant background component.Source: A. Background separation and initial… · p. 5
  • For tiny, well-separated regions of high absorption, the non-blurred sinogram is modeled as a finite union of sine curves with varying amplitudes, phases, offsets, and gray scales.Each curve has the form r = A sin(θ + ϕ) + β.Source: A. Background separation and initial… · p. 5
  • The model incorporates blur by convolving the sinusoidal structure with a kernel Kσ, whose scaling factor is assumed known.The illustrated secondary absorption sinogram resembles a blurred parallel-beam sinogram, with visible waves associated with highly absorbing parts.Source: A. Background separation and initial… · p. 5
  • Estimating curve gray values and peak locations at sampled angles is a Prony-type problem, but it requires well-separated peaks that are not guaranteed at every angle.Even a few violations of the separation assumption can prevent successful recovery.Source: A. Background separation and initial… · p. 6

B. Towards a more practical model

The section develops a continuous secondary-absorption sinogram model with finite-width sinusoids and a practical parameterization. It describes the discrete fitting objective and the constraints and computational benefits of the model.

B. Towards a more practical model

  • Convolving each sinusoid with a symmetric characteristic function gives it width, adding volume information omitted by the mass-point model.The convolution is introduced for non-blurred secondary absorption sinograms.Source: B. Towards a more practical model · p. 6
  • Replacing Dirac delta measures with χ[a,b) yields a continuous model whose asymmetric intervals can absorb imaging-center offset and need not center each sinusoid.The shift β is removed because [a_m, b_m) can include the imaging-center offset.Source: B. Towards a more practical model · p. 6
  • The model constrains 0 ∈ [a_m, b_m] to represent a small but nonzero β.Source: B. Towards a more practical model · p. 6
  • After coordinate translation, the model is expressed discretely and fitted by minimizing square loss, written as a Frobenius norm over sampled arrays.Source: B. Towards a more practical model · p. 6
  • The fitting uses a Cauchy kernel, which is smooth for L-BFGS-B optimization and has an explicit primitive that can accelerate optimization.The paper expects the approach not to depend specifically on the Cauchy-kernel choice.Source: B. Towards a more practical model · pp. 6–7
  • The practical model uses 5M parameters, accommodates incomplete sinograms or large β, and jointly handles θ and r rather than separately recovering parameters.Source: B. Towards a more practical model · p. 6

C. Constant background

Directly subtracting an interpolated secondary absorption sinogram from the background-only image can turn localized background errors into reconstruction line artifacts. The method therefore uses that model to select candidate rows, then restores background information in a refined intensity-sinogram model.

C. Constant background

  • Localized errors in the secondary background-only image can produce line artifacts when the interpolated absorption sinogram is subtracted from its row values.Source: C. Constant background · p. 7
  • The absorption model is used to identify candidate rows; after choosing a starting row, the method reintroduces background information into the model.Source: C. Constant background · p. 7
  • The refined model fits secondary intensity sinograms with background parameters d_k bounded by the maximum secondary background-only intensity.The initial d_k values come from corresponding gray values in the secondary background-only image; neighboring-level parameters initialize most other variables.Source: C. Constant background · p. 7

D. Incorporating 3D information

To recover sinograms across all levels, the method uses continuity in the h direction to regularize model parameters and initializes each level from its neighbor. It leaves the constant background values unregularized because they transition more sharply near the top and bottom.

D. Incorporating 3D information

  • Neighboring-level parameter initialization and regularization exploit the continuous change of sinusoids along h while discouraging abrupt parameter changes.The current level uses the previous level's model as its initialization and the partially observed sinogram as its observation.Source: D. Incorporating 3D information · p. 7
  • The regularization penalizes differences between neighboring levels' a parameters using an ℓ2 norm summed over m.Source: D. Incorporating 3D information · p. 7
  • The loss with these regularization terms can be minimized by L-BFGS-B because the terms are smooth.Source: D. Incorporating 3D information · p. 7

E. Multi-level sinogram interpolation algorithm

The algorithm selects a feasible starting level near the center, then optimizes neighboring levels in both directions using previously fitted parameters. It reconstructs sinograms at arbitrary angles and notes that multiple starts may help on other data.

E. Multi-level sinogram interpolation algorithm

  • The procedure takes secondary intensity sinograms at H + 1 levels, initializes from the central level, and returns optimized parameters for every level.Source: E. Multi-level sinogram interpolation… · p. 7
  • Starting from the selected level, the algorithm optimizes downward and upward, initializing each level from its already processed neighbor.Source: E. Multi-level sinogram interpolation… · p. 7
  • The fitted parameters are integrated into the model to compute interpolated sinograms at arbitrary angles.Source: E. Multi-level sinogram interpolation… · p. 7
  • Candidate starting levels near the center are tested for nonvanishing sine-wave parameters and noncollapsed intervals, and the floor of the feasible-level median is selected.The center provides the initialization; the paper allows stronger feasibility requirements, such as requiring more nonvanishing sine waves.Source: E. Multi-level sinogram interpolation… · pp. 7–8
  • One starting level suffices for the paper's data, though multiple starts may help across neighboring levels; the selection procedure is not guaranteed to find a feasible start.The authors suggest increasing ζ or tuning the base-level initialization as possible remedies.Source: E. Multi-level sinogram interpolation… · p. 8

IV. NUMERICAL RESULTS

On a real micro-CT coil dataset, interpolated secondary scans with N = 25 eliminated the secondary-radiation artifact, while reconstruction quality degraded from N = 5 to 4 for this dataset. The minimum scan count may vary with the object and measured data.

IV. NUMERICAL RESULTS

  • The real-data test used primary and secondary scans of a 1.6×0.8×0.8 mm coil acquired by a micro-CT system.The processed data included 1001 equidistant primary and original secondary scans, plus subsampled secondary scans for reconstruction.Source: §IV · p. 8, A. Recovery with multi-level sinogram… · p. 8
  • Because the first and last secondary scans share an angle, the scan at 2π was omitted from sinogram reconstruction.Source: §IV · p. 8

A. Recovery with multi-level sinogram interpolation algorithm

V. CONCLUSIONS

The paper presents a continuous model and interpolation algorithm for recovering secondary sinograms in micro-CT. It identifies automated parameter initialization and extension to more difficult settings as future work.

V. CONCLUSIONS

  • The paper’s contributions are a continuous model for secondary sinograms and a corresponding interpolation algorithm.Source: §V · p. 9
  • The approach reconstructs sparsely sampled secondary sinograms using simple box constraints and classical optimization methods with few parameters.Source: §V · p. 9
  • Future work includes automating parameter initialization and developing the method for ill-posed interpolation without object-attenuation information.The authors also identify anisotropic objects with more complex sinusoids whose gray values and width vary with θ as a direction to explore.Source: §V · p. 9
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