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Gradient-Based Trajectory Optimisation over Continuous Poses for Sparse-View Cone-Beam CT

Linda-Sophie Schneider, Simon Wittl, Gabriel Herl, Andreas Maier

arXiv:2610.09579v1cs.CV

TL;DR

Sparse-view CT trajectory design must choose source positions, but fixed candidate pools constrain off-grid refinement and require new pools for new acquisition manifolds. The paper continuously optimises each pose on a scanner kinematic manifold using an objective that combines coverage, attenuation, and view information. Continuous selection reaches discrete-search quality on the free sphere, while measured robot scans demonstrate feasibility without consistent metric gains over uniform band sampling.

  • Problem

    Fixed candidate pools restrict pose selection to predefined positions and require a new pool for each acquisition manifold, while population-based refinement becomes costly as each view adds variables.

    Source: §I · p. 1

  • Method

    The method optimises individual continuous source poses on a kinematic manifold using a differentiable objective that combines soft-Tuy coverage, attenuation-aware bundle penalty, and continuous VCL information.

    Source: §I · p. 1

  • Results

    Continuous selection reaches discrete swap-search quality on the free sphere; off-grid refinement adds four to five decibels on the asymmetric flange but less than one on the near-axisymmetric nozzle.

    Source: §VI · p. 10

  • Takeaways & Limitations

    The robot study establishes acquisition feasibility, while consistent measured image-quality gains remain to be demonstrated.

    Source: §VII · p. 11

    The optimiser is local and the objective is degenerate at the studied budgets; virtual photon-noise tests use one idealised monochromatic fluence and omit scatter, detector effects, and beam hardening.

    Source: §VI · p. 10

Abstract

from arXiv · show

Trajectory optimisation for cone-beam computed tomography (CT) determines which information sparse-view scans acquire. Fixed candidate pools prevent off-grid refinement and require new object-specific precomputation for each acquisition manifold. We make every source pose an individual continuous variable and move all poses jointly by gradient ascent on the scanner's kinematic manifold. The objective combines soft-Tuy plane coverage, continuous View Covariance Loss, and an analytic attenuation-aware ray-bundle penalty. The same optimiser handles circular, limited C-arm, two-axis, and freesphere parametrisations. On a Defrise flange, continuous selection recovers laminar defects invisible to a circular orbit, matches discrete swap search on the free sphere at the sparser budget, and leads at the denser one, with the same objective evaluated in every arm. A moderate elevation band already recovers most of the free-sphere gain at the defects, so the same optimiser transfers to bounded scanner envelopes. Photon noise preserves the ordering on the flange and compresses it on a dense fuel nozzle. Sparseprescan planning benefits from matching prescan and planned acquisition manifolds. Selection takes seconds rather than minutes without an object-specific reconstruction basis. Prescan-planned poses were executed on a robot CT bench and reconstructed in a common frame, demonstrating feasibility but no consistent metric gain over uniform band sampling. Continuous pose optimisation incorporates attenuation and scanner constraints directly into sparse-view acquisition design.

I. INTRODUCTION

Sparse-view CT makes source placement consequential, but fixed candidate pools constrain off-grid refinement and require new pools for new acquisition manifolds. This paper makes individual source poses continuous variables in a differentiable objective, combining geometric coverage, attenuation, and reconstruction information.

II. RELATED WORK

  • Prior selectors include discrete-pool coverage and VCLS, which uses a reconstruction-error surrogate but requires object-specific precomputation and does not model photon starvation.The paper positions its approach as retaining geometric and reconstruction-driven perspectives while making source positions differentiable.Source: §I · p. 1, §II · p. 2

I. INTRODUCTION

  • Fixed candidate pools restrict pose refinement to predefined positions and must be rebuilt for each acquisition manifold; population-based searches become costly as individual views add variables.The introduction identifies the missing ingredient as an analytic search direction requiring one objective evaluation per step rather than per sampled perturbation.Source: §I · p. 1
  • The method jointly refines each source pose as a continuous variable on a differentiable objective, avoiding candidate-pool restrictions.The approach targets off-grid refinement, richer-than-circular manifolds, and attenuation-aware planning.Source: §I · pp. 1–2
  • One objective combines soft-Tuy geometric coverage, an analytic attenuation-aware bundle term, and continuous VCL information for noise-free data.The terms respectively model angular coverage, ray attenuation under monochromatic Beer–Lambert transmission, and reconstruction-relevant information.Source: §I · pp. 1–2

II. RELATED WORK

  • The framework starts from greedy soft-Tuy selection and supports circular, limited C-arm, two-axis gantry, and free-sphere manifolds.Related work also includes continuous optimisation of global orbit parameters, whereas the stated gap is gradient-based fixed-budget selection of the source poses themselves.Source: §I · p. 2, §II · p. 2

III. METHODS

The method treats source positions as continuous variables on a scanner kinematic manifold and optimises a differentiable objective. Its formulation accommodates multiple acquisition geometries, including bounded scanner envelopes.

III. METHODS

  • Source positions are optimised as continuous variables on a kinematic manifold, with coverage, absorption, and view information represented as differentiable functions.Adam maximises the combined objective with a cosine schedule and manifold-aware refinement.Source: §III · pp. 2–3

A. Problem Formulation

  • Each source pose is generated by a differentiable kinematic map from its parameter vector, while sampled Radon-plane normals define the coverage directions.The formulation builds on the discrete soft-Tuy framework but changes the search variables to the source positions themselves.Source: A. Problem Formulation · p. 3
  • The same formulation represents a single-axis circle, two-axis gantry, and limited C-arm, with the limited C-arm using bounded charts on both axes.For the C-arm, the stated ranges are θ ∈ [−110°, 110°] and φ ∈ [−45°, 45°].Source: A. Problem Formulation · p. 3
  • The free-sphere model constrains each source to a sphere around the ROI centre and retracts ambient updates back onto that sphere.Envelope charts keep every iterate admissible by construction; detector geometry remains fixed during optimisation.Source: A. Problem Formulation · p. 3

B. Differentiable Soft Coverage

The soft coverage surrogate smooths plane-coverage scoring, while its closed-form source-position gradient moves tangentially to the viewing sphere. The measured study averages coverage over sampled ROI points, with near-identical attained coverage to the centred form under the stated settings.

B. Differentiable Soft Coverage

  • The smooth soft-Tuy surrogate replaces a hinge score that is nondifferentiable at its tolerance boundary and provides no gradient beyond it.It accumulates Gaussian-kernel contributions from views for sampled Radon-plane normals, then applies a smooth saturation and averages across normals.Source: B. Differentiable Soft Coverage · p. 3

C. Closed-Form Tangential Coverage Gradient

  • The coverage score depends on viewing direction rather than source-to-ROI distance, so radial motion alone does not change it.The unit-direction Jacobian projects onto the tangent plane at the viewing direction and scales by inverse source-to-ROI distance.Source: C. Closed-Form Tangential Coverage… · p. 3
  • The closed-form coverage gradient is tangential to the viewing sphere, mapping each source-position gradient into kinematic coordinates by the chain rule.Its direction factor lies in the tangent plane, and the saturation factor reduces further investment in already well-covered directions.Source: C. Closed-Form Tangential Coverage… · p. 4
  • At a 5 mm ROI and Δγ = 15° smoothing, the measured study’s sampled-point coverage and centred coverage agree within 10^-3 in attained coverage.The measured construction averages the same coverage surrogate over sampled ROI points instead of using only the centre.Source: C. Closed-Form Tangential Coverage… · p. 4

D. Analytic Bundle Absorption Penalty

The analytic bundle penalty adds object-aware attenuation to geometric view selection by evaluating source-to-ROI ray bundles. It uses differentiable line-integral estimates to penalize directions with low transmitted intensity.

D. Analytic Bundle Absorption Penalty

  • The bundle penalty complements geometry-only coverage by penalizing source directions whose ray bundles encounter strong attenuation.It uses a small bundle of rays per source and provides a closed-form source gradient without differentiating through a general volume projector.Source: D. Analytic Bundle Absorption Penalty · pp. 4–5
  • Each source’s square bundle targets lie in a plane perpendicular to its viewing direction, and the penalty uses the bundle-mean line integral.The attenuation field comes from the reference volume available at planning time.Source: D. Analytic Bundle Absorption Penalty · pp. 4–5
  • The ray integral is sampled only inside the reference volume, using trilinear attenuation interpolation to retain gradients as source positions move.The quadrature uses Nt = 256; its integrals, source gradients, and view rankings were reported converged against a 4096-sample reference.Source: D. Analytic Bundle Absorption Penalty · p. 5
  • The bundle mean is differentiated end-to-end, but the hard reference-axis branch makes the construction piecewise differentiable with a possible discontinuity at its threshold.The raw attenuation score is used because a bounded exponential gate can saturate on highly absorbing directions and lose gradient there.Source: D. Analytic Bundle Absorption Penalty · p. 5

E. Continuous View Covariance Loss

The continuous View Covariance Loss extends an information criterion used in discrete view selection to continuous pose optimisation. Its score uses per-view reconstruction responses, with central differences estimating their geometry derivatives.

E. Continuous View Covariance Loss

  • The VCL approximates the normalized mean-squared error of linear filtered-backprojection reconstruction at a candidate view set.Its per-view response uses the cone-beam projector, row-wise ramp filter, and matched-footprint adjoint.Source: E. Continuous View Covariance Loss · p. 5
  • The continuous VCL score extends the discrete VCLS information criterion from candidate-pool swap search to continuous optimisation.In the unregularized limit, the score is the squared cosine between the sampled reference and its projection onto the span of per-view responses.Source: E. Continuous View Covariance Loss · pp. 5–6
  • Because the available backprojector lacks a complete geometry VJP, the method estimates each normalized response’s geometry derivative by central differences through the full pipeline.Each view can be perturbed along the same coordinate simultaneously, requiring six additional basis evaluations per Adam step, independent of k.Source: E. Continuous View Covariance Loss · p. 6
  • The outer derivative of the VCL quadratic form is computed with a custom VJP rather than autograd through an explicit matrix inverse.It uses one linear solve and one outer product, without materializing a k × k inverse.Source: E. Continuous View Covariance Loss · p. 6

F. Composite Objective and Manifold Refinement

The method adds coverage, continuous VCL, and attenuation penalties into a weighted ascent objective. It optimizes poses through the chosen kinematic map with Adam.

F. Composite Objective and Manifold Refinement

  • The composite objective adds weighted soft-Tuy coverage, continuous VCL information, and an explicit absorption penalty.The VCL score carries no noise model, while the bundle term acts as a monochromatic photon-starvation proxy.Source: F. Composite Objective and Manifold… · p. 6
  • Inactive objective terms are disabled by setting their weights to zero, and weights remain fixed during each optimization run.A discrete swap search can score the same weighted sum, but each candidate swap requires a full evaluation of every active term.Source: F. Composite Objective and Manifold… · p. 6
  • The optimizer differentiates through the chosen kinematic map and updates pose parameters with Adam, cosine learning-rate decay, and patience-based early stopping.On the circle and C-arm, the map enforces feasibility; on the free sphere, each source is projected back to the viewing sphere after each step.Source: F. Composite Objective and Manifold… · p. 6
  • All reported runs initialize from a discrete soft-coverage greedy solution, although the optimizer can also start from a converged discrete VCLS solution.The two initializations place the optimizer in different objective-landscape basins.Source: F. Composite Objective and Manifold… · p. 6

IV. EXPERIMENTAL SETUP

The experiments compare continuous and discrete selection at matched view budgets, alongside tests of noise, optimization choices, constraints, planning, cost, and physical acquisition.

IV. EXPERIMENTAL SETUP

  • The evaluation compares continuous and discrete selection at matched view budgets with and without photon noise, and tests optimizer choices, constraints, planning, selection cost, and robot-bench acquisition.Source: §IV · p. 6

A. Virtual Benchmarks and Test Objects

The virtual benchmarks use the Defrise flange to test missing-elevation defects and the dense, near-axisymmetric fuel nozzle as a counterpart. Photon-noise evaluation is also reported for both objects.

A. Virtual Benchmarks and Test Objects

  • The Defrise flange tests whether trajectories recover off-plane lack-of-fusion voids that an equatorial orbit misses, with tilted cracks and central pores as controls.Steel inserts create attenuation shadows.Source: A. Virtual Benchmarks and Test Objects · p. 6
  • The ORNL fuel nozzle is a near-axisymmetric dense-metal counterpart, with an FDK volume supplying its attenuation reference.Virtual comparisons use matched forward projections and 15 SART iterations with relaxation 0.9 and non-negativity.Source: A. Virtual Benchmarks and Test Objects · p. 6
  • Photon-noise evaluation on both objects is reported with the complete results in the Supplementary Material.Source: A. Virtual Benchmarks and Test Objects · p. 6

B. Measured Robot-Based CT Study

The robot CT bench scans a digital camera over a full azimuth and bounded elevation range. The study uses a circular prescan for planning and evaluates against a separate, withheld circular FDK reference.

B. Measured Robot-Based CT Study

  • The camera detector is mean-pooled by a factor of four before reconstruction.Source: Table I · p. 7
  • A 120-view circular prescan serves as the planning prior; a separate 1200-view circular FDK reference is withheld until evaluation.Source: B. Measured Robot-Based CT Study · pp. 6–7
  • Measured sparse arms use equal-budget circular-reference subsets or uniform-on-band views, with TV-regularised ASD-POCS reconstructions.The planned arms use the historical 32-sample bundle rule, and reconstructions are intensity-matched to the reference before scoring.Source: B. Measured Robot-Based CT Study · p. 7
  • The custom robot bench has two-axis source and detector manipulators, and scans the camera over 360° azimuth and ±30° elevation.Source: B. Measured Robot-Based CT Study · p. 6, Figure 4 · p. 7

C. Compared Methods and Evaluation

The evaluations compare continuous selection with discrete selectors on virtual objects, then test how reachable geometry and prescan quality affect results. The same composite objective is used across the kinematic study.

C. Compared Methods and Evaluation

  • The comparison includes discrete VCLS and soft-Tuy selectors alongside greedy-initialised continuous coverage, VCL, bundle, and combined arms, scored with PSNR, SSIM, and HFEN.PSNR and SSIM are higher-is-better; HFEN is lower-is-better, with defect-resolved spherical ROIs on the flange.Source: C. Compared Methods and Evaluation · p. 7

D. Optimisation Settings and Implementation

  • Virtual selection uses 2,000 Radon normals and 100 Adam steps; measured planning uses 150 steps and a 128³ VCL model.The virtual cosine schedule starts at η0 = 0.05; measured VCL uses voxel sampling fraction r1 = 10^-3, ridge λR = 10^-3, and εvcl = 0.5 mm.Source: D. Optimisation Settings and… · p. 7

A. Noise-Free Benchmarks

  • On the nozzle, coverage plus VCL exceeds the discrete sphere baseline at both budgets, while coverage plus absorption alone falls behind by more than 3 dB at the larger budget.The three-term arm is best on all three metrics at the sparse budget, but at the larger budget absorption costs about 2 dB against coverage plus VCL.Source: A. Noise-Free Benchmarks · p. 8, Table II · p. 8
  • On the flange, the best continuous arm gains 4 dB over soft-Tuy greedy at the sparse budget and 5 dB at the larger budget, cutting HFEN by about half.On the nozzle, the same off-grid step buys less than a decibel; photon noise preserves selector ordering on the flange and compresses it on the dense nozzle.Source: A. Noise-Free Benchmarks · p. 8, Figure 5 · p. 8

B. Constrained Acquisition

  • At the LoF stacks, the equatorial circle trails out-of-plane sets by 5–9 dB ROI-PSNR at k = 40 and by more than 10 dB at k = 80.The deficit indicates missing Radon data; the bench band recovers most of the full-sphere gain and matches the larger C-arm envelope at the defects, while the full sphere is strongest at both budgets.Source: B. Constrained Acquisition · p. 8, Table III · p. 9, Figure 6 · p. 9
  • On the free sphere, 11 of 80 poses lie at the LoF-stack heights of 20–27 mm and 7 remain above 60° elevation; these out-of-plane poses carry the defect gain.Across two-dimensional reachable sets, poses move off-grid by median great-circle angles of roughly one to three candidate spacings.Source: B. Constrained Acquisition · pp. 8–9

C. Scout Robustness and Selection Cost

Prescan geometry affects planning performance on the Defrise flange, while selection cost is reported separately from reconstruction. The selection objective scales mainly with view budget rather than reconstruction-grid size.

C. Scout Robustness and Selection Cost

  • At equal measurement cost, circular prescans cost the three-term selector several decibels, and additional views do not reliably recover the loss.A band prescan removes most of the penalty at 8 and 64 views, less at 32, and none at 16; the flange is described as a deliberately extreme case.Source: C. Scout Robustness and Selection Cost · p. 9
  • The measured camera experiment is evaluated against a quantitative 1200-view FDK reference, with circular subsets sharing that reference acquisition.Source: Table VI · p. 10
  • The objective scales mainly with view budget, whereas VCLS requires an object-specific candidate-basis precompute tied to reconstruction-grid size.The reported selection wall-clock comparison excludes reconstruction and separates VCLS precomputation from its swap loop.Source: Table V · p. 9, C. Scout Robustness and Selection Cost · p. 10

D. Measured Robot-Based CT

The robot study demonstrates physical acquisition and reconstruction, but reports no consistent metric gain over uniform band sampling. Discussion benchmarks and analyses identify reachable geometry, planning priors, and objective choice as key factors in the results.

D. Measured Robot-Based CT

  • On the measured prescan, coverage saturates within about thirty steps; at k = 400 the bundle integral falls roughly a third while coverage changes by less than one percent.These are geometric findings; the acquired scans determine whether image quality improves.Source: D. Measured Robot-Based CT · p. 10

VI. DISCUSSION

  • Continuous selection reaches discrete swap-search quality; off-grid refinement adds four to five decibels on the asymmetric flange but less than one on the near-axisymmetric nozzle.Photon noise compresses the margins, and the estimator ablation attributes final quality to the objective and selection cost to the gradient.Source: §VI · p. 10
  • Leaving the equatorial plane recovers the designed laminae, while a moderate bench band captures most of the free-sphere gain; bounded envelopes can lower global performance relative to the full sphere.Matching the prescan and planned acquisition manifolds removes most of the circular-prescan penalty at most budgets.Source: §VI · p. 10

D. Measured Robot-Based CT

  • The three-term arm reconstructs the fine pins most completely, but the measured study shows no consistent metric gain over uniform band sampling.Fig. 7 compares the measured central slices and pin-region enlargements.Source: §VI · p. 10, Figure 7 · p. 11

VI. DISCUSSION

  • Geometry-only coverage is the default without a prior or under tight selection cost, while the bundle term is preferable when a prior exists and attenuation or occlusion matters.On the Defrise phantom, adding the information term keeps attenuation-driven poses out of unfavourable geometry.Source: §VI · p. 10
  • The claims are bounded by local optimisation, idealised monochromatic noise modelling, measured-session confounds, use of the historical bundle estimator, and planning for the scanned object.The prescan study therefore bounds prior sparsity, not mismatch between the prior and the part.Source: §VI · p. 10
  • Future work could impose continuity and velocity constraints by optimising trajectory coefficients, or use measured projections to replan remaining poses with task-specific image losses.The latter would progressively replace the planning prior with data from the object itself.Source: §VI · pp. 10–11

VII. CONCLUSION

  • Kinematic-manifold optimisation offers an alternative to fixed candidate pools and object-specific precomputation, with robot acquisition feasibility demonstrated but consistent measured image-quality gains still unestablished.The continuous selector reaches discrete-search quality on the free sphere while incorporating attenuation and scanner limits into selection.Source: §VII · p. 11
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