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Analyzing Reconstruction Artifacts from Arbitrary Incomplete X-ray CT Data

Leise Borg, Jurgen Frikel, Jakob Sauer Jorgensen, Eric Todd Quinto

arXiv:1707.03055v6math.FAmath.APphysics.ins-det

TL;DR

Incomplete CT data are common, and reconstructions from such data with filtered backprojection (FBP) usually contain artifacts. Using microlocal analysis, the paper characterizes these artifacts and shows that their forms depend on the geometry of the data-set boundary.

  • Problem

    Incomplete CT reconstructions with FBP usually suffer from artifacts because measurements cover only a proper subset of lines.

    Source: §1 · p. 1, §1.2 · p. 3

  • Method

    The paper uses microlocal analysis to characterize artifacts caused by arbitrary incomplete-data sets and establish which singular artifacts can occur for FBP in the continuous case.

    Source: §1.2 · p. 3

  • Results

    Object-dependent artifacts can extend along lines when object singularities lie on boundary-associated lines, while object-independent artifacts can occur on lines at nonsmooth data-set boundaries or along curves near smooth boundaries.

    Source: §1.2 · p. 3

  • Takeaways & Limitations

    The geometric characterization covers classical incomplete-data problems and emerging imaging situations with incomplete data.

    Source: §1.2 · p. 3

    The theory treats continuous data; discretization and finite resolution can introduce or alter visible artifacts, so their interplay with the continuous theory needs further analysis.

    Source: §8.1 · p. 18

Abstract

from arXiv · show

This article provides a mathematical analysis of singular (nonsmooth) artifacts added to reconstructions by filtered backprojection (FBP) type algorithms for X-ray CT with arbitrary incomplete data. We prove that these singular artifacts arise from points at the boundary of the data set. Our results show that, depending on the geometry of this boundary, two types of artifacts can arise: object-dependent and object-independent artifacts. Object-dependent artifacts are generated by singularities of the object being scanned and these artifacts can extend along lines. They generalize the streak artifacts observed in limited-angle tomography. Object-independent artifacts, on the other hand, are essentially independent of the object and take one of two forms: streaks on lines if the boundary of the data set is not smooth at a point and curved artifacts if the boundary is smooth locally. We prove that these streak and curve artifacts are the only singular artifacts that can occur for FBP in the continuous case. In addition to the geometric description of artifacts, the article provides characterizations of their strength in Sobolev scale in certain cases. The results of this article apply to the well-known incomplete data problems, including limited-angle and region-of-interest tomography, as well as to unconventional X-ray CT imaging setups that arise in new practical applications. Reconstructions from simulated and real data are analyzed to illustrate our theorems, including the reconstruction that motivated this work---a synchrotron data set in which artifacts appear on lines that have no relation to the object.

1. Introduction

Incomplete X-ray CT data can make FBP reconstructions unreliable, while newer imaging setups raise artifact patterns not explained by earlier theory. The paper develops a microlocal account of artifacts from arbitrary incomplete-data boundaries and characterizes their forms.

1. Introduction

  • Incomplete data make CT reconstruction severely ill-posed, and resulting artifacts can impair image interpretation.Incomplete data may result from dose reduction or practical scanner constraints.Source: §1 · pp. 1–2

1.2. Basic mathematical setup and our results.

  • The paper traces singular artifacts to data-set boundary points, distinguishing object-dependent streaks from object-independent lines or curves according to boundary geometry.Object-independent lines arise at nonsmooth boundary points, while smooth boundary regions can generate curves; the authors state their geometric account describes all singular artifacts for continuous-case FBP.Source: §1.2 · p. 3

1.1. Related research in the mathematical literature.

  • Earlier microlocal analyses mainly explained object-singularity-generated streaks in specific settings; the authors report no prior microlocal explanation for ROI ring artifacts or nonstandard setups.The ring artifact was already known to practitioners, and limited-angle streaks had been analyzed.Source: §1.2 · pp. 3–4

1.3. Organization of the article.

  • The article moves from notation and main theorems to applications, Sobolev-strength estimates, artifact reduction, the synchrotron experiment, and generalizations.Technical theorems and proofs appear in the appendix.Source: §1.3 · p. 4

2. Mathematical basis

The mathematical setup models CT data as line integrals over parameterized lines and restricts the object functions considered. It then uses wavefront sets to locate and describe singularities for the artifact analysis.

2. Mathematical basis

  • The framework considers Lebesgue-measurable functions, a setup the authors describe as realistic and simpler than the general-distribution case.The theory can be made rigorous for some compactly supported distributions, but this treatment considers only Lebesgue-measurable functions.Source: §2 · p. 4

2.1. Notation.

  • Lines are parameterized by (θ, p), and each X-ray measurement is the integral of the object function over the corresponding line.The Radon line transform denotes this integral, with the parametrization satisfying Rf(θ, p) = Rf(−θ, −p).Source: §2.1 · p. 4

2.2. Wavefront sets.

  • The wavefront set records singularity location and direction, providing the framework used to analyze artifacts in incomplete-data reconstructions.For a line, WFL(f) selects wavefront directions based on that line that are normal to it.Source: §2.2 · pp. 5–6
  • A wavefront singularity is a failure of smoothness in a specified direction, characterized through failure of rapid Fourier decay after localization.The wavefront set collects these singularities; its base point gives their location.Source: §2.2 · p. 5

3. Main results

The results characterize singularities and artifacts in incomplete-data FBP through the geometry of the measured-data region and the object’s singularities. Artifacts arise only at the data-set boundary and are limited to object-dependent streaks, object-independent streaks, or object-independent curves.

3. Main results

  • Incomplete-data FBP models measurements by restricting the Radon data to a proper, symmetric region A and applying the standard FBP reconstruction.The analysis covers arbitrary incomplete data, including classical limited-data problems as special cases.Source: §3 · p. 6

3.1. Singularities and artifacts.

  • For lines parameterized inside A, object singularities are visible without artifacts; outside the measured support, singularities are invisible, and artifacts can occur only at bd(A).If every line through a point lies in int(A), all singularities of the object at that point are visible in the reconstruction.Source: §3.1 · p. 7

3.2. Analyzing singular artifacts.

  • The classification has two types: object-dependent artifacts are streaks generated by object singularities, while object-independent artifacts depend primarily on bd(A) and can be streaks or curves.Theorems 3.5 and 3.7 classify these as the only singular artifacts for the reconstruction considered.Source: §3.2 · pp. 8–9
  • A smooth data boundary with finite slope can generate an object-independent artifact along the curve x_b(θ)=p(θ)θ+p′(θ)θ⊥, which is curved unless it is a point.The curve can occur even when the object is smooth normal to the corresponding line; a vertical smooth boundary generates no object-independent artifact there.Source: §3.2 · pp. 8–9
  • A streak can arise from an object singularity normal to a boundary line, while a nonsmooth data boundary can produce an object-independent streak on that line.For a corner boundary and nonzero Radon data, the object-independent streak does occur; a smooth vertical boundary instead permits only object-dependent streaks.Source: §3.2 · pp. 8–9
  • The artifact conditions are necessary, not sufficient: an example satisfies the stated conditions yet reconstructs without artifacts.The example has LAf=f, so the reconstruction contains no artifacts.Source: §3.2 · p. 9

4. Numerical illustrations of our theoretical results

The numerical illustrations apply the paper’s theorems to classical and unconventional incomplete-data geometries. Across the examples, artifact form and location track the data-set boundary, while object-dependent streaks also require object singularities.

4.1. Limited-angle tomography.

  • Limited-angle data produce object-dependent streaks on boundary lines tangent to object singularities; the smooth vertical boundary produces no object-independent artifacts.The examples generalize the limited-angle pattern to cutoff boundaries with nonvertical tangents.Source: §4.1 · p. 10

4.2. Smooth boundary with finite slope.

  • Changing the slope of a smooth curved cutoff boundary can move its xb-curve from outside the reconstruction region to intersecting it, where the calculated curve matches the artifact.Object-dependent streaks can also appear where a boundary line is tangent to the skull.Source: §4.2 · p. 11

4.3. Region-of-interest (ROI) tomography.

  • In ROI tomography, the xb-curve is a circle tracing the ROI boundary, while object-dependent streaks arise on lines tangent to object singularities.The artifact circle does not extend through regions where the Radon data vanish; for a strictly convex ROI with smooth boundary, the curve traces the ROI boundary.Source: §4.3 · pp. 12–13

4.4. The general case.

  • In the general example, horizontal sinogram boundaries generate circular arcs, while corners generate line artifacts that meet the arcs tangentially.Additional object-dependent streaks correspond to vertical boundaries in the sinogram.Source: §4.4 · p. 13

4.5. Summary.

  • The examples classify boundary-generated artifacts as object-dependent line streaks, object-independent line streaks at nonsmooth boundary points, or object-independent curves from smooth, small-slope boundary portions.Object-dependent streaks require a smooth data boundary and an object singularity normal to the corresponding line.Source: §4.5 · pp. 13–14

5. Strength of added artifacts

This section characterizes the Sobolev strength of added singularities under specified assumptions. It defines the Sobolev wavefront set and gives regularity estimates for smooth and cornered data boundaries.

Sobolev wavefront set

  • The Sobolev wavefront set records where and in which direction a distribution fails to have local Sobolev regularity of order s.Membership means the localized, microlocalized Sobolev seminorm is not finite for any suitable cutoff and cone.Source: §5 · p. 14

Smooth boundary

  • Under the theorem’s assumptions, a smooth nonvertical data boundary yields an added singularity at xb with Sobolev order below 0 and membership in WF0.The stated assumptions include Rf(θ0,p0)≠0 and smoothness of f normal to the corresponding line.Source: §5 · p. 15

Corner boundary

  • At a corner, the reconstruction is in WF1 along the conormal directions, while it is Hs for s<1 except at at most two points on the corresponding line.If one one-sided tangent is vertical, there is only one such exceptional point; under smoothness of f, a vertical data-boundary tangent causes no added artifacts normal to the line.Source: §5 · p. 15

6. Artifact reduction

Hard truncation of incomplete data can create boundary jumps and added artifacts. The section describes smooth data tapering as a way to avoid adding singular artifacts, while noting that it can smooth existing ones.

Hard truncation

  • Zero-padding incomplete data creates jumps at the data-set boundary, which explain added artifacts and are stronger singularities than those of Rf when f is in L2(D).The stated comparison is with Rf∈H1/2(S1×R).Source: §6 · p. 15

Smooth cutoff

  • Replacing the sharp indicator with a smooth symmetric cutoff that tapers near the boundary yields a reconstruction operator that adds no artifacts.Because the cutoff is smooth, the resulting reconstruction operator is a standard pseudodifferential operator.Source: §6 · p. 15

Wavefront behavior

  • Where ψ(θ,x·θ) is nonzero, the reconstructed image and the object have the same wavefront singularities in that direction.Source: §6 · p. 15

Practical caveat

  • The smoothing technique can turn singular artifacts into smooth artifacts, including by smoothing xb-curves; illustrations show its use on simulated and real synchrotron data.Source: §6 · p. 16

7. Application: a synchrotron experiment

A synchrotron scan of a chalk sample contains object-independent streaks associated with the data boundary. The reconstruction’s sharp cutoff produces severe streaks, while smoothing the cutoff essentially eliminates them.

7. Application: a synchrotron experiment

  • Figure 7 pairs the truncated attenuation sinogram with an enlargement of its data-boundary segment and a zoom of the corresponding reconstruction.The sinogram is shown after processing to obtain Radon transform data.Source: Figure 7 · p. 16
  • The chalk-sample reconstruction contains dramatic streaks described as independent of the object.The data were acquired in a synchrotron experiment.Source: §7 · pp. 16–17
  • The attenuation sinogram shows vertical and horizontal staircasing caused by X-rays blocked by four metal bars stabilizing the pressurized sample holder.Figure 8 shows the data-acquisition setup.Source: §7 · p. 17, Figure 8 · p. 17
  • The sharp cutoff produces severe streak artifacts, while a smooth cutoff at the data boundary essentially eliminates them.The authors attribute each line artifact to a corner of the sinogram boundary.Source: §7 · p. 17

8. Discussion

The discussion relates streaks to boundary conditions and describes where the theory and simple smoothing approach apply. It also identifies discretization and imaging-geometry qualifications on these results.

8. Discussion

  • A data-boundary point can generate a streak along its corresponding line when the truncated Radon data have a full wavefront set there; the Sobolev analogue assumes a boundary corner.For weaker boundary singularities, the wavefront’s Sobolev strength must also be accounted for.Source: §8 · p. 17

8.1. Observations.

  • The simple artifact-reduction method works well on the synchrotron data, but may be less efficient for other incomplete-data problems; abrupt cutoffs should be avoided.More elaborate methods for this particular problem are described in.Source: §8.1 · p. 17
  • The discussion also points to data-completion and artifact-reduction approaches for other incomplete-data problems, including quantitative susceptibility mapping and metal artifacts.The cited examples include,,, and.Source: §8.1 · p. 17
  • The theory treats continuous data, whereas real data are discrete; undersampling and finite resolution may create or de-emphasize artifacts, leaving their interplay with the continuous theory unresolved.Source: §8.1 · p. 18

8.2. Generalizations.

  • The artifact theorems extend to symmetry-invariant pseudodifferential filters satisfying the stated ellipticity condition, though Sobolev results depend on filter order.For Lambda CT, the order-two filter yields reconstructions one Sobolev degree less smooth than those for LA.Source: §8.2 · pp. 18–19
  • The results apply to fan-beam data with a smooth convex source curve and compactly supported interior object, subject to checking Assumption 3.1 for the equivalent parallel-beam data set.The theorems also hold verbatim for generalized Radon transforms with smooth measures on lines in R2.Source: §8.2 · p. 19
  • Higher-dimensional artifacts may occupy only proper subsets of the data surface; cone-beam reconstruction can add artifacts even with complete data.Analogous results are anticipated for cone-beam CT, but that setting is more subtle.Source: §8.2 · p. 19

Appendix A. Proofs

The appendix introduces microlocal analysis as the framework for proving the paper’s results, using wavefront sets to track singularities.

Appendix A. Proofs

  • The appendix introduces basic microlocal analysis before using it to prove the paper’s theorems.Source: Appendix A · p. 19
  • It treats wavefront sets as subsets of cotangent spaces.Source: Appendix A · p. 19
  • The text directs readers to elementary tomography presentations and standard references.Source: Appendix A · p. 19

A.1. Building blocks.

This subsection assembles wavefront-set facts and microlocal correspondences used to analyze how singularities pass through the X-ray transform, data truncation, filtering, and backprojection.

A.1. Building blocks.

  • The appendix states basic wavefront-set properties, including locality and the effect of multiplying a distribution by a smooth function.Source: §A.1 · p. 19
  • The line parametrization imposes a symmetry relating data at (θ, p) and (−θ, −p), which is used in the wavefront-set identifications.Source: §A.1 · pp. 19–20
  • The X-ray transform and its dual are treated as elliptic Fourier integral operators, with canonical relations used to describe singularity correspondence.The article identifies this correspondence as a main technical theorem.Source: §A.1 · p. 20
  • The proof uses the global Bolker assumption for the X-ray transform to establish the stated wavefront correspondence.Source: §A.1 · p. 20
  • A distribution-product criterion based on non-cancellation of wavefront directions is described, but it does not apply to the product of the data indicator and Radon data in the cases considered.The authors therefore work with f in L2(D), for which that product remains an L2 function.Source: §A.1 · p. 20
  • The filter Lambda is elliptic in the cotangent directions relevant after dual-transform composition, so the theorems extend to symmetry-invariant pseudodifferential operators satisfying that condition.The Sobolev results depend on the operator’s order.Source: §A.1 · p. 21

A.2. Proof of Theorems 3.5, 3.7, and 3.9.

The proofs trace reconstruction singularities to the data boundary and distinguish artifacts caused by object singularities from those caused by the data-set boundary. Boundary geometry and the local Radon data determine where these artifacts can occur.

A.2. Proof of Theorems 3.5, 3.7, and 3.9.

  • Artifacts conormal to a line can be added only when its corresponding data point lies on the boundary of the data set.The proofs attribute them either to singularities of the object’s Radon data or to singularities of the data-set indicator; these cases account for all singular artifacts in the continuous problem.Source: §A.2 · pp. 21–22
  • For a smooth data boundary, the possible object-independent artifact lies at the boundary-generated point x_b(θ0) on the corresponding image line.That point is the x-projection of C_t applied to the conormal of the data boundary.Source: §A.2 · p. 21
  • The x_b curve can lie in a line only when the data boundary is locally formed by lines through a point, in which case the curve reduces to that single point.The boundary is then locally given by p(θ) = x0 · θ.Source: §A.2 · pp. 21–22
  • Object singularities can generate artifacts only on the line to which those singularities are conormal, while a nonsmooth data boundary can generate artifacts conormal to its corresponding line.The object singularity contributes through the Radon data at the corresponding data point.Source: §A.2 · p. 22
  • At a boundary corner, smooth object data with nonzero Radon value can produce a reconstruction singular along the full conormal to the corresponding line.If the Radon data vanish near the point, the reconstruction is smooth conormal to that line; if they are smooth and nonzero there, the corner generates the full conormal set.Source: §A.2 · p. 22

A.3. Proof of Theorem 5.2.

The proof uses Sobolev wavefront correspondence to track how boundary singularities pass through the reconstruction operators. It derives different regularity and artifact-location conclusions for smooth boundaries and corners, with a special case when a corner side is vertical.

A.3. Proof of Theorem 5.2.

  • Sobolev wavefront correspondence for R and R* transfers local regularity information between object and data spaces, with R* mapping H^s continuously into H_loc^(s+1/2).The proof uses the ellipticity of Λ, except in the irrelevant dθ direction, in establishing the correspondence.Source: §A.3 · pp. 22–23
  • For a smooth finite-slope boundary, 1_A has threshold regularity H^s for s < 1/2 at its conormal directions, and the resulting reconstruction artifact is localized at x_b(θ0).No other covectors conormal to the line belong to the reconstruction wavefront set.Source: §A.3 · p. 23
  • At a corner, the two boundary-normal directions yield points where the reconstruction is more singular; the localized Fourier transform decays at order −1 in those directions and −2 in others.The proof assumes the Radon data are smooth and nonzero near the corner.Source: §A.3 · p. 23
  • If one side of the corner is vertical, one transformed direction has empty C_t image, leaving only one point on the line where the object is more singular.Source: §A.3 · p. 24
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