A practical global distortion correction method for an image intensifier based x-ray fluoroscopy system

Abstract

X-ray images acquired on systems with image intensifiers (II) exhibit characteristic distortion which is due to both external and internal factors. The distortion is dependent on the orientation of the II, a fact particularly relevant to II's mounted on C arms which have several degrees of freedom of motion. Previous descriptions of distortion correction strategies have relied on a dense sampling of the C-arm orientation space, and as such have been limited mostly to a single arc of the primary angle, alpha. We present a new method which smooths the trajectories of the segmented vertices of the grid phantom as a function of a prior to solving the two-dimensional warping problem. It also shows that the same residual errors of distortion correction could be achieved without fitting the trajectories of the grid vertices, but instead applying the previously described global method of distortion correction, followed by directly smoothing the values of the polynomial coefficients as functions of the C-arm orientation parameters. When this technique was applied to a series of test images at arbitrary alpha, the root-mean-square (RMS) residual error was 0.22 pixels. The new method was extended to three degrees of freedom of the C-arm motion: the primary angle, alpha; the secondary angle, beta; and the source-to-intensifier distance, lambda. Only 75 images were used to characterize the distortion for the following ranges: alpha, +/- 45 degrees (Deltaalpha = 22.5 degrees); beta, +/- 36 degrees (Deltabeta = 18 degrees); lambda, 98-118 cm (Deltalambda = 10 cm). When evaluated on a series of test images acquired at arbitrary (alpha, beta, lambda), the RMS residual error was 0.33 pixels. This method is targeted at applications such as guidance of catheter-based interventions and treatment planning for brachytherapy, which require distortion-corrected images over a large range of C-arm orientations.

Figures

Fig. 1

Schematic diagram of the steps of the three methods of distortion correction. Method 1, which is the previously published method of global distortion correction, solves the coefficients of the distortion model by inverting the Vandermonde matrices of Eq. (1); the model returns the modeled ring locations, (1û, 1v̂). Method 2 modifies this by smoothing the input ring locations with 3rd-order polynomials of α. Method 3 applies the smoothing as a function of α directly to the coefficients determined in Method 1. Note that the smoothing functions in Methods 2 and 3 are of the same form (dashed box), though one acts on the grid coordinates in the image domain, while the other acts on polynomial coefficients.

Fig. 2

The angular dependence of II distortion can be visualized by taking a minimum intensity projection of images acquired every 5° along an arc of primary angles from −50° to 50°. On any individual image, the rings of the grid phantom do not fall on a regular grid; additionally, images at different C-arm orientations have different amounts of in-plane rotation and translation, resulting in the pattern seen here.

Fig. 3

(A) A closer look at the highlighted part of the image in Fig. 2 shows the smooth shape of the angular-dependent distortion pattern. The circles (°) show the ring locations (u⃗, v⃗) as they were segmented in each image, and the line is a two-dimensional 3rd-order polynomial (s⃗, t⃗) fitted to these points. (B) The RMS (solid line) and maximum (dashed line) errors of the ring locations as a function of the order ℓ of the trajectory-fitting polynomials σ⃗ and τ⃗. (C) The RMS error of the trajectory-fitting polynomials as a function of primary angle. Note that the 2nd-order polynomial shows a distinct pattern, suggesting that it does not fully characterize the ring trajectories, while the higher order polynomials are virtually equivalent, with RMS errors <0.05 pixels.

Fig. 4

The 36 coefficients of the distortion correction polynomials in the u direction (A) and v direction (B) determined using Method 1 (thin line) and Method 2 (thick line). In all cases, Method 2 results in coefficients that exhibit dramatically decreased variability as a function of primary angle (the abscissa of each axis represents the primary angle, from −50° to 50°). (C) The residual errors of the modeled ring locations using the distortion correction polynomials shown in panels A–B.

Fig. 5

(A) The 3rd-order polynomial fits (s⃗, t⃗) of the ring trajectories are shown for all six sampling frequencies, Δα={5°,10°,15°,20°,25°,30°}. With the exception of the Δα=30° curve, the trajectories are nearly indistinguishable. The circles (°) represent the segmented ring locations (u⃗, v⃗). The inset highlights 1 pixel containing a ring location which shows an occasional deviation from the underlying trend, but this is partly corrected by the trajectory-fitting polynomials. (B) The modeled ring locations (2û, 2v̂) determined with Method 2 were calculated for the six sampling frequencies (solid lines). Again, the curves are very similar. This panel also shows the modeled ring locations (1û, 1v̂) determined by Method 1 (dashed line). This curve matches the others in most places, except that it follows the deviation (inset). (C) The curves of (3û, 3v̂) are shown as for panel B.

Fig. 6

The residual errors of the modeled ring locations for all three methods of distortion correction are presented, including all six sampling densities evaluated for Methods 2 and 3. Note that at this scale, the results for Methods 2 and 3 are identical.

Fig. 7

(A) The minimum intensity projection of distortion-corrected test images is shown for the Δα=30° sampling density. The relative rotation and translation of the images due to the angular dependent distortion can be seen to be corrected. Also, the superimposed lines show that the rings now fall on a regular grid. (B) The RMS residual errors of the distortion-corrected test images are shown for all six sampling densities, and the curve is nearly flat, with the RMS errors about 0.22 pixels.

Fig. 8

The minimum intensity projection of the ten test images acquired at arbitrary (α, β, λ), before (A) and after (B) distortion correction. Panel A shows the larger trajectory of the uncorrected rings compared to the images acquired over a single arc of α [see Fig. 2(A)]. After distortion correction based on only 75 images, the angular dependence is removed and the rings fall on a regular grid. The RMS residual error of distortion correction was 0.33 pixels.