Example of calculation of 2D model image with one shape. (a) shows a contour that was generated using Eq. (4) with Fourier terms. In (b), all pixels with centers within the contour were given the intensity ; the other pixels intensity . (c) was obtained by blurring image (b) with a 2D Gaussian with . Each image measures .
[(a)–(d)] The two phantoms mimicking the neck with four arteries near the center. [(a) and (c)] Axial images of phantoms 1 and 2 with holes with diameters of 4.2, 2.4, 1.2, and 0.6 mm and 2.8, 1.6, 0.8, and 0.4 mm, respectively. The holes are filled with a water diluted contrast agent. [(b) and (d)] Maximum intensity projections at the sagittal plane of both phantoms, showing at the left the holes with diameters between (b) 4.2 and 0.6 mm and (d) 2.8 and 0.4 mm, and at the right the reference holes of (b) 6.0 and (d) 4.0 mm. The smallest holes are nearly invisible. (e) A composition of sagittal images through the center of the holes with sizes used in this study. The 4.2 mm hole was not used. The window center is 200 HU; the window width is 400 HU.
Average estimated values of the diameter standard deviation using the cylindrical model with six parameters. The line of identity is also shown. In order to facilitate comparison with Fig. 5, measurements up to 3 mm are shown. Estimated values for 4 and 6 mm can be found in Table III.
Relation between estimated values of (a) the diameter and sigma and (b) the diameter and the intensity of the cylindrical hole with a diameter of 1.2 mm.
Average estimated values of the diameter standard deviation, using the three-parameter cylindrical model, in which prior knowledge of , , and is used. The upper curve shows the diameters when the full-width at half-maximum criterion is used, the lower curve displays the diameters as determined using second derivative zero-order crossing. The line of identity is also shown.
Registered surfaces of CTA images and 3DRA images of both patients. The centers of the stenotic segment are indicated with an arrow head and the centers of the reference segments are indicated with a triangle.
Cross-sections of the common carotid artery of patient 1, second image of the proximal section (Fig. 9, at 0.5 mm). (a) shows the CTA image and (d) the 3DRA image. The other figures show the fitted model images (middle column) and contours (last column) for CTA (upper) and 3DRA (lower) using four Fourier terms. The contours bound the (unblurred) shapes. Each image measures . The window center is 160 HU and the window width is 400 HU for images (a) and (b); the window width and level in the images (d) and (e) were chosen to obtain approximately the same gray values for the artery and background.
CTA image of the common carotid of patient 2, ninth image of the distal section (Fig. 10 at 29 mm), with a high-intensity structure in the periphery of the ROI (at the lower left). (a) shows the fitted contours with for the central artery and for the peripheral shape. (b)–(d) show the effect of disregarding the peripheral shape; in (b), the same number of Fourier terms was use as in (a); in (c) and in (d) . Each image measures . The window center is 200 HU; the window width is 500 HU.
Estimated equivalent diameters of the carotid of patient 1. On the left are the equivalent diameters of both reference segments. CTA measurements are indicated with circles; 3DRA measurements with squares. On the right are the equivalent diameters of the stenosed segment. For the CTA and 3DRA measurements without use of prior knowledge, the same symbols are used as at left. Rotated squares and triangles indicate CTA measurements with the use of prior knowledge of three and two parameters, respectively (see text). In the reference sections, Fourier terms were used. In the stenosed section, was used for the arteries (i.e., ellipses were fitted) and was used for the calcifications and other peripheral densities, except when they were very small (area ). In the last case, was used. At the horizontal axis, the distance of the first proximal cross-section is indicated (in mm); at the vertical axis, the equivalent diameter (in mm).
Estimated equivalent diameters of the carotid artery of patient 2. Details as in Fig. 9.
Cross-sections of the common carotid of patient 2, image 16 of the distal reference segment (Fig. 10, at 32.5 mm) (a) shows the CTA image and (c) the 3DRA image. (b) and (d) show the fitted model images, using four Fourier terms. Note the severe streak artifacts in the 3DRA image (c), with background values varying between −600 and 600, with the intensity of the artery in the order of 1800 (all arbitrary units) and edge enhancement of the artery. These artifacts, probably in combination with some mismatch, cause deviations in the shape of the artery [see (c) in comparison with (a)] and a poor fit (d). As a consequence, the estimated equivalent diameter in (d) is much larger than that in (b) (see Fig. 10). Each image measures . The window center is 170 HU and the window width is 600 HU for images (a) and (b); the window width and level in the images (c) and (d) were chosen to obtain approximately the same gray values for the artery and background.
(a) First CTA image in the proximal part of the stenosed segment of patient 1 with calcification (Fig. 9, at 19 mm); the lumen is indicated with an L and the calcification with a C; (b) ROI with initial segmentation of the artery (light grey), border (dark grey), part of calcification within border (white), and very small part of calcification at top right (light gray); the shape of the calcification in (b) does not correspond exactly with that in (a) because of the 3D dilatation in a preprocessing step (see text); (c) ROI; (d) initialization with (ellipse) for the artery, for the calcification, and for the very small part of the calcification at top right; (e) fitted contours and (f) fitted model images, using prior knowledge of , , and . (Table VI, patient 1). Use of prior knowledge of and only gave virtually the same fit. The window center is 180 HU and the window width is 500 HU for images (a) and (c)–(f). Each image measures
(a) CTA image at the site of maximal stenosis of patient 1 (Fig. 9, 13th image of stenosis segment at 25 mm); the lumen is indicated with an L and the calcification with a C; (b, c, and f) fitted images using all parameters (b) and prior knowledge of three (c) and two (f) parameters. (d) 3DRA image at the same site with nearly invisible calcification; note the streak artifacts. (e) fitted 3DRA image. Each image measures . The window center is 200 HU and the window width is 400 HU for (a)–(c) and (f); for (d) and (e), settings were chosen to obtain approximately the same gray values for artery and background.
(a) Cross-section just above the bifurcation of the carotid of patient 2 (first image of the stenosed segment Fig. 10 at 15 mm); [(b) and (c)] ROIs with fitted contours without prior knowledge and with prior knowledge of and , respectively (Table VI, patient 2). For the internal and part of the external carotid (middle and right), one and two Fourier terms were used, respectively. Each image measures . The window center is 200 HU; the window width is 500 HU.
Cross-sections of the 2D intensity profiles of a disk-shaped detail with a diameter of 1 mm and intensity (left) and of the same detail blurred with a Gaussian PSF with a of 0.4 mm (right). The integrated intensity of the unblurred detail is the same as that of the blurred detail. is the mean intensity within the FWHM. In this example, . The integrated intensity of the blurred detail can be approximated with the integrated intensity within the FWHM multiplied by a factor to allow for the tails that are not included in the integration. For a Gaussian profile, which is a good approximation to the true intensity distribution in this example, .
The initial step sizes used in the simplex minimization method.
The thresholds used in the segmentation to obtain initial contours for the extended model in the patient data.
Average estimated values of the diameter , , , and and their standard deviations using the cylindrical model with six parameters. The values of and in the two phantoms are slightly different. Av.: Average; SD: Standard deviation.
Average estimated value of the diameter and standard deviation using the cylindrical model with three free parameters and prior knowledge of , , and . The average number of restarts for convergence was between 1.1 and 1.2. Av.: Average; SD: Standard deviation.
Details of the fit of a model image to the CTA image of Fig. 7(a). The number of Fourier terms increases from 0 (i.e., a circle) to 6. In Fig. 7(b), a fitted image with is shown. EFD: Elliptical Fourier descriptors; RMS err.: Root mean square error.
Mean values and standard deviations of , , and for the reference segments of the CTA images of patients 1 and 2. These data are used as prior information in the segments with a stenosis.
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