The parameter a in Eq. (4) is inversely proportional to the width of the filter and the amount of smoothing that is performed. We set a to 1.0 for ^ and and to 0.7 for j^, since the second-order derivatives require greater smoothing to obtain better noise immunity .
The coefficients c0, c1; c2, c3 are chosen to normalize the filters in Eq. (4). For the discrete implementation, the normalization is done using
for integer sampling . The coefficients need to be adjusted for the case of noninteger or anisotropic sampling. We use filters of finite width (for example, seven or nine voxels), compute the values of the functions in Eq. (4) and normalize the values using Eq. (6). Since the vertices forming the surface do not necessarily lie on voxel boundaries, we used linear interpolation to compute voxel intensities.
The curvature computation is used to identify surface patches with a common curvature in order to segment the overall surface. Vertices having elliptical curvature of the peak subtype are considered to be within a potential lesion and are colored red to distinguish them from the others and assist visual inspection. Other curvature types describe normal airway surfaces. For example, hyperbolic curvature describes saddle points at bifurcations, cartilaginous rings, and haustral folds; cylindrical curvature and elliptical curvature of the "pit subtype" describe normal airway and colon (ulcerations fit into the latter category but are ignored because of overlap with normal shape). The next step is to cluster these vertices using region growing to achieve a minimum lesion size. Size criteria offer some immunity to noise and excessive false positive detections, for example, by ignoring isolated vertices and "lesions" smaller than some threshold. The minimum size criterion is preferably expressed in millimeters rather than number of vertices, since vertex density can vary depending upon the voxel size.
This method for computing differential geometric quantities for surfaces has a number of features that make it preferable to an alternative method that fits bicubic spline patches at each surface point on the isosurface [23,24]. First, it is very fast, since curve fitting is unnecessary. Second, it works well for highly curved surfaces (such as small airways and vessels). Like the patch-fitting method, it can be used to color the original surface to indicate areas of different curvature.
3.1 Clinical Application of Shape-Based Lesion Detection
We have applied this curvature-based technique to identify lesions of the airway. In a study of 18 virtual bronchoscopy patient examinations, we found sensitivities of 55 to 100% and specificities of 63 to 82% for detecting airway lesions 5 mm in diameter or larger . The results varied depending upon the choice of an adjustable parameter (the mean principal curvature). Other potential applications of this method are to detect pulmonary emboli and colonic lesions (Figs. 2 and 3) [29,30]. In Fig. 2, a tiny but physiologically significant embolus in a pulmonary artery branch of a pig is detected using curvature analysis. In Fig. 3, nodular lesions of the colonic mucosa are automatically detected. A software application which permits rapid inspection of the potential lesion sites has been shown to improve efficiency of interpretation as much as 77% .
Limitations of these methods include an inability to detect stenoses and a large number of false positive detections. Additional criteria (such as wall thickness) may be necessary to reduce the number of false positive detections [32,33].
Another type of endoluminal surface abnormality potentially detectible with VE is surface roughness. Biomedical surface texture is usually thought of in the context of biomaterials (i.e., orthopedic prostheses, dental implants), but the same concepts can be applied to endoluminal surfaces. This is a new concept, and the "normal" roughness of endoluminal surfaces is not well characterized. Surface texture depends on the choice of scale of observation. Typically, normal endoluminal surfaces are smooth on gross examination. On a microscopic scale the intestinal mucosa is rough, consisting of millions of villi (fine finger-like projections) per square centimeter. We confine ourselves to macroscopic scales on the order of the voxel dimensions. Based on our clinical experience, abnormal endoluminal surface texture (i.e., too smooth or rough) may occur under a variety of circumstances, including inflammation, atherosclerosis, or invasion of the wall by tumor. Hypothetically, swelling (edema) of the wall of a lumen could present as a smoother surface and atherosclerosis and tumor as a rougher surface compared to the baseline state.
There are a number of ways to measure surface roughness and develop a numeric index of roughness. These include fractal analysis, Fourier descriptors, variation of the surface normal, and the difference between either a fitted spline patch or a smoothed version of the surface and the original [34-36]. For example, as a surface becomes rougher the relative weighting of high-frequency to low-frequency components in its Fourier spectrum increases; the variance of the direction of the surface normal increases; and the disparity between the smooth and unsmoothed data increases. We chose to investigate fractal methods for quantitating roughness because of their widespread use in a variety of medical settings. For example, fractal analysis has been used to quantitate roughness of dental implants, orthopedic prostheses, and osteoarthritic joints [37-39].
Fractals have the property of being self-similar over a wide range of scale and are a natural way to describe roughness. One accepted method of using fractal analysis to quantitate roughness is to compute the fractal dimension (D), a nonintegral number that lies between the corresponding ideal topological dimension and the Euclidean dimension of the space that contains the structure . Points, curves, surfaces, and volumes have topological dimensions of 0, 1, 2, and 3, respectively. In contrast, fractal curves have a fractal dimension between 1 and 2 and fractal surfaces have a fractal dimension between 2 and 3. Figure 4 shows a curve with a fractal dimension that increases from left to right. As the fractal dimension increases, the curve becomes more chaotic and fills more space. If one imagines sliding a small disk (for example, of diameter 0.01 unit in this case) along the curve, the area swept out by the disk would be greatest for the most chaotic portion. This idea of measuring the area swept out by a shape or template (such as a disk, sphere, or higher dimensional analogue) forms the foundation for many methods of measuring fractal dimension. Figure 5 shows prototypical fractal surfaces generated using the midpoint displacement method . The surfaces in the figure are similar in texture to anatomic surfaces, we have studied. The rougher the surface the greater the fractal dimension.
There are several methods for computing the fractal dimension of experimental data. Examples include box-counting, Fourier power spectral density, variation, and Minkowski-Bouligand sausage, of which box-counting is probably the most familiar [42-45]. These are fraught with error and there is some controversy over the best method [46,47].
We implemented the variation method that is a modified version of the Minkowski sausage [25,48]. For a real-valued nonconstant function f(x, y) defined on the interval 0 < x, y < 1, the e-oscillation of the function f in an e-neighborhood of the point (x, y) is defined to be
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