An illustration of the relationship among prior class distributions and class uncertainty for a two-class problem. It may be noted that class uncertainty is maximum around the threshold selected under minimum error criterion. Image points with intensity values of either or are classified as object points. However, the class uncertainty associated with points having intensity is significantly higher than that for points with intensity .
An illustration of the relationship between class uncertainty and tissue interfaces under different conditions of thresholding. (a) An image slice from a CT data of a patient’s lower abdomen. (b) Image intensity histogram for (a) with three thresholds marked as , , and . The first two thresholds are manually selected to segment proper tissue regions while the third one is intentionally picked as a bad threshold. (c,d) Thresholded tissue regions and class uncertainty image for the threshold . (e,f), (g,h) Same as (c,d) but for thresholds and , respectively. Note that class uncertainty images in (d) and (f) depict respective tissue boundaries while the same in (h) fails to indicate any tissue interface and spills out into the entire soft tissue region.
An illustration of intrinsic basins on an energy line . Different colors are used to indicate different intrinsic basins; however, one intrinsic basin may include multiple colors. All invalid valley points are marked in a different color than valid valley points.
An illustration of different types of optimum locations on an energy surface/function. The energy function is rendered using a 3D matlab display function with color indicating the energy value. Here, valley lines are shown with Types I (pit) and II optimum locations on the energy surface are denoted by hollow circles of different colors.
A graphical illustration of the error measure between a selected threshold and the true threshold . Essentially, it computes the pixel/voxel density weighted distance (the area of the grey region) between the two thresholds and normalize by image size, i.e., the total area under the histogram.
Results of application of different thresholding methods on a CT image slice of lower abdomen. (a) Original CT image slice. (b) Optimum thresholds (red lines) derived by the new method. (c) The energy surface with valley lines and optimum threshold and gradient parameters (hollow circles). (d) Thresholded regions in different colors as applied to the original image. (e) Same as (d) but applied to a smoothed image. (f) Object class uncertainty maps at different optimum thresholds. Note that the class uncertainty image highlights different tissue interface at different optimum thresholds. (g,h,i) Same as (b,d,e), respectively, but for Otsu’s method. (j,k,l) Results of thresholding as obtained by the MSII, ME, MHUE algorithms, respectively. One of the merits of the proposed method is that by incorporating spatial information, the new method succeeds in dealing with all three examples of Figs. 6–8; however, Otsu’s method succeeds in Fig. 6 but clearly fails in Fig. 8.
Same as Fig. 6, but for a CT image slice of upper abdomen.
Same as Fig. 6, but for a 3D CT image of a cadaveric ankle specimen.
Results of quantitative analyses of image threshold errors by different methods. For each ankle CT image, indicated by a number between one and twelve on the horizontal axis, height of the bar indicates the percent image threshold error. The line one each bar denotes the standard deviation (in percentage scale) of threshold errors for different interfaces in an image.
Results of thresholding on MR brain phantom images at different noise levels: 0% (a), 3% (b), 5% (c), 7% (d), and 9% (e). (f–j) Results of thresholding by the new method for (a–e), respectively, at different noise levels. (k–o) Class uncertainty images at optimum thresholds of (f–j), respectively. (p–y) Same as (f–j) but for Otsu’s (p–t) and the MHUE (u–y) methods, respectively.
Same as Fig. 10 but at a different slice location.
Same as Fig. 10 but at different nonuniformity levels: 10% (a), 15% (b), 20% (c), and 25% (d) intensity nonuniformity. Here, a display-intensity window different from Figs. 10 and 11 was used to illustrate the intensity nonuniformity.
Results of quantitative comparison on MR brain phantom images at varying levels of noise, slice thickness and intensity nonuniformity. (a) Errors for different image thresholding algorithms as functions of noise level. (b) Same as (a) but without MSII and ME. (c, d) Same as (a) but as functions of slice thickness (c) and intensity nonuniformity (d).
Results of reproducibility analysis and intraclass coefficient (ICC) of threshold values in repeat CT scans. (a) ICC for threshold values of different tissue interfaces of four specimens in three repeat scans using the new method. (b) Same as (a) but for Otsu’s method. (c,d) Same as (a,b) but for ICC value of thresholds for matching interfaces in different specimens using the first CT scan.
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