The iterative process of the bias field correction algorithm. The original image is denoted as . The N3 corrected image is denoted as , which still shows inhomogeneous intensity in the posterior breast. Then, the FCM is applied to to generate the corrected image, denoted as . Although the bias field shown in is removed in , but the intensity of some anterior fibroglandular tissues is brightened too much and appears as the fatty tissue. The bias field is estimated by calculating the difference between and in log space (illustrated in Fig. 2). Then, the Gaussian kernel and the B-spline surface fitting is used to smooth the bias field, so the problem of erroneously changing contrast in the anterior breast can be minimized. By the deconvolution of new smoothed bias field from , the bias field corrected image is generated, denoted as . In order to keep the dynamic range in the entire intensity spectrum increasing, the intensity of each pixel on the is compared to the corrected image in the previous iteration (i.e., the original image for the first iteration), and the higher intensity is used to form the corrected image after the first iteration. This process is repeated until the stopping criteria are met. For this example, ten iterations are needed. The corrected image after the first, sixth, and tenth iterations, , , and are shown.
The calculation of the bias field for the example shown in Fig. 1 during the first iteration. (a) The N3 corrected image . (b) The FCM corrected image using the N3 corrected image as the input. (c) The calculated bias field by taking exponential of using Eq. (3), with . The dark area in the anterior breast indicates the erroneous change in contrast that makes fibroglandular tissues appear as fatty tissues. (d) Smoothing of (c) using a Gaussian kernel. (e) Smoothing of (d) using B-spline surface fitting to obtain the estimated bias field . The dark area in the anterior breast shown in (c) and (d) is removed after the B-spline surface fitting, and the homogeneity correction is mainly seen in the posterior breast. (f) The corrected image calculated by the deconvolution of from . Note that all displayed images are in a relative scale.
The subtraction image between the pair of images before and after each iteration. Ten images from iterations 1 to 10 are shown using a normalized scale indicated by the color bar. They demonstrate that the area showing intensity difference between two iterations is shrinking. The lateral posterior breast presents the strongest field inhomogeneity, and the correction effect is clearly seen after each iteration. The iteration will stop when the number of pixels showing changes is smaller than 5% of the total number of pixels in the whole breast.
Comparison of the fibroglandular tissue segmentation quality based on images corrected using four methods. The top row from left to right shows original image, FCM corrected image, N3 corrected image, CLIC corrected image, and corrected image. The middle row from left to right shows the truth fibroglandular tissue delineated by a radiologist and the segmentation results based on FCM, N3, CLIC, and corrected images. It can be seen that both CLIC and produce the most accurate results close to the truth outlined by the radiologist, and FCM has the worst segmentation quality. The bottom row shows the corresponding histograms from pixels in the radiologist outline fibroglandular and fatty tissues on each image. The two curves denote the histograms of fibroglandular tissue and fatty tissue, respectively. It clearly shows that both CLIC and the proposed algorithms increase the dynamic range and widen the separation between the histogram peaks of fatty tissue and fibroglandular tissue.
A case example of “.” It can be seen that both the CLIC and correction brighten the signal of fatty tissues in the medial posterior breast and allow the correct classification of pixels in that area as fatty tissues. N3 did not completely correct the bias field, and some tissues in that area are misclassified as dense tissues. The FCM gives the worst performance. Not only that some fatty tissues in the medial posterior breast are misclassified as dense tissues, but also some dense tissues in the anterior breast close to the nipple are misclassified as fatty tissues.
A case example of “.” Although the N3, CLIC, and show slightly different results, all are acceptable, and their performances are rated equal. The same problems indicated in Fig. 5 for FCM corrected images (fatty tissues misclassified as dense tissues and dense tissues misclassified as fatty tissues) are seen, and that makes the performance of FCM inferior to the other three methods.
A case example of “.” The segmentation based on all four methods yields similar results, and their performances are rated equal. Note that this breast is fatter (with a smaller protruding depth into the coil), and there is no visually discernable strong bias field. For this case, the correction is probably not needed, and all four methods perform equally well.
Radiologist’s visual ranking of the fibroglandular tissue segmentation quality based on images corrected by these three methods. “” means superior quality and “” means equal quality. The two reading sessions are 1 month apart, performed independently.
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