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To segment a fetal head, we modified the Iterative Randomized Hough Transform (IRHT) [87] so that the fetal skull can be efficiently and accurately detected with little user involvement. In this dissertation, the ellipsoid is used to model the fetal head of a given 3D image volume because the ellipsoid is intuitively close to the shape of a fetal head and has been used in a number of studies [107,108,109]. As shown in Fig. 6-5, the skull in every 2D image of a given 3D image volume is identified and modeled by a set ellipses with different axes lengths, and then these ellipses are used to construct an ellipsoid using the least-square-fit method.

Fig. 6-5. The procedure of modelling a fetal head.

Image pre-processing

Before applying the IRHT, the 2D/3D images are pre-processed, as shown in Fig. 6-6. The 3D image volume containing the fetal head is first processed using a low-pass filter to reduce speckle noise. Then each single 2D slice is sent through a white top-hat filter to enhance image contrast. The filtered slices are individually processed in a binary fashion using the Otsu threshold method [110] that distinguishes the foreground (skull and bright objects) from the background (other tissue structures). Following that, every 2D image is processed by two consecutive morphological operations, open and close, to remove small objects and holes on the fetal skull. Finally, all foreground objects in each binary slice are skeletonized and then processed by a custom filter.

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Fig. 6-6. The pre-processing of a fetal head image: (a) the image processed by the low pass filter; (b) the image processed by the white top-hat filter; (c) the image processed by the Otsu method; (d) the image processed by the morphological operations; (e) the image processed by the skeletonization filter; (f) the image processed by the custom filter.

The custom filter, as described in this paragraph, is intended to decrease the number of iterations for finding the parameters of the best ellipse by further removing small objects that are not likely to be a part of the fetal skull. After a single 2D image is processed by the skeletonization filter, a list of the foreground pixels’ coordinates is obtained. Starting with first pixel (P) in the list, the custom filter searches the eight neighbors of the pixel P. If it has one or more neighbors, the filter pushes the neighbor

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pixel(s) into a container and then repeats searching the neighbors of the pixels in the container until the container is empty. Finally, the custom filter generates a set of separate objects, each of which is a set of connected pixels, and then removes the small objects based on an experimentally determined threshold.

Fetal head detection and modelling

An quadratic form of an ellipse is given in eq. (6-3) and can be described with a parameter set s = (m, n, xc, yc, ψ) as shown in eq. (6-4), where 0 ≤ θ < 2π; (x, y) represents a point on the ellipse; (xc, yc) is the center of the ellipse; m and n are the long and short semi-axes and R is a rotation matrix derived from the rotation angle ψ as shown in eq. (6-5).

U + •+ + U+ + U + p+ + Õ = 0 (6-3)

ÖU_{+× = ∗ Ö}† cos

! sin × + ÖU+VV× (6-4)

= Ø )1Ù −1 !Ù_{1 !Ù )1Ù Ú} (6-5) To start the IRHT, the region of interest (ROI) is set to the whole 2D slice containing the thalami. In addition, all elements of the parameter set s are set to zero. The IRHT has been implemented in the following iterative steps. sk is the parameter set s in kth _iteration,

k =1,2,….

1. If the input parameter set s exists, let sk = s (k=1), go to step 4. Otherwise continue with step 2.

2. Randomly select five pixels from the list and solve eq. (6-4) to obtain a parameter set s’. If it is valid, s’ is pushed into a five-column parameter array. Repeat until the size of the parameter array is less than 1000.

3. Find ten most frequent elements in each column of the parameter array and average them to generate a new parameter set sk

4. Adjust the ROI based on the center and radii of an ellipse defined by sk. 5. Eliminate the pixels that are out of ROI from the slice.

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6. Compute the difference vector t =Û _IÜ _I = (‚_IM− ‚_IMF Ý/‚_IM, = 1 ) 5ß if ‚_IMF exists. If the number of iterations is less than two or any element in t is bigger than 0.05, go to step 2.

As illustrated in Fig. 6-7 (d), the best-fit ellipse of a given 2D slice is highlighted with blue color. We can see that the ellipse accurately follows the shape of the fetal skull. Similarly, the IRHT is sequentially applied to the rest of the slices containing the fetal skull except that the input parameter set s, inherited from the previous slice, are provided to the current slice. This method can reduce the calculation for new ellipse parameters by limiting the size of an initial ROI and increase the robustness of the IRHT.

Fig. 6-7. The process of finding the best fit ellipse of the fetal head: (a) the original image; (b) the image processed by the pre-processing filters; (c) the image processed by IRHT; (d) the ellipse fitting to the fetal skull.

After the IRHT is applied to all 2D images containing the fetal skull, along the x and z axes of the 3D image volume, a set of best-fit ellipses are obtained, which are composed of a number of voxels, (¼ , ¼ , … , ¼_f), of the 3D image volume. The reason to apply the IRHT to the 2D images along two axes is to obtain sufficient number of fetal skull voxels for generating the ellipsoid. Finally, the Non-Linear Newton-Gauss (NGNL) is applied to calculate the parameters of the ellipsoid that is fit for these voxels, as shown in Fig. 6-8. The NGNL has been detailed in Chapter 4 where we used it to generate the

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virtual cylinder and virtual ellipsoid models, so we will not describe the NGNL in this chapter.

Fig. 6-8. The ellipsoid fitting to the fetal head