Enhanced interpreter-aided salt-boundary extraction using shape deformation

Method

Our approach is mainly based on the idea of Wang et al. (2001), described briefly here. Let us define the reference image slice that has been properly segmented as the template image, and define the image to which we want to propagate the segmentation result as the input image. The segmentation result in the template image is characterized by a set of contours. For simplicity, we assume that the template image contains only a single closed contour. Nonetheless, the extension of this method to handle multiple contours is straightforward.

We represent the known contour on the template image as a set of landmark points, $V={v_1,v_2,..., v_n}$ where $v_i = (x_i,y_i)$ (in 2-D Cartesian coordinates). Wang et al. (2001) describe the skeleton of this algorithm as follows:

``For each landmark $v_i$ , the proposed method first identifies a set of possible corresponding landmark points $B_i = \{v_i^{(j)}, j=1,2,...,n_i\}$ on the input image, where $v_i^{(j)}=(x_i^{(j)},y_i^{(j)})$ . Then conceptually the deformation is solved in two major steps:

1. Identify the best landmark point $v_i'$ from the landmark set $B_i$ such that $V' = \{v_1',v_2',..., v_n'\}$ is located in or near the true object boundary in the input image.
2. Deform the prior shape $V$ to match $V'$ while keeping the general shape characteristics of $V$ . ''

The cartoon in Figure 2 illustrates the idea of landmark-based contour deformation.

Fig2-deform Figure 2. Landmark-based shape deformation, from Wang et al. (2001).

For the first step, we search the candidate points in set $B_i$ along a short line segment that centers around point $v_i$ and aligns to the contour’s normal direction $\mathbf{n_i}$ . It is difficult to determine the best landmark point $v_i'$ from all candidates in $B_i$ at the first try. Therefore, we just choose randomly an element in each $B_i$ to form the initial set $V'$ and iterate this selection process a few times. During each iteration, we update the set $V'$ such that $V'$ more likely contains the correct corresponding landmarks.

The next deformation step is formulated by finding the optimal solution to an objective function which takes into account both the goal of deforming the points in $V$ into the current landmark set $V'$ and the goal of preserving the prior shape $V$ (using the bending-energy formula from (Bookstein, 1989)). The optimization goal is

$$\min_{V',\mathbf{t}} \left\{ \frac{1}{n}\sum_{i=1}Q(v'_i,\mathbf{t}(v_i)) + \lambda \phi(\mathbf{t}) \right\}$$ (1)

in which $\mathbf{t}$ defines the deformation from $V$ to $V'$ as a mapping; i.e. $\mathbf{t}: (x,y)\rightarrow(f(x,y),g(x,y))=(x',y')$ . Function $Q$ describes the term that penalizes the mismatch between $V'$ (the landmarks we found on the input image) and the mapping defined by $\mathbf{t}(V)$ . The $Q$ term corresponds to the first goal, deforming the landmarks in set $V$ to those in $V'$ . Function $\phi(\mathbf{t})$ is a regularization term that tries to force the mapping $\mathbf{t}$ to be smooth, in other words, preserving the global shape information of the original $V$ . We add a $\lambda$ parameter to balance the weights of the two terms, Q and $\phi$ . The choice of $\lambda$ is up to the user's judgment. After mathematical simplification, this optimization can be solved easily using the classical SVM(Support Vector Machine) regression technique (as a quadratic programming problem of size $n$ ). Moreover, the badly fitted components in set $V'$ are identified as the support-vectors. We update the set $V'$ by replacing those support-vectors with other candidates in $B_i$ , such that the new set $V'$ would achieve better fitting.

Enhanced interpreter-aided salt-boundary extraction using shape deformation