Full Multiresolution Active Shape Models

Abstract

The incorporation of a multiresolution image approach is one of the most popular variants of Active Shape Models (ASMs), providing a more robust algorithm and minimizing its initialization dependency. Using the wavelet transform, the present paper extends the multiresolution analysis to the shape space, developing a novel multiresolution shape framework, capable of being incorporated into most of ASM variants. The tests performed with two different types of images, face images (AR database) and chest radiographs (JSRT database), demonstrate how this new generation of algorithms significantly reduce the computational cost, more than halving it, while maintaining the same levels of accuracy.

Appendix: Starting Point Dependency in Closed Contours

Invariance, uniqueness and stability are desirable properties of shape descriptors [44]. Chuang and Kuox [9] successfully derived these properties when using wavelets as shape descriptors, under the assumption of a well defined starting point for the curves. While this condition is easily satisfied in the case of open contours, there exists an inherent uncertainty when dealing with closed ones. In the absence of a prior registration, the problem of starting point of the contour must be handled ad hoc, since different starting points of the same shape will result in different wavelet coefficients. Lohakan et al. [37] solve this problem by defining the point of maximum curvature as starting point, while Zhang et al. [55] propose a starting point location strategy based on half-axes-angles, i.e., the least counterclockwise angles formed by the line segments connecting the centroid to the nearest and farthest point of the contour. However, the PDM requires that the landmarks are placed in the same order and in the same places along the object’s contour, ensuring thus a point correspondence, and eliminating the need for a consistent localization of the starting point in every shape. Nevertheless, there still exists an uncertainty about whether or not the chosen starting point, and thus the corresponding coarser versions of the shape, significantly affects the final segmentation accuracy.

Suppose (14) is the vector expression of a contour. If the contour is closed, this notation remains valid assuming an implicit periodicity of its control points [19]

$$ \bigl(x_{K_{0}+i}^{0}, x_{K_{0}+i}^{0}\bigr) = \bigl(x_{i}^{0}, x_{i}^{0}\bigr), \quad i=1,\ldots,k $$

(28)

for a uniform k-th degree B-spline closed curve. This periodicity condition, depending on whether the contour is closed or open, is also reflected in the set of filtering matrices. In the case of open contours, and following the block notation proposed by Samavati [46], the form of these matrices for a B-spline wavelet decomposition can be expressed as

$$ \mathbf{M}^{j} = \begin{pmatrix} \mathbf{M}_{s}^{j}\\ \mathbf{M}_{r}^{j}\\ \mathbf{M}_{e}^{j} \end{pmatrix} $$

(29)

where M ^j can be A ^j, B ^j, F ^j or G ^j. $\mathbf{M}_{s}^{j}$ and $\mathbf{M}_{e}^{j}$ represent special submatrices for the start and end of the curve, required to guarantee the end-point interpolation, while $\mathbf{M}_{r}^{j}$ corresponds to the regular portion of the curve. In the case of closed contours, this block notation is simplified thanks to the implicit periodicity, $\mathbf{M}^{j}=\mathbf{M}_{r}^{j}$. Focusing our attention on the analysis filter A ^j, the elements of each row of its regular structure, $\mathbf{A}_{r}^{j}$, are obtained by shifting right by two positions of the elements of the previous row. An immediate consequence of this particular structure is the invariance of the wavelet coefficients to the displacements of the starting point that are a multiple of two. Suppose $\mathbf{x}_{t}^{j-1}$ represents the vector form of the closed contour x ^j−1 whose starting point has been shifted t positions to the right. With this notation

$$ \mathbf{A}^{j}\mathbf{x}_{t}^{j-1} = \mathbf{A}^{j}\mathbf{x}_{t+2n}^{j-1}, \quad n\in\mathbb{Z} $$

(30)

That is, given a closed contour at the j-th level of resolution, there are 2^L possible coarser versions at the level j+L-th, with L∈ℕ (see Fig. 9). This raises an interesting issue when incorporating a multiresolution shape approach to the segmentation algorithm, since a new degree-of-freedom has been added to the algorithm and thus, it should be properly optimized.

Fig. 9

Starting point dependency in closed contours. (a) Contour of a right lung in highest resolution (level 0); (b), (c) and (d) show the possible contours obtained in levels 1, 2 and 3, respectively, when shifting the starting point