Robust Statistical Shape Models

⎜

⎝

Σ^k_X u^k_X u^k⊺_X v^k_X

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⎟

⎠

(4.6)

More exactly, the covariance matricesΣ^j_X,∀j≤kare leading principal submatrices ofΣ^k+1_X . Given these remarks on the eigenvalues and eigenvectors, we preferred to compute for each level an SSM in order to preserve its specificity and to give the possibility to use a different align trans-form at each level.

SSM Locality Indeed, at each mesh resolution levelk, a different alignment transform can be used to build the SSM. The alignment transform affects what we call the “locality” of the SSM.

The locality is an intuitive notion: rigid SSMs capture global shape changes while affine or similarity SSMs better express local variations as depicted in Fig. 4.12. A multi-resolution SSM adopts different mesh and locality levels based on multi-resolution alignment schemes, such as the rigid-affine-affine (RAA) and similarity for all levels (SSS) schemes. RAA means that the SSM at resolution 1 is rigid, at resolution 2 affine, and so on. The appropriate selection and impact of the multi-resolution alignment will be studied in the segmentation context in next Chapter.

(a)−3.0√

λ0 (b) mean shape (c) 3.0√

λ0 (d)−3.0√

λ0 (e) mean shape (f) 3.0√ λ0

Figure 4.12: Variation modes of femur shape with different alignment schemes. (a)-(c): first mode of variation of a rigid SSM. This mode clearly captures the length of the femur.(d)-(f): first mode of variation of a similarity SSM. This mode mainly expresses the thickness of the femur.

4.3.2 Robust Statistical Shape Models

4.3.2.1 SSMs for Images with Varying FOV

We have presented in Sec. 3.3.4the direct relationship between image field of view (FOV) and resolution. It is quite common to have images with different FOV in order to image the structures

of interest with a desired image resolution. In these images, structures of interest are often only partially visible in the image, especially long bones such as the femurs which cover large body areas.

There are two kinds of segmentation results from images with small FOVs (Fig.4.13):

– incompleteshapes, i.e. shapes that only represent the visible parts of the bone in the image.

– corruptedshapes, i.e. complete shapes with some points that are not reliable or not part of the image region to be segmented. These unreliable points are hereafter referred to as erroneous points.

Figure 4.13:Possible shapes resulting from segmentation on images with small FOVs. Center: MRI image with a small FOV and with hip bone segmentation overlaid (white); Left:corruptedshape, where only shape vertices in the image FOV (bright wire-framed part) are reliable; Right: incompleteshape, where the shape is cropped to only represent reliable parts.

We also previously mentioned in Sec. 2.5.6the fact that SSMs need to be built with a certain amount of shapes to yield satisfactory results. However, the collection of data in 3D is in practice a problematic and tedious task [HM09], usually due to the lack of sufficient segmented images.

Thus it is advantageous to be able to exploit segmentation results from a wide range of images, including those with small FOVs, in order to build and improve the SSMs.

Hence, when we deal with images in which the structures of interest are partially visible, there are two things to consider:

1. How to build an adapted SSM to be used to segment these images.

2. How to re-exploit the segmented results (complete, incomplete or corrupted shapes) from these images.

A common approach to tackle the SSM construction for these images is by creating and using incomplete shapes depending on the image FOV [DLH07,DGZ07]. As reported by Kainmüller et al.[KLZH09], this approach is time consuming and often prevents automation, because unseen images which cover a larger part than those of the training shapes cannot be segmented with

the same SSM. It could be argued that it would be sufficient to build one SSM with complete shapes but we just mentioned that there are often too few (complete) training samples and it is hence a real “waste” not to reuse the segmentation results from images with different FOV. Our solution to address these two aspects is to build robust statistical shape models from complete andcorrupted data as explained in the next Section.

4.3.2.2 Robust Models based on Corrupted and Complete Shapes

Incomplete shapes cannot be directly mixed together because they do not necessarily present identical topology and same number of points. The idea instead is to always use complete shapes with deformable models to segment images with different FOV. However, the resulting shapes are usually corrupted with some erroneous points. Hence, a major challenge is combining these corrupted shapes with complete ones into the same SSM while preserving the SSM’s Generality and Specificity, conventional construction approaches being very sensitive to erroneous points (e.g., PCA sensitivity to outliers depicted in Fig. 2.11). We proposed to address this problem as follows.

Robust Alignment Alignment is necessary when applying the generalized Procrustes during SSM construction (Sec.4.3.1.2). In the case of shapes with erroneous points, a standard alignment is affected since the least square minimization is not robust to outliers. A common way to find the optimal alignment transform T^⋆ between a current estimate of the mean training shape X= {x¯₁, . . . , ¯xN}and a training shapeXjis to adopt a weighted formulation: the degree of reliability of each shape point.

As a side remark, this weighted alignment can also be used to regulate a shapeX^l_jat a resolution level l with an SSM built from shapes of a higher resolution levelk, l < k. Indeed, the shape regularization (Sec. 4.3.1.3) uses an alignment step which can also be weighted for our needs.

The regularization is done here by creating first an artificial shape ̃

X^k_j with N^k points, where

X^k_j, the vertices of the regulated shape at resolution l are simply updated from the firstN^l vertices̃

x^k_ij.

Robust PCA The robust PCA is then computed as follows. Let us consider the training set S as the union of non-corruptedS^●and corruptedS^○shape sets, whereS^●might be empty. Before any robust PCA approach, the training shapes are aligned with a weighted version of the generalized Procrustes alignment. This is achieved by replacing the classic mean and alignment calculations by their respective weighted versions (i.e., ¯xi= ∑_jwijxij/ ∑_jwijfor mean, and the abovementioned weighted alignment (4.7)). The missing data approach (mdEM) of [SLB07] is then utilized as the

robust PCA method. This method extends the iterative expectation-maximization (EM) algorithm for PCA [Row98, TB99] by allowing it to support binary weights. In the E step, the shape parameters of the training data are estimated based on the current estimation of the principal directions. In the M step, principal directions are estimated by computing a classic PCA on corrected data in which missing information is replaced by reconstructed values. This process thus estimates the “non-corrupted” value of the corrupted points based on reconstructed values.

This robust approach was chosen based on the comparison of various robust PCA approaches that will be reported in Sec.4.5.4.

During both alignment and PCA, a weightwijis set to 0 if pointiis an erroneous point in shape j, and to 1 otherwise. We did not need to use any technique to detect and correct erroneous points as in [RG02] or [BBLS05], since in our case these points were known in advance. In fact, we only flag points out the image FOV as erroneous.

It must be stressed that this robust PCA method usually converges to a local minimum. Tipping and Bishop [TB99] showed that in the case of the EM algorithm the only stable local solution spanned the principal subspace, but this property is not necessarily valid for a weighted EM algorithm. The initialization of the EM algorithm is thus very important. A standard PCA could be used to begin with, but for ∣S^●∣ ≥0, we proposed the initialization obtained from a PCA as follows:

1. Build a PCAP^●fromS^●.

2. Replace corrupted points of shapes inS^○with reconstructed points fromP^●. 3. Rebuild a final PCA fromS^●and correctedS^○.