External Forces

5.3.2.1 Image Forces

External image forces attract the shape towards the desired boundaries by using image features.

Given the point position x_i, the target position ˜x_i is sought along the normal directionn_i, such

that an image energy E(y^j_i) is minimized. In practice, the energy is computed at L = 2D+ 1 positions y^j_i, regularly spaced by a step S, symmetrically with respect to xi (i.e. D times inside and outside the mesh). Parameters D and S are named search depth and search step, respectively. Depending on the image features and chosen strategies, different image energies with corresponding image forces can be devised.

Gradient Direction Force The gradient energyE^g has been already presented in a regularized and normalized form (Eq. (5.4)) in Sec. 5.2.3.3to allow the correct summation of energy terms.

A regularized and normalized gradient force f^ng= α^ng(x˜−x)is thus simply derived from this energy. In the context of force computation, the regularized normalization can be removed as the edge strength is also a strong indicator of the presence of the structure boundaries, yielding a non-normalized gradient force f^g=α^g(x˜−x). In both cases, the gradient direction is maintained as a force only based on gradient magnitude will not be able to distinguish the right interface between tissues (e.g., bone boundary in our MRI images is darker than surrounding muscular tissue, i.e. gradients point outward like our mesh normals).

Intensity Profile Force Based on intensity profiles (IP, Sec.4.4), we consider two types of image energies. The first energy E^ip denotes the energy computed with the NCC similarity measure, recalled in Sec. 5.2.3.3. For each point x_i, this energy compares a reference profile q^ref_i with L candidate profiles p_i^j, j∈ [1,L]based on the NCC measure. The NCC has the nice property to be invariant under intensity affine changes, which confers some robustness against variations in the local appearance of structures across individuals and protocols. We chose this similarity measure as it outperformed various common measures in the context of MRI musculoskeletal segmentation [Gil07].

The second energyE^mgdis based on the multivariate Gaussian distribution (MGD) formulation which exploits the Mahalanobis distance as described in Sec.4.4.3. In particular, the PCA-based Mahalanobis formulation (4.11) is preferred as it reduces the memory footprint and minimizes the influence of the noise in the data. We denote the forces based on IP and MGD as f^ip = α^ip(x˜−x)and f^mgd=α^mgd(x˜−x), respectively.

5.3.2.2 Constraint Point-based Force

The segmentation evolution can be controlled by using what is referred to as constraint points (CP) [Gil07]. Three types of CPs are available:

1. Internal points: the model evolves to include the points in its interior.

2. External points: the model evolves to exclude the points from its interior.

3. Frontier points: the model evolves to have the points lying exactly on its surface.

In practice, constraint points attract or repel meshes by creating forces on some vertices. An example of an internal CP Pis depicted in Fig. 5.8(a), where the closest face of the mesh, here represented by two vertices P₀ and P₁, is attracted under the action of two forces f₁^cp and f₂^cp applied on P0andP1, respectively. Forces become inactive when the CP is in the interior of the mesh (Fig. 5.8(b)).

(a) (b)

Figure 5.8:Illustration of the forces based on an internal constraint point (CP)P.(a)The closest face (P0, P1) to the CPPis attracted by creating 2 forcesf₁^cpandf₂^cponP₀andP₁respectively, whose calculation depends onPand its projectionP_⊥on the face.(b)End of evolution in whichPis located inside the model.

5.3.2.3 Symmetric Force

Another type of prior knowledge that can be exploited in the segmentation context is the sym-metry property. Indeed, a structure of interest can have a point, an axis or a plane of symsym-metry which can be used to regulate its deformation. For instance, the pelvis is composed of a right and left hip bones which are mostly symmetric with respect to a sagittal plane. This symmetry is commonly observed in healthy subjects.

The use of two independent models to segment each hip bone generally confers a greater flexibil-ity in the segmentation since each model can evolve freely. In fact, consider the situation where very narrow pelvic models are used as training shapes to built an SSM. In the case where a sub-ject presents a wide pelvis, the segmentation might not be able to correctly segment the bones due to over-constraining shape priors which prevent the hip bones to separate too much from each other. In this situation, two independent deformable models of hip bones with associated SSMs would have been more adapted.

In most of our experiments, segmented hip bones appeared as symmetric at the end of the segmentation. However, in some cases asymmetry was observed as depicted in Fig. 5.9(a)and (b). The left hip bone is clearly erroneous, especially at the level of the pubic arc. Strong image artifacts in this image area could be at the origin of this deviation from the correct result.

To tackle this issue, we devised a force which enforces a symmetry between the two structures.

This force has to regulate the evolution of each shape until a symmetry is reached with respect to a symmetry plane. The two shapes Sa = {P₁^a, . . . ,P_n^a}and S_b = {P₁^b, . . . ,P_n^b} must be in point correspondence and have the same of number of points n. These requirements are fulfilled by our hip bone shapes. Given a symmetry planeP with normaln, the symmetry property implies two conditions:

1. the middle pointC_iof the segment[P_i^aP_i^b]lies on the planeP. 2. vectorP_i^aP_i^bis parallel ton.

To satisfy these conditions, we propose the following procedure to compute target positions ̃P_i^a and̃P_i^b:

(a) (b) (c) (d)

Figure 5.9: Symmetric force illustration. Subfigures(a)and(b) illustrate segmented hip bones as mesh overlays and 3d models, respectively, that are not symmetric. Based on our force which enforces symmetry, the segmentation results can be corrected for this subject as shown in(c)and(d).

1. the planeP is estimated by performing a PCA with all current middle points 0.5(P_i^a+P_i^b).

The principal direction with smaller variance is chosen as the normalnto the plane and an arbitrary point on the planePmis selected (Fig.5.10(a)).

2. For each pair of points(P_i^a, P_i^b), the middle pointC_i is projected on the plane asC_i^⊥, and new points P_i^⋆a and P_i^⋆b are computed as the projections ofP_i^a and P_i^b on the line passing throughCiand directed byn, respectively (Fig.5.10(b)). These points are simply computed with the following formulas:

C^⊥_i =C_i+ <CiPm∣n>n (5.6) P_i^⋆x=C_i+ <CiP_i^x∣n>n, ∀x∈ {a,b} (5.7) 3. Target points̃P_i^a and̃

P_i^b are eventually computed as (Fig.5.10(c)):

P̃_i^x=P_i^x+P_i^xP^⋆x+CiC_i^⊥, ∀x∈ {a,b} (5.8) As shown in Fig. 5.10(c), the final points do satisfy the abovementioned symmetry conditions. A corrected result based on this symmetric force is visible in Figs. 5.9(c)and(d), where the left hip bone is now clearly symmetric and the segmentation is of satisfactory quality.