Radial constraints

Soft-tissues of the musculoskeletal system are, most of the time, modelled and simulated using 1D or 2D medial representations accounting for both position and thickness. Indeed, muscles generally have a smooth and tubular shape, that can be represented by an underlying piece-wise action line [ND98] [AT01] [TSB⁺05], where isotonic contractions are modelled through action line shrinking/ stretching, and isometric contrac-tions through radial constraints applied to a wrapped surface (Section 2.5). However, muscles with large attachment areas and/ or several origins/ insertions require many action lines [NTH00] [AT01]. Ligaments are thin tissues effectively represented by FEM shells [WGL⁺05], multiple lines being not accurate to assess load transfer [BH91] (Section 2.7.2). Cartilages are mostly analysed through their thickness [WTGW03]

(Section 2.6.2). Looking at the different approaches presented in the literature, we see that authors use medial representations both to abstract objects (to decrease the number of parameters to represent them) and reversely, to constrain them (functional aspect of medial representations). However, representations that are used arenot reversible, meaning that they are not able to capture the shape and to replicate the function accurately. Moreover, their construction is subjected to user interaction (no unique representation). On the contrary, the medial axis, in the geometric sense, is unique and is able to reconstruct exactly any object.

The medial axis transform (MAT) has been introduced in 1964 by H. Blum as the computation of maximal balls inside on object [Blu64]. The medial axis is the union of these balls (center and radius). Three main properties characterise the medial axis:

• Homotopy equivalence: An object and its medial axis are connected the same way.

• Good localisation: each point of the medial axis is equidistant from at least two points of the surface.

• Reversibility: An object is reconstructed exactly by the union of maximal balls of its medial axis.

In this section, we propose a new method to compute medial surfaces of musculoskeletal tissues automatically, which will better obey to these properties than previous methods. It leads to enhanced mechanisms for constraining surfaces (continuous constraints) and approximating/ characterising shapes. The medial axis transform is not straightforward to compute and several methods have been presented for approximating it, based on Voronoi diagrams [ACK01], on distance maps [Bor84], or on thinning (see [ABE07] for a complete review of these methods). In fact medial axis are very sensitive to the smoothness of the surface (instability of the medial axis transform): small bumps create new branches and spikes on the medial axis (see Figure 4.15). This is quite undesirable as the goal is precisely to have a simpler representation of an object thanks to its medial axis. To tackle this well known problem for noisy surfaces, two general approaches have been explored to create minimal medial axis:

• Pruning [ACK01] [DGB03] [ABE07]: An accurate (complex) medial axis is iteratively simplified by suppressing redundant or irrelevant parts. The resulting medial axis is generally no more homotopy equivalent.

• Shape constraints[TWK87] [PFJ⁺03] [FVMO04]: A medial axis with a predefined (simple) shape is iteratively fitted to the center of the object. Extra parameters may be added to the medial axis (excursion parameters).

While the first approach makes no assumptions about the final structure of the medial axis, the second starts from a ”desired” shape and is therefore less demanding. For muscles, ligaments and cartilages, we can make the hypothesis that medial sheets will be homeomorphic to planes (no branching); and this will be corroborated by the results. We will consequently focus on the second approach. In [TWK87] [PFJ⁺03], medial curves and planes (M-reps) are used to represent complex objects. In addition to radius parameters, they add shape perturbations to better approximate objects. We will show that such parameters are not useful for our purpose since our objects are much more regular

To approximate its true medial axis, our goal is to iteratively fit a plane (the medial axis) to the center of an object (the model) through forces. These forces aim at minimising the distance between the reconstructed surface from the medial axis and the model. Reciprocally, forces will be applied to the model in order to constrain it radially. The model is a closed 2-simplex surface without any hole (g = 0 and H = 0 in the Euler-Poincar´e formula). The medial axis is a 2-simplex surface with genus zero and one hole (g = 0 and H = 1). A radius is associated to each one of its vertices, the reconstructed surface being the boundary of the union of the resulting balls. Because our medial axis is discrete, we linearly interpolate the radii along its surface in order to reconstruct a smooth and continuous model (see Figure 4.12). We have previously shown in [DGB03] that radius interpolation is suited and efficient for representing objects with a minimal number of medial points.

Figure 4.12: A 2D illustration of the discrete medial axis and the reconstructed model (in grey) without radius interpolation (left) and with radius interpolation (right)

The goal is to reach a state where model vertices Pi (i indexes model vertices) lie on the surface of the reconstructed model (see Figure 4.13). In other words, we want that the distance between Pi and its orthogonal projection on the medial axis Pi⊥ corresponds to the interpolated radius ri. Let Qj be the medial axis vertices of radius Rj (j indexes medial vertices) and wij the barycentric coordinates of the projection ofPi. We havePi⊥=P

jwij.Qj andri =P

jwij.Rj.

In a deformed state, we want to displace medial axis vertices or model vertices to reach the desired state.

The error can be calculated by err =< |ri−PiPi⊥| >i (the < . >i operator denotes averaging over all points). If a desired reference state has been defined before, projections and radii (wij andRj) are known;

otherwise, weights have to be recomputed at each iteration by projectingPiorthogonally, and radii need to be approximated. We have tested different methods for estimating radii, minimising the error:

• Closest point: Rj =QjP_i^∗ whereP^∗_i is the closest model vertex toQj.

• Weighted mean: Rj =P

iwij.PiPi⊥/P

iwij.

Figure 4.13: A 2D illustration of the desired state (left) and deformed state (right). In red, the model; in black, the medial axis

• Iterative minimisation of the error: For a random vertex Qj, we compute the derivative of the error∂err/∂Rj =P

isign(ri−PiPi⊥)wij . If it is positive, we decreaseRj by an infinetisimal value ε, otherwise we increase Rj byε. We iterate the process until convergence.

In practice, we use theweighted mean method which gives very close results to the optimal ones (iterative minimisationtechnique) without excessive computational costs. The closest point technique was found to be too inaccurate. Now, given that allwij andRj are known, we have to derive the desired positions ˜QjofQj

(approximation of the true medial axis) and/ or ˜PiofPi(radial regularisation of the model) that minimise the error. Medial axis vertex positions ˜Qj are, in fact, defined indirectly through the desired displacement ofPi⊥:

δPi⊥= ˜Pi⊥−Pi⊥ =X

j

wij.( ˜Qj−Qj) =X

j

wij.δQj

We will see in Section 4.8 that when applying forces on the set of particles Qj, we must comply with the momentum conservation principle. Assuming that all forcesFjare collinear and proportional toPi⊥P˜i⊥, we getδQj =wijPi⊥/P

jw²_ij. This is true if we consider only one projection. However, since each Qj shares multiple Pi⊥, we average their contribution. The final formula for calculating the desired medial vertex positions is thus:

Q˜j=Qj+< wijPi⊥P˜i⊥/X

j

w²_ij)>i

We have investigated several methods for determining desired displacements that are summarised in Table 4.22. The Closest point method is based on the straightforward projection on medial spheres, without using radius interpolation. Contrary to the two other methods, this leads to undesirable distortions of the surfaces as well as inaccurate medial axis localisation and object reconstruction (see Figure 4.15). With theradius method, the goal is to seek interpolated reference radii; whereas with theradius and barycentric coordinates method, the goal is to seek reference radii and positions. The last method does not allow any sliding of model vertices along the reconstructed surface. For model boundaries, because the direction of the projection is not normal, this method is not applicable and theradius method must be used. We could remedy this through angular parameters, as used by Pizer et al. [PFJ⁺03]. However, we have experienced that smoothing forces favourably prevents from excessive sliding (without requiring extra parameters). Note that these two methods are equivalent when no reference state is available, since orthogonal projections are performed at each time-step. When using a reference state, theradius and barycentric coordinates method requires an extra parameter λi for each model vertex: the side with regards to the medial surface normal on which they are projected. This side determines the sign in the expressions of Table 4.22. However, in

Method Illustrations Medial axis constraints Model constraints

Closest point

Q˜j=P^∗i +RjP^∗_iQ_j P_i^∗Q_j

P^∗i: closest point toQj.

P˜i=Q^∗_j +R^∗_j^Q_Q^∗j∗^Pi jP_i

P^∗_i: closest point toQj

Radius

Q˜j=Qj+< wijPP_i⊥P˜_i⊥

jw²_ij) >i

=Qj+< wij

P_i⊥P_i+r_i^PiPi⊥

PiPi⊥

P

jw_ij² >i

P˜i=Pi⊥+riP_i⊥P_i P_i⊥P_i

Radius and barycentric coordinates

Q˜j=Qj+< wijPP_i⊥P˜_i⊥

jw²_ij) >i

=Qj+< wijP_i⊥PP_i+λ_ir_in_i⊥

jw²_ij >i

P˜i=Pi⊥+λirini⊥

Table 4.22: Radial constraints With:

• Pi⊥=P

jwij.Qj

• ni⊥=P

jwij.nj/kP

jwij.njkwhere nj is the normal atQj.

• ri=P

jwij.Rj

• λi=sign(Pi⊥Pi.ni⊥) in the reference configuration (side parameter).

practice, this parameter is also used in theradiusmethod. The reason is that we experienced some flipping when radii are small, even if reference radii are reached (see Figure 4.14). Using theside parameter, we can detect those undesirable auto-intersections and remove them through reflection with regards to the medial axis. The reflection is performed by: δP˜_i = 2(P_i⊥P_i.n_i⊥)n_i⊥ (note that this is also valid for boundary vertices).

Figure 4.14: Auto-intersections happen when not using theside parameter

We will see that the application of the technique to the musculoskeletal system (Section 5.4.3) is valuable, since organ shapes can be well and quickly approximated by the medial axis (err'0.6mm). The localisation is good because no extra excursion or perturbation parameter (that would bump the surface) is necessary, and our technique preserves homotopy (a plane is homotopic to a closed surface). We will test more in details the two methods we have presented. Their main difference lies in the way to constrain the overlying model.

We will see if smoothing forces can recover the shape (without sliding). This would mean that barycentric

coordinates of projected model vertices do not need to be enforced.

Figure 4.15: From left to right: original model; its medial axis constructed with the powercrust method [ACK01]; medial axis constructed with our methodclosest point; medial axis constructed with our method radius. Radii are color mapped (blue=max, red=min)