Minimum Slice Perimeter Function
3.2 Minimum Perimeter Slice Refinement
Figure 3.3: The normal of minimum perimeter slices that is projected onto the surface.
Figure 3.4: MSP function for the cat model in different poses.
number of slices higher than or equal to eight, the MSP values are almost the same. A typical range of the number of slices is from four to eight for trading off between quality and efficiency.
Some highlights of MSP value can be observed at the tips of object parts (such as the chest of horse, shoulder of nepture, and head of dancing children) in Fig. 3.2. This is due to the improper orientation of minimum perimeter slices, where the slice planes in such tip regions are almost parallel to the local symmetric axis. Moreover, the slicing along the normal direction may be sensitive to the surface noise. Fig. 3.6 illustrates two examples for the MSP function on noisy surfaces, where the raptor model with high resolution has details on its surface and the horse model is produced by applying a turbulence function on the model surface. Non-smooth MSP value can be observed, particularly at small regions where MSP values change dramatically.
3.2 Minimum Perimeter Slice Refinement
A good minimum perimeter slice should be capable of representing the shape in the slab of its local region well. Therefore, a good minimum perimeter slice can be considered as
3.2 Minimum Perimeter Slice Refinement 20
Figure 3.5: MSP function for the raptor model in different resolutions.
Figure 3.6: The MSP function of noisy surfaces.
the slice on which all the surface points along the slice yield similar minimum perimeter.
Based on this observation, to measure how well a minimum perimeter slice is, we define the slice-deviation error for the minimum perimeter slice of a surface point p as follows:
cdev(p) = 1 kS (p) k
Z
q∈S(p)cos−1 np· nq dq, (3.2)
where S (p) is the minimum perimeter slice at the surface point p, np and nq are the unit plane normals of the minimum perimeter slices at p and q, respectively. Fig. 3.7 depicts the slice-deviation error of the models shown in the top row of Fig. 3.2. High slice-deviation er-ror can be observed at the tips of object parts and the junctions, where improper orientation may be used to derive the minimum perimeter slices.
According to the definition in [79], the short-cut is not necessary to be perpendicular to the surface normal, slicing along the surface normal may be too restrict for deriving a good minimum perimeter slice for approximating the short cut. It is reasonable to argue that an ideal minimum perimeter slice at the surface point p should be minimal for both the slice perimeter and the slice deviation error. However, such a minimum perimeter slice may not always exist. Instead, we refine the orientation of the minimum perimeter slice at
3.2 Minimum Perimeter Slice Refinement 21
Figure 3.7: The slice deviation error of 3D models.
the surface point p by minimizing the objective function described in Eq. 3.3.
Op(n) =kpl (n, p) ∩ Mk
2πrM +cdev(p)
π /2 , (3.3)
where the plane normal n is not restricted to be perpendicular to the surface normal at point p, and rM denotes the half of the diagonal of the object’s bounding box, which is used to normalize the slice perimeter.
Refining the minimum perimeter slice using Eq. 3.3 is, however, difficult due to the intercorrelation between slice normal and slice deviation error. Instead, we use an iteration-based method that takes the minimum perimeter slice computed using method described by Eq. 3.1 as an initial guess. For each iteration, the new slice that minimizes the combination of the slice perimeter and the slice deviation error is taken. The searching space for the new slice plane should cover the entire hemisphere centered at p, which is too large. For the minimum perimeter slice S (p) of the surface point p, we consider the unit plane normal nq of the minimum perimeter slice of each point q along S (p). The average of those normal nq weighted by using the reciprocal of slice-deviation error at q gives a reasonable orientation of the object part, and hence offers a candidate slice-plane normal for computing a better minimum perimeter slice; as shown in Eq. 3.4.
navg(S (p) , p) = Z
q∈S(p)
nq
cdev(q)dq, (3.4)
Hence, the search of the slice-plane normal in the entire hemisphere space is reduced to the search in the range bounded by the normal of the minimum perimeter slice plane derived in previous iteration and the candidate slice-plane normal navg. The iterative process is
3.3 Comparisons 22
Figure 3.8: The refined MSP function of the 3D models.
Figure 3.9: The refined MSP function of noisy surfaces.
performed until the normal difference between the new minimum perimeter slice plane and the previous one is below a user-specified threshold.
Fig. 3.8 illustrates the refined MSP function for models shown in the top row of Fig. 3.2.
The highlights at the chest of horse, shoulder of nepture, and head of dancing children models are eliminated. Fig. 3.9 shows another refinement result on the noisy surfaces.
3.3 Comparisons
Both of MSP and SDF [73] reveal the local volume of the object, but by different defi-nitions. SDF is an approximation of the medial axis which describes the local volume of object at the surface point by the maximum inscribed ball attaching to the surface point.
Ray casting through the interior of object is applied to approximate the diameter of the maximum inscribed ball. Both MSP and SDF can reveal the local volume well for
cylin-3.3 Comparisons 23
(a) MSP (b) SDF
(c) Gradient of MSP (d) Gradient of SDF
Figure 3.10: The distributions of MSP and SDF and their gradients.
drical parts, but MSP is capable of capturing better volume information than the SDF for those object parts that are non-cylindrical, such as the palm, as shown in Fig. 3.10. For non-cylindrical models or parts, the SDF value reflects only a portion of the internal vol-ume information and measures the local thickness or local diameter of the model. The red color around the side of the palm means that the internal volume is much larger at the side of palm than at the center. Instead of MSP function, the minimum perimeter slice carries additional shape information about the object part, such as the slice shape and the orientation. The SDF, however, is unable to derive such information.
The surface function which describes the internal volume of the object is useful for part-based mesh segmentation. The averaging computation in SDF formulation can alleviate the problem induced by surface details or noise. It, however, tends to blur the change of volume, leading to some difficulties in deriving the part boundary, especially for the surface regions with dramatic change in local volume. Fig. 3.10(c) and Fig. 3.10(d) depict that the
3.4 Limitations 24
(a) MSP (b) SDF
Figure 3.11: Vertex distribution of the shrunk mesh using MSP and SDF.
gradient of MSP changes significantly across the object parts, e.g. the regions between the fingers and the palm and the gradient of SDF does not change obviously across the object parts.
For skeletonization application, the centroid of the minimum perimeter slice has better representative for the curve skeleton, compared to the transformed vertices derived using the half-depth toward the inverse normal direction of SDF, as shown in Fig. 3.11.
3.4 Limitations
In some cases, the minimum perimeter slice does not faithfully capture the local shape of a part; e.g., at the feet of the dancing-children model in Fig. 3.2 where some planar slices pass across multiple parts and yield relatively larger MSP value than they should be.