Chapter III: Fitting Tractable Convex Sets to Support Function Evaluations
3.5 Numerical Experiments
(a) 20 noiseless measure- ments
(b) 50 noiseless measure- ments
(c) 200 noiseless measure- ments
(d) 20 noisy measurements (e) 50 noisy measurements (f) 200 noisy measure- ments
Figure 3.8: Reconstruction of the unit `1-ball in R3 from noiseless (first row) and noisy (second row) support function measurements. The reconstructions obtained using our method (withC= ∆6in (3.2)) are the on the left of every subfigure, while the LSE reconstructions are on the right of every subfigure.
the LSE only begins to resemble the`2-ball with n= 200 measurements (and even then, the reconstruction is a polyhedral approximation).
Turning our attention next to the noisy case, the contrast between the results obtained using our framework and those of the LSE approach is even more stark. For both the `1-ball and the `2-ball, the LSE reconstructions bear little resemblance to the underlying convex set, unlike the estimates produced using our method. Notice that the reconstructions of the`2-ball using our algorithm are not even ellipsoidal when the number of measurements is small (e.g., whenn = 20), as linear images of the free spectrahedron O3may be non-ellipsoidal in general and need not even consist of smooth boundaries. Nonetheless, as the number of measurements available to our algorithm increases, the estimates improve in quality and offer improved reconstructions – with smooth boundaries – of the`2-ball.
In summary, these synthetic examples demonstrate that our framework is much more effective than the LSE in terms of robustness to noise, accuracy of reconstruction given a small number of measurements, and in settings in which the underlying set is non-polyhedral.
3.5.2 Reconstruction via Linear Images of the Free Spectrahedron
In the next series of synthetic experiments, we consider reconstructions of convex sets with non-smooth boundaries via linear images of the free spectrahedron. In
(a) 20 noiseless measure- ments
(b) 50 noiseless measure- ments
(c) 200 noiseless measure- ments
(d) 20 noisy measurements (e) 50 noisy measurements (f) 200 noisy measure- ments
Figure 3.9: Reconstruction of the unit `2-ball in R3 from noiseless (first row) and noisy (second row) support function measurements. The reconstructions obtained using our method (withC= O3in (3.2)) are the on the left of every subfigure, while the LSE reconstructions are on the right of every subfigure.
these illustrations, we consider sets in R2 and in R3 for which noiseless support function evaluations are obtained and supplied as input to the problem (3.2), with C equal to a free spectrahedron Oq in some larger-dimensional space q. For the examples in R2, the support function evaluations are obtained at 1000 equally spaced points on the unit circleS1. For the examples inR3, the support function evaluations are obtained at 2562 regularly spaced points on the unit sphereS2based on an icosphere discretization.
We consider reconstruction of the`1-ball inRd. Figure 3.10 shows the output from our algorithm whend =2 forq ∈ {2,3,4}, and the reconstruction is exact forq= 4.
Figure 3.11 shows the output from our algorithm when d = 3 for q ∈ {3,4,5,6}.
Interestingly, when d = 3 the computed solution for q = 5 does not contain any isolated extreme point (i.e., vertices) even though such features are expressible as projections of the free spectrahedronO5.
As our next illustration, we consider the following projection ofO4: UPillow=
(x,y,z)0: X ∈ O4,X12 = X21 = x,X23 = X32 = y,X34 = X43 = z ⊂ R3. (3.16) We term this convex set as the ‘uncomfortable pillow’ and it contains both smooth and non-smooth components in its boundary. Figure 3.12 shows the reconstruction of UPillow as linear images of O3 and O4 computed using our algorithm. The reconstruction based onO4is exact, while the reconstruction based onO3smooths
Figure 3.10: Approximating the `1-ball in R2 as the projection of the free- spectrahedron inS2(left),S3(center), andS4(right).
Figure 3.11: Approximating the`1-ball inR3as the projection of free spectrahedron inS3,S4,S5, andS6(from left to right).
Figure 3.12: Reconstructions of K? (defined in (3.16)) as the projection of O3 (top row) and O4 (bottom row). The figures in each row are different views of a single reconstruction, and are orientated in the(0,0,1),(0,1,0),(1,0,1), and(1,1,0) directions (from left to right) respectively.
out some of the ‘pointy’ features of UPillow; see for example the reconstructions based onO3and onO4viewed in the(0,1,0)direction in Figure 3.12.
3.5.3 Polyhedral Approximations of the`2-ball and the Tammes Problem In the third set of synthetic experiments, we consider reconstructions of the`2-ball in R3 via linear images of the simplex; i.e., polytopes. The experimental set-up is similar to the previous series of experiments: we supply 2562 regularly-spaced noiseless support function measurements of the`2-ball as an input to (3.2), and we selectCto be the simplex∆qin some larger-dimensional spaceq, and forq over a range of values.
The purpose of our experiment is to explore a specific instance of the broader question of approximating the`2-ball inR3as a polytope. Such a problem has been widely studied in different contexts and varying forms. For instance, the Tammes problem seeks the optimal placement of q points on S2 so as to maximize the minimum pairwise distance, and is inspired by pollen patterns [142].6 A separate body of work studies polyhedral approximations of general compact convex bodies in the asymptotics [25]. Yet another piece of work arising from optimization is that of computing polyhedral approximations of the second-order cone [14] – in particular, the approach in [14] leads to an approximation that is based on expressing the`2-ball via a nested hierarchy of planar spherical constraints and subsequently approximate these constraints with regular polygons.
Figure 3.13 shows the optimal solutions computed using our method for q ∈ {4,5, . . . ,12}. We remark that the configurations in our solutions are similar to those of the Tammes Problem [41, 125] in some instances:
argmax
{aj}qj=1⊂Sd−1
1≤k<l≤qmin dist(ak,al)= argmin
{aj}qj=1⊂Sd−1
1≤k<l≤qmax hak,ali. (3.17) Specifically, the face lattice (as a graph) of our solutions is isomorphic to that of the Tammes for q ∈ {4,5,6,7,12}, which suggests that these configurations are stable and optimal for a broader class of objectives. We are currently not aware if the difference between solutions to both sets of problems for q ∈ {8,9,10,11} arises because our method recovers a solution that is locally optimal but not globally optimal due to a lack of random initializations (in generating these results, we apply 500 initializations for each instance ofq), or is inherently due to the different objectives that both problems seek to optimize. We conjecture that the difference for q =8 is due to the latter reason we raised, as an initialization using a configuration that is isomorphic to the Tammes solution led to a suboptimal local minimum.
3.5.4 Reconstruction of a Human Lung
In the final set of experiments we apply our algorithm to reconstruct a convex mesh of a human lung. The purpose of this experiment is to demonstrate the utility of our algorithm in a setting in which the underlying object is not convex.
Indeed, in many applications in practice of reconstruction from support function evaluations, the underlying set of interest is not convex; however, due to the nature of the measurements available, one seeks a reconstruction of the convex hull of the
6Tammes Problem is a special case of Thompson’s Problem, as well as Smale’s 7th Problem [133].
Figure 3.13: Approximating the `2-ball in R3 as the projection of ∆q for q ∈ {4,5, . . . ,12}(from left to right, top to bottom).
underlying set. In the present example, the set of interest is obtained from the CT scan of the left lung of a healthy individual [59]. We note that a priori it is unclear whether the convex hull of the lung is well-approximated as the linear image of a low-dimensional simplex or free spectrahedron.
We first obtainn =50 noiseless support function evaluations of the lung (note that this object lies inR3) in directions that are generated uniformly at random over the sphereS2. In the top row of Figure 3.14 we show the reconstructions as projections of Oq for q ∈ {3,4,5,6}, and we contrast these with the LSE. We repeat the same experiment with n = 300 measurements, with the reconstructions shown in the bottom row of Figure 3.14.
To concretely compare the results obtained using our framework and those based on the LSE, we contrast the description complexity – the number of parameters used to specify the reconstruction – of the estimates obtained from both frameworks.
An estimator computed using our approach is specified by a projection map A ∈ L(Rq,Rd), and hence it requires dq parameters; while the LSE proposed by the algorithm in [60] assigns a vertex to every measurement, and hence it requires dn parameters. The LSE usingn= 300 measurements requires 3×300 parameters to specify whereas the estimates obtained using our framework – these are specified as projections ofO5andO6– require 3×15 and 3×21 parameters respectively. Despite requiring significantly fewer parameters to specify, the estimates obtained using our method offer comparable quality to the LSE. This substantial discrepancy highlights the drawback of using polyhedral sets of growing complexity to approximate non-
(a)O3 (b)O4 (c)O5 (d)O6 (e) LSE
(f)O3 (g)O4 (h)O5 (i)O6 (j) LSE
Figure 3.14: Reconstructions of the left lung from 50 support function measure- ments (top row) and 300 support function measurements (bottom row). Subfigures (a),(b),(c),(d),(f),(g),(h), and (i) are projections of free spectrahedra with dimensions as indicated, and subfigures (e) and (j) are LSEs.
polyhedral objects in higher dimensions.