In many CAD-based applications, complex geometries are defined by a high number of design parameters. This leads to high-dimensional design spaces that are challenging for downstream engineering processes like simulations, optimization, and design exploration tasks. Therefore, dimension reduction methods such as principal component analysis (PCA) are used. The PCA identifies dominant modes of geometric variation and yields a compact representation of the geometry. While classical PCA excels in the compact representation part, it does not directly recover underlying design parameters of a generated geometry. In this work, we deal with the problem of estimating design parameters from PCA-based representations. Analyzing a recent modification of the PCA dedicated to our field of application, we show that the results are actually identical to the standard PCA. We investigate limitations of this approach and present reasonable conditions under which accurate, interpretable parameter estimation can be obtained. With the help of dedicated experiments, we take a more in-depth look at every stage of the PCA and the possible changes of the geometry during these processes.
Figures & tables
Figure 1: The rectangle and rectangular cuboid geometry classes. In the first and third image from the left, the mean geometries are plotted. In the second and fourth image, there are the corresponding centered geometries. For both classes, there is no change in geometry class. We have a good interpretable knowledge representation for both classes.
Figure 2: The simplified helix and helix geometry classes. In the first and third image from the left, the mean geometries are plotted. In the second and fourth image, there are the corresponding centered geometries. The simplified helix class does not change, but the helix class does. Therefore, the interpretability of the knowledge representation of the simplified helix is better than that of the helix.
Figure 3: The fan blade and tube geometry classes. In the first and third column from the left, the mean geometries are plotted from two different viewpoints. In the second and fourth column, there are the corresponding centered geometries, plotted again in two viewpoints. Noticeably, both geometry classes get changed during the centering process. Both geometries have a knowledge representation that is challenging to interpret.
Geometry class
CRVt=0.9
CRVt=0.95
CRVt=0.99
k
first set
rectangle
2
2
2
2
rectangular cuboid
3
3
3
3
simple helix
2
2
2
2
second set
helix
10
11
12
3
fan blade
3
4
9
12
tube
2
3
6
14
Table 1: Here, the number of eigenvalues t needed to get a ratio of 0.9=90% , 0.95=95% , and 0.99=99% for the geometry classes is presented. The number of generating parameters k is incorporated into the table for the purpose of comparison.
Figure 4: The first three eigenvectors of selected geometry classes from the two sets are visualized. In the first two columns, the classes rectangular cuboid and simplified helix, as representatives of the first set, are presented. In the last two columns, the helix and tube geometries from the second set are presented. The first row shows the first eigenvectors, the second row the second ones, and in the third row the third eigenvectors can be seen, respectively.
Figure 5: The mean of the absolute difference between the estimated and original parameters for each parameter for different numbers of used eigenvectors r . The red square denotes that the number of eigenvectors equals the number of design parameters, r=k . With the blue triangle, we denote the number of eigenvectors equals the number of eigenvalues that stores 95% of the total information, r=t0.95 . And finally, the black circle denotes the use of 200 eigenvectors, r=200 . For classes, where k=t0.95 only r=k is presented. The tube geometry class is logarithmically scaled, because the 10th parameter is much larger than the other ones. In the first row, the classes from the first set are presented. The second row shows the geometry classes from the second set.
Figure 6: The eigenvalues for all geometry classes. In the top row, the geometry classes from the first set are presented. The second row, shows the classes from the second set. For the fan blade and tube class, the y axis is logarithmically scaled and limited downward at 10−16 .
Principal component analysis (PCA) finds orthogonal directions of greatest variance for data compression, visualization, denoising, and feature discovery.