Quiz

The Karhunen-Loève (KL) expansion and the Principal Component Analysis (PCA) are of great importance in shape modelling. Here you can check if you have understood the concepts and why these representations are important.

Question 1

The KL expansion is a representation of a Gaussian Process and hence probabilistic. Which part makes the model probabilistic?

The eigenfunctions are probabilistic quantities.

The eigenvalues are probabilistic quantities.

The expansion coefficients are probabilistic quantities.

Question 2

What are the advantages of the finite, parametric representation? Multiple answers are possible.

We have a compact description of any shape in terms of a (often small) number of coefficients

We can easily compute probabilities.

The mathematics are easier for finite dimensional quantities and we have more algorithms available.

Question 3

When is it not feasible (or not very useful) to obtain a good, finite rank approximation of a Gaussian Process in terms of the leading eigenfunctions?

When the number of points of a model is huge.

When the eigenvalues do not decay rapidly.

Question 4

What are the differences between the PCA and the KL representation? Multiple answers are possible.

PCA always assumes that the covariance function /covariance matrix is estimated from data while the KL representation does not make this assumption.

PCA always assumes that the shape representation is discrete while for the KL representation it can be continuous.

PCA learns the main axis of variation while the KL representation learns statistically independent components.