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This content is taken from the University of Basel's online course, Statistical Shape Modelling: Computing the Human Anatomy. Join the course to learn more.
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## University of Basel

Skip to 0 minutes and 7 seconds Welcome back. We have now seen that we can model shape variation by modelling deformations using a Gaussian Process. In this video, we’ll show you how you can use example data to estimate the mean and the covariance function of such a Gaussian Process.

Skip to 0 minutes and 29 seconds We have started with the reference shape gamma r.

Skip to 0 minutes and 36 seconds We have seen how any shape gamma can be obtained by moving the points of this reference shape.

Skip to 0 minutes and 46 seconds We notice that by construction, the target shape gamma will be described by the same number of points as the reference shape. Furthermore, we find, for each point of the reference shape, the corresponding point in the shape gamma. We say that the shapes are in correspondence or that u establishes correspondence between the two shapes.

Skip to 1 minute and 14 seconds If we have a set of shapes that are in correspondence, we can easily compare them and use them to compute statistics. Consider, for example, the point at the tip of the thumb. We can find that point in all the example shapes.

Skip to 1 minute and 40 seconds And we can observe its deformation in all the examples.

Skip to 1 minute and 56 seconds Now, what we can do is we can compute the average deformation at this point. We could even go further and we could estimate correlations from these examples. This can be done by taking a second point.

Skip to 2 minutes and 27 seconds Similarly, we can take these deformation and estimate from them the correlations.

Skip to 2 minutes and 36 seconds With these ideas, we can compute our first mean and covariance function for the Gaussian Process model. If we are given n deformations here, we can estimate from these deformations the mean and the covariance using the standard formulas for the sample mean and the sample covariance from basic statistics.

Skip to 3 minutes and 4 seconds Let us summarise how we build a shape model. We first select a reference shape. We find deformations from the hand to the example data. This gives us correspondence.

Skip to 3 minutes and 21 seconds We then estimate the mean and the covariance function. We define a Gaussian Process using the estimated parameters.

Skip to 3 minutes and 32 seconds As the u here in this formula is a probabilistic quantity, so is the shape that it generates. So this gamma in this formula is actually a probabilistic description of a shape. Congratulations. Now, you know how to build a shape model. As a first result, we see here the average hand shape.

Skip to 4 minutes and 1 second This simple procedure is everything that is needed to build the most important class of statistical shape model as they are currently used in research and industry.

Skip to 4 minutes and 14 seconds The only important point that we have not discussed here is how we get these deformations that relates the reference shape to the example shapes. We will come back to this question towards the end of the course.

# Learning a model from example data

It is very difficult to explicitly model the shape variations that define a shape family. Fortunately, we can still obtain very powerful shape models, by learning the typical deformations that occur within a shape family from a set of normal example shapes.

In this video we will explain what it means for shapes to be in correspondence, and how this helps us to learn shape variations from data. At the end of this video you will be familiar with the most popular type of shape models currently used in science and industry.